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1，等于什么都没说。",[11,564,565],{},"这两个事实放在一起，说明「参数个数」和「容量」是两件不同的事。VC 维（Vapnik–Chervonenkis dimension）就是 Vapnik 与 Chervonenkis 在 1971 年为了把后者说清楚而发明的量：它不数参数，不假设分布，只看一个假设类作为「函数集合」的组合结构有多丰富。",[11,567,568],{},"本文的路线是：第一至四节把定义、Sauer–Shelah 引理、一致收敛定理这条逻辑链完整搭起来；第五至七节讨论它在深度学习里的处境——为什么界会空、空界是否意味着理论失效、今天用什么替代它；第八节清理一批常见误解。",[570,571,573],"h2",{"id":572},"一学习问题","一、学习问题",[575,576,578],"h3",{"id":577},"_11-设定","1.1 设定",[11,580,581,582,616,617,696,697,701,702,800,801,804,805,836,837,891,892,922],{},"设输入空间为 ",[15,583,585,601],{"className":584,"translate":19},[23],[15,586,588],{"className":587},[27],[29,589,590],{"xmlns":31},[34,591,592,598],{},[37,593,594],{},[43,595,597],{"mathvariant":596},"script","X",[109,599,600],{"encoding":111},"\\mathcal X",[15,602,604],{"className":603,"ariaHidden":72},[116],[15,605,607,611],{"className":606},[120],[15,608],{"className":609,"style":610},[124],"height:0.6833em;",[15,612,597],{"className":613,"style":615},[129,614],"mathcal","margin-right:0.1464em;","，标签空间为 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",[15,1286,1288,1301],{"className":1287,"translate":19},[23],[15,1289,1291],{"className":1290},[27],[29,1292,1293],{"xmlns":31},[34,1294,1295,1299],{},[37,1296,1297],{},[43,1298,906],{"mathvariant":596},[109,1300,909],{"encoding":111},[15,1302,1304],{"className":1303,"ariaHidden":72},[116],[15,1305,1307,1310],{"className":1306},[120],[15,1308],{"className":1309,"style":610},[124],[15,1311,906],{"className":1312,"style":170},[129,614],"。",[11,1315,1316,1317,1345,1346,1349,1350,1353],{},"假设 ",[15,1318,1320,1333],{"className":1319,"translate":19},[23],[15,1321,1323],{"className":1322},[27],[29,1324,1325],{"xmlns":31},[34,1326,1327,1331],{},[37,1328,1329],{},[43,1330,45],{},[109,1332,45],{"encoding":111},[15,1334,1336],{"className":1335,"ariaHidden":72},[116],[15,1337,1339,1342],{"className":1338},[120],[15,1340],{"className":1341,"style":522},[124],[15,1343,45],{"className":1344},[129,133]," 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y_i\\bigr].",[15,1499,1501,1569,1708,1790,2049],{"className":1500,"ariaHidden":72},[116],[15,1502,1504,1507,1551,1554,1557,1560,1563,1566],{"className":1503},[120],[15,1505],{"className":1506,"style":125},[124],[15,1508,1510,1513],{"className":1509},[129],[15,1511,1373],{"className":1512},[129,133],[15,1514,1516],{"className":1515},[137],[15,1517,1519,1543],{"className":1518},[141,142],[15,1520,1522,1540],{"className":1521},[146],[15,1523,1526],{"className":1524,"style":1525},[150],"height:0.3283em;",[15,1527,1528,1531],{"style":154},[15,1529],{"className":1530,"style":159},[158],[15,1532,1534],{"className":1533},[163,164,165,166],[15,1535,1537],{"className":1536},[129,166],[15,1538,906],{"className":1539,"style":170},[129,614,166],[15,1541,175],{"className":1542},[174],[15,1544,1546],{"className":1545},[146],[15,1547,1549],{"className":1548,"style":182},[150],[15,1550],{},[15,1552,53],{"className":1553},[188],[15,1555,45],{"className":1556},[129,133],[15,1558,59],{"className":1559},[195],[15,1561],{"className":1562,"style":199},[93],[15,1564,62],{"className":1565},[203],[15,1567],{"className":1568,"style":199},[93],[15,1570,1572,1576,1640,1646,1649,1652,1655,1658,1661,1705],{"className":1571},[120],[15,1573],{"className":1574,"style":1575},[124],"height:1.2052em;vertical-align:-0.3552em;",[15,1577,1579,1582],{"className":1578},[129],[15,1580,1388],{"className":1581},[129,285],[15,1583,1585],{"className":1584},[137],[15,1586,1588,1631],{"className":1587},[141,142],[15,1589,1591,1628],{"className":1590},[146],[15,1592,1595],{"className":1593,"style":1594},[150],"height:0.3448em;",[15,1596,1598,1601],{"style":1597},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[15,1599],{"className":1600,"style":159},[158],[15,1602,1604],{"className":1603},[163,164,165,166],[15,1605,1607,1610,1613,1616,1619,1622,1625],{"className":1606},[129,166],[15,1608,53],{"className":1609},[188,166],[15,1611,56],{"className":1612},[129,133,166],[15,1614,91],{"className":1615},[254,166],[15,1617,857],{"className":1618,"style":887},[129,133,166],[15,1620,59],{"className":1621},[195,166],[15,1623,991],{"className":1624},[203,166],[15,1626,906],{"className":1627,"style":170},[129,614,166],[15,1629,175],{"className":1630},[174],[15,1632,1634],{"className":1633},[146],[15,1635,1638],{"className":1636,"style":1637},[150],"height:0.3552em;",[15,1639],{},[15,1641,1643],{"className":1642},[188],[15,1644,1407],{"className":1645},[228,229],[15,1647,45],{"className":1648},[129,133],[15,1650,53],{"className":1651},[188],[15,1653,56],{"className":1654},[129,133],[15,1656,59],{"className":1657},[195],[15,1659],{"className":1660,"style":199},[93],[15,1662,1664,1698,1702],{"className":1663},[203],[15,1665,1667],{"className":1666},[203],[15,1668,1671],{"className":1669},[129,1670],"vbox",[15,1672,1675],{"className":1673},[1674],"thinbox",[15,1676,1679,1683,1694],{"className":1677},[1678],"rlap",[15,1680],{"className":1681,"style":1682},[124],"height:0.8889em;vertical-align:-0.1944em;",[15,1684,1687],{"className":1685},[1686],"inner",[15,1688,1690],{"className":1689},[129],[15,1691,1693],{"className":1692},[203],"",[15,1695],{"className":1696},[1697],"fix",[15,1699],{"className":1700},[93,1701],"nobreak",[15,1703,62],{"className":1704},[203],[15,1706],{"className":1707,"style":199},[93],[15,1709,1711,1714,1717,1723,1726,1729,1732,1772,1775,1778,1781,1784,1787],{"className":1710},[120],[15,1712],{"className":1713,"style":213},[124],[15,1715,857],{"className":1716,"style":887},[129,133],[15,1718,1720],{"className":1719},[195],[15,1721,1423],{"className":1722},[228,229],[15,1724,91],{"className":1725},[254],[15,1727],{"className":1728,"style":258},[93],[15,1730],{"className":1731,"style":262},[93],[15,1733,1735,1738],{"className":1734},[129],[15,1736,1373],{"className":1737},[129,133],[15,1739,1741],{"className":1740},[137],[15,1742,1744,1764],{"className":1743},[141,142],[15,1745,1747,1761],{"className":1746},[146],[15,1748,1750],{"className":1749,"style":1525},[150],[15,1751,1752,1755],{"style":154},[15,1753],{"className":1754,"style":159},[158],[15,1756,1758],{"className":1757},[163,164,165,166],[15,1759,940],{"className":1760,"style":1013},[129,133,166],[15,1762,175],{"className":1763},[174],[15,1765,1767],{"className":1766},[146],[15,1768,1770],{"className":1769,"style":182},[150],[15,1771],{},[15,1773,53],{"className":1774},[188],[15,1776,45],{"className":1777},[129,133],[15,1779,59],{"className":1780},[195],[15,1782],{"className":1783,"style":199},[93],[15,1785,62],{"className":1786},[203],[15,1788],{"className":1789,"style":199},[93],[15,1791,1793,1797,1868,1871,1944,1947,1951,1957,1960,1963,2004,2007,2010,2046],{"className":1792},[120],[15,1794],{"className":1795,"style":1796},[124],"height:2.9291em;vertical-align:-1.2777em;",[15,1798,1800,1804,1865],{"className":1799},[129],[15,1801],{"className":1802},[188,1803],"nulldelimiter",[15,1805,1807],{"className":1806},[1443],[15,1808,1810,1856],{"className":1809},[141,142],[15,1811,1813,1853],{"className":1812},[146],[15,1814,1817,1830,1841],{"className":1815,"style":1816},[150],"height:1.3214em;",[15,1818,1820,1824],{"style":1819},"top:-2.314em;",[15,1821],{"className":1822,"style":1823},[158],"height:3em;",[15,1825,1827],{"className":1826},[129],[15,1828,348],{"className":1829},[129,133],[15,1831,1833,1836],{"style":1832},"top:-3.23em;",[15,1834],{"className":1835,"style":1823},[158],[15,1837],{"className":1838,"style":1840},[1839],"frac-line","border-bottom-width:0.04em;",[15,1842,1844,1847],{"style":1843},"top:-3.677em;",[15,1845],{"className":1846,"style":1823},[158],[15,1848,1850],{"className":1849},[129],[15,1851,318],{"className":1852},[129],[15,1854,175],{"className":1855},[174],[15,1857,1859],{"className":1858},[146],[15,1860,1863],{"className":1861,"style":1862},[150],"height:0.686em;",[15,1864],{},[15,1866],{"className":1867},[195,1803],[15,1869],{"className":1870,"style":262},[93],[15,1872,1875],{"className":1873},[217,1874],"op-limits",[15,1876,1878,1935],{"className":1877},[141,142],[15,1879,1881,1932],{"className":1880},[146],[15,1882,1885,1907,1920],{"className":1883,"style":1884},[150],"height:1.6514em;",[15,1886,1888,1892],{"style":1887},"top:-1.8723em;margin-left:0em;",[15,1889],{"className":1890,"style":1891},[158],"height:3.05em;",[15,1893,1895],{"className":1894},[163,164,165,166],[15,1896,1898,1901,1904],{"className":1897},[129,166],[15,1899,1459],{"className":1900},[129,133,166],[15,1902,62],{"className":1903},[203,166],[15,1905,318],{"className":1906},[129,166],[15,1908,1910,1913],{"style":1909},"top:-3.05em;",[15,1911],{"className":1912,"style":1891},[158],[15,1914,1915],{},[15,1916,1454],{"className":1917},[217,1918,1919],"op-symbol","large-op",[15,1921,1923,1926],{"style":1922},"top:-4.3em;margin-left:0em;",[15,1924],{"className":1925,"style":1891},[158],[15,1927,1929],{"className":1928},[163,164,165,166],[15,1930,348],{"className":1931},[129,133,166],[15,1933,175],{"className":1934},[174],[15,1936,1938],{"className":1937},[146],[15,1939,1942],{"className":1940,"style":1941},[150],"height:1.2777em;",[15,1943],{},[15,1945],{"className":1946,"style":262},[93],[15,1948,318],{"className":1949},[129,1950],"mathbf",[15,1952,1954],{"className":1953},[188],[15,1955,1407],{"className":1956},[228,229],[15,1958,45],{"className":1959},[129,133],[15,1961,53],{"className":1962},[188],[15,1964,1966,1969],{"className":1965},[129],[15,1967,56],{"className":1968},[129,133],[15,1970,1972],{"className":1971},[137],[15,1973,1975,1996],{"className":1974},[141,142],[15,1976,1978,1993],{"className":1977},[146],[15,1979,1982],{"className":1980,"style":1981},[150],"height:0.3117em;",[15,1983,1984,1987],{"style":154},[15,1985],{"className":1986,"style":159},[158],[15,1988,1990],{"className":1989},[163,164,165,166],[15,1991,1459],{"className":1992},[129,133,166],[15,1994,175],{"className":1995},[174],[15,1997,1999],{"className":1998},[146],[15,2000,2002],{"className":2001,"style":182},[150],[15,2003],{},[15,2005,59],{"className":2006},[195],[15,2008],{"className":2009,"style":199},[93],[15,2011,2013,2040,2043],{"className":2012},[203],[15,2014,2016],{"className":2015},[203],[15,2017,2019],{"className":2018},[129,1670],[15,2020,2022],{"className":2021},[1674],[15,2023,2025,2028,2037],{"className":2024},[1678],[15,2026],{"className":2027,"style":1682},[124],[15,2029,2031],{"className":2030},[1686],[15,2032,2034],{"className":2033},[129],[15,2035,1693],{"className":2036},[203],[15,2038],{"className":2039},[1697],[15,2041],{"className":2042},[93,1701],[15,2044,62],{"className":2045},[203],[15,2047],{"className":2048,"style":199},[93],[15,2050,2052,2055,2095,2101],{"className":2051},[120],[15,2053],{"className":2054,"style":213},[124],[15,2056,2058,2061],{"className":2057},[129],[15,2059,857],{"className":2060,"style":887},[129,133],[15,2062,2064],{"className":2063},[137],[15,2065,2067,2087],{"className":2066},[141,142],[15,2068,2070,2084],{"className":2069},[146],[15,2071,2073],{"className":2072,"style":1981},[150],[15,2074,2075,2078],{"style":1100},[15,2076],{"className":2077,"style":159},[158],[15,2079,2081],{"className":2080},[163,164,165,166],[15,2082,1459],{"className":2083},[129,133,166],[15,2085,175],{"className":2086},[174],[15,2088,2090],{"className":2089},[146],[15,2091,2093],{"className":2092,"style":182},[150],[15,2094],{},[15,2096,2098],{"className":2097},[195],[15,2099,1423],{"className":2100},[228,229],[15,2102,107],{"className":2103},[129],[11,2105,2106,2107,2110],{},"前者是我们真正关心但看不见的量，后者是能算出来的量。",[698,2108,2109],{},"经验风险最小化","（ERM）只做一件事：",[15,2112,2114],{"className":2113,"translate":19},[18],[15,2115,2117,2174],{"className":2116,"translate":19},[23],[15,2118,2120],{"className":2119},[27],[29,2121,2122],{"xmlns":31,"display":32},[34,2123,2124,2171],{},[37,2125,2126,2132,2134,2137,2139,2157,2163,2165,2167,2169],{},[40,2127,2128,2130],{},[43,2129,45],{},[43,2131,940],{},[50,2133,100],{},[43,2135,2136],{},"arg",[50,2138,69],{},[2140,2141,2142,2149],"munder",{},[37,2143,2144,2147],{},[43,2145,2146],{},"min",[50,2148,69],{},[37,2150,2151,2153,2155],{},[43,2152,45],{},[50,2154,100],{},[43,2156,819],{"mathvariant":596},[40,2158,2159,2161],{},[43,2160,1373],{},[43,2162,940],{},[50,2164,53],{"stretchy":52},[43,2166,45],{},[50,2168,59],{"stretchy":52},[43,2170,107],{"mathvariant":65},[109,2172,2173],{"encoding":111},"h_S\\in\\arg\\min_{h\\in\\mathcal H}L_S(h).",[15,2175,2177,2233],{"className":2176,"ariaHidden":72},[116],[15,2178,2180,2184,2224,2227,2230],{"className":2179},[120],[15,2181],{"className":2182,"style":2183},[124],"height:0.8444em;vertical-align:-0.15em;",[15,2185,2187,2190],{"className":2186},[129],[15,2188,45],{"className":2189},[129,133],[15,2191,2193],{"className":2192},[137],[15,2194,2196,2216],{"className":2195},[141,142],[15,2197,2199,2213],{"className":2198},[146],[15,2200,2202],{"className":2201,"style":1525},[150],[15,2203,2204,2207],{"style":154},[15,2205],{"className":2206,"style":159},[158],[15,2208,2210],{"className":2209},[163,164,165,166],[15,2211,940],{"className":2212,"style":1013},[129,133,166],[15,2214,175],{"className":2215},[174],[15,2217,2219],{"className":2218},[146],[15,2220,2222],{"className":2221,"style":182},[150],[15,2223],{},[15,2225],{"className":2226,"style":199},[93],[15,2228,100],{"className":2229},[203],[15,2231],{"className":2232,"style":199},[93],[15,2234,2236,2240,2248,2251,2308,2311,2351,2354,2357,2360],{"className":2235},[120],[15,2237],{"className":2238,"style":2239},[124],"height:1.5295em;vertical-align:-0.7795em;",[15,2241,2243,2244],{"className":2242},[217],"ar",[15,2245,2247],{"style":2246},"margin-right:0.0139em;","g",[15,2249],{"className":2250,"style":262},[93],[15,2252,2254],{"className":2253},[217,1874],[15,2255,2257,2299],{"className":2256},[141,142],[15,2258,2260,2296],{"className":2259},[146],[15,2261,2264,2285],{"className":2262,"style":2263},[150],"height:0.6679em;",[15,2265,2267,2270],{"style":2266},"top:-2.3479em;margin-left:0em;",[15,2268],{"className":2269,"style":1823},[158],[15,2271,2273],{"className":2272},[163,164,165,166],[15,2274,2276,2279,2282],{"className":2275},[129,166],[15,2277,45],{"className":2278},[129,133,166],[15,2280,100],{"className":2281},[203,166],[15,2283,819],{"className":2284,"style":835},[129,614,166],[15,2286,2288,2291],{"style":2287},"top:-3em;",[15,2289],{"className":2290,"style":1823},[158],[15,2292,2293],{},[15,2294,2146],{"className":2295},[217],[15,2297,175],{"className":2298},[174],[15,2300,2302],{"className":2301},[146],[15,2303,2306],{"className":2304,"style":2305},[150],"height:0.7795em;",[15,2307],{},[15,2309],{"className":2310,"style":262},[93],[15,2312,2314,2317],{"className":2313},[129],[15,2315,1373],{"className":2316},[129,133],[15,2318,2320],{"className":2319},[137],[15,2321,2323,2343],{"className":2322},[141,142],[15,2324,2326,2340],{"className":2325},[146],[15,2327,2329],{"className":2328,"style":1525},[150],[15,2330,2331,2334],{"style":154},[15,2332],{"className":2333,"style":159},[158],[15,2335,2337],{"className":2336},[163,164,165,166],[15,2338,940],{"className":2339,"style":1013},[129,133,166],[15,2341,175],{"className":2342},[174],[15,2344,2346],{"className":2345},[146],[15,2347,2349],{"className":2348,"style":182},[150],[15,2350],{},[15,2352,53],{"className":2353},[188],[15,2355,45],{"className":2356},[129,133],[15,2358,59],{"className":2359},[195],[15,2361,107],{"className":2362},[129],[575,2364,2366],{"id":2365},"_12-误差分解","1.2 误差分解",[11,2368,2369,2370,2455,2456,2585],{},"对任意 ",[15,2371,2373,2396],{"className":2372,"translate":19},[23],[15,2374,2376],{"className":2375},[27],[29,2377,2378],{"xmlns":31},[34,2379,2380,2393],{},[37,2381,2382,2389,2391],{},[305,2383,2384,2386],{},[43,2385,45],{},[50,2387,2388],{},"∗",[50,2390,100],{},[43,2392,819],{"mathvariant":596},[109,2394,2395],{"encoding":111},"h^*\\in\\mathcal H",[15,2397,2399,2446],{"className":2398,"ariaHidden":72},[116],[15,2400,2402,2406,2437,2440,2443],{"className":2401},[120],[15,2403],{"className":2404,"style":2405},[124],"height:0.7335em;vertical-align:-0.0391em;",[15,2407,2409,2412],{"className":2408},[129],[15,2410,45],{"className":2411},[129,133],[15,2413,2415],{"className":2414},[137],[15,2416,2418],{"className":2417},[141],[15,2419,2421],{"className":2420},[146],[15,2422,2425],{"className":2423,"style":2424},[150],"height:0.6887em;",[15,2426,2427,2430],{"style":383},[15,2428],{"className":2429,"style":159},[158],[15,2431,2433],{"className":2432},[163,164,165,166],[15,2434,2388],{"className":2435},[2436,166],"mbin",[15,2438],{"className":2439,"style":199},[93],[15,2441,100],{"className":2442},[203],[15,2444],{"className":2445,"style":199},[93],[15,2447,2449,2452],{"className":2448},[120],[15,2450],{"className":2451,"style":610},[124],[15,2453,819],{"className":2454,"style":835},[129,614],"，把 ",[15,2457,2459,2487],{"className":2458,"translate":19},[23],[15,2460,2462],{"className":2461},[27],[29,2463,2464],{"xmlns":31},[34,2465,2466,2484],{},[37,2467,2468,2474,2476,2482],{},[40,2469,2470,2472],{},[43,2471,1373],{},[43,2473,906],{"mathvariant":596},[50,2475,53],{"stretchy":52},[40,2477,2478,2480],{},[43,2479,45],{},[43,2481,940],{},[50,2483,59],{"stretchy":52},[109,2485,2486],{"encoding":111},"L_{\\mathcal D}(h_S)",[15,2488,2490],{"className":2489,"ariaHidden":72},[116],[15,2491,2493,2496,2539,2542,2582],{"className":2492},[120],[15,2494],{"className":2495,"style":125},[124],[15,2497,2499,2502],{"className":2498},[129],[15,2500,1373],{"className":2501},[129,133],[15,2503,2505],{"className":2504},[137],[15,2506,2508,2531],{"className":2507},[141,142],[15,2509,2511,2528],{"className":2510},[146],[15,2512,2514],{"className":2513,"style":1525},[150],[15,2515,2516,2519],{"style":154},[15,2517],{"className":2518,"style":159},[158],[15,2520,2522],{"className":2521},[163,164,165,166],[15,2523,2525],{"className":2524},[129,166],[15,2526,906],{"className":2527,"style":170},[129,614,166],[15,2529,175],{"className":2530},[174],[15,2532,2534],{"className":2533},[146],[15,2535,2537],{"className":2536,"style":182},[150],[15,2538],{},[15,2540,53],{"className":2541},[188],[15,2543,2545,2548],{"className":2544},[129],[15,2546,45],{"className":2547},[129,133],[15,2549,2551],{"className":2550},[137],[15,2552,2554,2574],{"className":2553},[141,142],[15,2555,2557,2571],{"className":2556},[146],[15,2558,2560],{"className":2559,"style":1525},[150],[15,2561,2562,2565],{"style":154},[15,2563],{"className":2564,"style":159},[158],[15,2566,2568],{"className":2567},[163,164,165,166],[15,2569,940],{"className":2570,"style":1013},[129,133,166],[15,2572,175],{"className":2573},[174],[15,2575,2577],{"className":2576},[146],[15,2578,2580],{"className":2579,"style":182},[150],[15,2581],{},[15,2583,59],{"className":2584},[195]," 拆开：",[15,2587,2589],{"className":2588,"translate":19},[18],[15,2590,2592,2979],{"className":2591,"translate":19},[23],[15,2593,2595],{"className":2594},[27],[29,2596,2597],{"xmlns":31,"display":32},[34,2598,2599,2976],{},[2600,2601,2605,2699,2774,2888],"mtable",{"rowspacing":2602,"columnalign":2603,"columnspacing":2604},"0.25em","right left","0em",[2606,2607,2608,2632],"mtr",{},[2609,2610,2611],"mtd",{},[2612,2613,2614],"mstyle",{"scriptlevel":541,"displaystyle":72},[37,2615,2616,2622,2624,2630],{},[40,2617,2618,2620],{},[43,2619,1373],{},[43,2621,906],{"mathvariant":596},[50,2623,53],{"stretchy":52},[40,2625,2626,2628],{},[43,2627,45],{},[43,2629,940],{},[50,2631,59],{"stretchy":52},[2609,2633,2634],{},[2612,2635,2636],{"scriptlevel":541,"displaystyle":72},[37,2637,2638,2640,2642,2644,2650,2652,2658,2660,2662,2668,2670,2676,2678,2680,2683,2689,2691,2697],{},[37,2639],{},[50,2641,62],{},[50,2643,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2645,2646,2648],{},[43,2647,1373],{},[43,2649,906],{"mathvariant":596},[50,2651,53],{"stretchy":52},[40,2653,2654,2656],{},[43,2655,45],{},[43,2657,940],{},[50,2659,59],{"stretchy":52},[50,2661,315],{},[40,2663,2664,2666],{},[43,2665,1373],{},[43,2667,940],{},[50,2669,53],{"stretchy":52},[40,2671,2672,2674],{},[43,2673,45],{},[43,2675,940],{},[50,2677,59],{"stretchy":52},[50,2679,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2681,2682],{},"+",[40,2684,2685,2687],{},[43,2686,1373],{},[43,2688,940],{},[50,2690,53],{"stretchy":52},[40,2692,2693,2695],{},[43,2694,45],{},[43,2696,940],{},[50,2698,59],{"stretchy":52},[2606,2700,2701,2707],{},[2609,2702,2703],{},[2612,2704,2705],{"scriptlevel":541,"displaystyle":72},[37,2706],{},[2609,2708,2709],{},[2612,2710,2711],{"scriptlevel":541,"displaystyle":72},[37,2712,2713,2715,2718,2720,2726,2728,2734,2736,2738,2744,2746,2752,2754,2756,2758,2764,2766,2772],{},[37,2714],{},[50,2716,2717],{},"≤",[50,2719,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2721,2722,2724],{},[43,2723,1373],{},[43,2725,906],{"mathvariant":596},[50,2727,53],{"stretchy":52},[40,2729,2730,2732],{},[43,2731,45],{},[43,2733,940],{},[50,2735,59],{"stretchy":52},[50,2737,315],{},[40,2739,2740,2742],{},[43,2741,1373],{},[43,2743,940],{},[50,2745,53],{"stretchy":52},[40,2747,2748,2750],{},[43,2749,45],{},[43,2751,940],{},[50,2753,59],{"stretchy":52},[50,2755,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2757,2682],{},[40,2759,2760,2762],{},[43,2761,1373],{},[43,2763,940],{},[50,2765,53],{"stretchy":52},[305,2767,2768,2770],{},[43,2769,45],{},[50,2771,2388],{},[50,2773,59],{"stretchy":52},[2606,2775,2776,2782],{},[2609,2777,2778],{},[2612,2779,2780],{"scriptlevel":541,"displaystyle":72},[37,2781],{},[2609,2783,2784],{},[2612,2785,2786],{"scriptlevel":541,"displaystyle":72},[37,2787,2788,2790,2792,2794,2800,2802,2808,2810,2812,2818,2820,2826,2828,2830,2832,2834,2840,2842,2848,2850,2852,2858,2860,2866,2868,2870,2872,2878,2880,2886],{},[37,2789],{},[50,2791,62],{},[50,2793,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2795,2796,2798],{},[43,2797,1373],{},[43,2799,906],{"mathvariant":596},[50,2801,53],{"stretchy":52},[40,2803,2804,2806],{},[43,2805,45],{},[43,2807,940],{},[50,2809,59],{"stretchy":52},[50,2811,315],{},[40,2813,2814,2816],{},[43,2815,1373],{},[43,2817,940],{},[50,2819,53],{"stretchy":52},[40,2821,2822,2824],{},[43,2823,45],{},[43,2825,940],{},[50,2827,59],{"stretchy":52},[50,2829,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2831,2682],{},[50,2833,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,2835,2836,2838],{},[43,2837,1373],{},[43,2839,940],{},[50,2841,53],{"stretchy":52},[305,2843,2844,2846],{},[43,2845,45],{},[50,2847,2388],{},[50,2849,59],{"stretchy":52},[50,2851,315],{},[40,2853,2854,2856],{},[43,2855,1373],{},[43,2857,906],{"mathvariant":596},[50,2859,53],{"stretchy":52},[305,2861,2862,2864],{},[43,2863,45],{},[50,2865,2388],{},[50,2867,59],{"stretchy":52},[50,2869,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,2871,2682],{},[40,2873,2874,2876],{},[43,2875,1373],{},[43,2877,906],{"mathvariant":596},[50,2879,53],{"stretchy":52},[305,2881,2882,2884],{},[43,2883,45],{},[50,2885,2388],{},[50,2887,59],{"stretchy":52},[2606,2889,2890,2896],{},[2609,2891,2892],{},[2612,2893,2894],{"scriptlevel":541,"displaystyle":72},[37,2895],{},[2609,2897,2898],{},[2612,2899,2900],{"scriptlevel":541,"displaystyle":72},[37,2901,2902,2904,2906,2912,2914,2920,2922,2924,2926,2943,2946,2952,2954,2956,2958,2960,2966,2968,2970,2972,2974],{},[37,2903],{},[50,2905,2717],{},[40,2907,2908,2910],{},[43,2909,1373],{},[43,2911,906],{"mathvariant":596},[50,2913,53],{"stretchy":52},[305,2915,2916,2918],{},[43,2917,45],{},[50,2919,2388],{},[50,2921,59],{"stretchy":52},[50,2923,2682],{},[308,2925,310],{},[2140,2927,2928,2935],{},[37,2929,2930,2933],{},[43,2931,2932],{},"sup",[50,2934,69],{},[37,2936,2937,2939,2941],{},[43,2938,45],{},[50,2940,100],{},[43,2942,819],{"mathvariant":596},[50,2944,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},"∣",[40,2947,2948,2950],{},[43,2949,1373],{},[43,2951,906],{"mathvariant":596},[50,2953,53],{"stretchy":52},[43,2955,45],{},[50,2957,59],{"stretchy":52},[50,2959,315],{},[40,2961,2962,2964],{},[43,2963,1373],{},[43,2965,940],{},[50,2967,53],{"stretchy":52},[43,2969,45],{},[50,2971,59],{"stretchy":52},[50,2973,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,2975,107],{"mathvariant":65},[109,2977,2978],{"encoding":111},"\\begin{aligned}\nL_{\\mathcal D}(h_S)\n&=\\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+L_S(h_S)\\\\\n&\\le \\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+L_S(h^*)\\\\\n&=\\bigl[L_{\\mathcal D}(h_S)-L_S(h_S)\\bigr]+\\bigl[L_S(h^*)-L_{\\mathcal D}(h^*)\\bigr]+L_{\\mathcal D}(h^*)\\\\\n&\\le L_{\\mathcal D}(h^*)+2\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal 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convergence）。",[11,5370,5371,5403,5404,5432,5433,5461,5462,5532,5533,5561],{},[15,5372,5374,5390],{"className":5373,"translate":19},[23],[15,5375,5377],{"className":5376},[27],[29,5378,5379],{"xmlns":31},[34,5380,5381,5387],{},[37,5382,5383,5385],{},[43,5384,2932],{},[50,5386,69],{},[109,5388,5389],{"encoding":111},"\\sup",[15,5391,5393],{"className":5392,"ariaHidden":72},[116],[15,5394,5396,5400],{"className":5395},[120],[15,5397],{"className":5398,"style":5399},[124],"height:0.625em;vertical-align:-0.1944em;",[15,5401,2932],{"className":5402},[217]," 在概率里面，这个顺序很关键：不是「对每个 ",[15,5405,5407,5420],{"className":5406,"translate":19},[23],[15,5408,5410],{"className":5409},[27],[29,5411,5412],{"xmlns":31},[34,5413,5414,5418],{},[37,5415,5416],{},[43,5417,45],{},[109,5419,45],{"encoding":111},[15,5421,5423],{"className":5422,"ariaHidden":72},[116],[15,5424,5426,5429],{"className":5425},[120],[15,5427],{"className":5428,"style":522},[124],[15,5430,45],{"className":5431},[129,133]," 分别收敛」，而是「所有 ",[15,5434,5436,5449],{"className":5435,"translate":19},[23],[15,5437,5439],{"className":5438},[27],[29,5440,5441],{"xmlns":31},[34,5442,5443,5447],{},[37,5444,5445],{},[43,5446,45],{},[109,5448,45],{"encoding":111},[15,5450,5452],{"className":5451,"ariaHidden":72},[116],[15,5453,5455,5458],{"className":5454},[120],[15,5456],{"className":5457,"style":522},[124],[15,5459,45],{"className":5460},[129,133]," 同时收敛」。前者由大数定律自动成立，后者才是学习所需要的——因为 ",[15,5463,5465,5483],{"className":5464,"translate":19},[23],[15,5466,5468],{"className":5467},[27],[29,5469,5470],{"xmlns":31},[34,5471,5472,5480],{},[37,5473,5474],{},[40,5475,5476,5478],{},[43,5477,45],{},[43,5479,940],{},[109,5481,5482],{"encoding":111},"h_S",[15,5484,5486],{"className":5485,"ariaHidden":72},[116],[15,5487,5489,5492],{"className":5488},[120],[15,5490],{"className":5491,"style":2183},[124],[15,5493,5495,5498],{"className":5494},[129],[15,5496,45],{"className":5497},[129,133],[15,5499,5501],{"className":5500},[137],[15,5502,5504,5524],{"className":5503},[141,142],[15,5505,5507,5521],{"className":5506},[146],[15,5508,5510],{"className":5509,"style":1525},[150],[15,5511,5512,5515],{"style":154},[15,5513],{"className":5514,"style":159},[158],[15,5516,5518],{"className":5517},[163,164,165,166],[15,5519,940],{"className":5520,"style":1013},[129,133,166],[15,5522,175],{"className":5523},[174],[15,5525,5527],{"className":5526},[146],[15,5528,5530],{"className":5529,"style":182},[150],[15,5531],{}," 本身依赖数据，你不能先固定 ",[15,5534,5536,5549],{"className":5535,"translate":19},[23],[15,5537,5539],{"className":5538},[27],[29,5540,5541],{"xmlns":31},[34,5542,5543,5547],{},[37,5544,5545],{},[43,5546,45],{},[109,5548,45],{"encoding":111},[15,5550,5552],{"className":5551,"ariaHidden":72},[116],[15,5553,5555,5558],{"className":5554},[120],[15,5556],{"className":5557,"style":522},[124],[15,5559,45],{"className":5560},[129,133]," 再用大数定律。",[11,5563,5564,5565,5593],{},"于是问题变成：什么样的 ",[15,5566,5568,5581],{"className":5567,"translate":19},[23],[15,5569,5571],{"className":5570},[27],[29,5572,5573],{"xmlns":31},[34,5574,5575,5579],{},[37,5576,5577],{},[43,5578,819],{"mathvariant":596},[109,5580,822],{"encoding":111},[15,5582,5584],{"className":5583,"ariaHidden":72},[116],[15,5585,5587,5590],{"className":5586},[120],[15,5588],{"className":5589,"style":610},[124],[15,5591,819],{"className":5592,"style":835},[129,614]," 具有一致收敛性？答案由 VC 维给出。",[570,5595,5597],{"id":5596},"二打散与-vc-维","二、打散与 VC 维",[575,5599,5601],{"id":5600},"_21-二分","2.1 二分",[11,5603,5604,5605,1313,5803,5831,5832,5860,5861,5864],{},"固定一个有限的点集 ",[15,5606,5608,5653],{"className":5607,"translate":19},[23],[15,5609,5611],{"className":5610},[27],[29,5612,5613],{"xmlns":31},[34,5614,5615,5650],{},[37,5616,5617,5620,5622,5624,5631,5633,5635,5637,5643,5645,5648],{},[43,5618,5619],{},"C",[50,5621,62],{},[50,5623,636],{"stretchy":52},[40,5625,5626,5629],{},[43,5627,5628],{},"c",[308,5630,318],{},[50,5632,91],{"separator":72},[50,5634,335],{},[50,5636,91],{"separator":72},[40,5638,5639,5641],{},[43,5640,5628],{},[43,5642,348],{},[50,5644,645],{"stretchy":52},[50,5646,5647],{},"⊆",[43,5649,597],{"mathvariant":596},[109,5651,5652],{"encoding":111},"C=\\{c_1,\\dots,c_m\\}\\subseteq\\mathcal X",[15,5654,5656,5675,5794],{"className":5655,"ariaHidden":72},[116],[15,5657,5659,5662,5666,5669,5672],{"className":5658},[120],[15,5660],{"className":5661,"style":610},[124],[15,5663,5619],{"className":5664,"style":5665},[129,133],"margin-right:0.0715em;",[15,5667],{"className":5668,"style":199},[93],[15,5670,62],{"className":5671},[203],[15,5673],{"className":5674,"style":199},[93],[15,5676,5678,5681,5684,5724,5727,5730,5733,5736,5739,5742,5782,5785,5788,5791],{"className":5677},[120],[15,5679],{"className":5680,"style":125},[124],[15,5682,636],{"className":5683},[188],[15,5685,5687,5690],{"className":5686},[129],[15,5688,5628],{"className":5689},[129,133],[15,5691,5693],{"className":5692},[137],[15,5694,5696,5716],{"className":5695},[141,142],[15,5697,5699,5713],{"className":5698},[146],[15,5700,5702],{"className":5701,"style":1051},[150],[15,5703,5704,5707],{"style":154},[15,5705],{"className":5706,"style":159},[158],[15,5708,5710],{"className":5709},[163,164,165,166],[15,5711,318],{"className":5712},[129,166],[15,5714,175],{"className":5715},[174],[15,5717,5719],{"className":5718},[146],[15,5720,5722],{"className":5721,"style":182},[150],[15,5723],{},[15,5725,91],{"className":5726},[254],[15,5728],{"className":5729,"style":262},[93],[15,5731,335],{"className":5732},[449],[15,5734],{"className":5735,"style":262},[93],[15,5737,91],{"className":5738},[254],[15,5740],{"className":5741,"style":262},[93],[15,5743,5745,5748],{"className":5744},[129],[15,5746,5628],{"className":5747},[129,133],[15,5749,5751],{"className":5750},[137],[15,5752,5754,5774],{"className":5753},[141,142],[15,5755,5757,5771],{"className":5756},[146],[15,5758,5760],{"className":5759,"style":1163},[150],[15,5761,5762,5765],{"style":154},[15,5763],{"className":5764,"style":159},[158],[15,5766,5768],{"className":5767},[163,164,165,166],[15,5769,348],{"className":5770},[129,133,166],[15,5772,175],{"className":5773},[174],[15,5775,5777],{"className":5776},[146],[15,5778,5780],{"className":5779,"style":182},[150],[15,5781],{},[15,5783,645],{"className":5784},[195],[15,5786],{"className":5787,"style":199},[93],[15,5789,5647],{"className":5790},[203],[15,5792],{"className":5793,"style":199},[93],[15,5795,5797,5800],{"className":5796},[120],[15,5798],{"className":5799,"style":610},[124],[15,5801,597],{"className":5802,"style":615},[129,614],[15,5804,5806,5819],{"className":5805,"translate":19},[23],[15,5807,5809],{"className":5808},[27],[29,5810,5811],{"xmlns":31},[34,5812,5813,5817],{},[37,5814,5815],{},[43,5816,819],{"mathvariant":596},[109,5818,822],{"encoding":111},[15,5820,5822],{"className":5821,"ariaHidden":72},[116],[15,5823,5825,5828],{"className":5824},[120],[15,5826],{"className":5827,"style":610},[124],[15,5829,819],{"className":5830,"style":835},[129,614]," 在 ",[15,5833,5835,5848],{"className":5834,"translate":19},[23],[15,5836,5838],{"className":5837},[27],[29,5839,5840],{"xmlns":31},[34,5841,5842,5846],{},[37,5843,5844],{},[43,5845,5619],{},[109,5847,5619],{"encoding":111},[15,5849,5851],{"className":5850,"ariaHidden":72},[116],[15,5852,5854,5857],{"className":5853},[120],[15,5855],{"className":5856,"style":610},[124],[15,5858,5619],{"className":5859,"style":5665},[129,133]," 上的",[698,5862,5863],{},"限制","（restriction）是",[15,5866,5868],{"className":5867,"translate":19},[18],[15,5869,5871,5957],{"className":5870,"translate":19},[23],[15,5872,5874],{"className":5873},[27],[29,5875,5876],{"xmlns":31,"display":32},[34,5877,5878,5954],{},[37,5879,5880,5882,5888,5890,5892,5894,5896,5898,5904,5906,5908,5910,5912,5914,5916,5922,5924,5926,5928,5930,5932,5934,5936,5938,5940,5942,5944,5946,5952],{},[43,5881,819],{"mathvariant":596},[40,5883,5884,5886],{},[43,5885,2945],{"mathvariant":65},[43,5887,5619],{},[50,5889,62],{},[50,5891,636],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,5893,53],{"stretchy":52},[43,5895,45],{},[50,5897,53],{"stretchy":52},[40,5899,5900,5902],{},[43,5901,5628],{},[308,5903,318],{},[50,5905,59],{"stretchy":52},[50,5907,91],{"separator":72},[50,5909,335],{},[50,5911,91],{"separator":72},[43,5913,45],{},[50,5915,53],{"stretchy":52},[40,5917,5918,5920],{},[43,5919,5628],{},[43,5921,348],{},[50,5923,59],{"stretchy":52},[50,5925,59],{"stretchy":52},[50,5927,718],{},[43,5929,45],{},[50,5931,100],{},[43,5933,819],{"mathvariant":596},[50,5935,645],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,5937,5647],{},[50,5939,636],{"stretchy":52},[308,5941,541],{},[50,5943,91],{"separator":72},[308,5945,318],{},[305,5947,5948,5950],{},[50,5949,645],{"stretchy":52},[43,5951,348],{},[43,5953,107],{"mathvariant":65},[109,5955,5956],{"encoding":111},"\\mathcal H|_C=\\bigl\\{(h(c_1),\\dots,h(c_m)):h\\in\\mathcal H\\bigr\\}\\subseteq\\{0,1\\}^m.",[15,5958,5960,6018,6159,6177,6201],{"className":5959,"ariaHidden":72},[116],[15,5961,5963,5966,5969,6009,6012,6015],{"className":5962},[120],[15,5964],{"className":5965,"style":125},[124],[15,5967,819],{"className":5968,"style":835},[129,614],[15,5970,5972,5975],{"className":5971},[129],[15,5973,2945],{"className":5974},[129],[15,5976,5978],{"className":5977},[137],[15,5979,5981,6001],{"className":5980},[141,142],[15,5982,5984,5998],{"className":5983},[146],[15,5985,5987],{"className":5986,"style":1525},[150],[15,5988,5989,5992],{"style":154},[15,5990],{"className":5991,"style":159},[158],[15,5993,5995],{"className":5994},[163,164,165,166],[15,5996,5619],{"className":5997,"style":5665},[129,133,166],[15,5999,175],{"className":6000},[174],[15,6002,6004],{"className":6003},[146],[15,6005,6007],{"className":6006,"style":182},[150],[15,6008],{},[15,6010],{"className":6011,"style":199},[93],[15,6013,62],{"className":6014},[203],[15,6016],{"className":6017,"style":199},[93],[15,6019,6021,6024,6030,6033,6036,6039,6079,6082,6085,6088,6091,6094,6097,6100,6103,6106,6146,6150,6153,6156],{"className":6020},[120],[15,6022],{"className":6023,"style":213},[124],[15,6025,6027],{"className":6026},[188],[15,6028,636],{"className":6029},[228,229],[15,6031,53],{"className":6032},[188],[15,6034,45],{"className":6035},[129,133],[15,6037,53],{"className":6038},[188],[15,6040,6042,6045],{"className":6041},[129],[15,6043,5628],{"className":6044},[129,133],[15,6046,6048],{"className":6047},[137],[15,6049,6051,6071],{"className":6050},[141,142],[15,6052,6054,6068],{"className":6053},[146],[15,6055,6057],{"className":6056,"style":1051},[150],[15,6058,6059,6062],{"style":154},[15,6060],{"className":6061,"style":159},[158],[15,6063,6065],{"className":6064},[163,164,165,166],[15,6066,318],{"className":6067},[129,166],[15,6069,175],{"className":6070},[174],[15,6072,6074],{"className":6073},[146],[15,6075,6077],{"className":6076,"style":182},[150],[15,6078],{},[15,6080,59],{"className":6081},[195],[15,6083,91],{"className":6084},[254],[15,6086],{"className":6087,"style":262},[93],[15,6089,335],{"className":6090},[449],[15,6092],{"className":6093,"style":262},[93],[15,6095,91],{"className":6096},[254],[15,6098],{"className":6099,"style":262},[93],[15,6101,45],{"className":6102},[129,133],[15,6104,53],{"className":6105},[188],[15,6107,6109,6112],{"className":6108},[129],[15,6110,5628],{"className":6111},[129,133],[15,6113,6115],{"className":6114},[137],[15,6116,6118,6138],{"className":6117},[141,142],[15,6119,6121,6135],{"className":6120},[146],[15,6122,6124],{"className":6123,"style":1163},[150],[15,6125,6126,6129],{"style":154},[15,6127],{"className":6128,"style":159},[158],[15,6130,6132],{"className":6131},[163,164,165,166],[15,6133,348],{"className":6134},[129,133,166],[15,6136,175],{"className":6137},[174],[15,6139,6141],{"className":6140},[146],[15,6142,6144],{"className":6143,"style":182},[150],[15,6145],{},[15,6147,6149],{"className":6148},[195],"))",[15,6151],{"className":6152,"style":199},[93],[15,6154,718],{"className":6155},[203],[15,6157],{"className":6158,"style":199},[93],[15,6160,6162,6165,6168,6171,6174],{"className":6161},[120],[15,6163],{"className":6164,"style":2405},[124],[15,6166,45],{"className":6167},[129,133],[15,6169],{"className":6170,"style":199},[93],[15,6172,100],{"className":6173},[203],[15,6175],{"className":6176,"style":199},[93],[15,6178,6180,6183,6186,6192,6195,6198],{"className":6179},[120],[15,6181],{"className":6182,"style":213},[124],[15,6184,819],{"className":6185,"style":835},[129,614],[15,6187,6189],{"className":6188},[195],[15,6190,645],{"className":6191},[228,229],[15,6193],{"className":6194,"style":199},[93],[15,6196,5647],{"className":6197},[203],[15,6199],{"className":6200,"style":199},[93],[15,6202,6204,6207,6210,6213,6216,6219,6222,6251],{"className":6203},[120],[15,6205],{"className":6206,"style":125},[124],[15,6208,636],{"className":6209},[188],[15,6211,541],{"className":6212},[129],[15,6214,91],{"className":6215},[254],[15,6217],{"className":6218,"style":262},[93],[15,6220,318],{"className":6221},[129],[15,6223,6225,6228],{"className":6224},[195],[15,6226,645],{"className":6227},[195],[15,6229,6231],{"className":6230},[137],[15,6232,6234],{"className":6233},[141],[15,6235,6237],{"className":6236},[146],[15,6238,6240],{"className":6239,"style":1251},[150],[15,6241,6242,6245],{"style":1272},[15,6243],{"className":6244,"style":159},[158],[15,6246,6248],{"className":6247},[163,164,165,166],[15,6249,348],{"className":6250},[129,133,166],[15,6252,107],{"className":6253},[129],[11,6255,6256,6257,6260,6261,6346,6347,6407,6408,6466,6467,6495,6496,6528,6529,6557],{},"这个集合里的每个元素称为一个",[698,6258,6259],{},"二分","（dichotomy）。",[15,6262,6264,6288],{"className":6263,"translate":19},[23],[15,6265,6267],{"className":6266},[27],[29,6268,6269],{"xmlns":31},[34,6270,6271,6285],{},[37,6272,6273,6275,6277,6283],{},[43,6274,2945],{"mathvariant":65},[43,6276,819],{"mathvariant":596},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H|_C|",[15,6289,6291],{"className":6290,"ariaHidden":72},[116],[15,6292,6294,6297,6300,6303,6343],{"className":6293},[120],[15,6295],{"className":6296,"style":125},[124],[15,6298,2945],{"className":6299},[129],[15,6301,819],{"className":6302,"style":835},[129,614],[15,6304,6306,6309],{"className":6305},[129],[15,6307,2945],{"className":6308},[129],[15,6310,6312],{"className":6311},[137],[15,6313,6315,6335],{"className":6314},[141,142],[15,6316,6318,6332],{"className":6317},[146],[15,6319,6321],{"className":6320,"style":1525},[150],[15,6322,6323,6326],{"style":154},[15,6324],{"className":6325,"style":159},[158],[15,6327,6329],{"className":6328},[163,164,165,166],[15,6330,5619],{"className":6331,"style":5665},[129,133,166],[15,6333,175],{"className":6334},[174],[15,6336,6338],{"className":6337},[146],[15,6339,6341],{"className":6340,"style":182},[150],[15,6342],{},[15,6344,2945],{"className":6345},[129]," 最多是 ",[15,6348,6350,6368],{"className":6349,"translate":19},[23],[15,6351,6353],{"className":6352},[27],[29,6354,6355],{"xmlns":31},[34,6356,6357,6365],{},[37,6358,6359],{},[305,6360,6361,6363],{},[308,6362,310],{},[43,6364,348],{},[109,6366,6367],{"encoding":111},"2^m",[15,6369,6371],{"className":6370,"ariaHidden":72},[116],[15,6372,6374,6378],{"className":6373},[120],[15,6375],{"className":6376,"style":6377},[124],"height:0.6644em;",[15,6379,6381,6384],{"className":6380},[129],[15,6382,310],{"className":6383},[129],[15,6385,6387],{"className":6386},[137],[15,6388,6390],{"className":6389},[141],[15,6391,6393],{"className":6392},[146],[15,6394,6396],{"className":6395,"style":6377},[150],[15,6397,6398,6401],{"style":383},[15,6399],{"className":6400,"style":159},[158],[15,6402,6404],{"className":6403},[163,164,165,166],[15,6405,348],{"className":6406},[129,133,166],"。如果取到 ",[15,6409,6411,6428],{"className":6410,"translate":19},[23],[15,6412,6414],{"className":6413},[27],[29,6415,6416],{"xmlns":31},[34,6417,6418,6426],{},[37,6419,6420],{},[305,6421,6422,6424],{},[308,6423,310],{},[43,6425,348],{},[109,6427,6367],{"encoding":111},[15,6429,6431],{"className":6430,"ariaHidden":72},[116],[15,6432,6434,6437],{"className":6433},[120],[15,6435],{"className":6436,"style":6377},[124],[15,6438,6440,6443],{"className":6439},[129],[15,6441,310],{"className":6442},[129],[15,6444,6446],{"className":6445},[137],[15,6447,6449],{"className":6448},[141],[15,6450,6452],{"className":6451},[146],[15,6453,6455],{"className":6454,"style":6377},[150],[15,6456,6457,6460],{"style":383},[15,6458],{"className":6459,"style":159},[158],[15,6461,6463],{"className":6462},[163,164,165,166],[15,6464,348],{"className":6465},[129,133,166],"，说明对这 ",[15,6468,6470,6483],{"className":6469,"translate":19},[23],[15,6471,6473],{"className":6472},[27],[29,6474,6475],{"xmlns":31},[34,6476,6477,6481],{},[37,6478,6479],{},[43,6480,348],{},[109,6482,348],{"encoding":111},[15,6484,6486],{"className":6485,"ariaHidden":72},[116],[15,6487,6489,6492],{"className":6488},[120],[15,6490],{"className":6491,"style":4383},[124],[15,6493,348],{"className":6494},[129,133]," 个点的每一种 ",[15,6497,6499,6516],{"className":6498,"translate":19},[23],[15,6500,6502],{"className":6501},[27],[29,6503,6504],{"xmlns":31},[34,6505,6506,6514],{},[37,6507,6508,6510,6512],{},[308,6509,541],{},[43,6511,544],{"mathvariant":65},[308,6513,318],{},[109,6515,549],{"encoding":111},[15,6517,6519],{"className":6518,"ariaHidden":72},[116],[15,6520,6522,6525],{"className":6521},[120],[15,6523],{"className":6524,"style":125},[124],[15,6526,549],{"className":6527},[129]," 标注，",[15,6530,6532,6545],{"className":6531,"translate":19},[23],[15,6533,6535],{"className":6534},[27],[29,6536,6537],{"xmlns":31},[34,6538,6539,6543],{},[37,6540,6541],{},[43,6542,819],{"mathvariant":596},[109,6544,822],{"encoding":111},[15,6546,6548],{"className":6547,"ariaHidden":72},[116],[15,6549,6551,6554],{"className":6550},[120],[15,6552],{"className":6553,"style":610},[124],[15,6555,819],{"className":6556,"style":835},[129,614]," 里都有一个假设恰好实现它。",[6559,6560,6561],"blockquote",{},[11,6562,6563,6566,6567,6704,6705,804,6733,6736,6737,1313],{},[698,6564,6565],{},"定义 2.1（打散）"," 若 ",[15,6568,6570,6602],{"className":6569,"translate":19},[23],[15,6571,6573],{"className":6572},[27],[29,6574,6575],{"xmlns":31},[34,6576,6577,6599],{},[37,6578,6579,6581,6583,6589,6591,6593],{},[43,6580,2945],{"mathvariant":65},[43,6582,819],{"mathvariant":596},[40,6584,6585,6587],{},[43,6586,2945],{"mathvariant":65},[43,6588,5619],{},[43,6590,2945],{"mathvariant":65},[50,6592,62],{},[305,6594,6595,6597],{},[308,6596,310],{},[43,6598,348],{},[109,6600,6601],{"encoding":111},"|\\mathcal H|_C|=2^m",[15,6603,6605,6669],{"className":6604,"ariaHidden":72},[116],[15,6606,6608,6611,6614,6617,6657,6660,6663,6666],{"className":6607},[120],[15,6609],{"className":6610,"style":125},[124],[15,6612,2945],{"className":6613},[129],[15,6615,819],{"className":6616,"style":835},[129,614],[15,6618,6620,6623],{"className":6619},[129],[15,6621,2945],{"className":6622},[129],[15,6624,6626],{"className":6625},[137],[15,6627,6629,6649],{"className":6628},[141,142],[15,6630,6632,6646],{"className":6631},[146],[15,6633,6635],{"className":6634,"style":1525},[150],[15,6636,6637,6640],{"style":154},[15,6638],{"className":6639,"style":159},[158],[15,6641,6643],{"className":6642},[163,164,165,166],[15,6644,5619],{"className":6645,"style":5665},[129,133,166],[15,6647,175],{"className":6648},[174],[15,6650,6652],{"className":6651},[146],[15,6653,6655],{"className":6654,"style":182},[150],[15,6656],{},[15,6658,2945],{"className":6659},[129],[15,6661],{"className":6662,"style":199},[93],[15,6664,62],{"className":6665},[203],[15,6667],{"className":6668,"style":199},[93],[15,6670,6672,6675],{"className":6671},[120],[15,6673],{"className":6674,"style":6377},[124],[15,6676,6678,6681],{"className":6677},[129],[15,6679,310],{"className":6680},[129],[15,6682,6684],{"className":6683},[137],[15,6685,6687],{"className":6686},[141],[15,6688,6690],{"className":6689},[146],[15,6691,6693],{"className":6692,"style":6377},[150],[15,6694,6695,6698],{"style":383},[15,6696],{"className":6697,"style":159},[158],[15,6699,6701],{"className":6700},[163,164,165,166],[15,6702,348],{"className":6703},[129,133,166],"，称 ",[15,6706,6708,6721],{"className":6707,"translate":19},[23],[15,6709,6711],{"className":6710},[27],[29,6712,6713],{"xmlns":31},[34,6714,6715,6719],{},[37,6716,6717],{},[43,6718,819],{"mathvariant":596},[109,6720,822],{"encoding":111},[15,6722,6724],{"className":6723,"ariaHidden":72},[116],[15,6725,6727,6730],{"className":6726},[120],[15,6728],{"className":6729,"style":610},[124],[15,6731,819],{"className":6732,"style":835},[129,614],[698,6734,6735],{},"打散","（shatter）了 ",[15,6738,6740,6753],{"className":6739,"translate":19},[23],[15,6741,6743],{"className":6742},[27],[29,6744,6745],{"xmlns":31},[34,6746,6747,6751],{},[37,6748,6749],{},[43,6750,5619],{},[109,6752,5619],{"encoding":111},[15,6754,6756],{"className":6755,"ariaHidden":72},[116],[15,6757,6759,6762],{"className":6758},[120],[15,6760],{"className":6761,"style":610},[124],[15,6763,5619],{"className":6764,"style":5665},[129,133],[6559,6766,6767,6806,7138],{},[11,6768,6769,6772,6773,6801,6802,6805],{},[698,6770,6771],{},"定义 2.2（VC 维）"," 假设类 ",[15,6774,6776,6789],{"className":6775,"translate":19},[23],[15,6777,6779],{"className":6778},[27],[29,6780,6781],{"xmlns":31},[34,6782,6783,6787],{},[37,6784,6785],{},[43,6786,819],{"mathvariant":596},[109,6788,822],{"encoding":111},[15,6790,6792],{"className":6791,"ariaHidden":72},[116],[15,6793,6795,6798],{"className":6794},[120],[15,6796],{"className":6797,"style":610},[124],[15,6799,819],{"className":6800,"style":835},[129,614]," 的 ",[698,6803,6804],{},"VC 维","是它能打散的最大有限集的大小：",[15,6807,6809],{"className":6808,"translate":19},[18],[15,6810,6812,6900],{"className":6811,"translate":19},[23],[15,6813,6815],{"className":6814},[27],[29,6816,6817],{"xmlns":31,"display":32},[34,6818,6819,6897],{},[37,6820,6821,6824,6826,6828,6830,6832,6834,6836,6838,6840,6842,6844,6847,6849,6851,6853,6855,6859,6861,6863,6865,6867,6869,6871,6873,6875,6877,6883,6885,6887,6893,6895],{},[43,6822,6823],{"mathvariant":65},"VCdim",[50,6825,69],{},[50,6827,53],{"stretchy":52},[43,6829,819],{"mathvariant":596},[50,6831,59],{"stretchy":52},[50,6833,62],{},[43,6835,2932],{},[50,6837,69],{},[50,6839,636],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,6841,348],{},[50,6843,718],{},[43,6845,6846],{"mathvariant":65},"∃",[43,6848,5619],{},[50,6850,5647],{},[43,6852,597],{"mathvariant":596},[50,6854,91],{"separator":72},[6856,6857,6858],"mtext",{}," ",[43,6860,2945],{"mathvariant":65},[43,6862,5619],{},[43,6864,2945],{"mathvariant":65},[50,6866,62],{},[43,6868,348],{},[50,6870,91],{"separator":72},[6856,6872,6858],{},[50,6874,2945],{"stretchy":52},[43,6876,819],{"mathvariant":596},[40,6878,6879,6881],{},[43,6880,2945],{"mathvariant":65},[43,6882,5619],{},[50,6884,2945],{"stretchy":52},[50,6886,62],{},[305,6888,6889,6891],{},[308,6890,310],{},[43,6892,348],{},[50,6894,645],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,6896,107],{"mathvariant":65},[109,6898,6899],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)=\\sup\\bigl\\{m:\\exists C\\subseteq\\mathcal X,\\ |C|=m,\\ \\lvert\\mathcal H|_C\\rvert=2^m\\bigr\\}.",[15,6901,6903,6933,6960,6982,7018,7094],{"className":6902,"ariaHidden":72},[116],[15,6904,6906,6909,6915,6918,6921,6924,6927,6930],{"className":6905},[120],[15,6907],{"className":6908,"style":125},[124],[15,6910,6912],{"className":6911},[217],[15,6913,6823],{"className":6914},[129,221],[15,6916,53],{"className":6917},[188],[15,6919,819],{"className":6920,"style":835},[129,614],[15,6922,59],{"className":6923},[195],[15,6925],{"className":6926,"style":199},[93],[15,6928,62],{"className":6929},[203],[15,6931],{"className":6932,"style":199},[93],[15,6934,6936,6939,6942,6948,6951,6954,6957],{"className":6935},[120],[15,6937],{"className":6938,"style":213},[124],[15,6940,2932],{"className":6941},[217],[15,6943,6945],{"className":6944},[188],[15,6946,636],{"className":6947},[228,229],[15,6949,348],{"className":6950},[129,133],[15,6952],{"className":6953,"style":199},[93],[15,6955,718],{"className":6956},[203],[15,6958],{"className":6959,"style":199},[93],[15,6961,6963,6967,6970,6973,6976,6979],{"className":6962},[120],[15,6964],{"className":6965,"style":6966},[124],"height:0.8304em;vertical-align:-0.136em;",[15,6968,6846],{"className":6969},[129],[15,6971,5619],{"className":6972,"style":5665},[129,133],[15,6974],{"className":6975,"style":199},[93],[15,6977,5647],{"className":6978},[203],[15,6980],{"className":6981,"style":199},[93],[15,6983,6985,6988,6991,6994,6997,7000,7003,7006,7009,7012,7015],{"className":6984},[120],[15,6986],{"className":6987,"style":125},[124],[15,6989,597],{"className":6990,"style":615},[129,614],[15,6992,91],{"className":6993},[254],[15,6995,6858],{"className":6996},[93],[15,6998],{"className":6999,"style":262},[93],[15,7001,2945],{"className":7002},[129],[15,7004,5619],{"className":7005,"style":5665},[129,133],[15,7007,2945],{"className":7008},[129],[15,7010],{"className":7011,"style":199},[93],[15,7013,62],{"className":7014},[203],[15,7016],{"className":7017,"style":199},[93],[15,7019,7021,7024,7027,7030,7033,7036,7039,7042,7082,7085,7088,7091],{"className":7020},[120],[15,7022],{"className":7023,"style":125},[124],[15,7025,348],{"className":7026},[129,133],[15,7028,91],{"className":7029},[254],[15,7031,6858],{"className":7032},[93],[15,7034],{"className":7035,"style":262},[93],[15,7037,2945],{"className":7038},[188],[15,7040,819],{"className":7041,"style":835},[129,614],[15,7043,7045,7048],{"className":7044},[129],[15,7046,2945],{"className":7047},[129],[15,7049,7051],{"className":7050},[137],[15,7052,7054,7074],{"className":7053},[141,142],[15,7055,7057,7071],{"className":7056},[146],[15,7058,7060],{"className":7059,"style":1525},[150],[15,7061,7062,7065],{"style":154},[15,7063],{"className":7064,"style":159},[158],[15,7066,7068],{"className":7067},[163,164,165,166],[15,7069,5619],{"className":7070,"style":5665},[129,133,166],[15,7072,175],{"className":7073},[174],[15,7075,7077],{"className":7076},[146],[15,7078,7080],{"className":7079,"style":182},[150],[15,7081],{},[15,7083,2945],{"className":7084},[195],[15,7086],{"className":7087,"style":199},[93],[15,7089,62],{"className":7090},[203],[15,7092],{"className":7093,"style":199},[93],[15,7095,7097,7100,7129,7135],{"className":7096},[120],[15,7098],{"className":7099,"style":213},[124],[15,7101,7103,7106],{"className":7102},[129],[15,7104,310],{"className":7105},[129],[15,7107,7109],{"className":7108},[137],[15,7110,7112],{"className":7111},[141],[15,7113,7115],{"className":7114},[146],[15,7116,7118],{"className":7117,"style":1251},[150],[15,7119,7120,7123],{"style":1272},[15,7121],{"className":7122,"style":159},[158],[15,7124,7126],{"className":7125},[163,164,165,166],[15,7127,348],{"className":7128},[129,133,166],[15,7130,7132],{"className":7131},[195],[15,7133,645],{"className":7134},[228,229],[15,7136,107],{"className":7137},[129],[11,7139,7140,7141,1313],{},"若右侧无界，则记 ",[15,7142,7144,7170],{"className":7143,"translate":19},[23],[15,7145,7147],{"className":7146},[27],[29,7148,7149],{"xmlns":31},[34,7150,7151,7167],{},[37,7152,7153,7155,7157,7159,7161,7163,7165],{},[43,7154,6823],{"mathvariant":65},[50,7156,69],{},[50,7158,53],{"stretchy":52},[43,7160,819],{"mathvariant":596},[50,7162,59],{"stretchy":52},[50,7164,62],{},[43,7166,4909],{"mathvariant":65},[109,7168,7169],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)=\\infty",[15,7171,7173,7203],{"className":7172,"ariaHidden":72},[116],[15,7174,7176,7179,7185,7188,7191,7194,7197,7200],{"className":7175},[120],[15,7177],{"className":7178,"style":125},[124],[15,7180,7182],{"className":7181},[217],[15,7183,6823],{"className":7184},[129,221],[15,7186,53],{"className":7187},[188],[15,7189,819],{"className":7190,"style":835},[129,614],[15,7192,59],{"className":7193},[195],[15,7195],{"className":7196,"style":199},[93],[15,7198,62],{"className":7199},[203],[15,7201],{"className":7202,"style":199},[93],[15,7204,7206,7209],{"className":7205},[120],[15,7207],{"className":7208,"style":4383},[124],[15,7210,4909],{"className":7211},[129],[11,7213,7214,7215,7218,7219,7247,7248,804,7251,7309,7310,804,7312,7340,7341,7369],{},"请把定义里的量词读三遍：",[698,7216,7217],{},"存在","一个点集 ",[15,7220,7222,7235],{"className":7221,"translate":19},[23],[15,7223,7225],{"className":7224},[27],[29,7226,7227],{"xmlns":31},[34,7228,7229,7233],{},[37,7230,7231],{},[43,7232,5619],{},[109,7234,5619],{"encoding":111},[15,7236,7238],{"className":7237,"ariaHidden":72},[116],[15,7239,7241,7244],{"className":7240},[120],[15,7242],{"className":7243,"style":610},[124],[15,7245,5619],{"className":7246,"style":5665},[129,133],"，使得对",[698,7249,7250],{},"所有",[15,7252,7254,7271],{"className":7253,"translate":19},[23],[15,7255,7257],{"className":7256},[27],[29,7258,7259],{"xmlns":31},[34,7260,7261,7269],{},[37,7262,7263],{},[305,7264,7265,7267],{},[308,7266,310],{},[43,7268,348],{},[109,7270,6367],{"encoding":111},[15,7272,7274],{"className":7273,"ariaHidden":72},[116],[15,7275,7277,7280],{"className":7276},[120],[15,7278],{"className":7279,"style":6377},[124],[15,7281,7283,7286],{"className":7282},[129],[15,7284,310],{"className":7285},[129],[15,7287,7289],{"className":7288},[137],[15,7290,7292],{"className":7291},[141],[15,7293,7295],{"className":7294},[146],[15,7296,7298],{"className":7297,"style":6377},[150],[15,7299,7300,7303],{"style":383},[15,7301],{"className":7302,"style":159},[158],[15,7304,7306],{"className":7305},[163,164,165,166],[15,7307,348],{"className":7308},[129,133,166]," 种标注，都",[698,7311,7217],{},[15,7313,7315,7328],{"className":7314,"translate":19},[23],[15,7316,7318],{"className":7317},[27],[29,7319,7320],{"xmlns":31},[34,7321,7322,7326],{},[37,7323,7324],{},[43,7325,45],{},[109,7327,45],{"encoding":111},[15,7329,7331],{"className":7330,"ariaHidden":72},[116],[15,7332,7334,7337],{"className":7333},[120],[15,7335],{"className":7336,"style":522},[124],[15,7338,45],{"className":7339},[129,133],"。不是「对所有 ",[15,7342,7344,7357],{"className":7343,"translate":19},[23],[15,7345,7347],{"className":7346},[27],[29,7348,7349],{"xmlns":31},[34,7350,7351,7355],{},[37,7352,7353],{},[43,7354,5619],{},[109,7356,5619],{"encoding":111},[15,7358,7360],{"className":7359,"ariaHidden":72},[116],[15,7361,7363,7366],{"className":7362},[120],[15,7364],{"className":7365,"style":610},[124],[15,7367,5619],{"className":7368,"style":5665},[129,133],"」，也不是「对某一种标注」。",[11,7371,7372],{},"这个「存在」是 VC 维最容易被误读的地方，也是它作为最坏情况度量的根源：它问的是假设类在某个精心挑选的点集上最多能变出多少花样，而不是它在典型数据上会怎么做。",[575,7374,7376],{"id":7375},"_22-几个例子","2.2 几个例子",[11,7378,7379],{},[698,7380,7381,7382,7438],{},"例 1（半直线，",[15,7383,7385,7405],{"className":7384,"translate":19},[23],[15,7386,7388],{"className":7387},[27],[29,7389,7390],{"xmlns":31},[34,7391,7392,7402],{},[37,7393,7394,7396,7398,7400],{},[43,7395,6823],{"mathvariant":65},[50,7397,69],{},[50,7399,62],{},[308,7401,318],{},[109,7403,7404],{"encoding":111},"\\operatorname{VCdim}=1",[15,7406,7408,7429],{"className":7407,"ariaHidden":72},[116],[15,7409,7411,7414,7420,7423,7426],{"className":7410},[120],[15,7412],{"className":7413,"style":522},[124],[15,7415,7417],{"className":7416},[217],[15,7418,6823],{"className":7419},[129,221],[15,7421],{"className":7422,"style":199},[93],[15,7424,62],{"className":7425},[203],[15,7427],{"className":7428,"style":199},[93],[15,7430,7432,7435],{"className":7431},[120],[15,7433],{"className":7434,"style":4886},[124],[15,7436,318],{"className":7437},[129],"）",[11,7440,7441,7442,7610,7611,1313],{},"取 ",[15,7443,7445,7492],{"className":7444,"translate":19},[23],[15,7446,7448],{"className":7447},[27],[29,7449,7450],{"xmlns":31},[34,7451,7452,7489],{},[37,7453,7454,7456,7458,7460,7462,7465,7467,7469,7471,7474,7477,7479,7481,7483,7485,7487],{},[43,7455,819],{"mathvariant":596},[50,7457,62],{},[50,7459,636],{"stretchy":52},[43,7461,56],{},[50,7463,7464],{},"↦",[308,7466,318],{"mathvariant":1468},[50,7468,1407],{"stretchy":52},[43,7470,56],{},[50,7472,7473],{},"≥",[43,7475,7476],{},"t",[50,7478,1423],{"stretchy":52},[50,7480,718],{},[43,7482,7476],{},[50,7484,100],{},[43,7486,104],{"mathvariant":103},[50,7488,645],{"stretchy":52},[109,7490,7491],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\mathbf 1[x\\ge t]:t\\in\\mathbb R\\}",[15,7493,7495,7513,7534,7558,7579,7598],{"className":7494,"ariaHidden":72},[116],[15,7496,7498,7501,7504,7507,7510],{"className":7497},[120],[15,7499],{"className":7500,"style":610},[124],[15,7502,819],{"className":7503,"style":835},[129,614],[15,7505],{"className":7506,"style":199},[93],[15,7508,62],{"className":7509},[203],[15,7511],{"className":7512,"style":199},[93],[15,7514,7516,7519,7522,7525,7528,7531],{"className":7515},[120],[15,7517],{"className":7518,"style":125},[124],[15,7520,636],{"className":7521},[188],[15,7523,56],{"className":7524},[129,133],[15,7526],{"className":7527,"style":199},[93],[15,7529,7464],{"className":7530},[203],[15,7532],{"className":7533,"style":199},[93],[15,7535,7537,7540,7543,7546,7549,7552,7555],{"className":7536},[120],[15,7538],{"className":7539,"style":125},[124],[15,7541,318],{"className":7542},[129,1950],[15,7544,1407],{"className":7545},[188],[15,7547,56],{"className":7548},[129,133],[15,7550],{"className":7551,"style":199},[93],[15,7553,7473],{"className":7554},[203],[15,7556],{"className":7557,"style":199},[93],[15,7559,7561,7564,7567,7570,7573,7576],{"className":7560},[120],[15,7562],{"className":7563,"style":125},[124],[15,7565,7476],{"className":7566},[129,133],[15,7568,1423],{"className":7569},[195],[15,7571],{"className":7572,"style":199},[93],[15,7574,718],{"className":7575},[203],[15,7577],{"className":7578,"style":199},[93],[15,7580,7582,7586,7589,7592,7595],{"className":7581},[120],[15,7583],{"className":7584,"style":7585},[124],"height:0.6542em;vertical-align:-0.0391em;",[15,7587,7476],{"className":7588},[129,133],[15,7590],{"className":7591,"style":199},[93],[15,7593,100],{"className":7594},[203],[15,7596],{"className":7597,"style":199},[93],[15,7599,7601,7604,7607],{"className":7600},[120],[15,7602],{"className":7603,"style":125},[124],[15,7605,104],{"className":7606},[129,285],[15,7608,645],{"className":7609},[195],"，",[15,7612,7614,7632],{"className":7613,"translate":19},[23],[15,7615,7617],{"className":7616},[27],[29,7618,7619],{"xmlns":31},[34,7620,7621,7629],{},[37,7622,7623,7625,7627],{},[43,7624,597],{"mathvariant":596},[50,7626,62],{},[43,7628,104],{"mathvariant":103},[109,7630,7631],{"encoding":111},"\\mathcal X=\\mathbb R",[15,7633,7635,7653],{"className":7634,"ariaHidden":72},[116],[15,7636,7638,7641,7644,7647,7650],{"className":7637},[120],[15,7639],{"className":7640,"style":610},[124],[15,7642,597],{"className":7643,"style":615},[129,614],[15,7645],{"className":7646,"style":199},[93],[15,7648,62],{"className":7649},[203],[15,7651],{"className":7652,"style":199},[93],[15,7654,7656,7659],{"className":7655},[120],[15,7657],{"className":7658,"style":281},[124],[15,7660,104],{"className":7661},[129,285],[11,7663,7664,7665,7726,7727,7779,7780,7610,7808,7779,7859,1313],{},"一个点 ",[15,7666,7668,7690],{"className":7667,"translate":19},[23],[15,7669,7671],{"className":7670},[27],[29,7672,7673],{"xmlns":31},[34,7674,7675,7687],{},[37,7676,7677,7679,7681,7683,7685],{},[43,7678,5619],{},[50,7680,62],{},[50,7682,636],{"stretchy":52},[308,7684,541],{},[50,7686,645],{"stretchy":52},[109,7688,7689],{"encoding":111},"C=\\{0\\}",[15,7691,7693,7711],{"className":7692,"ariaHidden":72},[116],[15,7694,7696,7699,7702,7705,7708],{"className":7695},[120],[15,7697],{"className":7698,"style":610},[124],[15,7700,5619],{"className":7701,"style":5665},[129,133],[15,7703],{"className":7704,"style":199},[93],[15,7706,62],{"className":7707},[203],[15,7709],{"className":7710,"style":199},[93],[15,7712,7714,7717,7720,7723],{"className":7713},[120],[15,7715],{"className":7716,"style":125},[124],[15,7718,636],{"className":7719},[188],[15,7721,541],{"className":7722},[129],[15,7724,645],{"className":7725},[195]," 显然可以被打散：",[15,7728,7730,7748],{"className":7729,"translate":19},[23],[15,7731,7733],{"className":7732},[27],[29,7734,7735],{"xmlns":31},[34,7736,7737,7745],{},[37,7738,7739,7741,7743],{},[43,7740,7476],{},[50,7742,2717],{},[308,7744,541],{},[109,7746,7747],{"encoding":111},"t\\le 0",[15,7749,7751,7770],{"className":7750,"ariaHidden":72},[116],[15,7752,7754,7758,7761,7764,7767],{"className":7753},[120],[15,7755],{"className":7756,"style":7757},[124],"height:0.7719em;vertical-align:-0.136em;",[15,7759,7476],{"className":7760},[129,133],[15,7762],{"className":7763,"style":199},[93],[15,7765,2717],{"className":7766},[203],[15,7768],{"className":7769,"style":199},[93],[15,7771,7773,7776],{"className":7772},[120],[15,7774],{"className":7775,"style":4886},[124],[15,7777,541],{"className":7778},[129]," 输出 ",[15,7781,7783,7796],{"className":7782,"translate":19},[23],[15,7784,7786],{"className":7785},[27],[29,7787,7788],{"xmlns":31},[34,7789,7790,7794],{},[37,7791,7792],{},[308,7793,318],{},[109,7795,318],{"encoding":111},[15,7797,7799],{"className":7798,"ariaHidden":72},[116],[15,7800,7802,7805],{"className":7801},[120],[15,7803],{"className":7804,"style":4886},[124],[15,7806,318],{"className":7807},[129],[15,7809,7811,7829],{"className":7810,"translate":19},[23],[15,7812,7814],{"className":7813},[27],[29,7815,7816],{"xmlns":31},[34,7817,7818,7826],{},[37,7819,7820,7822,7824],{},[43,7821,7476],{},[50,7823,4852],{},[308,7825,541],{},[109,7827,7828],{"encoding":111},"t>0",[15,7830,7832,7850],{"className":7831,"ariaHidden":72},[116],[15,7833,7835,7838,7841,7844,7847],{"className":7834},[120],[15,7836],{"className":7837,"style":7585},[124],[15,7839,7476],{"className":7840},[129,133],[15,7842],{"className":7843,"style":199},[93],[15,7845,4852],{"className":7846},[203],[15,7848],{"className":7849,"style":199},[93],[15,7851,7853,7856],{"className":7852},[120],[15,7854],{"className":7855,"style":4886},[124],[15,7857,541],{"className":7858},[129],[15,7860,7862,7875],{"className":7861,"translate":19},[23],[15,7863,7865],{"className":7864},[27],[29,7866,7867],{"xmlns":31},[34,7868,7869,7873],{},[37,7870,7871],{},[308,7872,541],{},[109,7874,541],{"encoding":111},[15,7876,7878],{"className":7877,"ariaHidden":72},[116],[15,7879,7881,7884],{"className":7880},[120],[15,7882],{"className":7883,"style":4886},[124],[15,7885,541],{"className":7886},[129],[11,7888,7889,7890,8026,8027,8079,8080,8172,8173,8265,8266,8398,8399,1313],{},"两个点 ",[15,7891,7893,7920],{"className":7892,"translate":19},[23],[15,7894,7896],{"className":7895},[27],[29,7897,7898],{"xmlns":31},[34,7899,7900,7917],{},[37,7901,7902,7908,7911],{},[40,7903,7904,7906],{},[43,7905,56],{},[308,7907,318],{},[50,7909,7910],{},"\u003C",[40,7912,7913,7915],{},[43,7914,56],{},[308,7916,310],{},[109,7918,7919],{"encoding":111},"x_1\u003Cx_2",[15,7921,7923,7979],{"className":7922,"ariaHidden":72},[116],[15,7924,7926,7930,7970,7973,7976],{"className":7925},[120],[15,7927],{"className":7928,"style":7929},[124],"height:0.6891em;vertical-align:-0.15em;",[15,7931,7933,7936],{"className":7932},[129],[15,7934,56],{"className":7935},[129,133],[15,7937,7939],{"className":7938},[137],[15,7940,7942,7962],{"className":7941},[141,142],[15,7943,7945,7959],{"className":7944},[146],[15,7946,7948],{"className":7947,"style":1051},[150],[15,7949,7950,7953],{"style":154},[15,7951],{"className":7952,"style":159},[158],[15,7954,7956],{"className":7955},[163,164,165,166],[15,7957,318],{"className":7958},[129,166],[15,7960,175],{"className":7961},[174],[15,7963,7965],{"className":7964},[146],[15,7966,7968],{"className":7967,"style":182},[150],[15,7969],{},[15,7971],{"className":7972,"style":199},[93],[15,7974,7910],{"className":7975},[203],[15,7977],{"className":7978,"style":199},[93],[15,7980,7982,7986],{"className":7981},[120],[15,7983],{"className":7984,"style":7985},[124],"height:0.5806em;vertical-align:-0.15em;",[15,7987,7989,7992],{"className":7988},[129],[15,7990,56],{"className":7991},[129,133],[15,7993,7995],{"className":7994},[137],[15,7996,7998,8018],{"className":7997},[141,142],[15,7999,8001,8015],{"className":8000},[146],[15,8002,8004],{"className":8003,"style":1051},[150],[15,8005,8006,8009],{"style":154},[15,8007],{"className":8008,"style":159},[158],[15,8010,8012],{"className":8011},[163,164,165,166],[15,8013,310],{"className":8014},[129,166],[15,8016,175],{"className":8017},[174],[15,8019,8021],{"className":8020},[146],[15,8022,8024],{"className":8023,"style":182},[150],[15,8025],{}," 则不行：标注 ",[15,8028,8030,8052],{"className":8029,"translate":19},[23],[15,8031,8033],{"className":8032},[27],[29,8034,8035],{"xmlns":31},[34,8036,8037,8049],{},[37,8038,8039,8041,8043,8045,8047],{},[50,8040,53],{"stretchy":52},[308,8042,318],{},[50,8044,91],{"separator":72},[308,8046,541],{},[50,8048,59],{"stretchy":52},[109,8050,8051],{"encoding":111},"(1,0)",[15,8053,8055],{"className":8054,"ariaHidden":72},[116],[15,8056,8058,8061,8064,8067,8070,8073,8076],{"className":8057},[120],[15,8059],{"className":8060,"style":125},[124],[15,8062,53],{"className":8063},[188],[15,8065,318],{"className":8066},[129],[15,8068,91],{"className":8069},[254],[15,8071],{"className":8072,"style":262},[93],[15,8074,541],{"className":8075},[129],[15,8077,59],{"className":8078},[195]," 要求 ",[15,8081,8083,8105],{"className":8082,"translate":19},[23],[15,8084,8086],{"className":8085},[27],[29,8087,8088],{"xmlns":31},[34,8089,8090,8102],{},[37,8091,8092,8094,8096],{},[43,8093,7476],{},[50,8095,2717],{},[40,8097,8098,8100],{},[43,8099,56],{},[308,8101,318],{},[109,8103,8104],{"encoding":111},"t\\le x_1",[15,8106,8108,8126],{"className":8107,"ariaHidden":72},[116],[15,8109,8111,8114,8117,8120,8123],{"className":8110},[120],[15,8112],{"className":8113,"style":7757},[124],[15,8115,7476],{"className":8116},[129,133],[15,8118],{"className":8119,"style":199},[93],[15,8121,2717],{"className":8122},[203],[15,8124],{"className":8125,"style":199},[93],[15,8127,8129,8132],{"className":8128},[120],[15,8130],{"className":8131,"style":7985},[124],[15,8133,8135,8138],{"className":8134},[129],[15,8136,56],{"className":8137},[129,133],[15,8139,8141],{"className":8140},[137],[15,8142,8144,8164],{"className":8143},[141,142],[15,8145,8147,8161],{"className":8146},[146],[15,8148,8150],{"className":8149,"style":1051},[150],[15,8151,8152,8155],{"style":154},[15,8153],{"className":8154,"style":159},[158],[15,8156,8158],{"className":8157},[163,164,165,166],[15,8159,318],{"className":8160},[129,166],[15,8162,175],{"className":8163},[174],[15,8165,8167],{"className":8166},[146],[15,8168,8170],{"className":8169,"style":182},[150],[15,8171],{}," 且 ",[15,8174,8176,8198],{"className":8175,"translate":19},[23],[15,8177,8179],{"className":8178},[27],[29,8180,8181],{"xmlns":31},[34,8182,8183,8195],{},[37,8184,8185,8187,8189],{},[43,8186,7476],{},[50,8188,4852],{},[40,8190,8191,8193],{},[43,8192,56],{},[308,8194,310],{},[109,8196,8197],{"encoding":111},"t>x_2",[15,8199,8201,8219],{"className":8200,"ariaHidden":72},[116],[15,8202,8204,8207,8210,8213,8216],{"className":8203},[120],[15,8205],{"className":8206,"style":7585},[124],[15,8208,7476],{"className":8209},[129,133],[15,8211],{"className":8212,"style":199},[93],[15,8214,4852],{"className":8215},[203],[15,8217],{"className":8218,"style":199},[93],[15,8220,8222,8225],{"className":8221},[120],[15,8223],{"className":8224,"style":7985},[124],[15,8226,8228,8231],{"className":8227},[129],[15,8229,56],{"className":8230},[129,133],[15,8232,8234],{"className":8233},[137],[15,8235,8237,8257],{"className":8236},[141,142],[15,8238,8240,8254],{"className":8239},[146],[15,8241,8243],{"className":8242,"style":1051},[150],[15,8244,8245,8248],{"style":154},[15,8246],{"className":8247,"style":159},[158],[15,8249,8251],{"className":8250},[163,164,165,166],[15,8252,310],{"className":8253},[129,166],[15,8255,175],{"className":8256},[174],[15,8258,8260],{"className":8259},[146],[15,8261,8263],{"className":8262,"style":182},[150],[15,8264],{},"，与 ",[15,8267,8269,8294],{"className":8268,"translate":19},[23],[15,8270,8272],{"className":8271},[27],[29,8273,8274],{"xmlns":31},[34,8275,8276,8292],{},[37,8277,8278,8284,8286],{},[40,8279,8280,8282],{},[43,8281,56],{},[308,8283,318],{},[50,8285,7910],{},[40,8287,8288,8290],{},[43,8289,56],{},[308,8291,310],{},[109,8293,7919],{"encoding":111},[15,8295,8297,8352],{"className":8296,"ariaHidden":72},[116],[15,8298,8300,8303,8343,8346,8349],{"className":8299},[120],[15,8301],{"className":8302,"style":7929},[124],[15,8304,8306,8309],{"className":8305},[129],[15,8307,56],{"className":8308},[129,133],[15,8310,8312],{"className":8311},[137],[15,8313,8315,8335],{"className":8314},[141,142],[15,8316,8318,8332],{"className":8317},[146],[15,8319,8321],{"className":8320,"style":1051},[150],[15,8322,8323,8326],{"style":154},[15,8324],{"className":8325,"style":159},[158],[15,8327,8329],{"className":8328},[163,164,165,166],[15,8330,318],{"className":8331},[129,166],[15,8333,175],{"className":8334},[174],[15,8336,8338],{"className":8337},[146],[15,8339,8341],{"className":8340,"style":182},[150],[15,8342],{},[15,8344],{"className":8345,"style":199},[93],[15,8347,7910],{"className":8348},[203],[15,8350],{"className":8351,"style":199},[93],[15,8353,8355,8358],{"className":8354},[120],[15,8356],{"className":8357,"style":7985},[124],[15,8359,8361,8364],{"className":8360},[129],[15,8362,56],{"className":8363},[129,133],[15,8365,8367],{"className":8366},[137],[15,8368,8370,8390],{"className":8369},[141,142],[15,8371,8373,8387],{"className":8372},[146],[15,8374,8376],{"className":8375,"style":1051},[150],[15,8377,8378,8381],{"style":154},[15,8379],{"className":8380,"style":159},[158],[15,8382,8384],{"className":8383},[163,164,165,166],[15,8385,310],{"className":8386},[129,166],[15,8388,175],{"className":8389},[174],[15,8391,8393],{"className":8392},[146],[15,8394,8396],{"className":8395,"style":182},[150],[15,8397],{}," 矛盾。所以 VC 维是 ",[15,8400,8402,8415],{"className":8401,"translate":19},[23],[15,8403,8405],{"className":8404},[27],[29,8406,8407],{"xmlns":31},[34,8408,8409,8413],{},[37,8410,8411],{},[308,8412,318],{},[109,8414,318],{"encoding":111},[15,8416,8418],{"className":8417,"ariaHidden":72},[116],[15,8419,8421,8424],{"className":8420},[120],[15,8422],{"className":8423,"style":4886},[124],[15,8425,318],{"className":8426},[129],[11,8428,8429,8430,8458,8459,8487,8488,8540,8541,1313],{},"这个例子说明，VC 维和「能拟合多少点」不是一回事：半直线可以把任意多个点拟合成「前面全是 ",[15,8431,8433,8446],{"className":8432,"translate":19},[23],[15,8434,8436],{"className":8435},[27],[29,8437,8438],{"xmlns":31},[34,8439,8440,8444],{},[37,8441,8442],{},[308,8443,541],{},[109,8445,541],{"encoding":111},[15,8447,8449],{"className":8448,"ariaHidden":72},[116],[15,8450,8452,8455],{"className":8451},[120],[15,8453],{"className":8454,"style":4886},[124],[15,8456,541],{"className":8457},[129],"、后面全是 ",[15,8460,8462,8475],{"className":8461,"translate":19},[23],[15,8463,8465],{"className":8464},[27],[29,8466,8467],{"xmlns":31},[34,8468,8469,8473],{},[37,8470,8471],{},[308,8472,318],{},[109,8474,318],{"encoding":111},[15,8476,8478],{"className":8477,"ariaHidden":72},[116],[15,8479,8481,8484],{"className":8480},[120],[15,8482],{"className":8483,"style":4886},[124],[15,8485,318],{"className":8486},[129],"」的形状，但它只能实现 ",[15,8489,8491,8509],{"className":8490,"translate":19},[23],[15,8492,8494],{"className":8493},[27],[29,8495,8496],{"xmlns":31},[34,8497,8498,8506],{},[37,8499,8500,8502,8504],{},[43,8501,348],{},[50,8503,2682],{},[308,8505,318],{},[109,8507,8508],{"encoding":111},"m+1",[15,8510,8512,8531],{"className":8511,"ariaHidden":72},[116],[15,8513,8515,8519,8522,8525,8528],{"className":8514},[120],[15,8516],{"className":8517,"style":8518},[124],"height:0.6667em;vertical-align:-0.0833em;",[15,8520,348],{"className":8521},[129,133],[15,8523],{"className":8524,"style":3277},[93],[15,8526,2682],{"className":8527},[2436],[15,8529],{"className":8530,"style":3277},[93],[15,8532,8534,8537],{"className":8533},[120],[15,8535],{"className":8536,"style":4886},[124],[15,8538,318],{"className":8539},[129]," 种二分，远少于 ",[15,8542,8544,8561],{"className":8543,"translate":19},[23],[15,8545,8547],{"className":8546},[27],[29,8548,8549],{"xmlns":31},[34,8550,8551,8559],{},[37,8552,8553],{},[305,8554,8555,8557],{},[308,8556,310],{},[43,8558,348],{},[109,8560,6367],{"encoding":111},[15,8562,8564],{"className":8563,"ariaHidden":72},[116],[15,8565,8567,8570],{"className":8566},[120],[15,8568],{"className":8569,"style":6377},[124],[15,8571,8573,8576],{"className":8572},[129],[15,8574,310],{"className":8575},[129],[15,8577,8579],{"className":8578},[137],[15,8580,8582],{"className":8581},[141],[15,8583,8585],{"className":8584},[146],[15,8586,8588],{"className":8587,"style":6377},[150],[15,8589,8590,8593],{"style":383},[15,8591],{"className":8592,"style":159},[158],[15,8594,8596],{"className":8595},[163,164,165,166],[15,8597,348],{"className":8598},[129,133,166],[11,8600,8601],{},[698,8602,8603,8604,7438],{},"例 2（区间，",[15,8605,8607,8627],{"className":8606,"translate":19},[23],[15,8608,8610],{"className":8609},[27],[29,8611,8612],{"xmlns":31},[34,8613,8614,8624],{},[37,8615,8616,8618,8620,8622],{},[43,8617,6823],{"mathvariant":65},[50,8619,69],{},[50,8621,62],{},[308,8623,310],{},[109,8625,8626],{"encoding":111},"\\operatorname{VCdim}=2",[15,8628,8630,8651],{"className":8629,"ariaHidden":72},[116],[15,8631,8633,8636,8642,8645,8648],{"className":8632},[120],[15,8634],{"className":8635,"style":522},[124],[15,8637,8639],{"className":8638},[217],[15,8640,6823],{"className":8641},[129,221],[15,8643],{"className":8644,"style":199},[93],[15,8646,62],{"className":8647},[203],[15,8649],{"className":8650,"style":199},[93],[15,8652,8654,8657],{"className":8653},[120],[15,8655],{"className":8656,"style":4886},[124],[15,8658,310],{"className":8659},[129],[11,8661,7441,8662,1313],{},[15,8663,8665,8715],{"className":8664,"translate":19},[23],[15,8666,8668],{"className":8667},[27],[29,8669,8670],{"xmlns":31},[34,8671,8672,8712],{},[37,8673,8674,8676,8678,8680,8682,8684,8686,8688,8691,8693,8695,8697,8700,8702,8704,8706,8708,8710],{},[43,8675,819],{"mathvariant":596},[50,8677,62],{},[50,8679,636],{"stretchy":52},[43,8681,56],{},[50,8683,7464],{},[308,8685,318],{"mathvariant":1468},[50,8687,1407],{"stretchy":52},[43,8689,8690],{},"a",[50,8692,2717],{},[43,8694,56],{},[50,8696,2717],{},[43,8698,8699],{},"b",[50,8701,1423],{"stretchy":52},[50,8703,718],{},[43,8705,8690],{},[50,8707,2717],{},[43,8709,8699],{},[50,8711,645],{"stretchy":52},[109,8713,8714],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\mathbf 1[a\\le x\\le b]:a\\le 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b\\ge x_2",[15,9604,9606,9624,9743],{"className":9605,"ariaHidden":72},[116],[15,9607,9609,9612,9615,9618,9621],{"className":9608},[120],[15,9610],{"className":9611,"style":4867},[124],[15,9613,8690],{"className":9614},[129,133],[15,9616],{"className":9617,"style":199},[93],[15,9619,100],{"className":9620},[203],[15,9622],{"className":9623,"style":199},[93],[15,9625,9627,9630,9633,9673,9676,9679,9719,9722,9725,9728,9731,9734,9737,9740],{"className":9626},[120],[15,9628],{"className":9629,"style":125},[124],[15,9631,53],{"className":9632},[188],[15,9634,9636,9639],{"className":9635},[129],[15,9637,56],{"className":9638},[129,133],[15,9640,9642],{"className":9641},[137],[15,9643,9645,9665],{"className":9644},[141,142],[15,9646,9648,9662],{"className":9647},[146],[15,9649,9651],{"className":9650,"style":1051},[150],[15,9652,9653,9656],{"style":154},[15,9654],{"className":9655,"style":159},[158],[15,9657,9659],{"className":9658},[163,164,165,166],[15,9660,318],{"className":9661},[129,166],[15,9663,175],{"className":9664},[174],[15,9666,9668],{"className":9667},[146],[15,9669,9671],{"className":9670,"style":182},[150],[15,9672],{},[15,9674,91],{"className":9675},[254],[15,9677],{"className":9678,"style":262},[93],[15,9680,9682,9685],{"className":9681},[129],[15,9683,56],{"className":9684},[129,133],[15,9686,9688],{"className":9687},[137],[15,9689,9691,9711],{"className":9690},[141,142],[15,9692,9694,9708],{"className":9693},[146],[15,9695,9697],{"className":9696,"style":1051},[150],[15,9698,9699,9702],{"style":154},[15,9700],{"className":9701,"style":159},[158],[15,9703,9705],{"className":9704},[163,164,165,166],[15,9706,310],{"className":9707},[129,166],[15,9709,175],{"className":9710},[174],[15,9712,9714],{"className":9713},[146],[15,9715,9717],{"className":9716,"style":182},[150],[15,9718],{},[15,9720,1423],{"className":9721},[195],[15,9723,91],{"className":9724},[254],[15,9726,6858],{"className":9727},[93],[15,9729],{"className":9730,"style":262},[93],[15,9732,8699],{"className":9733},[129,133],[15,9735],{"className":9736,"style":199},[93],[15,9738,7473],{"className":9739},[203],[15,9741],{"className":9742,"style":199},[93],[15,9744,9746,9749],{"className":9745},[120],[15,9747],{"className":9748,"style":7985},[124],[15,9750,9752,9755],{"className":9751},[129],[15,9753,56],{"className":9754},[129,133],[15,9756,9758],{"className":9757},[137],[15,9759,9761,9781],{"className":9760},[141,142],[15,9762,9764,9778],{"className":9763},[146],[15,9765,9767],{"className":9766,"style":1051},[150],[15,9768,9769,9772],{"style":154},[15,9770],{"className":9771,"style":159},[158],[15,9773,9775],{"className":9774},[163,164,165,166],[15,9776,310],{"className":9777},[129,166],[15,9779,175],{"className":9780},[174],[15,9782,9784],{"className":9783},[146],[15,9785,9787],{"className":9786,"style":182},[150],[15,9788],{},[15,9790,9792,9814],{"className":9791,"translate":19},[23],[15,9793,9795],{"className":9794},[27],[29,9796,9797],{"xmlns":31},[34,9798,9799,9811],{},[37,9800,9801,9803,9805,9807,9809],{},[50,9802,53],{"stretchy":52},[308,9804,541],{},[50,9806,91],{"separator":72},[308,9808,541],{},[50,9810,59],{"stretchy":52},[109,9812,9813],{"encoding":111},"(0,0)",[15,9815,9817],{"className":9816,"ariaHidden":72},[116],[15,9818,9820,9823,9826,9829,9832,9835,9838],{"className":9819},[120],[15,9821],{"className":9822,"style":125},[124],[15,9824,53],{"className":9825},[188],[15,9827,541],{"className":9828},[129],[15,9830,91],{"className":9831},[254],[15,9833],{"className":9834,"style":262},[93],[15,9836,541],{"className":9837},[129],[15,9839,59],{"className":9840},[195]," 取一个避开两点的区间。",[11,9843,9844,9845,10042,10043,10108,10109,10179,10180,10250,10251,10321,10322,10350],{},"三个点 ",[15,9846,9848,9883],{"className":9847,"translate":19},[23],[15,9849,9851],{"className":9850},[27],[29,9852,9853],{"xmlns":31},[34,9854,9855,9880],{},[37,9856,9857,9863,9865,9871,9873],{},[40,9858,9859,9861],{},[43,9860,56],{},[308,9862,318],{},[50,9864,7910],{},[40,9866,9867,9869],{},[43,9868,56],{},[308,9870,310],{},[50,9872,7910],{},[40,9874,9875,9877],{},[43,9876,56],{},[308,9878,9879],{},"3",[109,9881,9882],{"encoding":111},"x_1\u003Cx_2\u003Cx_3",[15,9884,9886,9941,9996],{"className":9885,"ariaHidden":72},[116],[15,9887,9889,9892,9932,9935,9938],{"className":9888},[120],[15,9890],{"className":9891,"style":7929},[124],[15,9893,9895,9898],{"className":9894},[129],[15,9896,56],{"className":9897},[129,133],[15,9899,9901],{"className":9900},[137],[15,9902,9904,9924],{"className":9903},[141,142],[15,9905,9907,9921],{"className":9906},[146],[15,9908,9910],{"className":9909,"style":1051},[150],[15,9911,9912,9915],{"style":154},[15,9913],{"className":9914,"style":159},[158],[15,9916,9918],{"className":9917},[163,164,165,166],[15,9919,318],{"className":9920},[129,166],[15,9922,175],{"className":9923},[174],[15,9925,9927],{"className":9926},[146],[15,9928,9930],{"className":9929,"style":182},[150],[15,9931],{},[15,9933],{"className":9934,"style":199},[93],[15,9936,7910],{"className":9937},[203],[15,9939],{"className":9940,"style":199},[93],[15,9942,9944,9947,9987,9990,9993],{"className":9943},[120],[15,9945],{"className":9946,"style":7929},[124],[15,9948,9950,9953],{"className":9949},[129],[15,9951,56],{"className":9952},[129,133],[15,9954,9956],{"className":9955},[137],[15,9957,9959,9979],{"className":9958},[141,142],[15,9960,9962,9976],{"className":9961},[146],[15,9963,9965],{"className":9964,"style":1051},[150],[15,9966,9967,9970],{"style":154},[15,9968],{"className":9969,"style":159},[158],[15,9971,9973],{"className":9972},[163,164,165,166],[15,9974,310],{"className":9975},[129,166],[15,9977,175],{"className":9978},[174],[15,9980,9982],{"className":9981},[146],[15,9983,9985],{"className":9984,"style":182},[150],[15,9986],{},[15,9988],{"className":9989,"style":199},[93],[15,9991,7910],{"className":9992},[203],[15,9994],{"className":9995,"style":199},[93],[15,9997,9999,10002],{"className":9998},[120],[15,10000],{"className":10001,"style":7985},[124],[15,10003,10005,10008],{"className":10004},[129],[15,10006,56],{"className":10007},[129,133],[15,10009,10011],{"className":10010},[137],[15,10012,10014,10034],{"className":10013},[141,142],[15,10015,10017,10031],{"className":10016},[146],[15,10018,10020],{"className":10019,"style":1051},[150],[15,10021,10022,10025],{"style":154},[15,10023],{"className":10024,"style":159},[158],[15,10026,10028],{"className":10027},[163,164,165,166],[15,10029,9879],{"className":10030},[129,166],[15,10032,175],{"className":10033},[174],[15,10035,10037],{"className":10036},[146],[15,10038,10040],{"className":10039,"style":182},[150],[15,10041],{}," 不行：标注 ",[15,10044,10046,10072],{"className":10045,"translate":19},[23],[15,10047,10049],{"className":10048},[27],[29,10050,10051],{"xmlns":31},[34,10052,10053,10069],{},[37,10054,10055,10057,10059,10061,10063,10065,10067],{},[50,10056,53],{"stretchy":52},[308,10058,318],{},[50,10060,91],{"separator":72},[308,10062,541],{},[50,10064,91],{"separator":72},[308,10066,318],{},[50,10068,59],{"stretchy":52},[109,10070,10071],{"encoding":111},"(1,0,1)",[15,10073,10075],{"className":10074,"ariaHidden":72},[116],[15,10076,10078,10081,10084,10087,10090,10093,10096,10099,10102,10105],{"className":10077},[120],[15,10079],{"className":10080,"style":125},[124],[15,10082,53],{"className":10083},[188],[15,10085,318],{"className":10086},[129],[15,10088,91],{"className":10089},[254],[15,10091],{"className":10092,"style":262},[93],[15,10094,541],{"className":10095},[129],[15,10097,91],{"className":10098},[254],[15,10100],{"className":10101,"style":262},[93],[15,10103,318],{"className":10104},[129],[15,10106,59],{"className":10107},[195]," 要求区间同时包含 ",[15,10110,10112,10130],{"className":10111,"translate":19},[23],[15,10113,10115],{"className":10114},[27],[29,10116,10117],{"xmlns":31},[34,10118,10119,10127],{},[37,10120,10121],{},[40,10122,10123,10125],{},[43,10124,56],{},[308,10126,318],{},[109,10128,10129],{"encoding":111},"x_1",[15,10131,10133],{"className":10132,"ariaHidden":72},[116],[15,10134,10136,10139],{"className":10135},[120],[15,10137],{"className":10138,"style":7985},[124],[15,10140,10142,10145],{"className":10141},[129],[15,10143,56],{"className":10144},[129,133],[15,10146,10148],{"className":10147},[137],[15,10149,10151,10171],{"className":10150},[141,142],[15,10152,10154,10168],{"className":10153},[146],[15,10155,10157],{"className":10156,"style":1051},[150],[15,10158,10159,10162],{"style":154},[15,10160],{"className":10161,"style":159},[158],[15,10163,10165],{"className":10164},[163,164,165,166],[15,10166,318],{"className":10167},[129,166],[15,10169,175],{"className":10170},[174],[15,10172,10174],{"className":10173},[146],[15,10175,10177],{"className":10176,"style":182},[150],[15,10178],{}," 与 ",[15,10181,10183,10201],{"className":10182,"translate":19},[23],[15,10184,10186],{"className":10185},[27],[29,10187,10188],{"xmlns":31},[34,10189,10190,10198],{},[37,10191,10192],{},[40,10193,10194,10196],{},[43,10195,56],{},[308,10197,9879],{},[109,10199,10200],{"encoding":111},"x_3",[15,10202,10204],{"className":10203,"ariaHidden":72},[116],[15,10205,10207,10210],{"className":10206},[120],[15,10208],{"className":10209,"style":7985},[124],[15,10211,10213,10216],{"className":10212},[129],[15,10214,56],{"className":10215},[129,133],[15,10217,10219],{"className":10218},[137],[15,10220,10222,10242],{"className":10221},[141,142],[15,10223,10225,10239],{"className":10224},[146],[15,10226,10228],{"className":10227,"style":1051},[150],[15,10229,10230,10233],{"style":154},[15,10231],{"className":10232,"style":159},[158],[15,10234,10236],{"className":10235},[163,164,165,166],[15,10237,9879],{"className":10238},[129,166],[15,10240,175],{"className":10241},[174],[15,10243,10245],{"className":10244},[146],[15,10246,10248],{"className":10247,"style":182},[150],[15,10249],{},"，而区间是凸的，必然包含中间的 ",[15,10252,10254,10272],{"className":10253,"translate":19},[23],[15,10255,10257],{"className":10256},[27],[29,10258,10259],{"xmlns":31},[34,10260,10261,10269],{},[37,10262,10263],{},[40,10264,10265,10267],{},[43,10266,56],{},[308,10268,310],{},[109,10270,10271],{"encoding":111},"x_2",[15,10273,10275],{"className":10274,"ariaHidden":72},[116],[15,10276,10278,10281],{"className":10277},[120],[15,10279],{"className":10280,"style":7985},[124],[15,10282,10284,10287],{"className":10283},[129],[15,10285,56],{"className":10286},[129,133],[15,10288,10290],{"className":10289},[137],[15,10291,10293,10313],{"className":10292},[141,142],[15,10294,10296,10310],{"className":10295},[146],[15,10297,10299],{"className":10298,"style":1051},[150],[15,10300,10301,10304],{"style":154},[15,10302],{"className":10303,"style":159},[158],[15,10305,10307],{"className":10306},[163,164,165,166],[15,10308,310],{"className":10309},[129,166],[15,10311,175],{"className":10312},[174],[15,10314,10316],{"className":10315},[146],[15,10317,10319],{"className":10318,"style":182},[150],[15,10320],{},"，与中间的 ",[15,10323,10325,10338],{"className":10324,"translate":19},[23],[15,10326,10328],{"className":10327},[27],[29,10329,10330],{"xmlns":31},[34,10331,10332,10336],{},[37,10333,10334],{},[308,10335,541],{},[109,10337,541],{"encoding":111},[15,10339,10341],{"className":10340,"ariaHidden":72},[116],[15,10342,10344,10347],{"className":10343},[120],[15,10345],{"className":10346,"style":4886},[124],[15,10348,541],{"className":10349},[129]," 矛盾。",[11,10352,10353],{},[698,10354,10355,10356,7438],{},"例 3（仿射超平面，",[15,10357,10359,10384],{"className":10358,"translate":19},[23],[15,10360,10362],{"className":10361},[27],[29,10363,10364],{"xmlns":31},[34,10365,10366,10381],{},[37,10367,10368,10370,10372,10374,10377,10379],{},[43,10369,6823],{"mathvariant":65},[50,10371,69],{},[50,10373,62],{},[43,10375,10376],{},"d",[50,10378,2682],{},[308,10380,318],{},[109,10382,10383],{"encoding":111},"\\operatorname{VCdim}=d+1",[15,10385,10387,10408,10427],{"className":10386,"ariaHidden":72},[116],[15,10388,10390,10393,10399,10402,10405],{"className":10389},[120],[15,10391],{"className":10392,"style":522},[124],[15,10394,10396],{"className":10395},[217],[15,10397,6823],{"className":10398},[129,221],[15,10400],{"className":10401,"style":199},[93],[15,10403,62],{"className":10404},[203],[15,10406],{"className":10407,"style":199},[93],[15,10409,10411,10415,10418,10421,10424],{"className":10410},[120],[15,10412],{"className":10413,"style":10414},[124],"height:0.7778em;vertical-align:-0.0833em;",[15,10416,10376],{"className":10417},[129,133],[15,10419],{"className":10420,"style":3277},[93],[15,10422,2682],{"className":10423},[2436],[15,10425],{"className":10426,"style":3277},[93],[15,10428,10430,10433],{"className":10429},[120],[15,10431],{"className":10432,"style":4886},[124],[15,10434,318],{"className":10435},[129],[11,10437,7441,10438,7610,10520,1313],{},[15,10439,10441,10463],{"className":10440,"translate":19},[23],[15,10442,10444],{"className":10443},[27],[29,10445,10446],{"xmlns":31},[34,10447,10448,10460],{},[37,10449,10450,10452,10454],{},[43,10451,597],{"mathvariant":596},[50,10453,62],{},[305,10455,10456,10458],{},[43,10457,104],{"mathvariant":103},[43,10459,10376],{},[109,10461,10462],{"encoding":111},"\\mathcal X=\\mathbb R^d",[15,10464,10466,10484],{"className":10465,"ariaHidden":72},[116],[15,10467,10469,10472,10475,10478,10481],{"className":10468},[120],[15,10470],{"className":10471,"style":610},[124],[15,10473,597],{"className":10474,"style":615},[129,614],[15,10476],{"className":10477,"style":199},[93],[15,10479,62],{"className":10480},[203],[15,10482],{"className":10483,"style":199},[93],[15,10485,10487,10491],{"className":10486},[120],[15,10488],{"className":10489,"style":10490},[124],"height:0.8491em;",[15,10492,10494,10497],{"className":10493},[129],[15,10495,104],{"className":10496},[129,285],[15,10498,10500],{"className":10499},[137],[15,10501,10503],{"className":10502},[141],[15,10504,10506],{"className":10505},[146],[15,10507,10509],{"className":10508,"style":10490},[150],[15,10510,10511,10514],{"style":383},[15,10512],{"className":10513,"style":159},[158],[15,10515,10517],{"className":10516},[163,164,165,166],[15,10518,10376],{"className":10519},[129,133,166],[15,10521,10523,10567],{"className":10522,"translate":19},[23],[15,10524,10526],{"className":10525},[27],[29,10527,10528],{"xmlns":31},[34,10529,10530,10564],{},[37,10531,10532,10534,10536,10538,10540,10542,10544,10546,10548,10551,10554,10556,10558,10560,10562],{},[43,10533,819],{"mathvariant":596},[50,10535,62],{},[50,10537,636],{"stretchy":52},[43,10539,56],{},[50,10541,7464],{},[43,10543,66],{"mathvariant":65},[50,10545,69],{},[50,10547,53],{"stretchy":52},[43,10549,10550],{},"w",[50,10552,10553],{},"⋅",[43,10555,56],{},[50,10557,2682],{},[43,10559,8699],{},[50,10561,59],{"stretchy":52},[50,10563,645],{"stretchy":52},[109,10565,10566],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(w\\cdot x+b)\\}",[15,10568,10570,10588,10609,10637,10655],{"className":10569,"ariaHidden":72},[116],[15,10571,10573,10576,10579,10582,10585],{"className":10572},[120],[15,10574],{"className":10575,"style":610},[124],[15,10577,819],{"className":10578,"style":835},[129,614],[15,10580],{"className":10581,"style":199},[93],[15,10583,62],{"className":10584},[203],[15,10586],{"className":10587,"style":199},[93],[15,10589,10591,10594,10597,10600,10603,10606],{"className":10590},[120],[15,10592],{"className":10593,"style":125},[124],[15,10595,636],{"className":10596},[188],[15,10598,56],{"className":10599},[129,133],[15,10601],{"className":10602,"style":199},[93],[15,10604,7464],{"className":10605},[203],[15,10607],{"className":10608,"style":199},[93],[15,10610,10612,10615,10621,10624,10628,10631,10634],{"className":10611},[120],[15,10613],{"className":10614,"style":125},[124],[15,10616,10618],{"className":10617},[217],[15,10619,66],{"className":10620},[129,221],[15,10622,53],{"className":10623},[188],[15,10625,10550],{"className":10626,"style":10627},[129,133],"margin-right:0.0269em;",[15,10629],{"className":10630,"style":3277},[93],[15,10632,10553],{"className":10633},[2436],[15,10635],{"className":10636,"style":3277},[93],[15,10638,10640,10643,10646,10649,10652],{"className":10639},[120],[15,10641],{"className":10642,"style":8518},[124],[15,10644,56],{"className":10645},[129,133],[15,10647],{"className":10648,"style":3277},[93],[15,10650,2682],{"className":10651},[2436],[15,10653],{"className":10654,"style":3277},[93],[15,10656,10658,10661,10664],{"className":10657},[120],[15,10659],{"className":10660,"style":125},[124],[15,10662,8699],{"className":10663},[129,133],[15,10665,1235],{"className":10666},[195],[11,10668,10669,10672,10673,10859,10860,10911,10912,11026],{},[698,10670,10671],{},"下界","：取 ",[15,10674,10676,10719],{"className":10675,"translate":19},[23],[15,10677,10679],{"className":10678},[27],[29,10680,10681],{"xmlns":31},[34,10682,10683,10716],{},[37,10684,10685,10687,10689,10691,10693,10695,10702,10704,10706,10708,10714],{},[43,10686,5619],{},[50,10688,62],{},[50,10690,636],{"stretchy":52},[308,10692,541],{},[50,10694,91],{"separator":72},[40,10696,10697,10700],{},[43,10698,10699],{},"e",[308,10701,318],{},[50,10703,91],{"separator":72},[50,10705,335],{},[50,10707,91],{"separator":72},[40,10709,10710,10712],{},[43,10711,10699],{},[43,10713,10376],{},[50,10715,645],{"stretchy":52},[109,10717,10718],{"encoding":111},"C=\\{0,e_1,\\dots,e_d\\}",[15,10720,10722,10740],{"className":10721,"ariaHidden":72},[116],[15,10723,10725,10728,10731,10734,10737],{"className":10724},[120],[15,10726],{"className":10727,"style":610},[124],[15,10729,5619],{"className":10730,"style":5665},[129,133],[15,10732],{"className":10733,"style":199},[93],[15,10735,62],{"className":10736},[203],[15,10738],{"className":10739,"style":199},[93],[15,10741,10743,10746,10749,10752,10755,10758,10798,10801,10804,10807,10810,10813,10816,10856],{"className":10742},[120],[15,10744],{"className":10745,"style":125},[124],[15,10747,636],{"className":10748},[188],[15,10750,541],{"className":10751},[129],[15,10753,91],{"className":10754},[254],[15,10756],{"className":10757,"style":262},[93],[15,10759,10761,10764],{"className":10760},[129],[15,10762,10699],{"className":10763},[129,133],[15,10765,10767],{"className":10766},[137],[15,10768,10770,10790],{"className":10769},[141,142],[15,10771,10773,10787],{"className":10772},[146],[15,10774,10776],{"className":10775,"style":1051},[150],[15,10777,10778,10781],{"style":154},[15,10779],{"className":10780,"style":159},[158],[15,10782,10784],{"className":10783},[163,164,165,166],[15,10785,318],{"className":10786},[129,166],[15,10788,175],{"className":10789},[174],[15,10791,10793],{"className":10792},[146],[15,10794,10796],{"className":10795,"style":182},[150],[15,10797],{},[15,10799,91],{"className":10800},[254],[15,10802],{"className":10803,"style":262},[93],[15,10805,335],{"className":10806},[449],[15,10808],{"className":10809,"style":262},[93],[15,10811,91],{"className":10812},[254],[15,10814],{"className":10815,"style":262},[93],[15,10817,10819,10822],{"className":10818},[129],[15,10820,10699],{"className":10821},[129,133],[15,10823,10825],{"className":10824},[137],[15,10826,10828,10848],{"className":10827},[141,142],[15,10829,10831,10845],{"className":10830},[146],[15,10832,10834],{"className":10833,"style":151},[150],[15,10835,10836,10839],{"style":154},[15,10837],{"className":10838,"style":159},[158],[15,10840,10842],{"className":10841},[163,164,165,166],[15,10843,10376],{"className":10844},[129,133,166],[15,10846,175],{"className":10847},[174],[15,10849,10851],{"className":10850},[146],[15,10852,10854],{"className":10853,"style":182},[150],[15,10855],{},[15,10857,645],{"className":10858},[195],"，这是 ",[15,10861,10863,10881],{"className":10862,"translate":19},[23],[15,10864,10866],{"className":10865},[27],[29,10867,10868],{"xmlns":31},[34,10869,10870,10878],{},[37,10871,10872,10874,10876],{},[43,10873,10376],{},[50,10875,2682],{},[308,10877,318],{},[109,10879,10880],{"encoding":111},"d+1",[15,10882,10884,10902],{"className":10883,"ariaHidden":72},[116],[15,10885,10887,10890,10893,10896,10899],{"className":10886},[120],[15,10888],{"className":10889,"style":10414},[124],[15,10891,10376],{"className":10892},[129,133],[15,10894],{"className":10895,"style":3277},[93],[15,10897,2682],{"className":10898},[2436],[15,10900],{"className":10901,"style":3277},[93],[15,10903,10905,10908],{"className":10904},[120],[15,10906],{"className":10907,"style":4886},[124],[15,10909,318],{"className":10910},[129]," 个仿射无关的点。对任意标注 ",[15,10913,10915,10950],{"className":10914,"translate":19},[23],[15,10916,10918],{"className":10917},[27],[29,10919,10920],{"xmlns":31},[34,10921,10922,10947],{},[37,10923,10924,10926,10928,10930,10933,10935],{},[43,10925,857],{},[50,10927,100],{},[50,10929,636],{"stretchy":52},[50,10931,10932],{},"±",[308,10934,318],{},[305,10936,10937,10939],{},[50,10938,645],{"stretchy":52},[37,10940,10941,10943,10945],{},[43,10942,10376],{},[50,10944,2682],{},[308,10946,318],{},[109,10948,10949],{"encoding":111},"y\\in\\{\\pm1\\}^{d+1}",[15,10951,10953,10972],{"className":10952,"ariaHidden":72},[116],[15,10954,10956,10960,10963,10966,10969],{"className":10955},[120],[15,10957],{"className":10958,"style":10959},[124],"height:0.7335em;vertical-align:-0.1944em;",[15,10961,857],{"className":10962,"style":887},[129,133],[15,10964],{"className":10965,"style":199},[93],[15,10967,100],{"className":10968},[203],[15,10970],{"className":10971,"style":199},[93],[15,10973,10975,10979,10982,10985,10988],{"className":10974},[120],[15,10976],{"className":10977,"style":10978},[124],"height:1.0991em;vertical-align:-0.25em;",[15,10980,636],{"className":10981},[188],[15,10983,10932],{"className":10984},[129],[15,10986,318],{"className":10987},[129],[15,10989,10991,10994],{"className":10990},[195],[15,10992,645],{"className":10993},[195],[15,10995,10997],{"className":10996},[137],[15,10998,11000],{"className":10999},[141],[15,11001,11003],{"className":11002},[146],[15,11004,11006],{"className":11005,"style":10490},[150],[15,11007,11008,11011],{"style":383},[15,11009],{"className":11010,"style":159},[158],[15,11012,11014],{"className":11013},[163,164,165,166],[15,11015,11017,11020,11023],{"className":11016},[129,166],[15,11018,10376],{"className":11019},[129,133,166],[15,11021,2682],{"className":11022},[2436,166],[15,11024,318],{"className":11025},[129,166],"，方程组",[15,11028,11030],{"className":11029,"translate":19},[18],[15,11031,11033,11089],{"className":11032,"translate":19},[23],[15,11034,11036],{"className":11035},[27],[29,11037,11038],{"xmlns":31,"display":32},[34,11039,11040,11086],{},[37,11041,11042,11044,11046,11052,11054,11056,11058,11064,11066,11068,11070,11072,11074,11076,11078,11080,11082,11084],{},[43,11043,10550],{},[50,11045,10553],{},[40,11047,11048,11050],{},[43,11049,56],{},[43,11051,1459],{},[50,11053,2682],{},[43,11055,8699],{},[50,11057,62],{},[40,11059,11060,11062],{},[43,11061,857],{},[43,11063,1459],{},[50,11065,91],{"separator":72},[93,11067],{"width":95},[43,11069,1459],{},[50,11071,62],{},[308,11073,318],{},[50,11075,91],{"separator":72},[50,11077,335],{},[50,11079,91],{"separator":72},[43,11081,10376],{},[50,11083,2682],{},[308,11085,318],{},[109,11087,11088],{"encoding":111},"w\\cdot x_i+b=y_i,\\qquad i=1,\\dots,d+1",[15,11090,11092,11111,11167,11185,11253,11292],{"className":11091,"ariaHidden":72},[116],[15,11093,11095,11099,11102,11105,11108],{"className":11094},[120],[15,11096],{"className":11097,"style":11098},[124],"height:0.4445em;",[15,11100,10550],{"className":11101,"style":10627},[129,133],[15,11103],{"className":11104,"style":3277},[93],[15,11106,10553],{"className":11107},[2436],[15,11109],{"className":11110,"style":3277},[93],[15,11112,11114,11118,11158,11161,11164],{"className":11113},[120],[15,11115],{"className":11116,"style":11117},[124],"height:0.7333em;vertical-align:-0.15em;",[15,11119,11121,11124],{"className":11120},[129],[15,11122,56],{"className":11123},[129,133],[15,11125,11127],{"className":11126},[137],[15,11128,11130,11150],{"className":11129},[141,142],[15,11131,11133,11147],{"className":11132},[146],[15,11134,11136],{"className":11135,"style":1981},[150],[15,11137,11138,11141],{"style":154},[15,11139],{"className":11140,"style":159},[158],[15,11142,11144],{"className":11143},[163,164,165,166],[15,11145,1459],{"className":11146},[129,133,166],[15,11148,175],{"className":11149},[174],[15,11151,11153],{"className":11152},[146],[15,11154,11156],{"className":11155,"style":182},[150],[15,11157],{},[15,11159],{"className":11160,"style":3277},[93],[15,11162,2682],{"className":11163},[2436],[15,11165],{"className":11166,"style":3277},[93],[15,11168,11170,11173,11176,11179,11182],{"className":11169},[120],[15,11171],{"className":11172,"style":522},[124],[15,11174,8699],{"className":11175},[129,133],[15,11177],{"className":11178,"style":199},[93],[15,11180,62],{"className":11181},[203],[15,11183],{"className":11184,"style":199},[93],[15,11186,11188,11192,11232,11235,11238,11241,11244,11247,11250],{"className":11187},[120],[15,11189],{"className":11190,"style":11191},[124],"height:0.854em;vertical-align:-0.1944em;",[15,11193,11195,11198],{"className":11194},[129],[15,11196,857],{"className":11197,"style":887},[129,133],[15,11199,11201],{"className":11200},[137],[15,11202,11204,11224],{"className":11203},[141,142],[15,11205,11207,11221],{"className":11206},[146],[15,11208,11210],{"className":11209,"style":1981},[150],[15,11211,11212,11215],{"style":1100},[15,11213],{"className":11214,"style":159},[158],[15,11216,11218],{"className":11217},[163,164,165,166],[15,11219,1459],{"className":11220},[129,133,166],[15,11222,175],{"className":11223},[174],[15,11225,11227],{"className":11226},[146],[15,11228,11230],{"className":11229,"style":182},[150],[15,11231],{},[15,11233,91],{"className":11234},[254],[15,11236],{"className":11237,"style":258},[93],[15,11239],{"className":11240,"style":262},[93],[15,11242,1459],{"className":11243},[129,133],[15,11245],{"className":11246,"style":199},[93],[15,11248,62],{"className":11249},[203],[15,11251],{"className":11252,"style":199},[93],[15,11254,11256,11259,11262,11265,11268,11271,11274,11277,11280,11283,11286,11289],{"className":11255},[120],[15,11257],{"className":11258,"style":1682},[124],[15,11260,318],{"className":11261},[129],[15,11263,91],{"className":11264},[254],[15,11266],{"className":11267,"style":262},[93],[15,11269,335],{"className":11270},[449],[15,11272],{"className":11273,"style":262},[93],[15,11275,91],{"className":11276},[254],[15,11278],{"className":11279,"style":262},[93],[15,11281,10376],{"className":11282},[129,133],[15,11284],{"className":11285,"style":3277},[93],[15,11287,2682],{"className":11288},[2436],[15,11290],{"className":11291,"style":3277},[93],[15,11293,11295,11298],{"className":11294},[120],[15,11296],{"className":11297,"style":4886},[124],[15,11299,318],{"className":11300},[129],[11,11302,11303,11304,11397,11398,11514,11515,11543],{},"有唯一解，因为以 ",[15,11305,11307,11333],{"className":11306,"translate":19},[23],[15,11308,11310],{"className":11309},[27],[29,11311,11312],{"xmlns":31},[34,11313,11314,11330],{},[37,11315,11316,11318,11324,11326,11328],{},[50,11317,53],{"stretchy":52},[40,11319,11320,11322],{},[43,11321,56],{},[43,11323,1459],{},[50,11325,91],{"separator":72},[308,11327,318],{},[50,11329,59],{"stretchy":52},[109,11331,11332],{"encoding":111},"(x_i,1)",[15,11334,11336],{"className":11335,"ariaHidden":72},[116],[15,11337,11339,11342,11345,11385,11388,11391,11394],{"className":11338},[120],[15,11340],{"className":11341,"style":125},[124],[15,11343,53],{"className":11344},[188],[15,11346,11348,11351],{"className":11347},[129],[15,11349,56],{"className":11350},[129,133],[15,11352,11354],{"className":11353},[137],[15,11355,11357,11377],{"className":11356},[141,142],[15,11358,11360,11374],{"className":11359},[146],[15,11361,11363],{"className":11362,"style":1981},[150],[15,11364,11365,11368],{"style":154},[15,11366],{"className":11367,"style":159},[158],[15,11369,11371],{"className":11370},[163,164,165,166],[15,11372,1459],{"className":11373},[129,133,166],[15,11375,175],{"className":11376},[174],[15,11378,11380],{"className":11379},[146],[15,11381,11383],{"className":11382,"style":182},[150],[15,11384],{},[15,11386,91],{"className":11387},[254],[15,11389],{"className":11390,"style":262},[93],[15,11392,318],{"className":11393},[129],[15,11395,59],{"className":11396},[195]," 为列的 ",[15,11399,11401,11436],{"className":11400,"translate":19},[23],[15,11402,11404],{"className":11403},[27],[29,11405,11406],{"xmlns":31},[34,11407,11408,11433],{},[37,11409,11410,11412,11414,11416,11418,11420,11423,11425,11427,11429,11431],{},[50,11411,53],{"stretchy":52},[43,11413,10376],{},[50,11415,2682],{},[308,11417,318],{},[50,11419,59],{"stretchy":52},[50,11421,11422],{},"×",[50,11424,53],{"stretchy":52},[43,11426,10376],{},[50,11428,2682],{},[308,11430,318],{},[50,11432,59],{"stretchy":52},[109,11434,11435],{"encoding":111},"(d+1)\\times(d+1)",[15,11437,11439,11460,11481,11502],{"className":11438,"ariaHidden":72},[116],[15,11440,11442,11445,11448,11451,11454,11457],{"className":11441},[120],[15,11443],{"className":11444,"style":125},[124],[15,11446,53],{"className":11447},[188],[15,11449,10376],{"className":11450},[129,133],[15,11452],{"className":11453,"style":3277},[93],[15,11455,2682],{"className":11456},[2436],[15,11458],{"className":11459,"style":3277},[93],[15,11461,11463,11466,11469,11472,11475,11478],{"className":11462},[120],[15,11464],{"className":11465,"style":125},[124],[15,11467,318],{"className":11468},[129],[15,11470,59],{"className":11471},[195],[15,11473],{"className":11474,"style":3277},[93],[15,11476,11422],{"className":11477},[2436],[15,11479],{"className":11480,"style":3277},[93],[15,11482,11484,11487,11490,11493,11496,11499],{"className":11483},[120],[15,11485],{"className":11486,"style":125},[124],[15,11488,53],{"className":11489},[188],[15,11491,10376],{"className":11492},[129,133],[15,11494],{"className":11495,"style":3277},[93],[15,11497,2682],{"className":11498},[2436],[15,11500],{"className":11501,"style":3277},[93],[15,11503,11505,11508,11511],{"className":11504},[120],[15,11506],{"className":11507,"style":125},[124],[15,11509,318],{"className":11510},[129],[15,11512,59],{"className":11513},[195]," 矩阵可逆。于是每种标注都能被某个仿射函数实现，",[15,11516,11518,11531],{"className":11517,"translate":19},[23],[15,11519,11521],{"className":11520},[27],[29,11522,11523],{"xmlns":31},[34,11524,11525,11529],{},[37,11526,11527],{},[43,11528,5619],{},[109,11530,5619],{"encoding":111},[15,11532,11534],{"className":11533,"ariaHidden":72},[116],[15,11535,11537,11540],{"className":11536},[120],[15,11538],{"className":11539,"style":610},[124],[15,11541,5619],{"className":11542,"style":5665},[129,133]," 被打散。",[11,11545,11546,11549,11550,11609,11610,11661],{},[698,11547,11548],{},"上界","：",[15,11551,11553,11571],{"className":11552,"translate":19},[23],[15,11554,11556],{"className":11555},[27],[29,11557,11558],{"xmlns":31},[34,11559,11560,11568],{},[37,11561,11562],{},[305,11563,11564,11566],{},[43,11565,104],{"mathvariant":103},[43,11567,10376],{},[109,11569,11570],{"encoding":111},"\\mathbb R^d",[15,11572,11574],{"className":11573,"ariaHidden":72},[116],[15,11575,11577,11580],{"className":11576},[120],[15,11578],{"className":11579,"style":10490},[124],[15,11581,11583,11586],{"className":11582},[129],[15,11584,104],{"className":11585},[129,285],[15,11587,11589],{"className":11588},[137],[15,11590,11592],{"className":11591},[141],[15,11593,11595],{"className":11594},[146],[15,11596,11598],{"className":11597,"style":10490},[150],[15,11599,11600,11603],{"style":383},[15,11601],{"className":11602,"style":159},[158],[15,11604,11606],{"className":11605},[163,164,165,166],[15,11607,10376],{"className":11608},[129,133,166]," 中任意 ",[15,11611,11613,11631],{"className":11612,"translate":19},[23],[15,11614,11616],{"className":11615},[27],[29,11617,11618],{"xmlns":31},[34,11619,11620,11628],{},[37,11621,11622,11624,11626],{},[43,11623,10376],{},[50,11625,2682],{},[308,11627,310],{},[109,11629,11630],{"encoding":111},"d+2",[15,11632,11634,11652],{"className":11633,"ariaHidden":72},[116],[15,11635,11637,11640,11643,11646,11649],{"className":11636},[120],[15,11638],{"className":11639,"style":10414},[124],[15,11641,10376],{"className":11642},[129,133],[15,11644],{"className":11645,"style":3277},[93],[15,11647,2682],{"className":11648},[2436],[15,11650],{"className":11651,"style":3277},[93],[15,11653,11655,11658],{"className":11654},[120],[15,11656],{"className":11657,"style":4886},[124],[15,11659,310],{"className":11660},[129]," 个点都仿射相关，因此可以用 Radon 定理处理。",[6559,11663,11664],{},[11,11665,11666,804,11669,11609,11727,11777],{},[698,11667,11668],{},"定理（Radon，1921）",[15,11670,11672,11689],{"className":11671,"translate":19},[23],[15,11673,11675],{"className":11674},[27],[29,11676,11677],{"xmlns":31},[34,11678,11679,11687],{},[37,11680,11681],{},[305,11682,11683,11685],{},[43,11684,104],{"mathvariant":103},[43,11686,10376],{},[109,11688,11570],{"encoding":111},[15,11690,11692],{"className":11691,"ariaHidden":72},[116],[15,11693,11695,11698],{"className":11694},[120],[15,11696],{"className":11697,"style":10490},[124],[15,11699,11701,11704],{"className":11700},[129],[15,11702,104],{"className":11703},[129,285],[15,11705,11707],{"className":11706},[137],[15,11708,11710],{"className":11709},[141],[15,11711,11713],{"className":11712},[146],[15,11714,11716],{"className":11715,"style":10490},[150],[15,11717,11718,11721],{"style":383},[15,11719],{"className":11720,"style":159},[158],[15,11722,11724],{"className":11723},[163,164,165,166],[15,11725,10376],{"className":11726},[129,133,166],[15,11728,11730,11747],{"className":11729,"translate":19},[23],[15,11731,11733],{"className":11732},[27],[29,11734,11735],{"xmlns":31},[34,11736,11737,11745],{},[37,11738,11739,11741,11743],{},[43,11740,10376],{},[50,11742,2682],{},[308,11744,310],{},[109,11746,11630],{"encoding":111},[15,11748,11750,11768],{"className":11749,"ariaHidden":72},[116],[15,11751,11753,11756,11759,11762,11765],{"className":11752},[120],[15,11754],{"className":11755,"style":10414},[124],[15,11757,10376],{"className":11758},[129,133],[15,11760],{"className":11761,"style":3277},[93],[15,11763,2682],{"className":11764},[2436],[15,11766],{"className":11767,"style":3277},[93],[15,11769,11771,11774],{"className":11770},[120],[15,11772],{"className":11773,"style":4886},[124],[15,11775,310],{"className":11776},[129]," 个点都可以分成两个不相交的子集，使二者的凸包相交。",[11,11779,11780,11781,7610,11941,11991,11992,12065],{},"证明很短。把点写成齐次坐标 ",[15,11782,11784,11824],{"className":11783,"translate":19},[23],[15,11785,11787],{"className":11786},[27],[29,11788,11789],{"xmlns":31},[34,11790,11791,11821],{},[37,11792,11793,11795,11801,11803,11805,11807,11809],{},[50,11794,53],{"stretchy":52},[40,11796,11797,11799],{},[43,11798,56],{},[43,11800,1459],{},[50,11802,91],{"separator":72},[308,11804,318],{},[50,11806,59],{"stretchy":52},[50,11808,100],{},[305,11810,11811,11813],{},[43,11812,104],{"mathvariant":103},[37,11814,11815,11817,11819],{},[43,11816,10376],{},[50,11818,2682],{},[308,11820,318],{},[109,11822,11823],{"encoding":111},"(x_i,1)\\in\\mathbb R^{d+1}",[15,11825,11827,11897],{"className":11826,"ariaHidden":72},[116],[15,11828,11830,11833,11836,11876,11879,11882,11885,11888,11891,11894],{"className":11829},[120],[15,11831],{"className":11832,"style":125},[124],[15,11834,53],{"className":11835},[188],[15,11837,11839,11842],{"className":11838},[129],[15,11840,56],{"className":11841},[129,133],[15,11843,11845],{"className":11844},[137],[15,11846,11848,11868],{"className":11847},[141,142],[15,11849,11851,11865],{"className":11850},[146],[15,11852,11854],{"className":11853,"style":1981},[150],[15,11855,11856,11859],{"style":154},[15,11857],{"className":11858,"style":159},[158],[15,11860,11862],{"className":11861},[163,164,165,166],[15,11863,1459],{"className":11864},[129,133,166],[15,11866,175],{"className":11867},[174],[15,11869,11871],{"className":11870},[146],[15,11872,11874],{"className":11873,"style":182},[150],[15,11875],{},[15,11877,91],{"className":11878},[254],[15,11880],{"className":11881,"style":262},[93],[15,11883,318],{"className":11884},[129],[15,11886,59],{"className":11887},[195],[15,11889],{"className":11890,"style":199},[93],[15,11892,100],{"className":11893},[203],[15,11895],{"className":11896,"style":199},[93],[15,11898,11900,11903],{"className":11899},[120],[15,11901],{"className":11902,"style":10490},[124],[15,11904,11906,11909],{"className":11905},[129],[15,11907,104],{"className":11908},[129,285],[15,11910,11912],{"className":11911},[137],[15,11913,11915],{"className":11914},[141],[15,11916,11918],{"className":11917},[146],[15,11919,11921],{"className":11920,"style":10490},[150],[15,11922,11923,11926],{"style":383},[15,11924],{"className":11925,"style":159},[158],[15,11927,11929],{"className":11928},[163,164,165,166],[15,11930,11932,11935,11938],{"className":11931},[129,166],[15,11933,10376],{"className":11934},[129,133,166],[15,11936,2682],{"className":11937},[2436,166],[15,11939,318],{"className":11940},[129,166],[15,11942,11944,11961],{"className":11943,"translate":19},[23],[15,11945,11947],{"className":11946},[27],[29,11948,11949],{"xmlns":31},[34,11950,11951,11959],{},[37,11952,11953,11955,11957],{},[43,11954,10376],{},[50,11956,2682],{},[308,11958,310],{},[109,11960,11630],{"encoding":111},[15,11962,11964,11982],{"className":11963,"ariaHidden":72},[116],[15,11965,11967,11970,11973,11976,11979],{"className":11966},[120],[15,11968],{"className":11969,"style":10414},[124],[15,11971,10376],{"className":11972},[129,133],[15,11974],{"className":11975,"style":3277},[93],[15,11977,2682],{"className":11978},[2436],[15,11980],{"className":11981,"style":3277},[93],[15,11983,11985,11988],{"className":11984},[120],[15,11986],{"className":11987,"style":4886},[124],[15,11989,310],{"className":11990},[129]," 个向量线性相关，故存在不全为零的系数 ",[15,11993,11995,12014],{"className":11994,"translate":19},[23],[15,11996,11998],{"className":11997},[27],[29,11999,12000],{"xmlns":31},[34,12001,12002,12011],{},[37,12003,12004],{},[40,12005,12006,12009],{},[43,12007,12008],{},"α",[43,12010,1459],{},[109,12012,12013],{"encoding":111},"\\alpha_i",[15,12015,12017],{"className":12016,"ariaHidden":72},[116],[15,12018,12020,12023],{"className":12019},[120],[15,12021],{"className":12022,"style":7985},[124],[15,12024,12026,12030],{"className":12025},[129],[15,12027,12008],{"className":12028,"style":12029},[129,133],"margin-right:0.0037em;",[15,12031,12033],{"className":12032},[137],[15,12034,12036,12057],{"className":12035},[141,142],[15,12037,12039,12054],{"className":12038},[146],[15,12040,12042],{"className":12041,"style":1981},[150],[15,12043,12045,12048],{"style":12044},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[15,12046],{"className":12047,"style":159},[158],[15,12049,12051],{"className":12050},[163,164,165,166],[15,12052,1459],{"className":12053},[129,133,166],[15,12055,175],{"className":12056},[174],[15,12058,12060],{"className":12059},[146],[15,12061,12063],{"className":12062,"style":182},[150],[15,12064],{},"，使",[15,12067,12069],{"className":12068,"translate":19},[18],[15,12070,12072,12127],{"className":12071,"translate":19},[23],[15,12073,12075],{"className":12074},[27],[29,12076,12077],{"xmlns":31,"display":32},[34,12078,12079,12124],{},[37,12080,12081,12087,12093,12099,12101,12103,12105,12107,12113,12119,12121],{},[2140,12082,12083,12085],{},[50,12084,1454],{},[43,12086,1459],{},[40,12088,12089,12091],{},[43,12090,12008],{},[43,12092,1459],{},[40,12094,12095,12097],{},[43,12096,56],{},[43,12098,1459],{},[50,12100,62],{},[308,12102,541],{},[50,12104,91],{"separator":72},[93,12106],{"width":95},[2140,12108,12109,12111],{},[50,12110,1454],{},[43,12112,1459],{},[40,12114,12115,12117],{},[43,12116,12008],{},[43,12118,1459],{},[50,12120,62],{},[308,12122,12123],{},"0.",[109,12125,12126],{"encoding":111},"\\sum_i \\alpha_i x_i=0,\\qquad \\sum_i\\alpha_i=0.",[15,12128,12130,12274,12388],{"className":12129,"ariaHidden":72},[116],[15,12131,12133,12137,12182,12185,12225,12265,12268,12271],{"className":12132},[120],[15,12134],{"className":12135,"style":12136},[124],"height:2.3277em;vertical-align:-1.2777em;",[15,12138,12140],{"className":12139},[217,1874],[15,12141,12143,12174],{"className":12142},[141,142],[15,12144,12146,12171],{"className":12145},[146],[15,12147,12150,12161],{"className":12148,"style":12149},[150],"height:1.05em;",[15,12151,12152,12155],{"style":1887},[15,12153],{"className":12154,"style":1891},[158],[15,12156,12158],{"className":12157},[163,164,165,166],[15,12159,1459],{"className":12160},[129,133,166],[15,12162,12163,12166],{"style":1909},[15,12164],{"className":12165,"style":1891},[158],[15,12167,12168],{},[15,12169,1454],{"className":12170},[217,1918,1919],[15,12172,175],{"className":12173},[174],[15,12175,12177],{"className":12176},[146],[15,12178,12180],{"className":12179,"style":1941},[150],[15,12181],{},[15,12183],{"className":12184,"style":262},[93],[15,12186,12188,12191],{"className":12187},[129],[15,12189,12008],{"className":12190,"style":12029},[129,133],[15,12192,12194],{"className":12193},[137],[15,12195,12197,12217],{"className":12196},[141,142],[15,12198,12200,12214],{"className":12199},[146],[15,12201,12203],{"className":12202,"style":1981},[150],[15,12204,12205,12208],{"style":12044},[15,12206],{"className":12207,"style":159},[158],[15,12209,12211],{"className":12210},[163,164,165,166],[15,12212,1459],{"className":12213},[129,133,166],[15,12215,175],{"className":12216},[174],[15,12218,12220],{"className":12219},[146],[15,12221,12223],{"className":12222,"style":182},[150],[15,12224],{},[15,12226,12228,12231],{"className":12227},[129],[15,12229,56],{"className":12230},[129,133],[15,12232,12234],{"className":12233},[137],[15,12235,12237,12257],{"className":12236},[141,142],[15,12238,12240,12254],{"className":12239},[146],[15,12241,12243],{"className":12242,"style":1981},[150],[15,12244,12245,12248],{"style":154},[15,12246],{"className":12247,"style":159},[158],[15,12249,12251],{"className":12250},[163,164,165,166],[15,12252,1459],{"className":12253},[129,133,166],[15,12255,175],{"className":12256},[174],[15,12258,12260],{"className":12259},[146],[15,12261,12263],{"className":12262,"style":182},[150],[15,12264],{},[15,12266],{"className":12267,"style":199},[93],[15,12269,62],{"className":12270},[203],[15,12272],{"className":12273,"style":199},[93],[15,12275,12277,12280,12283,12286,12289,12292,12336,12339,12379,12382,12385],{"className":12276},[120],[15,12278],{"className":12279,"style":12136},[124],[15,12281,541],{"className":12282},[129],[15,12284,91],{"className":12285},[254],[15,12287],{"className":12288,"style":258},[93],[15,12290],{"className":12291,"style":262},[93],[15,12293,12295],{"className":12294},[217,1874],[15,12296,12298,12328],{"className":12297},[141,142],[15,12299,12301,12325],{"className":12300},[146],[15,12302,12304,12315],{"className":12303,"style":12149},[150],[15,12305,12306,12309],{"style":1887},[15,12307],{"className":12308,"style":1891},[158],[15,12310,12312],{"className":12311},[163,164,165,166],[15,12313,1459],{"className":12314},[129,133,166],[15,12316,12317,12320],{"style":1909},[15,12318],{"className":12319,"style":1891},[158],[15,12321,12322],{},[15,12323,1454],{"className":12324},[217,1918,1919],[15,12326,175],{"className":12327},[174],[15,12329,12331],{"className":12330},[146],[15,12332,12334],{"className":12333,"style":1941},[150],[15,12335],{},[15,12337],{"className":12338,"style":262},[93],[15,12340,12342,12345],{"className":12341},[129],[15,12343,12008],{"className":12344,"style":12029},[129,133],[15,12346,12348],{"className":12347},[137],[15,12349,12351,12371],{"className":12350},[141,142],[15,12352,12354,12368],{"className":12353},[146],[15,12355,12357],{"className":12356,"style":1981},[150],[15,12358,12359,12362],{"style":12044},[15,12360],{"className":12361,"style":159},[158],[15,12363,12365],{"className":12364},[163,164,165,166],[15,12366,1459],{"className":12367},[129,133,166],[15,12369,175],{"className":12370},[174],[15,12372,12374],{"className":12373},[146],[15,12375,12377],{"className":12376,"style":182},[150],[15,12378],{},[15,12380],{"className":12381,"style":199},[93],[15,12383,62],{"className":12384},[203],[15,12386],{"className":12387,"style":199},[93],[15,12389,12391,12394],{"className":12390},[120],[15,12392],{"className":12393,"style":4886},[124],[15,12395,12123],{"className":12396},[129],[11,12398,12399,12400,7610,12547,12695],{},"令 ",[15,12401,12403,12437],{"className":12402,"translate":19},[23],[15,12404,12406],{"className":12405},[27],[29,12407,12408],{"xmlns":31},[34,12409,12410,12434],{},[37,12411,12412,12414,12416,12418,12420,12422,12428,12430,12432],{},[43,12413,1388],{},[50,12415,62],{},[50,12417,636],{"stretchy":52},[43,12419,1459],{},[50,12421,718],{},[40,12423,12424,12426],{},[43,12425,12008],{},[43,12427,1459],{},[50,12429,4852],{},[308,12431,541],{},[50,12433,645],{"stretchy":52},[109,12435,12436],{"encoding":111},"P=\\{i:\\alpha_i>0\\}",[15,12438,12440,12459,12480,12535],{"className":12439,"ariaHidden":72},[116],[15,12441,12443,12446,12450,12453,12456],{"className":12442},[120],[15,12444],{"className":12445,"style":610},[124],[15,12447,1388],{"className":12448,"style":12449},[129,133],"margin-right:0.1389em;",[15,12451],{"className":12452,"style":199},[93],[15,12454,62],{"className":12455},[203],[15,12457],{"className":12458,"style":199},[93],[15,12460,12462,12465,12468,12471,12474,12477],{"className":12461},[120],[15,12463],{"className":12464,"style":125},[124],[15,12466,636],{"className":12467},[188],[15,12469,1459],{"className":12470},[129,133],[15,12472],{"className":12473,"style":199},[93],[15,12475,718],{"className":12476},[203],[15,12478],{"className":12479,"style":199},[93],[15,12481,12483,12486,12526,12529,12532],{"className":12482},[120],[15,12484],{"className":12485,"style":7929},[124],[15,12487,12489,12492],{"className":12488},[129],[15,12490,12008],{"className":12491,"style":12029},[129,133],[15,12493,12495],{"className":12494},[137],[15,12496,12498,12518],{"className":12497},[141,142],[15,12499,12501,12515],{"className":12500},[146],[15,12502,12504],{"className":12503,"style":1981},[150],[15,12505,12506,12509],{"style":12044},[15,12507],{"className":12508,"style":159},[158],[15,12510,12512],{"className":12511},[163,164,165,166],[15,12513,1459],{"className":12514},[129,133,166],[15,12516,175],{"className":12517},[174],[15,12519,12521],{"className":12520},[146],[15,12522,12524],{"className":12523,"style":182},[150],[15,12525],{},[15,12527],{"className":12528,"style":199},[93],[15,12530,4852],{"className":12531},[203],[15,12533],{"className":12534,"style":199},[93],[15,12536,12538,12541,12544],{"className":12537},[120],[15,12539],{"className":12540,"style":125},[124],[15,12542,541],{"className":12543},[129],[15,12545,645],{"className":12546},[195],[15,12548,12550,12585],{"className":12549,"translate":19},[23],[15,12551,12553],{"className":12552},[27],[29,12554,12555],{"xmlns":31},[34,12556,12557,12582],{},[37,12558,12559,12562,12564,12566,12568,12570,12576,12578,12580],{},[43,12560,12561],{},"N",[50,12563,62],{},[50,12565,636],{"stretchy":52},[43,12567,1459],{},[50,12569,718],{},[40,12571,12572,12574],{},[43,12573,12008],{},[43,12575,1459],{},[50,12577,7910],{},[308,12579,541],{},[50,12581,645],{"stretchy":52},[109,12583,12584],{"encoding":111},"N=\\{i:\\alpha_i\u003C0\\}",[15,12586,12588,12607,12628,12683],{"className":12587,"ariaHidden":72},[116],[15,12589,12591,12594,12598,12601,12604],{"className":12590},[120],[15,12592],{"className":12593,"style":610},[124],[15,12595,12561],{"className":12596,"style":12597},[129,133],"margin-right:0.109em;",[15,12599],{"className":12600,"style":199},[93],[15,12602,62],{"className":12603},[203],[15,12605],{"className":12606,"style":199},[93],[15,12608,12610,12613,12616,12619,12622,12625],{"className":12609},[120],[15,12611],{"className":12612,"style":125},[124],[15,12614,636],{"className":12615},[188],[15,12617,1459],{"className":12618},[129,133],[15,12620],{"className":12621,"style":199},[93],[15,12623,718],{"className":12624},[203],[15,12626],{"className":12627,"style":199},[93],[15,12629,12631,12634,12674,12677,12680],{"className":12630},[120],[15,12632],{"className":12633,"style":7929},[124],[15,12635,12637,12640],{"className":12636},[129],[15,12638,12008],{"className":12639,"style":12029},[129,133],[15,12641,12643],{"className":12642},[137],[15,12644,12646,12666],{"className":12645},[141,142],[15,12647,12649,12663],{"className":12648},[146],[15,12650,12652],{"className":12651,"style":1981},[150],[15,12653,12654,12657],{"style":12044},[15,12655],{"className":12656,"style":159},[158],[15,12658,12660],{"className":12659},[163,164,165,166],[15,12661,1459],{"className":12662},[129,133,166],[15,12664,175],{"className":12665},[174],[15,12667,12669],{"className":12668},[146],[15,12670,12672],{"className":12671,"style":182},[150],[15,12673],{},[15,12675],{"className":12676,"style":199},[93],[15,12678,7910],{"className":12679},[203],[15,12681],{"className":12682,"style":199},[93],[15,12684,12686,12689,12692],{"className":12685},[120],[15,12687],{"className":12688,"style":125},[124],[15,12690,541],{"className":12691},[129],[15,12693,645],{"className":12694},[195],"。两者都非空，且",[15,12697,12699],{"className":12698,"translate":19},[18],[15,12700,12702,12772],{"className":12701,"translate":19},[23],[15,12703,12705],{"className":12704},[27],[29,12706,12707],{"xmlns":31,"display":32},[34,12708,12709,12769],{},[37,12710,12711,12723,12729,12735,12737,12749,12751,12753,12759,12761,12767],{},[2140,12712,12713,12715],{},[50,12714,1454],{},[37,12716,12717,12719,12721],{},[43,12718,1459],{},[50,12720,100],{},[43,12722,1388],{},[40,12724,12725,12727],{},[43,12726,12008],{},[43,12728,1459],{},[40,12730,12731,12733],{},[43,12732,56],{},[43,12734,1459],{},[50,12736,62],{},[2140,12738,12739,12741],{},[50,12740,1454],{},[37,12742,12743,12745,12747],{},[43,12744,1459],{},[50,12746,100],{},[43,12748,12561],{},[50,12750,53],{"stretchy":52},[50,12752,315],{},[40,12754,12755,12757],{},[43,12756,12008],{},[43,12758,1459],{},[50,12760,59],{"stretchy":52},[40,12762,12763,12765],{},[43,12764,56],{},[43,12766,1459],{},[43,12768,107],{"mathvariant":65},[109,12770,12771],{"encoding":111},"\\sum_{i\\in P}\\alpha_i x_i=\\sum_{i\\in N}(-\\alpha_i)x_i.",[15,12773,12775,12929],{"className":12774,"ariaHidden":72},[116],[15,12776,12778,12782,12837,12840,12880,12920,12923,12926],{"className":12777},[120],[15,12779],{"className":12780,"style":12781},[124],"height:2.3717em;vertical-align:-1.3217em;",[15,12783,12785],{"className":12784},[217,1874],[15,12786,12788,12828],{"className":12787},[141,142],[15,12789,12791,12825],{"className":12790},[146],[15,12792,12794,12815],{"className":12793,"style":12149},[150],[15,12795,12797,12800],{"style":12796},"top:-1.8557em;margin-left:0em;",[15,12798],{"className":12799,"style":1891},[158],[15,12801,12803],{"className":12802},[163,164,165,166],[15,12804,12806,12809,12812],{"className":12805},[129,166],[15,12807,1459],{"className":12808},[129,133,166],[15,12810,100],{"className":12811},[203,166],[15,12813,1388],{"className":12814,"style":12449},[129,133,166],[15,12816,12817,12820],{"style":1909},[15,12818],{"className":12819,"style":1891},[158],[15,12821,12822],{},[15,12823,1454],{"className":12824},[217,1918,1919],[15,12826,175],{"className":12827},[174],[15,12829,12831],{"className":12830},[146],[15,12832,12835],{"className":12833,"style":12834},[150],"height:1.3217em;",[15,12836],{},[15,12838],{"className":12839,"style":262},[93],[15,12841,12843,12846],{"className":12842},[129],[15,12844,12008],{"className":12845,"style":12029},[129,133],[15,12847,12849],{"className":12848},[137],[15,12850,12852,12872],{"className":12851},[141,142],[15,12853,12855,12869],{"className":12854},[146],[15,12856,12858],{"className":12857,"style":1981},[150],[15,12859,12860,12863],{"style":12044},[15,12861],{"className":12862,"style":159},[158],[15,12864,12866],{"className":12865},[163,164,165,166],[15,12867,1459],{"className":12868},[129,133,166],[15,12870,175],{"className":12871},[174],[15,12873,12875],{"className":12874},[146],[15,12876,12878],{"className":12877,"style":182},[150],[15,12879],{},[15,12881,12883,12886],{"className":12882},[129],[15,12884,56],{"className":12885},[129,133],[15,12887,12889],{"className":12888},[137],[15,12890,12892,12912],{"className":12891},[141,142],[15,12893,12895,12909],{"className":12894},[146],[15,12896,12898],{"className":12897,"style":1981},[150],[15,12899,12900,12903],{"style":154},[15,12901],{"className":12902,"style":159},[158],[15,12904,12906],{"className":12905},[163,164,165,166],[15,12907,1459],{"className":12908},[129,133,166],[15,12910,175],{"className":12911},[174],[15,12913,12915],{"className":12914},[146],[15,12916,12918],{"className":12917,"style":182},[150],[15,12919],{},[15,12921],{"className":12922,"style":199},[93],[15,12924,62],{"className":12925},[203],[15,12927],{"className":12928,"style":199},[93],[15,12930,12932,12935,12988,12991,12994,13034,13037,13077],{"className":12931},[120],[15,12933],{"className":12934,"style":12781},[124],[15,12936,12938],{"className":12937},[217,1874],[15,12939,12941,12980],{"className":12940},[141,142],[15,12942,12944,12977],{"className":12943},[146],[15,12945,12947,12967],{"className":12946,"style":12149},[150],[15,12948,12949,12952],{"style":12796},[15,12950],{"className":12951,"style":1891},[158],[15,12953,12955],{"className":12954},[163,164,165,166],[15,12956,12958,12961,12964],{"className":12957},[129,166],[15,12959,1459],{"className":12960},[129,133,166],[15,12962,100],{"className":12963},[203,166],[15,12965,12561],{"className":12966,"style":12597},[129,133,166],[15,12968,12969,12972],{"style":1909},[15,12970],{"className":12971,"style":1891},[158],[15,12973,12974],{},[15,12975,1454],{"className":12976},[217,1918,1919],[15,12978,175],{"className":12979},[174],[15,12981,12983],{"className":12982},[146],[15,12984,12986],{"className":12985,"style":12834},[150],[15,12987],{},[15,12989,53],{"className":12990},[188],[15,12992,315],{"className":12993},[129],[15,12995,12997,13000],{"className":12996},[129],[15,12998,12008],{"className":12999,"style":12029},[129,133],[15,13001,13003],{"className":13002},[137],[15,13004,13006,13026],{"className":13005},[141,142],[15,13007,13009,13023],{"className":13008},[146],[15,13010,13012],{"className":13011,"style":1981},[150],[15,13013,13014,13017],{"style":12044},[15,13015],{"className":13016,"style":159},[158],[15,13018,13020],{"className":13019},[163,164,165,166],[15,13021,1459],{"className":13022},[129,133,166],[15,13024,175],{"className":13025},[174],[15,13027,13029],{"className":13028},[146],[15,13030,13032],{"className":13031,"style":182},[150],[15,13033],{},[15,13035,59],{"className":13036},[195],[15,13038,13040,13043],{"className":13039},[129],[15,13041,56],{"className":13042},[129,133],[15,13044,13046],{"className":13045},[137],[15,13047,13049,13069],{"className":13048},[141,142],[15,13050,13052,13066],{"className":13051},[146],[15,13053,13055],{"className":13054,"style":1981},[150],[15,13056,13057,13060],{"style":154},[15,13058],{"className":13059,"style":159},[158],[15,13061,13063],{"className":13062},[163,164,165,166],[15,13064,1459],{"className":13065},[129,133,166],[15,13067,175],{"className":13068},[174],[15,13070,13072],{"className":13071},[146],[15,13073,13075],{"className":13074,"style":182},[150],[15,13076],{},[15,13078,107],{"className":13079},[129],[11,13081,13082,13083,13136,13137,13165,13166,1313],{},"两边系数之和相同，记为 ",[15,13084,13086,13105],{"className":13085,"translate":19},[23],[15,13087,13089],{"className":13088},[27],[29,13090,13091],{"xmlns":31},[34,13092,13093,13102],{},[37,13094,13095,13098,13100],{},[43,13096,13097],{},"A",[50,13099,4852],{},[308,13101,541],{},[109,13103,13104],{"encoding":111},"A>0",[15,13106,13108,13127],{"className":13107,"ariaHidden":72},[116],[15,13109,13111,13115,13118,13121,13124],{"className":13110},[120],[15,13112],{"className":13113,"style":13114},[124],"height:0.7224em;vertical-align:-0.0391em;",[15,13116,13097],{"className":13117},[129,133],[15,13119],{"className":13120,"style":199},[93],[15,13122,4852],{"className":13123},[203],[15,13125],{"className":13126,"style":199},[93],[15,13128,13130,13133],{"className":13129},[120],[15,13131],{"className":13132,"style":4886},[124],[15,13134,541],{"className":13135},[129],"。除以 ",[15,13138,13140,13153],{"className":13139,"translate":19},[23],[15,13141,13143],{"className":13142},[27],[29,13144,13145],{"xmlns":31},[34,13146,13147,13151],{},[37,13148,13149],{},[43,13150,13097],{},[109,13152,13097],{"encoding":111},[15,13154,13156],{"className":13155,"ariaHidden":72},[116],[15,13157,13159,13162],{"className":13158},[120],[15,13160],{"className":13161,"style":610},[124],[15,13163,13097],{"className":13164},[129,133],"，得到同一点的两个凸组合表示，于是 ",[15,13167,13169,13210],{"className":13168,"translate":19},[23],[15,13170,13172],{"className":13171},[27],[29,13173,13174],{"xmlns":31},[34,13175,13176,13207],{},[37,13177,13178,13181,13183,13185,13187,13189,13192,13194,13196,13198,13200,13202,13204],{},[43,13179,13180],{"mathvariant":65},"conv",[50,13182,69],{},[50,13184,53],{"stretchy":52},[43,13186,1388],{},[50,13188,59],{"stretchy":52},[50,13190,13191],{},"∩",[43,13193,13180],{"mathvariant":65},[50,13195,69],{},[50,13197,53],{"stretchy":52},[43,13199,12561],{},[50,13201,59],{"stretchy":52},[50,13203,1418],{"mathvariant":65},[43,13205,13206],{"mathvariant":65},"∅",[109,13208,13209],{"encoding":111},"\\operatorname{conv}(P)\\cap\\operatorname{conv}(N)\\neq\\varnothing",[15,13211,13213,13243,13306],{"className":13212,"ariaHidden":72},[116],[15,13214,13216,13219,13225,13228,13231,13234,13237,13240],{"className":13215},[120],[15,13217],{"className":13218,"style":125},[124],[15,13220,13222],{"className":13221},[217],[15,13223,13180],{"className":13224,"style":2246},[129,221],[15,13226,53],{"className":13227},[188],[15,13229,1388],{"className":13230,"style":12449},[129,133],[15,13232,59],{"className":13233},[195],[15,13235],{"className":13236,"style":3277},[93],[15,13238,13191],{"className":13239},[2436],[15,13241],{"className":13242,"style":3277},[93],[15,13244,13246,13249,13255,13258,13261,13264,13267,13303],{"className":13245},[120],[15,13247],{"className":13248,"style":125},[124],[15,13250,13252],{"className":13251},[217],[15,13253,13180],{"className":13254,"style":2246},[129,221],[15,13256,53],{"className":13257},[188],[15,13259,12561],{"className":13260,"style":12597},[129,133],[15,13262,59],{"className":13263},[195],[15,13265],{"className":13266,"style":199},[93],[15,13268,13270,13297,13300],{"className":13269},[203],[15,13271,13273],{"className":13272},[203],[15,13274,13276],{"className":13275},[129,1670],[15,13277,13279],{"className":13278},[1674],[15,13280,13282,13285,13294],{"className":13281},[1678],[15,13283],{"className":13284,"style":1682},[124],[15,13286,13288],{"className":13287},[1686],[15,13289,13291],{"className":13290},[129],[15,13292,1693],{"className":13293},[203],[15,13295],{"className":13296},[1697],[15,13298],{"className":13299},[93,1701],[15,13301,62],{"className":13302},[203],[15,13304],{"className":13305,"style":199},[93],[15,13307,13309,13313],{"className":13308},[120],[15,13310],{"className":13311,"style":13312},[124],"height:0.6633em;vertical-align:-0.0817em;",[15,13314,13206],{"className":13315},[129,13316],"amsrm",[11,13318,13319,13320,13348,13349,13377,13378,13406,13407,13456,13457,13485,13486,13535,13536,13586,13587,1313],{},"现在取标注「",[15,13321,13323,13336],{"className":13322,"translate":19},[23],[15,13324,13326],{"className":13325},[27],[29,13327,13328],{"xmlns":31},[34,13329,13330,13334],{},[37,13331,13332],{},[43,13333,1388],{},[109,13335,1388],{"encoding":111},[15,13337,13339],{"className":13338,"ariaHidden":72},[116],[15,13340,13342,13345],{"className":13341},[120],[15,13343],{"className":13344,"style":610},[124],[15,13346,1388],{"className":13347,"style":12449},[129,133]," 为正、",[15,13350,13352,13365],{"className":13351,"translate":19},[23],[15,13353,13355],{"className":13354},[27],[29,13356,13357],{"xmlns":31},[34,13358,13359,13363],{},[37,13360,13361],{},[43,13362,12561],{},[109,13364,12561],{"encoding":111},[15,13366,13368],{"className":13367,"ariaHidden":72},[116],[15,13369,13371,13374],{"className":13370},[120],[15,13372],{"className":13373,"style":610},[124],[15,13375,12561],{"className":13376,"style":12597},[129,133]," 为负」（系数为零的点任意归入一侧，它们不影响下面的凸组合）。任何包含 ",[15,13379,13381,13394],{"className":13380,"translate":19},[23],[15,13382,13384],{"className":13383},[27],[29,13385,13386],{"xmlns":31},[34,13387,13388,13392],{},[37,13389,13390],{},[43,13391,1388],{},[109,13393,1388],{"encoding":111},[15,13395,13397],{"className":13396,"ariaHidden":72},[116],[15,13398,13400,13403],{"className":13399},[120],[15,13401],{"className":13402,"style":610},[124],[15,13404,1388],{"className":13405,"style":12449},[129,133]," 的半空间都包含 ",[15,13408,13410,13432],{"className":13409,"translate":19},[23],[15,13411,13413],{"className":13412},[27],[29,13414,13415],{"xmlns":31},[34,13416,13417,13429],{},[37,13418,13419,13421,13423,13425,13427],{},[43,13420,13180],{"mathvariant":65},[50,13422,69],{},[50,13424,53],{"stretchy":52},[43,13426,1388],{},[50,13428,59],{"stretchy":52},[109,13430,13431],{"encoding":111},"\\operatorname{conv}(P)",[15,13433,13435],{"className":13434,"ariaHidden":72},[116],[15,13436,13438,13441,13447,13450,13453],{"className":13437},[120],[15,13439],{"className":13440,"style":125},[124],[15,13442,13444],{"className":13443},[217],[15,13445,13180],{"className":13446,"style":2246},[129,221],[15,13448,53],{"className":13449},[188],[15,13451,1388],{"className":13452,"style":12449},[129,133],[15,13454,59],{"className":13455},[195],"；任何不包含 ",[15,13458,13460,13473],{"className":13459,"translate":19},[23],[15,13461,13463],{"className":13462},[27],[29,13464,13465],{"xmlns":31},[34,13466,13467,13471],{},[37,13468,13469],{},[43,13470,12561],{},[109,13472,12561],{"encoding":111},[15,13474,13476],{"className":13475,"ariaHidden":72},[116],[15,13477,13479,13482],{"className":13478},[120],[15,13480],{"className":13481,"style":610},[124],[15,13483,12561],{"className":13484,"style":12597},[129,133]," 的半空间，其补集是凸的，因而也不包含 ",[15,13487,13489,13511],{"className":13488,"translate":19},[23],[15,13490,13492],{"className":13491},[27],[29,13493,13494],{"xmlns":31},[34,13495,13496,13508],{},[37,13497,13498,13500,13502,13504,13506],{},[43,13499,13180],{"mathvariant":65},[50,13501,69],{},[50,13503,53],{"stretchy":52},[43,13505,12561],{},[50,13507,59],{"stretchy":52},[109,13509,13510],{"encoding":111},"\\operatorname{conv}(N)",[15,13512,13514],{"className":13513,"ariaHidden":72},[116],[15,13515,13517,13520,13526,13529,13532],{"className":13516},[120],[15,13518],{"className":13519,"style":125},[124],[15,13521,13523],{"className":13522},[217],[15,13524,13180],{"className":13525,"style":2246},[129,221],[15,13527,53],{"className":13528},[188],[15,13530,12561],{"className":13531,"style":12597},[129,133],[15,13533,59],{"className":13534},[195],"。两者相交，矛盾。所以 ",[15,13537,13539,13556],{"className":13538,"translate":19},[23],[15,13540,13542],{"className":13541},[27],[29,13543,13544],{"xmlns":31},[34,13545,13546,13554],{},[37,13547,13548,13550,13552],{},[43,13549,10376],{},[50,13551,2682],{},[308,13553,310],{},[109,13555,11630],{"encoding":111},[15,13557,13559,13577],{"className":13558,"ariaHidden":72},[116],[15,13560,13562,13565,13568,13571,13574],{"className":13561},[120],[15,13563],{"className":13564,"style":10414},[124],[15,13566,10376],{"className":13567},[129,133],[15,13569],{"className":13570,"style":3277},[93],[15,13572,2682],{"className":13573},[2436],[15,13575],{"className":13576,"style":3277},[93],[15,13578,13580,13583],{"className":13579},[120],[15,13581],{"className":13582,"style":4886},[124],[15,13584,310],{"className":13585},[129]," 个点无法被打散，",[15,13588,13590,13613],{"className":13589,"translate":19},[23],[15,13591,13593],{"className":13592},[27],[29,13594,13595],{"xmlns":31},[34,13596,13597,13611],{},[37,13598,13599,13601,13603,13605,13607,13609],{},[43,13600,6823],{"mathvariant":65},[50,13602,69],{},[50,13604,62],{},[43,13606,10376],{},[50,13608,2682],{},[308,13610,318],{},[109,13612,10383],{"encoding":111},[15,13614,13616,13637,13655],{"className":13615,"ariaHidden":72},[116],[15,13617,13619,13622,13628,13631,13634],{"className":13618},[120],[15,13620],{"className":13621,"style":522},[124],[15,13623,13625],{"className":13624},[217],[15,13626,6823],{"className":13627},[129,221],[15,13629],{"className":13630,"style":199},[93],[15,13632,62],{"className":13633},[203],[15,13635],{"className":13636,"style":199},[93],[15,13638,13640,13643,13646,13649,13652],{"className":13639},[120],[15,13641],{"className":13642,"style":10414},[124],[15,13644,10376],{"className":13645},[129,133],[15,13647],{"className":13648,"style":3277},[93],[15,13650,2682],{"className":13651},[2436],[15,13653],{"className":13654,"style":3277},[93],[15,13656,13658,13661],{"className":13657},[120],[15,13659],{"className":13660,"style":4886},[124],[15,13662,318],{"className":13663},[129],[11,13665,13666],{},"这个证明值得记住：它把「线性分类器的容量」归结为「仿射无关点的最大数目」，而后者正是线性代数意义上的维数加一。VC 维里的「维」字，在这里有了具体的所指。",[11,13668,13669],{},[698,13670,13671,13672,7438],{},"例 4（轴对齐矩形，",[15,13673,13675,13697],{"className":13674,"translate":19},[23],[15,13676,13678],{"className":13677},[27],[29,13679,13680],{"xmlns":31},[34,13681,13682,13694],{},[37,13683,13684,13686,13688,13690,13692],{},[43,13685,6823],{"mathvariant":65},[50,13687,69],{},[50,13689,62],{},[308,13691,310],{},[43,13693,10376],{},[109,13695,13696],{"encoding":111},"\\operatorname{VCdim}=2d",[15,13698,13700,13721],{"className":13699,"ariaHidden":72},[116],[15,13701,13703,13706,13712,13715,13718],{"className":13702},[120],[15,13704],{"className":13705,"style":522},[124],[15,13707,13709],{"className":13708},[217],[15,13710,6823],{"className":13711},[129,221],[15,13713],{"className":13714,"style":199},[93],[15,13716,62],{"className":13717},[203],[15,13719],{"className":13720,"style":199},[93],[15,13722,13724,13727,13730],{"className":13723},[120],[15,13725],{"className":13726,"style":522},[124],[15,13728,310],{"className":13729},[129],[15,13731,10376],{"className":13732},[129,133],[11,13734,13735,13763,13764,13822,13823,13851],{},[15,13736,13738,13751],{"className":13737,"translate":19},[23],[15,13739,13741],{"className":13740},[27],[29,13742,13743],{"xmlns":31},[34,13744,13745,13749],{},[37,13746,13747],{},[43,13748,819],{"mathvariant":596},[109,13750,822],{"encoding":111},[15,13752,13754],{"className":13753,"ariaHidden":72},[116],[15,13755,13757,13760],{"className":13756},[120],[15,13758],{"className":13759,"style":610},[124],[15,13761,819],{"className":13762,"style":835},[129,614]," 是 ",[15,13765,13767,13784],{"className":13766,"translate":19},[23],[15,13768,13770],{"className":13769},[27],[29,13771,13772],{"xmlns":31},[34,13773,13774,13782],{},[37,13775,13776],{},[305,13777,13778,13780],{},[43,13779,104],{"mathvariant":103},[43,13781,10376],{},[109,13783,11570],{"encoding":111},[15,13785,13787],{"className":13786,"ariaHidden":72},[116],[15,13788,13790,13793],{"className":13789},[120],[15,13791],{"className":13792,"style":10490},[124],[15,13794,13796,13799],{"className":13795},[129],[15,13797,104],{"className":13798},[129,285],[15,13800,13802],{"className":13801},[137],[15,13803,13805],{"className":13804},[141],[15,13806,13808],{"className":13807},[146],[15,13809,13811],{"className":13810,"style":10490},[150],[15,13812,13813,13816],{"style":383},[15,13814],{"className":13815,"style":159},[158],[15,13817,13819],{"className":13818},[163,164,165,166],[15,13820,10376],{"className":13821},[129,133,166]," 中所有轴对齐矩形（即 ",[15,13824,13826,13839],{"className":13825,"translate":19},[23],[15,13827,13829],{"className":13828},[27],[29,13830,13831],{"xmlns":31},[34,13832,13833,13837],{},[37,13834,13835],{},[43,13836,10376],{},[109,13838,10376],{"encoding":111},[15,13840,13842],{"className":13841,"ariaHidden":72},[116],[15,13843,13845,13848],{"className":13844},[120],[15,13846],{"className":13847,"style":522},[124],[15,13849,10376],{"className":13850},[129,133]," 个区间的乘积）的指示函数。",[11,13853,13854,13855,13911,13912,13940,13941,13971,13972,14044,14045,14073,14074,14143,14144,1313],{},"上界：任取 ",[15,13856,13858,13878],{"className":13857,"translate":19},[23],[15,13859,13861],{"className":13860},[27],[29,13862,13863],{"xmlns":31},[34,13864,13865,13875],{},[37,13866,13867,13869,13871,13873],{},[308,13868,310],{},[43,13870,10376],{},[50,13872,2682],{},[308,13874,318],{},[109,13876,13877],{"encoding":111},"2d+1",[15,13879,13881,13902],{"className":13880,"ariaHidden":72},[116],[15,13882,13884,13887,13890,13893,13896,13899],{"className":13883},[120],[15,13885],{"className":13886,"style":10414},[124],[15,13888,310],{"className":13889},[129],[15,13891,10376],{"className":13892},[129,133],[15,13894],{"className":13895,"style":3277},[93],[15,13897,2682],{"className":13898},[2436],[15,13900],{"className":13901,"style":3277},[93],[15,13903,13905,13908],{"className":13904},[120],[15,13906],{"className":13907,"style":4886},[124],[15,13909,318],{"className":13910},[129]," 个点。若存在一点 ",[15,13913,13915,13928],{"className":13914,"translate":19},[23],[15,13916,13918],{"className":13917},[27],[29,13919,13920],{"xmlns":31},[34,13921,13922,13926],{},[37,13923,13924],{},[43,13925,11],{},[109,13927,11],{"encoding":111},[15,13929,13931],{"className":13930,"ariaHidden":72},[116],[15,13932,13934,13937],{"className":13933},[120],[15,13935],{"className":13936,"style":5399},[124],[15,13938,11],{"className":13939},[129,133],"，使它在每个坐标上都不是严格极值——即对每个坐标 ",[15,13942,13944,13958],{"className":13943,"translate":19},[23],[15,13945,13947],{"className":13946},[27],[29,13948,13949],{"xmlns":31},[34,13950,13951,13956],{},[37,13952,13953],{},[43,13954,13955],{},"j",[109,13957,13955],{"encoding":111},[15,13959,13961],{"className":13960,"ariaHidden":72},[116],[15,13962,13964,13967],{"className":13963},[120],[15,13965],{"className":13966,"style":11191},[124],[15,13968,13955],{"className":13969,"style":13970},[129,133],"margin-right:0.0572em;","，都有别的点在 ",[15,13973,13975,13993],{"className":13974,"translate":19},[23],[15,13976,13978],{"className":13977},[27],[29,13979,13980],{"xmlns":31},[34,13981,13982,13990],{},[37,13983,13984],{},[40,13985,13986,13988],{},[43,13987,11],{},[43,13989,13955],{},[109,13991,13992],{"encoding":111},"p_j",[15,13994,13996],{"className":13995,"ariaHidden":72},[116],[15,13997,13999,14003],{"className":13998},[120],[15,14000],{"className":14001,"style":14002},[124],"height:0.7167em;vertical-align:-0.2861em;",[15,14004,14006,14009],{"className":14005},[129],[15,14007,11],{"className":14008},[129,133],[15,14010,14012],{"className":14011},[137],[15,14013,14015,14035],{"className":14014},[141,142],[15,14016,14018,14032],{"className":14017},[146],[15,14019,14021],{"className":14020,"style":1981},[150],[15,14022,14023,14026],{"style":154},[15,14024],{"className":14025,"style":159},[158],[15,14027,14029],{"className":14028},[163,164,165,166],[15,14030,13955],{"className":14031,"style":13970},[129,133,166],[15,14033,175],{"className":14034},[174],[15,14036,14038],{"className":14037},[146],[15,14039,14042],{"className":14040,"style":14041},[150],"height:0.2861em;",[15,14043],{}," 的左边和右边——那么标注「",[15,14046,14048,14061],{"className":14047,"translate":19},[23],[15,14049,14051],{"className":14050},[27],[29,14052,14053],{"xmlns":31},[34,14054,14055,14059],{},[37,14056,14057],{},[43,14058,11],{},[109,14060,11],{"encoding":111},[15,14062,14064],{"className":14063,"ariaHidden":72},[116],[15,14065,14067,14070],{"className":14066},[120],[15,14068],{"className":14069,"style":5399},[124],[15,14071,11],{"className":14072},[129,133]," 为负、其余为正」无法实现：任何包含其余所有点的矩形，在每个坐标上的区间都必须覆盖 ",[15,14075,14077,14094],{"className":14076,"translate":19},[23],[15,14078,14080],{"className":14079},[27],[29,14081,14082],{"xmlns":31},[34,14083,14084,14092],{},[37,14085,14086],{},[40,14087,14088,14090],{},[43,14089,11],{},[43,14091,13955],{},[109,14093,13992],{"encoding":111},[15,14095,14097],{"className":14096,"ariaHidden":72},[116],[15,14098,14100,14103],{"className":14099},[120],[15,14101],{"className":14102,"style":14002},[124],[15,14104,14106,14109],{"className":14105},[129],[15,14107,11],{"className":14108},[129,133],[15,14110,14112],{"className":14111},[137],[15,14113,14115,14135],{"className":14114},[141,142],[15,14116,14118,14132],{"className":14117},[146],[15,14119,14121],{"className":14120,"style":1981},[150],[15,14122,14123,14126],{"style":154},[15,14124],{"className":14125,"style":159},[158],[15,14127,14129],{"className":14128},[163,164,165,166],[15,14130,13955],{"className":14131,"style":13970},[129,133,166],[15,14133,175],{"className":14134},[174],[15,14136,14138],{"className":14137},[146],[15,14139,14141],{"className":14140,"style":14041},[150],[15,14142],{}," 两侧的值，因而必然包含 ",[15,14145,14147,14160],{"className":14146,"translate":19},[23],[15,14148,14150],{"className":14149},[27],[29,14151,14152],{"xmlns":31},[34,14153,14154,14158],{},[37,14155,14156],{},[43,14157,11],{},[109,14159,11],{"encoding":111},[15,14161,14163],{"className":14162,"ariaHidden":72},[116],[15,14164,14166,14169],{"className":14165},[120],[15,14167],{"className":14168,"style":5399},[124],[15,14170,11],{"className":14171},[129,133],[11,14173,14174,14175,14203,14204,14232,14233,14261,14262,14296,14297,14352,14353,14386],{},"这样的 ",[15,14176,14178,14191],{"className":14177,"translate":19},[23],[15,14179,14181],{"className":14180},[27],[29,14182,14183],{"xmlns":31},[34,14184,14185,14189],{},[37,14186,14187],{},[43,14188,11],{},[109,14190,11],{"encoding":111},[15,14192,14194],{"className":14193,"ariaHidden":72},[116],[15,14195,14197,14200],{"className":14196},[120],[15,14198],{"className":14199,"style":5399},[124],[15,14201,11],{"className":14202},[129,133]," 一定存在。否则每个点都在某个坐标上严格小于所有其他点、或严格大于所有其他点。把「坐标 ",[15,14205,14207,14220],{"className":14206,"translate":19},[23],[15,14208,14210],{"className":14209},[27],[29,14211,14212],{"xmlns":31},[34,14213,14214,14218],{},[37,14215,14216],{},[43,14217,13955],{},[109,14219,13955],{"encoding":111},[15,14221,14223],{"className":14222,"ariaHidden":72},[116],[15,14224,14226,14229],{"className":14225},[120],[15,14227],{"className":14228,"style":11191},[124],[15,14230,13955],{"className":14231,"style":13970},[129,133]," 的最小值」和「坐标 ",[15,14234,14236,14249],{"className":14235,"translate":19},[23],[15,14237,14239],{"className":14238},[27],[29,14240,14241],{"xmlns":31},[34,14242,14243,14247],{},[37,14244,14245],{},[43,14246,13955],{},[109,14248,13955],{"encoding":111},[15,14250,14252],{"className":14251,"ariaHidden":72},[116],[15,14253,14255,14258],{"className":14254},[120],[15,14256],{"className":14257,"style":11191},[124],[15,14259,13955],{"className":14260,"style":13970},[129,133]," 的最大值」看作 ",[15,14263,14265,14281],{"className":14264,"translate":19},[23],[15,14266,14268],{"className":14267},[27],[29,14269,14270],{"xmlns":31},[34,14271,14272,14278],{},[37,14273,14274,14276],{},[308,14275,310],{},[43,14277,10376],{},[109,14279,14280],{"encoding":111},"2d",[15,14282,14284],{"className":14283,"ariaHidden":72},[116],[15,14285,14287,14290,14293],{"className":14286},[120],[15,14288],{"className":14289,"style":522},[124],[15,14291,310],{"className":14292},[129],[15,14294,10376],{"className":14295},[129,133]," 个位置，每个点占据其中一个；若两个点占据同一个位置，立刻矛盾（不可能同时严格小于对方）。",[15,14298,14300,14319],{"className":14299,"translate":19},[23],[15,14301,14303],{"className":14302},[27],[29,14304,14305],{"xmlns":31},[34,14306,14307,14317],{},[37,14308,14309,14311,14313,14315],{},[308,14310,310],{},[43,14312,10376],{},[50,14314,2682],{},[308,14316,318],{},[109,14318,13877],{"encoding":111},[15,14320,14322,14343],{"className":14321,"ariaHidden":72},[116],[15,14323,14325,14328,14331,14334,14337,14340],{"className":14324},[120],[15,14326],{"className":14327,"style":10414},[124],[15,14329,310],{"className":14330},[129],[15,14332,10376],{"className":14333},[129,133],[15,14335],{"className":14336,"style":3277},[93],[15,14338,2682],{"className":14339},[2436],[15,14341],{"className":14342,"style":3277},[93],[15,14344,14346,14349],{"className":14345},[120],[15,14347],{"className":14348,"style":4886},[124],[15,14350,318],{"className":14351},[129]," 个点放进 ",[15,14354,14356,14371],{"className":14355,"translate":19},[23],[15,14357,14359],{"className":14358},[27],[29,14360,14361],{"xmlns":31},[34,14362,14363,14369],{},[37,14364,14365,14367],{},[308,14366,310],{},[43,14368,10376],{},[109,14370,14280],{"encoding":111},[15,14372,14374],{"className":14373,"ariaHidden":72},[116],[15,14375,14377,14380,14383],{"className":14376},[120],[15,14378],{"className":14379,"style":522},[124],[15,14381,310],{"className":14382},[129],[15,14384,10376],{"className":14385},[129,133]," 个位置，必有一对冲突。所以上界成立。",[11,14388,14389,14390,14423,14424,14475,14476,14637,14638,1313],{},"下界是初等的但需要一点构造：确实存在 ",[15,14391,14393,14408],{"className":14392,"translate":19},[23],[15,14394,14396],{"className":14395},[27],[29,14397,14398],{"xmlns":31},[34,14399,14400,14406],{},[37,14401,14402,14404],{},[308,14403,310],{},[43,14405,10376],{},[109,14407,14280],{"encoding":111},[15,14409,14411],{"className":14410,"ariaHidden":72},[116],[15,14412,14414,14417,14420],{"className":14413},[120],[15,14415],{"className":14416,"style":522},[124],[15,14418,310],{"className":14419},[129],[15,14421,10376],{"className":14422},[129,133]," 个点可以被矩形打散（",[15,14425,14427,14445],{"className":14426,"translate":19},[23],[15,14428,14430],{"className":14429},[27],[29,14431,14432],{"xmlns":31},[34,14433,14434,14442],{},[37,14435,14436,14438,14440],{},[43,14437,10376],{},[50,14439,62],{},[308,14441,310],{},[109,14443,14444],{"encoding":111},"d=2",[15,14446,14448,14466],{"className":14447,"ariaHidden":72},[116],[15,14449,14451,14454,14457,14460,14463],{"className":14450},[120],[15,14452],{"className":14453,"style":522},[124],[15,14455,10376],{"className":14456},[129,133],[15,14458],{"className":14459,"style":199},[93],[15,14461,62],{"className":14462},[203],[15,14464],{"className":14465,"style":199},[93],[15,14467,14469,14472],{"className":14468},[120],[15,14470],{"className":14471,"style":4886},[124],[15,14473,310],{"className":14474},[129]," 时，",[15,14477,14479,14538],{"className":14478,"translate":19},[23],[15,14480,14482],{"className":14481},[27],[29,14483,14484],{"xmlns":31},[34,14485,14486,14535],{},[37,14487,14488,14490,14492,14494,14496,14498,14500,14502,14504,14506,14509,14511,14513,14515,14517,14519,14521,14523,14525,14527,14529,14531,14533],{},[50,14489,53],{"stretchy":52},[308,14491,318],{},[50,14493,91],{"separator":72},[308,14495,310],{},[50,14497,59],{"stretchy":52},[50,14499,91],{"separator":72},[50,14501,53],{"stretchy":52},[308,14503,310],{},[50,14505,91],{"separator":72},[308,14507,14508],{},"4",[50,14510,59],{"stretchy":52},[50,14512,91],{"separator":72},[50,14514,53],{"stretchy":52},[308,14516,9879],{},[50,14518,91],{"separator":72},[308,14520,318],{},[50,14522,59],{"stretchy":52},[50,14524,91],{"separator":72},[50,14526,53],{"stretchy":52},[308,14528,14508],{},[50,14530,91],{"separator":72},[308,14532,9879],{},[50,14534,59],{"stretchy":52},[109,14536,14537],{"encoding":111},"(1,2),(2,4),(3,1),(4,3)",[15,14539,14541],{"className":14540,"ariaHidden":72},[116],[15,14542,14544,14547,14550,14553,14556,14559,14562,14565,14568,14571,14574,14577,14580,14583,14586,14589,14592,14595,14598,14601,14604,14607,14610,14613,14616,14619,14622,14625,14628,14631,14634],{"className":14543},[120],[15,14545],{"className":14546,"style":125},[124],[15,14548,53],{"className":14549},[188],[15,14551,318],{"className":14552},[129],[15,14554,91],{"className":14555},[254],[15,14557],{"className":14558,"style":262},[93],[15,14560,310],{"className":14561},[129],[15,14563,59],{"className":14564},[195],[15,14566,91],{"className":14567},[254],[15,14569],{"className":14570,"style":262},[93],[15,14572,53],{"className":14573},[188],[15,14575,310],{"className":14576},[129],[15,14578,91],{"className":14579},[254],[15,14581],{"className":14582,"style":262},[93],[15,14584,14508],{"className":14585},[129],[15,14587,59],{"className":14588},[195],[15,14590,91],{"className":14591},[254],[15,14593],{"className":14594,"style":262},[93],[15,14596,53],{"className":14597},[188],[15,14599,9879],{"className":14600},[129],[15,14602,91],{"className":14603},[254],[15,14605],{"className":14606,"style":262},[93],[15,14608,318],{"className":14609},[129],[15,14611,59],{"className":14612},[195],[15,14614,91],{"className":14615},[254],[15,14617],{"className":14618,"style":262},[93],[15,14620,53],{"className":14621},[188],[15,14623,14508],{"className":14624},[129],[15,14626,91],{"className":14627},[254],[15,14629],{"className":14630,"style":262},[93],[15,14632,9879],{"className":14633},[129],[15,14635,59],{"className":14636},[195]," 这四个点就可以；注意不能取正方形的四个角）。于是 ",[15,14639,14641,14662],{"className":14640,"translate":19},[23],[15,14642,14644],{"className":14643},[27],[29,14645,14646],{"xmlns":31},[34,14647,14648,14660],{},[37,14649,14650,14652,14654,14656,14658],{},[43,14651,6823],{"mathvariant":65},[50,14653,69],{},[50,14655,62],{},[308,14657,310],{},[43,14659,10376],{},[109,14661,13696],{"encoding":111},[15,14663,14665,14686],{"className":14664,"ariaHidden":72},[116],[15,14666,14668,14671,14677,14680,14683],{"className":14667},[120],[15,14669],{"className":14670,"style":522},[124],[15,14672,14674],{"className":14673},[217],[15,14675,6823],{"className":14676},[129,221],[15,14678],{"className":14679,"style":199},[93],[15,14681,62],{"className":14682},[203],[15,14684],{"className":14685,"style":199},[93],[15,14687,14689,14692,14695],{"className":14688},[120],[15,14690],{"className":14691,"style":522},[124],[15,14693,310],{"className":14694},[129],[15,14696,10376],{"className":14697},[129,133],[11,14699,14700],{},[698,14701,14702,14703,7438],{},"例 5（有限类，",[15,14704,14706,14741],{"className":14705,"translate":19},[23],[15,14707,14709],{"className":14708},[27],[29,14710,14711],{"xmlns":31},[34,14712,14713,14738],{},[37,14714,14715,14717,14719,14721,14732,14734,14736],{},[43,14716,6823],{"mathvariant":65},[50,14718,69],{},[50,14720,2717],{},[40,14722,14723,14730],{},[37,14724,14725,14728],{},[43,14726,14727],{},"log",[50,14729,69],{},[308,14731,310],{},[43,14733,2945],{"mathvariant":65},[43,14735,819],{"mathvariant":596},[43,14737,2945],{"mathvariant":65},[109,14739,14740],{"encoding":111},"\\operatorname{VCdim}\\le\\log_2|\\mathcal H|",[15,14742,14744,14765],{"className":14743,"ariaHidden":72},[116],[15,14745,14747,14750,14756,14759,14762],{"className":14746},[120],[15,14748],{"className":14749,"style":6966},[124],[15,14751,14753],{"className":14752},[217],[15,14754,6823],{"className":14755},[129,221],[15,14757],{"className":14758,"style":199},[93],[15,14760,2717],{"className":14761},[203],[15,14763],{"className":14764,"style":199},[93],[15,14766,14768,14771,14817,14820,14823,14826],{"className":14767},[120],[15,14769],{"className":14770,"style":125},[124],[15,14772,14774,14780],{"className":14773},[217],[15,14775,14777,14778],{"className":14776},[217],"lo",[15,14779,2247],{"style":2246},[15,14781,14783],{"className":14782},[137],[15,14784,14786,14808],{"className":14785},[141,142],[15,14787,14789,14805],{"className":14788},[146],[15,14790,14793],{"className":14791,"style":14792},[150],"height:0.207em;",[15,14794,14796,14799],{"style":14795},"top:-2.4559em;margin-right:0.05em;",[15,14797],{"className":14798,"style":159},[158],[15,14800,14802],{"className":14801},[163,164,165,166],[15,14803,310],{"className":14804},[129,166],[15,14806,175],{"className":14807},[174],[15,14809,14811],{"className":14810},[146],[15,14812,14815],{"className":14813,"style":14814},[150],"height:0.2441em;",[15,14816],{},[15,14818],{"className":14819,"style":262},[93],[15,14821,2945],{"className":14822},[129],[15,14824,819],{"className":14825,"style":835},[129,614],[15,14827,2945],{"className":14828},[129],[11,14830,14831,14832,14860,14861,14889,14890,15075,15076,1313],{},"若 ",[15,14833,14835,14848],{"className":14834,"translate":19},[23],[15,14836,14838],{"className":14837},[27],[29,14839,14840],{"xmlns":31},[34,14841,14842,14846],{},[37,14843,14844],{},[43,14845,819],{"mathvariant":596},[109,14847,822],{"encoding":111},[15,14849,14851],{"className":14850,"ariaHidden":72},[116],[15,14852,14854,14857],{"className":14853},[120],[15,14855],{"className":14856,"style":610},[124],[15,14858,819],{"className":14859,"style":835},[129,614]," 有限且能打散 ",[15,14862,14864,14877],{"className":14863,"translate":19},[23],[15,14865,14867],{"className":14866},[27],[29,14868,14869],{"xmlns":31},[34,14870,14871,14875],{},[37,14872,14873],{},[43,14874,5619],{},[109,14876,5619],{"encoding":111},[15,14878,14880],{"className":14879,"ariaHidden":72},[116],[15,14881,14883,14886],{"className":14882},[120],[15,14884],{"className":14885,"style":610},[124],[15,14887,5619],{"className":14888,"style":5665},[129,133],"，则 ",[15,14891,14893,14939],{"className":14892,"translate":19},[23],[15,14894,14896],{"className":14895},[27],[29,14897,14898],{"xmlns":31},[34,14899,14900,14936],{},[37,14901,14902,14914,14916,14918,14920,14926,14928,14930,14932,14934],{},[305,14903,14904,14906],{},[308,14905,310],{},[37,14907,14908,14910,14912],{},[43,14909,2945],{"mathvariant":65},[43,14911,5619],{},[43,14913,2945],{"mathvariant":65},[50,14915,62],{},[43,14917,2945],{"mathvariant":65},[43,14919,819],{"mathvariant":596},[40,14921,14922,14924],{},[43,14923,2945],{"mathvariant":65},[43,14925,5619],{},[43,14927,2945],{"mathvariant":65},[50,14929,2717],{},[43,14931,2945],{"mathvariant":65},[43,14933,819],{"mathvariant":596},[43,14935,2945],{"mathvariant":65},[109,14937,14938],{"encoding":111},"2^{|C|}=|\\mathcal H|_C|\\le|\\mathcal H|",[15,14940,14942,14996,15060],{"className":14941,"ariaHidden":72},[116],[15,14943,14945,14949,14987,14990,14993],{"className":14944},[120],[15,14946],{"className":14947,"style":14948},[124],"height:0.888em;",[15,14950,14952,14955],{"className":14951},[129],[15,14953,310],{"className":14954},[129],[15,14956,14958],{"className":14957},[137],[15,14959,14961],{"className":14960},[141],[15,14962,14964],{"className":14963},[146],[15,14965,14967],{"className":14966,"style":14948},[150],[15,14968,14969,14972],{"style":383},[15,14970],{"className":14971,"style":159},[158],[15,14973,14975],{"className":14974},[163,164,165,166],[15,14976,14978,14981,14984],{"className":14977},[129,166],[15,14979,2945],{"className":14980},[129,166],[15,14982,5619],{"className":14983,"style":5665},[129,133,166],[15,14985,2945],{"className":14986},[129,166],[15,14988],{"className":14989,"style":199},[93],[15,14991,62],{"className":14992},[203],[15,14994],{"className":14995,"style":199},[93],[15,14997,14999,15002,15005,15008,15048,15051,15054,15057],{"className":14998},[120],[15,15000],{"className":15001,"style":125},[124],[15,15003,2945],{"className":15004},[129],[15,15006,819],{"className":15007,"style":835},[129,614],[15,15009,15011,15014],{"className":15010},[129],[15,15012,2945],{"className":15013},[129],[15,15015,15017],{"className":15016},[137],[15,15018,15020,15040],{"className":15019},[141,142],[15,15021,15023,15037],{"className":15022},[146],[15,15024,15026],{"className":15025,"style":1525},[150],[15,15027,15028,15031],{"style":154},[15,15029],{"className":15030,"style":159},[158],[15,15032,15034],{"className":15033},[163,164,165,166],[15,15035,5619],{"className":15036,"style":5665},[129,133,166],[15,15038,175],{"className":15039},[174],[15,15041,15043],{"className":15042},[146],[15,15044,15046],{"className":15045,"style":182},[150],[15,15047],{},[15,15049,2945],{"className":15050},[129],[15,15052],{"className":15053,"style":199},[93],[15,15055,2717],{"className":15056},[203],[15,15058],{"className":15059,"style":199},[93],[15,15061,15063,15066,15069,15072],{"className":15062},[120],[15,15064],{"className":15065,"style":125},[124],[15,15067,2945],{"className":15068},[129],[15,15070,819],{"className":15071,"style":835},[129,614],[15,15073,2945],{"className":15074},[129],"，所以 ",[15,15077,15079,15115],{"className":15078,"translate":19},[23],[15,15080,15082],{"className":15081},[27],[29,15083,15084],{"xmlns":31},[34,15085,15086,15112],{},[37,15087,15088,15090,15092,15094,15096,15106,15108,15110],{},[43,15089,2945],{"mathvariant":65},[43,15091,5619],{},[43,15093,2945],{"mathvariant":65},[50,15095,2717],{},[40,15097,15098,15104],{},[37,15099,15100,15102],{},[43,15101,14727],{},[50,15103,69],{},[308,15105,310],{},[43,15107,2945],{"mathvariant":65},[43,15109,819],{"mathvariant":596},[43,15111,2945],{"mathvariant":65},[109,15113,15114],{"encoding":111},"|C|\\le\\log_2|\\mathcal H|",[15,15116,15118,15142],{"className":15117,"ariaHidden":72},[116],[15,15119,15121,15124,15127,15130,15133,15136,15139],{"className":15120},[120],[15,15122],{"className":15123,"style":125},[124],[15,15125,2945],{"className":15126},[129],[15,15128,5619],{"className":15129,"style":5665},[129,133],[15,15131,2945],{"className":15132},[129],[15,15134],{"className":15135,"style":199},[93],[15,15137,2717],{"className":15138},[203],[15,15140],{"className":15141,"style":199},[93],[15,15143,15145,15148,15190,15193,15196,15199],{"className":15144},[120],[15,15146],{"className":15147,"style":125},[124],[15,15149,15151,15156],{"className":15150},[217],[15,15152,14777,15154],{"className":15153},[217],[15,15155,2247],{"style":2246},[15,15157,15159],{"className":15158},[137],[15,15160,15162,15182],{"className":15161},[141,142],[15,15163,15165,15179],{"className":15164},[146],[15,15166,15168],{"className":15167,"style":14792},[150],[15,15169,15170,15173],{"style":14795},[15,15171],{"className":15172,"style":159},[158],[15,15174,15176],{"className":15175},[163,164,165,166],[15,15177,310],{"className":15178},[129,166],[15,15180,175],{"className":15181},[174],[15,15183,15185],{"className":15184},[146],[15,15186,15188],{"className":15187,"style":14814},[150],[15,15189],{},[15,15191],{"className":15192,"style":262},[93],[15,15194,2945],{"className":15195},[129],[15,15197,819],{"className":15198,"style":835},[129,614],[15,15200,2945],{"className":15201},[129],[11,15203,15204],{},[698,15205,15206,15207,7438],{},"例 6（正弦类，",[15,15208,15210,15230],{"className":15209,"translate":19},[23],[15,15211,15213],{"className":15212},[27],[29,15214,15215],{"xmlns":31},[34,15216,15217,15227],{},[37,15218,15219,15221,15223,15225],{},[43,15220,6823],{"mathvariant":65},[50,15222,69],{},[50,15224,62],{},[43,15226,4909],{"mathvariant":65},[109,15228,15229],{"encoding":111},"\\operatorname{VCdim}=\\infty",[15,15231,15233,15254],{"className":15232,"ariaHidden":72},[116],[15,15234,15236,15239,15245,15248,15251],{"className":15235},[120],[15,15237],{"className":15238,"style":522},[124],[15,15240,15242],{"className":15241},[217],[15,15243,6823],{"className":15244},[129,221],[15,15246],{"className":15247,"style":199},[93],[15,15249,62],{"className":15250},[203],[15,15252],{"className":15253,"style":199},[93],[15,15255,15257,15260],{"className":15256},[120],[15,15258],{"className":15259,"style":4383},[124],[15,15261,4909],{"className":15262},[129],[11,15264,7441,15265,7610,15431,15481,15482,15510,15511,15644,15645,15741],{},[15,15266,15268,15320],{"className":15267,"translate":19},[23],[15,15269,15271],{"className":15270},[27],[29,15272,15273],{"xmlns":31},[34,15274,15275,15317],{},[37,15276,15277,15279,15281,15283,15285,15287,15289,15291,15293,15295,15297,15299,15301,15303,15305,15307,15309,15311,15313,15315],{},[43,15278,819],{"mathvariant":596},[50,15280,62],{},[50,15282,636],{"stretchy":52},[43,15284,56],{},[50,15286,7464],{},[43,15288,66],{"mathvariant":65},[50,15290,69],{},[50,15292,53],{"stretchy":52},[43,15294,76],{},[50,15296,69],{},[50,15298,53],{"stretchy":52},[43,15300,48],{},[43,15302,56],{},[50,15304,59],{"stretchy":52},[50,15306,59],{"stretchy":52},[50,15308,718],{},[43,15310,48],{},[50,15312,4852],{},[308,15314,541],{},[50,15316,645],{"stretchy":52},[109,15318,15319],{"encoding":111},"\\mathcal H=\\{x\\mapsto\\operatorname{sign}(\\sin(\\theta x)):\\theta>0\\}",[15,15321,15323,15341,15362,15401,15419],{"className":15322,"ariaHidden":72},[116],[15,15324,15326,15329,15332,15335,15338],{"className":15325},[120],[15,15327],{"className":15328,"style":610},[124],[15,15330,819],{"className":15331,"style":835},[129,614],[15,15333],{"className":15334,"style":199},[93],[15,15336,62],{"className":15337},[203],[15,15339],{"className":15340,"style":199},[93],[15,15342,15344,15347,15350,15353,15356,15359],{"className":15343},[120],[15,15345],{"className":15346,"style":125},[124],[15,15348,636],{"className":15349},[188],[15,15351,56],{"className":15352},[129,133],[15,15354],{"className":15355,"style":199},[93],[15,15357,7464],{"className":15358},[203],[15,15360],{"className":15361,"style":199},[93],[15,15363,15365,15368,15374,15377,15380,15383,15386,15389,15392,15395,15398],{"className":15364},[120],[15,15366],{"className":15367,"style":125},[124],[15,15369,15371],{"className":15370},[217],[15,15372,66],{"className":15373},[129,221],[15,15375,53],{"className":15376},[188],[15,15378,76],{"className":15379},[217],[15,15381,53],{"className":15382},[188],[15,15384,48],{"className":15385,"style":170},[129,133],[15,15387,56],{"className":15388},[129,133],[15,15390,6149],{"className":15391},[195],[15,15393],{"className":15394,"style":199},[93],[15,15396,718],{"className":15397},[203],[15,15399],{"className":15400,"style":199},[93],[15,15402,15404,15407,15410,15413,15416],{"className":15403},[120],[15,15405],{"className":15406,"style":2405},[124],[15,15408,48],{"className":15409,"style":170},[129,133],[15,15411],{"className":15412,"style":199},[93],[15,15414,4852],{"className":15415},[203],[15,15417],{"className":15418,"style":199},[93],[15,15420,15422,15425,15428],{"className":15421},[120],[15,15423],{"className":15424,"style":125},[124],[15,15426,541],{"className":15427},[129],[15,15429,645],{"className":15430},[195],[15,15432,15434,15451],{"className":15433,"translate":19},[23],[15,15435,15437],{"className":15436},[27],[29,15438,15439],{"xmlns":31},[34,15440,15441,15449],{},[37,15442,15443,15445,15447],{},[43,15444,597],{"mathvariant":596},[50,15446,62],{},[43,15448,104],{"mathvariant":103},[109,15450,7631],{"encoding":111},[15,15452,15454,15472],{"className":15453,"ariaHidden":72},[116],[15,15455,15457,15460,15463,15466,15469],{"className":15456},[120],[15,15458],{"className":15459,"style":610},[124],[15,15461,597],{"className":15462,"style":615},[129,614],[15,15464],{"className":15465,"style":199},[93],[15,15467,62],{"className":15468},[203],[15,15470],{"className":15471,"style":199},[93],[15,15473,15475,15478],{"className":15474},[120],[15,15476],{"className":15477,"style":281},[124],[15,15479,104],{"className":15480},[129,285],"。对任意 ",[15,15483,15485,15498],{"className":15484,"translate":19},[23],[15,15486,15488],{"className":15487},[27],[29,15489,15490],{"xmlns":31},[34,15491,15492,15496],{},[37,15493,15494],{},[43,15495,348],{},[109,15497,348],{"encoding":111},[15,15499,15501],{"className":15500,"ariaHidden":72},[116],[15,15502,15504,15507],{"className":15503},[120],[15,15505],{"className":15506,"style":4383},[124],[15,15508,348],{"className":15509},[129,133],"，取 ",[15,15512,15514,15544],{"className":15513,"translate":19},[23],[15,15515,15517],{"className":15516},[27],[29,15518,15519],{"xmlns":31},[34,15520,15521,15541],{},[37,15522,15523,15529,15531],{},[40,15524,15525,15527],{},[43,15526,56],{},[43,15528,1459],{},[50,15530,62],{},[305,15532,15533,15535],{},[308,15534,310],{},[37,15536,15537,15539],{},[50,15538,315],{},[43,15540,1459],{},[109,15542,15543],{"encoding":111},"x_i=2^{-i}",[15,15545,15547,15602],{"className":15546,"ariaHidden":72},[116],[15,15548,15550,15553,15593,15596,15599],{"className":15549},[120],[15,15551],{"className":15552,"style":7985},[124],[15,15554,15556,15559],{"className":15555},[129],[15,15557,56],{"className":15558},[129,133],[15,15560,15562],{"className":15561},[137],[15,15563,15565,15585],{"className":15564},[141,142],[15,15566,15568,15582],{"className":15567},[146],[15,15569,15571],{"className":15570,"style":1981},[150],[15,15572,15573,15576],{"style":154},[15,15574],{"className":15575,"style":159},[158],[15,15577,15579],{"className":15578},[163,164,165,166],[15,15580,1459],{"className":15581},[129,133,166],[15,15583,175],{"className":15584},[174],[15,15586,15588],{"className":15587},[146],[15,15589,15591],{"className":15590,"style":182},[150],[15,15592],{},[15,15594],{"className":15595,"style":199},[93],[15,15597,62],{"className":15598},[203],[15,15600],{"className":15601,"style":199},[93],[15,15603,15605,15609],{"className":15604},[120],[15,15606],{"className":15607,"style":15608},[124],"height:0.8247em;",[15,15610,15612,15615],{"className":15611},[129],[15,15613,310],{"className":15614},[129],[15,15616,15618],{"className":15617},[137],[15,15619,15621],{"className":15620},[141],[15,15622,15624],{"className":15623},[146],[15,15625,15627],{"className":15626,"style":15608},[150],[15,15628,15629,15632],{"style":383},[15,15630],{"className":15631,"style":159},[158],[15,15633,15635],{"className":15634},[163,164,165,166],[15,15636,15638,15641],{"className":15637},[129,166],[15,15639,315],{"className":15640},[129,166],[15,15642,1459],{"className":15643},[129,133,166],"，给定任意标注 ",[15,15646,15648,15676],{"className":15647,"translate":19},[23],[15,15649,15651],{"className":15650},[27],[29,15652,15653],{"xmlns":31},[34,15654,15655,15673],{},[37,15656,15657,15659,15661,15663,15665,15667],{},[43,15658,857],{},[50,15660,100],{},[50,15662,636],{"stretchy":52},[50,15664,10932],{},[308,15666,318],{},[305,15668,15669,15671],{},[50,15670,645],{"stretchy":52},[43,15672,348],{},[109,15674,15675],{"encoding":111},"y\\in\\{\\pm1\\}^m",[15,15677,15679,15697],{"className":15678,"ariaHidden":72},[116],[15,15680,15682,15685,15688,15691,15694],{"className":15681},[120],[15,15683],{"className":15684,"style":10959},[124],[15,15686,857],{"className":15687,"style":887},[129,133],[15,15689],{"className":15690,"style":199},[93],[15,15692,100],{"className":15693},[203],[15,15695],{"className":15696,"style":199},[93],[15,15698,15700,15703,15706,15709,15712],{"className":15699},[120],[15,15701],{"className":15702,"style":125},[124],[15,15704,636],{"className":15705},[188],[15,15707,10932],{"className":15708},[129],[15,15710,318],{"className":15711},[129],[15,15713,15715,15718],{"className":15714},[195],[15,15716,645],{"className":15717},[195],[15,15719,15721],{"className":15720},[137],[15,15722,15724],{"className":15723},[141],[15,15725,15727],{"className":15726},[146],[15,15728,15730],{"className":15729,"style":6377},[150],[15,15731,15732,15735],{"style":383},[15,15733],{"className":15734,"style":159},[158],[15,15736,15738],{"className":15737},[163,164,165,166],[15,15739,348],{"className":15740},[129,133,166],"，令",[15,15743,15745],{"className":15744,"translate":19},[18],[15,15746,15748,15814],{"className":15747,"translate":19},[23],[15,15749,15751],{"className":15750},[27],[29,15752,15753],{"xmlns":31,"display":32},[34,15754,15755,15811],{},[37,15756,15757,15759,15761,15764,15809],{},[43,15758,48],{},[50,15760,62],{},[43,15762,15763],{},"π",[37,15765,15766,15768,15774,15776,15801,15807],{},[50,15767,53],{"fence":72},[1443,15769,15770,15772],{},[308,15771,318],{},[308,15773,310],{},[50,15775,2682],{},[2140,15777,15778,15780],{},[50,15779,1454],{},[37,15781,15782,15784,15786,15789,15795,15797,15799],{},[43,15783,1459],{},[50,15785,718],{},[6856,15787,15788],{}," ",[40,15790,15791,15793],{},[43,15792,857],{},[43,15794,1459],{},[50,15796,62],{},[50,15798,315],{},[308,15800,318],{},[305,15802,15803,15805],{},[308,15804,310],{},[43,15806,1459],{},[50,15808,59],{"fence":72},[43,15810,107],{"mathvariant":65},[109,15812,15813],{"encoding":111},"\\theta=\\pi\\left(\\frac12+\\sum_{i:\\,y_i=-1}2^i\\right).",[15,15815,15817,15835],{"className":15816,"ariaHidden":72},[116],[15,15818,15820,15823,15826,15829,15832],{"className":15819},[120],[15,15821],{"className":15822,"style":522},[124],[15,15824,48],{"className":15825,"style":170},[129,133],[15,15827],{"className":15828,"style":199},[93],[15,15830,62],{"className":15831},[203],[15,15833],{"className":15834,"style":199},[93],[15,15836,15838,15842,15845,15848,16077,16080],{"className":15837},[120],[15,15839],{"className":15840,"style":15841},[124],"height:3.1638em;vertical-align:-1.4138em;",[15,15843,15763],{"className":15844,"style":887},[129,133],[15,15846],{"className":15847,"style":262},[93],[15,15849,15851,15858,15920,15923,15926,15929,16038,16041,16071],{"className":15850},[449],[15,15852,15854],{"className":15853,"style":5052},[188,5051],[15,15855,53],{"className":15856},[228,15857],"size4",[15,15859,15861,15864,15917],{"className":15860},[129],[15,15862],{"className":15863},[188,1803],[15,15865,15867],{"className":15866},[1443],[15,15868,15870,15909],{"className":15869},[141,142],[15,15871,15873,15906],{"className":15872},[146],[15,15874,15876,15887,15895],{"className":15875,"style":1816},[150],[15,15877,15878,15881],{"style":1819},[15,15879],{"className":15880,"style":1823},[158],[15,15882,15884],{"className":15883},[129],[15,15885,310],{"className":15886},[129],[15,15888,15889,15892],{"style":1832},[15,15890],{"className":15891,"style":1823},[158],[15,15893],{"className":15894,"style":1840},[1839],[15,15896,15897,15900],{"style":1843},[15,15898],{"className":15899,"style":1823},[158],[15,15901,15903],{"className":15902},[129],[15,15904,318],{"className":15905},[129],[15,15907,175],{"className":15908},[174],[15,15910,15912],{"className":15911},[146],[15,15913,15915],{"className":15914,"style":1862},[150],[15,15916],{},[15,15918],{"className":15919},[195,1803],[15,15921],{"className":15922,"style":3277},[93],[15,15924,2682],{"className":15925},[2436],[15,15927],{"className":15928,"style":3277},[93],[15,15930,15932],{"className":15931},[217,1874],[15,15933,15935,16029],{"className":15934},[141,142],[15,15936,15938,16026],{"className":15937},[146],[15,15939,15941,16016],{"className":15940,"style":12149},[150],[15,15942,15943,15946],{"style":1887},[15,15944],{"className":15945,"style":1891},[158],[15,15947,15949],{"className":15948},[163,164,165,166],[15,15950,15952,15955,15958,15962,16007,16010,16013],{"className":15951},[129,166],[15,15953,1459],{"className":15954},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",[15,16087,16089,16102],{"className":16088,"translate":19},[23],[15,16090,16092],{"className":16091},[27],[29,16093,16094],{"xmlns":31},[34,16095,16096,16100],{},[37,16097,16098],{},[43,16099,1459],{},[109,16101,1459],{"encoding":111},[15,16103,16105],{"className":16104,"ariaHidden":72},[116],[15,16106,16108,16112],{"className":16107},[120],[15,16109],{"className":16110,"style":16111},[124],"height:0.6595em;",[15,16113,1459],{"className":16114},[129,133],[15,16116,16118,16138],{"className":16117,"translate":19},[23],[15,16119,16121],{"className":16120},[27],[29,16122,16123],{"xmlns":31},[34,16124,16125,16135],{},[37,16126,16127,16129],{},[43,16128,48],{},[40,16130,16131,16133],{},[43,16132,56],{},[43,16134,1459],{},[109,16136,16137],{"encoding":111},"\\theta x_i",[15,16139,16141],{"className":16140,"ariaHidden":72},[116],[15,16142,16144,16147,16150],{"className":16143},[120],[15,16145],{"className":16146,"style":2183},[124],[15,16148,48],{"className":16149,"style":170},[129,133],[15,16151,16153,16156],{"className":16152},[129],[15,16154,56],{"className":16155},[129,133],[15,16157,16159],{"className":16158},[137],[15,16160,16162,16182],{"className":16161},[141,142],[15,16163,16165,16179],{"className":16164},[146],[15,16166,16168],{"className":16167,"style":1981},[150],[15,16169,16170,16173],{"style":154},[15,16171],{"className":16172,"style":159},[158],[15,16174,16176],{"className":16175},[163,164,165,166],[15,16177,1459],{"className":16178},[129,133,166],[15,16180,175],{"className":16181},[174],[15,16183,16185],{"className":16184},[146],[15,16186,16188],{"className":16187,"style":182},[150],[15,16189],{}," 对 ",[15,16192,16194,16210],{"className":16193,"translate":19},[23],[15,16195,16197],{"className":16196},[27],[29,16198,16199],{"xmlns":31},[34,16200,16201,16207],{},[37,16202,16203,16205],{},[308,16204,310],{},[43,16206,15763],{},[109,16208,16209],{"encoding":111},"2\\pi",[15,16211,16213],{"className":16212,"ariaHidden":72},[116],[15,16214,16216,16219,16222],{"className":16215},[120],[15,16217],{"className":16218,"style":4886},[124],[15,16220,310],{"className":16221},[129],[15,16223,15763],{"className":16224,"style":887},[129,133]," 取模。",[15,16227,16229,16247],{"className":16228,"translate":19},[23],[15,16230,16232],{"className":16231},[27],[29,16233,16234],{"xmlns":31},[34,16235,16236,16244],{},[37,16237,16238,16240,16242],{},[43,16239,13955],{},[50,16241,4852],{},[43,16243,1459],{},[109,16245,16246],{"encoding":111},"j>i",[15,16248,16250,16268],{"className":16249,"ariaHidden":72},[116],[15,16251,16253,16256,16259,16262,16265],{"className":16252},[120],[15,16254],{"className":16255,"style":11191},[124],[15,16257,13955],{"className":16258,"style":13970},[129,133],[15,16260],{"className":16261,"style":199},[93],[15,16263,4852],{"className":16264},[203],[15,16266],{"className":16267,"style":199},[93],[15,16269,16271,16274],{"className":16270},[120],[15,16272],{"className":16273,"style":16111},[124],[15,16275,1459],{"className":16276},[129,133]," 的项贡献 ",[15,16279,16281,16296],{"className":16280,"translate":19},[23],[15,16282,16284],{"className":16283},[27],[29,16285,16286],{"xmlns":31},[34,16287,16288,16294],{},[37,16289,16290,16292],{},[308,16291,310],{},[43,16293,15763],{},[109,16295,16209],{"encoding":111},[15,16297,16299],{"className":16298,"ariaHidden":72},[116],[15,16300,16302,16305,16308],{"className":16301},[120],[15,16303],{"className":16304,"style":4886},[124],[15,16306,310],{"className":16307},[129],[15,16309,15763],{"className":16310,"style":887},[129,133]," 的整数倍，可以丢掉；",[15,16313,16315,16333],{"className":16314,"translate":19},[23],[15,16316,16318],{"className":16317},[27],[29,16319,16320],{"xmlns":31},[34,16321,16322,16330],{},[37,16323,16324,16326,16328],{},[43,16325,13955],{},[50,16327,62],{},[43,16329,1459],{},[109,16331,16332],{"encoding":111},"j=i",[15,16334,16336,16354],{"className":16335,"ariaHidden":72},[116],[15,16337,16339,16342,16345,16348,16351],{"className":16338},[120],[15,16340],{"className":16341,"style":11191},[124],[15,16343,13955],{"className":16344,"style":13970},[129,133],[15,16346],{"className":16347,"style":199},[93],[15,16349,62],{"className":16350},[203],[15,16352],{"className":16353,"style":199},[93],[15,16355,16357,16360],{"className":16356},[120],[15,16358],{"className":16359,"style":16111},[124],[15,16361,1459],{"className":16362},[129,133],[15,16364,16366,16380],{"className":16365,"translate":19},[23],[15,16367,16369],{"className":16368},[27],[29,16370,16371],{"xmlns":31},[34,16372,16373,16377],{},[37,16374,16375],{},[43,16376,15763],{},[109,16378,16379],{"encoding":111},"\\pi",[15,16381,16383],{"className":16382,"ariaHidden":72},[116],[15,16384,16386,16389],{"className":16385},[120],[15,16387],{"className":16388,"style":4383},[124],[15,16390,15763],{"className":16391,"style":887},[129,133],"（当 ",[15,16394,16396,16420],{"className":16395,"translate":19},[23],[15,16397,16399],{"className":16398},[27],[29,16400,16401],{"xmlns":31},[34,16402,16403,16417],{},[37,16404,16405,16411,16413,16415],{},[40,16406,16407,16409],{},[43,16408,857],{},[43,16410,1459],{},[50,16412,62],{},[50,16414,315],{},[308,16416,318],{},[109,16418,16419],{"encoding":111},"y_i=-1",[15,16421,16423,16478],{"className":16422,"ariaHidden":72},[116],[15,16424,16426,16429,16469,16472,16475],{"className":16425},[120],[15,16427],{"className":16428,"style":5399},[124],[15,16430,16432,16435],{"className":16431},[129],[15,16433,857],{"className":16434,"style":887},[129,133],[15,16436,16438],{"className":16437},[137],[15,16439,16441,16461],{"className":16440},[141,142],[15,16442,16444,16458],{"className":16443},[146],[15,16445,16447],{"className":16446,"style":1981},[150],[15,16448,16449,16452],{"style":1100},[15,16450],{"className":16451,"style":159},[158],[15,16453,16455],{"className":16454},[163,164,165,166],[15,16456,1459],{"className":16457},[129,133,166],[15,16459,175],{"className":16460},[174],[15,16462,16464],{"className":16463},[146],[15,16465,16467],{"className":16466,"style":182},[150],[15,16468],{},[15,16470],{"className":16471,"style":199},[93],[15,16473,62],{"className":16474},[203],[15,16476],{"className":16477,"style":199},[93],[15,16479,16481,16485,16488],{"className":16480},[120],[15,16482],{"className":16483,"style":16484},[124],"height:0.7278em;vertical-align:-0.0833em;",[15,16486,315],{"className":16487},[129],[15,16489,318],{"className":16490},[129],"）；",[15,16493,16495,16513],{"className":16494,"translate":19},[23],[15,16496,16498],{"className":16497},[27],[29,16499,16500],{"xmlns":31},[34,16501,16502,16510],{},[37,16503,16504,16506,16508],{},[43,16505,13955],{},[50,16507,7910],{},[43,16509,1459],{},[109,16511,16512],{"encoding":111},"j\u003Ci",[15,16514,16516,16534],{"className":16515,"ariaHidden":72},[116],[15,16517,16519,16522,16525,16528,16531],{"className":16518},[120],[15,16520],{"className":16521,"style":11191},[124],[15,16523,13955],{"className":16524,"style":13970},[129,133],[15,16526],{"className":16527,"style":199},[93],[15,16529,7910],{"className":16530},[203],[15,16532],{"className":16533,"style":199},[93],[15,16535,16537,16540],{"className":16536},[120],[15,16538],{"className":16539,"style":16111},[124],[15,16541,1459],{"className":16542},[129,133]," 的项合并为 ",[15,16545,16547,16576],{"className":16546,"translate":19},[23],[15,16548,16550],{"className":16549},[27],[29,16551,16552],{"xmlns":31},[34,16553,16554,16573],{},[37,16555,16556,16558,16565,16567],{},[43,16557,15763],{},[40,16559,16560,16563],{},[43,16561,16562],{},"k",[43,16564,1459],{},[43,16566,544],{"mathvariant":65},[305,16568,16569,16571],{},[308,16570,310],{},[43,16572,1459],{},[109,16574,16575],{"encoding":111},"\\pi k_i\u002F2^i",[15,16577,16579],{"className":16578,"ariaHidden":72},[116],[15,16580,16582,16586,16589,16631,16634],{"className":16581},[120],[15,16583],{"className":16584,"style":16585},[124],"height:1.0747em;vertical-align:-0.25em;",[15,16587,15763],{"className":16588,"style":887},[129,133],[15,16590,16592,16596],{"className":16591},[129],[15,16593,16562],{"className":16594,"style":16595},[129,133],"margin-right:0.0315em;",[15,16597,16599],{"className":16598},[137],[15,16600,16602,16623],{"className":16601},[141,142],[15,16603,16605,16620],{"className":16604},[146],[15,16606,16608],{"className":16607,"style":1981},[150],[15,16609,16611,16614],{"style":16610},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[15,16612],{"className":16613,"style":159},[158],[15,16615,16617],{"className":16616},[163,164,165,166],[15,16618,1459],{"className":16619},[129,133,166],[15,16621,175],{"className":16622},[174],[15,16624,16626],{"className":16625},[146],[15,16627,16629],{"className":16628,"style":182},[150],[15,16630],{},[15,16632,544],{"className":16633},[129],[15,16635,16637,16640],{"className":16636},[129],[15,16638,310],{"className":16639},[129],[15,16641,16643],{"className":16642},[137],[15,16644,16646],{"className":16645},[141],[15,16647,16649],{"className":16648},[146],[15,16650,16652],{"className":16651,"style":15608},[150],[15,16653,16654,16657],{"style":383},[15,16655],{"className":16656,"style":159},[158],[15,16658,16660],{"className":16659},[163,164,165,166],[15,16661,1459],{"className":16662},[129,133,166],"，其中 ",[15,16665,16667,16701],{"className":16666,"translate":19},[23],[15,16668,16670],{"className":16669},[27],[29,16671,16672],{"xmlns":31},[34,16673,16674,16698],{},[37,16675,16676,16678,16680,16686,16688,16694,16696],{},[308,16677,541],{},[50,16679,2717],{},[40,16681,16682,16684],{},[43,16683,16562],{},[43,16685,1459],{},[50,16687,2717],{},[305,16689,16690,16692],{},[308,16691,310],{},[43,16693,1459],{},[50,16695,315],{},[308,16697,318],{},[109,16699,16700],{"encoding":111},"0\\le k_i\\le 2^i-1",[15,16702,16704,16723,16778,16823],{"className":16703,"ariaHidden":72},[116],[15,16705,16707,16711,16714,16717,16720],{"className":16706},[120],[15,16708],{"className":16709,"style":16710},[124],"height:0.7804em;vertical-align:-0.136em;",[15,16712,541],{"className":16713},[129],[15,16715],{"className":16716,"style":199},[93],[15,16718,2717],{"className":16719},[203],[15,16721],{"className":16722,"style":199},[93],[15,16724,16726,16729,16769,16772,16775],{"className":16725},[120],[15,16727],{"className":16728,"style":2183},[124],[15,16730,16732,16735],{"className":16731},[129],[15,16733,16562],{"className":16734,"style":16595},[129,133],[15,16736,16738],{"className":16737},[137],[15,16739,16741,16761],{"className":16740},[141,142],[15,16742,16744,16758],{"className":16743},[146],[15,16745,16747],{"className":16746,"style":1981},[150],[15,16748,16749,16752],{"style":16610},[15,16750],{"className":16751,"style":159},[158],[15,16753,16755],{"className":16754},[163,164,165,166],[15,16756,1459],{"className":16757},[129,133,166],[15,16759,175],{"className":16760},[174],[15,16762,16764],{"className":16763},[146],[15,16765,16767],{"className":16766,"style":182},[150],[15,16768],{},[15,16770],{"className":16771,"style":199},[93],[15,16773,2717],{"className":16774},[203],[15,16776],{"className":16777,"style":199},[93],[15,16779,16781,16785,16814,16817,16820],{"className":16780},[120],[15,16782],{"className":16783,"style":16784},[124],"height:0.908em;vertical-align:-0.0833em;",[15,16786,16788,16791],{"className":16787},[129],[15,16789,310],{"className":16790},[129],[15,16792,16794],{"className":16793},[137],[15,16795,16797],{"className":16796},[141],[15,16798,16800],{"className":16799},[146],[15,16801,16803],{"className":16802,"style":15608},[150],[15,16804,16805,16808],{"style":383},[15,16806],{"className":16807,"style":159},[158],[15,16809,16811],{"className":16810},[163,164,165,166],[15,16812,1459],{"className":16813},[129,133,166],[15,16815],{"className":16816,"style":3277},[93],[15,16818,315],{"className":16819},[2436],[15,16821],{"className":16822,"style":3277},[93],[15,16824,16826,16829],{"className":16825},[120],[15,16827],{"className":16828,"style":4886},[124],[15,16830,318],{"className":16831},[129],"。于是",[15,16834,16836],{"className":16835,"translate":19},[18],[15,16837,16839,16947],{"className":16838,"translate":19},[23],[15,16840,16842],{"className":16841},[27],[29,16843,16844],{"xmlns":31,"display":32},[34,16845,16846,16944],{},[37,16847,16848,16850,16856,16859,16861,16917,16919,16922,16924,16933,16936,16938,16940,16942],{},[43,16849,48],{},[40,16851,16852,16854],{},[43,16853,56],{},[43,16855,1459],{},[50,16857,16858],{},"≡",[43,16860,15763],{},[37,16862,16863,16865,16879,16881,16883,16885,16891,16893,16895,16897,16899,16901,16915],{},[50,16864,53],{"fence":72},[305,16866,16867,16869],{},[308,16868,310],{},[37,16870,16871,16873,16875,16877],{},[50,16872,315],{},[43,16874,1459],{},[50,16876,315],{},[308,16878,318],{},[50,16880,2682],{},[308,16882,318],{"mathvariant":1468},[50,16884,1407],{"stretchy":52},[40,16886,16887,16889],{},[43,16888,857],{},[43,16890,1459],{},[50,16892,62],{},[50,16894,315],{},[308,16896,318],{},[50,16898,1423],{"stretchy":52},[50,16900,2682],{},[1443,16902,16903,16909],{},[40,16904,16905,16907],{},[43,16906,16562],{},[43,16908,1459],{},[305,16910,16911,16913],{},[308,16912,310],{},[43,16914,1459],{},[50,16916,59],{"fence":72},[93,16918],{},[93,16920],{"width":16921},"1em",[50,16923,53],{"stretchy":52},[37,16925,16926,16928,16931],{},[43,16927,348],{"mathvariant":65},[43,16929,16930],{"mathvariant":65},"o",[43,16932,10376],{"mathvariant":65},[93,16934],{"width":16935},"0.3333em",[308,16937,310],{},[43,16939,15763],{},[50,16941,59],{"stretchy":52},[43,16943,107],{"mathvariant":65},[109,16945,16946],{"encoding":111},"\\theta x_i\\equiv \\pi\\left(2^{-i-1}+\\mathbf 1[y_i=-1]+\\frac{k_i}{2^i}\\right)\\pmod{2\\pi}.",[15,16948,16950,17008,17298],{"className":16949,"ariaHidden":72},[116],[15,16951,16953,16956,16959,16999,17002,17005],{"className":16952},[120],[15,16954],{"className":16955,"style":2183},[124],[15,16957,48],{"className":16958,"style":170},[129,133],[15,16960,16962,16965],{"className":16961},[129],[15,16963,56],{"className":16964},[129,133],[15,16966,16968],{"className":16967},[137],[15,16969,16971,16991],{"className":16970},[141,142],[15,16972,16974,16988],{"className":16973},[146],[15,16975,16977],{"className":16976,"style":1981},[150],[15,16978,16979,16982],{"style":154},[15,16980],{"className":16981,"style":159},[158],[15,16983,16985],{"className":16984},[163,164,165,166],[15,16986,1459],{"className":16987},[129,133,166],[15,16989,175],{"className":16990},[174],[15,16992,16994],{"className":16993},[146],[15,16995,16997],{"className":16996,"style":182},[150],[15,16998],{},[15,17000],{"className":17001,"style":199},[93],[15,17003,16858],{"className":17004},[203],[15,17006],{"className":17007,"style":199},[93],[15,17009,17011,17015,17018,17021,17287,17291,17295],{"className":17010},[120],[15,17012],{"className":17013,"style":17014},[124],"height:2.4em;vertical-align:-0.95em;",[15,17016,15763],{"className":17017,"style":887},[129,133],[15,17019],{"className":17020,"style":262},[93],[15,17022,17024,17030,17071,17074,17077,17080,17083,17086,17126,17129,17132,17135,17138,17141,17144,17147,17150,17153,17281],{"className":17023},[449],[15,17025,17027],{"className":17026,"style":5052},[188,5051],[15,17028,53],{"className":17029},[228,165],[15,17031,17033,17036],{"className":17032},[129],[15,17034,310],{"className":17035},[129],[15,17037,17039],{"className":17038},[137],[15,17040,17042],{"className":17041},[141],[15,17043,17045],{"className":17044},[146],[15,17046,17048],{"className":17047,"style":16059},[150],[15,17049,17050,17053],{"style":1272},[15,17051],{"className":17052,"style":159},[158],[15,17054,17056],{"className":17055},[163,164,165,166],[15,17057,17059,17062,17065,17068],{"className":17058},[129,166],[15,17060,315],{"className":17061},[129,166],[15,17063,1459],{"className":17064},[129,133,166],[15,17066,315],{"className":17067},[2436,166],[15,17069,318],{"className":17070},[129,166],[15,17072],{"className":17073,"style":3277},[93],[15,17075,2682],{"className":17076},[2436],[15,17078],{"className":17079,"style":3277},[93],[15,17081,318],{"className":17082},[129,1950],[15,17084,1407],{"className":17085},[188],[15,17087,17089,17092],{"className":17088},[129],[15,17090,857],{"className":17091,"style":887},[129,133],[15,17093,17095],{"className":17094},[137],[15,17096,17098,17118],{"className":17097},[141,142],[15,17099,17101,17115],{"className":17100},[146],[15,17102,17104],{"className":17103,"style":1981},[150],[15,17105,17106,17109],{"style":1100},[15,17107],{"className":17108,"style":159},[158],[15,17110,17112],{"className":17111},[163,164,165,166],[15,17113,1459],{"className":17114},[129,133,166],[15,17116,175],{"className":17117},[174],[15,17119,17121],{"className":17120},[146],[15,17122,17124],{"className":17123,"style":182},[150],[15,17125],{},[15,17127],{"className":17128,"style":199},[93],[15,17130,62],{"className":17131},[203],[15,17133],{"className":17134,"style":199},[93],[15,17136,315],{"className":17137},[129],[15,17139,318],{"className":17140},[129],[15,17142,1423],{"className":17143},[195],[15,17145],{"className":17146,"style":3277},[93],[15,17148,2682],{"className":17149},[2436],[15,17151],{"className":17152,"style":3277},[93],[15,17154,17156,17159,17278],{"className":17155},[129],[15,17157],{"className":17158},[188,1803],[15,17160,17162],{"className":17161},[1443],[15,17163,17165,17270],{"className":17164},[141,142],[15,17166,17168,17267],{"className":17167},[146],[15,17169,17172,17211,17219],{"className":17170,"style":17171},[150],"height:1.3714em;",[15,17173,17174,17177],{"style":1819},[15,17175],{"className":17176,"style":1823},[158],[15,17178,17180],{"className":17179},[129],[15,17181,17183,17186],{"className":17182},[129],[15,17184,310],{"className":17185},[129],[15,17187,17189],{"className":17188},[137],[15,17190,17192],{"className":17191},[141],[15,17193,17195],{"className":17194},[146],[15,17196,17199],{"className":17197,"style":17198},[150],"height:0.7507em;",[15,17200,17202,17205],{"style":17201},"top:-2.989em;margin-right:0.05em;",[15,17203],{"className":17204,"style":159},[158],[15,17206,17208],{"className":17207},[163,164,165,166],[15,17209,1459],{"className":17210},[129,133,166],[15,17212,17213,17216],{"style":1832},[15,17214],{"className":17215,"style":1823},[158],[15,17217],{"className":17218,"style":1840},[1839],[15,17220,17221,17224],{"style":1843},[15,17222],{"className":17223,"style":1823},[158],[15,17225,17227],{"className":17226},[129],[15,17228,17230,17233],{"className":17229},[129],[15,17231,16562],{"className":17232,"style":16595},[129,133],[15,17234,17236],{"className":17235},[137],[15,17237,17239,17259],{"className":17238},[141,142],[15,17240,17242,17256],{"className":17241},[146],[15,17243,17245],{"className":17244,"style":1981},[150],[15,17246,17247,17250],{"style":16610},[15,17248],{"className":17249,"style":159},[158],[15,17251,17253],{"className":17252},[163,164,165,166],[15,17254,1459],{"className":17255},[129,133,166],[15,17257,175],{"className":17258},[174],[15,17260,17262],{"className":17261},[146],[15,17263,17265],{"className":17264,"style":182},[150],[15,17266],{},[15,17268,175],{"className":17269},[174],[15,17271,17273],{"className":17272},[146],[15,17274,17276],{"className":17275,"style":1862},[150],[15,17277],{},[15,17279],{"className":17280},[195,1803],[15,17282,17284],{"className":17283,"style":5052},[195,5051],[15,17285,59],{"className":17286},[228,165],[15,17288],{"className":17289},[93,17290],"allowbreak",[15,17292],{"className":17293,"style":17294},[93],"margin-right:1em;",[15,17296],{"className":17297,"style":262},[93],[15,17299,17301,17304,17307,17317,17321,17324,17327,17330],{"className":17300},[120],[15,17302],{"className":17303,"style":125},[124],[15,17305,53],{"className":17306},[188],[15,17308,17310],{"className":17309},[129],[15,17311,17313],{"className":17312},[129],[15,17314,17316],{"className":17315},[129,221],"mod",[15,17318],{"className":17319,"style":17320},[93],"margin-right:0.3333em;",[15,17322,310],{"className":17323},[129],[15,17325,15763],{"className":17326,"style":887},[129,133],[15,17328,59],{"className":17329},[195],[15,17331,107],{"className":17332},[129],[11,17334,17335,17336,17433,17434,17485,17486,17582,17583,17635,17636,17808,17809,17837],{},"当 ",[15,17337,17339,17363],{"className":17338,"translate":19},[23],[15,17340,17342],{"className":17341},[27],[29,17343,17344],{"xmlns":31},[34,17345,17346,17360],{},[37,17347,17348,17354,17356,17358],{},[40,17349,17350,17352],{},[43,17351,857],{},[43,17353,1459],{},[50,17355,62],{},[50,17357,2682],{},[308,17359,318],{},[109,17361,17362],{"encoding":111},"y_i=+1",[15,17364,17366,17421],{"className":17365,"ariaHidden":72},[116],[15,17367,17369,17372,17412,17415,17418],{"className":17368},[120],[15,17370],{"className":17371,"style":5399},[124],[15,17373,17375,17378],{"className":17374},[129],[15,17376,857],{"className":17377,"style":887},[129,133],[15,17379,17381],{"className":17380},[137],[15,17382,17384,17404],{"className":17383},[141,142],[15,17385,17387,17401],{"className":17386},[146],[15,17388,17390],{"className":17389,"style":1981},[150],[15,17391,17392,17395],{"style":1100},[15,17393],{"className":17394,"style":159},[158],[15,17396,17398],{"className":17397},[163,164,165,166],[15,17399,1459],{"className":17400},[129,133,166],[15,17402,175],{"className":17403},[174],[15,17405,17407],{"className":17406},[146],[15,17408,17410],{"className":17409,"style":182},[150],[15,17411],{},[15,17413],{"className":17414,"style":199},[93],[15,17416,62],{"className":17417},[203],[15,17419],{"className":17420,"style":199},[93],[15,17422,17424,17427,17430],{"className":17423},[120],[15,17425],{"className":17426,"style":16484},[124],[15,17428,2682],{"className":17429},[129],[15,17431,318],{"className":17432},[129]," 时，括号内的值落在 ",[15,17435,17437,17458],{"className":17436,"translate":19},[23],[15,17438,17440],{"className":17439},[27],[29,17441,17442],{"xmlns":31},[34,17443,17444,17456],{},[37,17445,17446,17448,17450,17452,17454],{},[50,17447,53],{"stretchy":52},[308,17449,541],{},[50,17451,91],{"separator":72},[308,17453,318],{},[50,17455,59],{"stretchy":52},[109,17457,9524],{"encoding":111},[15,17459,17461],{"className":17460,"ariaHidden":72},[116],[15,17462,17464,17467,17470,17473,17476,17479,17482],{"className":17463},[120],[15,17465],{"className":17466,"style":125},[124],[15,17468,53],{"className":17469},[188],[15,17471,541],{"className":17472},[129],[15,17474,91],{"className":17475},[254],[15,17477],{"className":17478,"style":262},[93],[15,17480,318],{"className":17481},[129],[15,17483,59],{"className":17484},[195],"；当 ",[15,17487,17489,17512],{"className":17488,"translate":19},[23],[15,17490,17492],{"className":17491},[27],[29,17493,17494],{"xmlns":31},[34,17495,17496,17510],{},[37,17497,17498,17504,17506,17508],{},[40,17499,17500,17502],{},[43,17501,857],{},[43,17503,1459],{},[50,17505,62],{},[50,17507,315],{},[308,17509,318],{},[109,17511,16419],{"encoding":111},[15,17513,17515,17570],{"className":17514,"ariaHidden":72},[116],[15,17516,17518,17521,17561,17564,17567],{"className":17517},[120],[15,17519],{"className":17520,"style":5399},[124],[15,17522,17524,17527],{"className":17523},[129],[15,17525,857],{"className":17526,"style":887},[129,133],[15,17528,17530],{"className":17529},[137],[15,17531,17533,17553],{"className":17532},[141,142],[15,17534,17536,17550],{"className":17535},[146],[15,17537,17539],{"className":17538,"style":1981},[150],[15,17540,17541,17544],{"style":1100},[15,17542],{"className":17543,"style":159},[158],[15,17545,17547],{"className":17546},[163,164,165,166],[15,17548,1459],{"className":17549},[129,133,166],[15,17551,175],{"className":17552},[174],[15,17554,17556],{"className":17555},[146],[15,17557,17559],{"className":17558,"style":182},[150],[15,17560],{},[15,17562],{"className":17563,"style":199},[93],[15,17565,62],{"className":17566},[203],[15,17568],{"className":17569,"style":199},[93],[15,17571,17573,17576,17579],{"className":17572},[120],[15,17574],{"className":17575,"style":16484},[124],[15,17577,315],{"className":17578},[129],[15,17580,318],{"className":17581},[129]," 时落在 ",[15,17584,17586,17608],{"className":17585,"translate":19},[23],[15,17587,17589],{"className":17588},[27],[29,17590,17591],{"xmlns":31},[34,17592,17593,17605],{},[37,17594,17595,17597,17599,17601,17603],{},[50,17596,53],{"stretchy":52},[308,17598,318],{},[50,17600,91],{"separator":72},[308,17602,310],{},[50,17604,59],{"stretchy":52},[109,17606,17607],{"encoding":111},"(1,2)",[15,17609,17611],{"className":17610,"ariaHidden":72},[116],[15,17612,17614,17617,17620,17623,17626,17629,17632],{"className":17613},[120],[15,17615],{"className":17616,"style":125},[124],[15,17618,53],{"className":17619},[188],[15,17621,318],{"className":17622},[129],[15,17624,91],{"className":17625},[254],[15,17627],{"className":17628,"style":262},[93],[15,17630,310],{"className":17631},[129],[15,17633,59],{"className":17634},[195],"。所以 ",[15,17637,17639,17683],{"className":17638,"translate":19},[23],[15,17640,17642],{"className":17641},[27],[29,17643,17644],{"xmlns":31},[34,17645,17646,17680],{},[37,17647,17648,17650,17652,17654,17656,17658,17660,17662,17668,17670,17672,17674],{},[43,17649,66],{"mathvariant":65},[50,17651,69],{},[50,17653,53],{"stretchy":52},[43,17655,76],{},[50,17657,69],{},[50,17659,53],{"stretchy":52},[43,17661,48],{},[40,17663,17664,17666],{},[43,17665,56],{},[43,17667,1459],{},[50,17669,59],{"stretchy":52},[50,17671,59],{"stretchy":52},[50,17673,62],{},[40,17675,17676,17678],{},[43,17677,857],{},[43,17679,1459],{},[109,17681,17682],{"encoding":111},"\\operatorname{sign}(\\sin(\\theta x_i))=y_i",[15,17684,17686,17762],{"className":17685,"ariaHidden":72},[116],[15,17687,17689,17692,17698,17701,17704,17707,17710,17750,17753,17756,17759],{"className":17688},[120],[15,17690],{"className":17691,"style":125},[124],[15,17693,17695],{"className":17694},[217],[15,17696,66],{"className":17697},[129,221],[15,17699,53],{"className":17700},[188],[15,17702,76],{"className":17703},[217],[15,17705,53],{"className":17706},[188],[15,17708,48],{"className":17709,"style":170},[129,133],[15,17711,17713,17716],{"className":17712},[129],[15,17714,56],{"className":17715},[129,133],[15,17717,17719],{"className":17718},[137],[15,17720,17722,17742],{"className":17721},[141,142],[15,17723,17725,17739],{"className":17724},[146],[15,17726,17728],{"className":17727,"style":1981},[150],[15,17729,17730,17733],{"style":154},[15,17731],{"className":17732,"style":159},[158],[15,17734,17736],{"className":17735},[163,164,165,166],[15,17737,1459],{"className":17738},[129,133,166],[15,17740,175],{"className":17741},[174],[15,17743,17745],{"className":17744},[146],[15,17746,17748],{"className":17747,"style":182},[150],[15,17749],{},[15,17751,6149],{"className":17752},[195],[15,17754],{"className":17755,"style":199},[93],[15,17757,62],{"className":17758},[203],[15,17760],{"className":17761,"style":199},[93],[15,17763,17765,17768],{"className":17764},[120],[15,17766],{"className":17767,"style":5399},[124],[15,17769,17771,17774],{"className":17770},[129],[15,17772,857],{"className":17773,"style":887},[129,133],[15,17775,17777],{"className":17776},[137],[15,17778,17780,17800],{"className":17779},[141,142],[15,17781,17783,17797],{"className":17782},[146],[15,17784,17786],{"className":17785,"style":1981},[150],[15,17787,17788,17791],{"style":1100},[15,17789],{"className":17790,"style":159},[158],[15,17792,17794],{"className":17793},[163,164,165,166],[15,17795,1459],{"className":17796},[129,133,166],[15,17798,175],{"className":17799},[174],[15,17801,17803],{"className":17802},[146],[15,17804,17806],{"className":17805,"style":182},[150],[15,17807],{},"。任意 ",[15,17810,17812,17825],{"className":17811,"translate":19},[23],[15,17813,17815],{"className":17814},[27],[29,17816,17817],{"xmlns":31},[34,17818,17819,17823],{},[37,17820,17821],{},[43,17822,348],{},[109,17824,348],{"encoding":111},[15,17826,17828],{"className":17827,"ariaHidden":72},[116],[15,17829,17831,17834],{"className":17830},[120],[15,17832],{"className":17833,"style":4383},[124],[15,17835,348],{"className":17836},[129,133]," 个点都能被打散，VC 维无穷。",[11,17839,17840],{},"一个参数，无穷容量。这就是「参数量不等于容量」最干净的反例。",[11,17842,17843],{},[698,17844,17845,17846,7438],{},"例 7（最近邻，",[15,17847,17849,17868],{"className":17848,"translate":19},[23],[15,17850,17852],{"className":17851},[27],[29,17853,17854],{"xmlns":31},[34,17855,17856,17866],{},[37,17857,17858,17860,17862,17864],{},[43,17859,6823],{"mathvariant":65},[50,17861,69],{},[50,17863,62],{},[43,17865,4909],{"mathvariant":65},[109,17867,15229],{"encoding":111},[15,17869,17871,17892],{"className":17870,"ariaHidden":72},[116],[15,17872,17874,17877,17883,17886,17889],{"className":17873},[120],[15,17875],{"className":17876,"style":522},[124],[15,17878,17880],{"className":17879},[217],[15,17881,6823],{"className":17882},[129,221],[15,17884],{"className":17885,"style":199},[93],[15,17887,62],{"className":17888},[203],[15,17890],{"className":17891,"style":199},[93],[15,17893,17895,17898],{"className":17894},[120],[15,17896],{"className":17897,"style":4383},[124],[15,17899,4909],{"className":17900},[129],[11,17902,17903,17904,17932,17933,17961,17962,18032],{},"1-NN 没有参数：给定训练集，它把测试点判为最近训练点的标签。给定任意有限点集 ",[15,17905,17907,17920],{"className":17906,"translate":19},[23],[15,17908,17910],{"className":17909},[27],[29,17911,17912],{"xmlns":31},[34,17913,17914,17918],{},[37,17915,17916],{},[43,17917,5619],{},[109,17919,5619],{"encoding":111},[15,17921,17923],{"className":17922,"ariaHidden":72},[116],[15,17924,17926,17929],{"className":17925},[120],[15,17927],{"className":17928,"style":610},[124],[15,17930,5619],{"className":17931,"style":5665},[129,133],"（点互不相同、没有距离并列）和任意标注，直接令训练集就是 ",[15,17934,17936,17949],{"className":17935,"translate":19},[23],[15,17937,17939],{"className":17938},[27],[29,17940,17941],{"xmlns":31},[34,17942,17943,17947],{},[37,17944,17945],{},[43,17946,5619],{},[109,17948,5619],{"encoding":111},[15,17950,17952],{"className":17951,"ariaHidden":72},[116],[15,17953,17955,17958],{"className":17954},[120],[15,17956],{"className":17957,"style":610},[124],[15,17959,5619],{"className":17960,"style":5665},[129,133]," 本身（标签按标注给）；对每个 ",[15,17963,17965,17983],{"className":17964,"translate":19},[23],[15,17966,17968],{"className":17967},[27],[29,17969,17970],{"xmlns":31},[34,17971,17972,17980],{},[37,17973,17974],{},[40,17975,17976,17978],{},[43,17977,56],{},[43,17979,1459],{},[109,17981,17982],{"encoding":111},"x_i",[15,17984,17986],{"className":17985,"ariaHidden":72},[116],[15,17987,17989,17992],{"className":17988},[120],[15,17990],{"className":17991,"style":7985},[124],[15,17993,17995,17998],{"className":17994},[129],[15,17996,56],{"className":17997},[129,133],[15,17999,18001],{"className":18000},[137],[15,18002,18004,18024],{"className":18003},[141,142],[15,18005,18007,18021],{"className":18006},[146],[15,18008,18010],{"className":18009,"style":1981},[150],[15,18011,18012,18015],{"style":154},[15,18013],{"className":18014,"style":159},[158],[15,18016,18018],{"className":18017},[163,164,165,166],[15,18019,1459],{"className":18020},[129,133,166],[15,18022,175],{"className":18023},[174],[15,18025,18027],{"className":18026},[146],[15,18028,18030],{"className":18029,"style":182},[150],[15,18031],{},"，最近邻是它自己，预测即标注。所以任意有限集都被打散，VC 维无穷。",[11,18034,18035],{},"这个例子说明，VC 维无穷并不妨碍一个算法在真实数据上表现良好；它只说明不存在一个对所有分布一致的泛化保证。",[11,18037,18038],{},"把上面几个例子汇总：",[18040,18041,18042,18056],"table",{},[18043,18044,18045],"thead",{},[18046,18047,18048,18051,18054],"tr",{},[18049,18050,803],"th",{},[18049,18052,18053],{},"输入空间",[18049,18055,6804],{},[18057,18058,18059,18126,18191,18308,18408,18525,18668,18801],"tbody",{},[18046,18060,18061,18065,18096],{},[18062,18063,18064],"td",{},"半直线",[18062,18066,18067],{},[15,18068,18070,18084],{"className":18069,"translate":19},[23],[15,18071,18073],{"className":18072},[27],[29,18074,18075],{"xmlns":31},[34,18076,18077,18081],{},[37,18078,18079],{},[43,18080,104],{"mathvariant":103},[109,18082,18083],{"encoding":111},"\\mathbb R",[15,18085,18087],{"className":18086,"ariaHidden":72},[116],[15,18088,18090,18093],{"className":18089},[120],[15,18091],{"className":18092,"style":281},[124],[15,18094,104],{"className":18095},[129,285],[18062,18097,18098],{},[15,18099,18101,18114],{"className":18100,"translate":19},[23],[15,18102,18104],{"className":18103},[27],[29,18105,18106],{"xmlns":31},[34,18107,18108,18112],{},[37,18109,18110],{},[308,18111,318],{},[109,18113,318],{"encoding":111},[15,18115,18117],{"className":18116,"ariaHidden":72},[116],[15,18118,18120,18123],{"className":18119},[120],[15,18121],{"className":18122,"style":4886},[124],[15,18124,318],{"className":18125},[129],[18046,18127,18128,18131,18161],{},[18062,18129,18130],{},"区间",[18062,18132,18133],{},[15,18134,18136,18149],{"className":18135,"translate":19},[23],[15,18137,18139],{"className":18138},[27],[29,18140,18141],{"xmlns":31},[34,18142,18143,18147],{},[37,18144,18145],{},[43,18146,104],{"mathvariant":103},[109,18148,18083],{"encoding":111},[15,18150,18152],{"className":18151,"ariaHidden":72},[116],[15,18153,18155,18158],{"className":18154},[120],[15,18156],{"className":18157,"style":281},[124],[15,18159,104],{"className":18160},[129,285],[18062,18162,18163],{},[15,18164,18166,18179],{"className":18165,"translate":19},[23],[15,18167,18169],{"className":18168},[27],[29,18170,18171],{"xmlns":31},[34,18172,18173,18177],{},[37,18174,18175],{},[308,18176,310],{},[109,18178,310],{"encoding":111},[15,18180,18182],{"className":18181,"ariaHidden":72},[116],[15,18183,18185,18188],{"className":18184},[120],[15,18186],{"className":18187,"style":4886},[124],[15,18189,310],{"className":18190},[129],[18046,18192,18193,18196,18256],{},[18062,18194,18195],{},"仿射超平面",[18062,18197,18198],{},[15,18199,18201,18218],{"className":18200,"translate":19},[23],[15,18202,18204],{"className":18203},[27],[29,18205,18206],{"xmlns":31},[34,18207,18208,18216],{},[37,18209,18210],{},[305,18211,18212,18214],{},[43,18213,104],{"mathvariant":103},[43,18215,10376],{},[109,18217,11570],{"encoding":111},[15,18219,18221],{"className":18220,"ariaHidden":72},[116],[15,18222,18224,18227],{"className":18223},[120],[15,18225],{"className":18226,"style":10490},[124],[15,18228,18230,18233],{"className":18229},[129],[15,18231,104],{"className":18232},[129,285],[15,18234,18236],{"className":18235},[137],[15,18237,18239],{"className":18238},[141],[15,18240,18242],{"className":18241},[146],[15,18243,18245],{"className":18244,"style":10490},[150],[15,18246,18247,18250],{"style":383},[15,18248],{"className":18249,"style":159},[158],[15,18251,18253],{"className":18252},[163,164,165,166],[15,18254,10376],{"className":18255},[129,133,166],[18062,18257,18258],{},[15,18259,18261,18278],{"className":18260,"translate":19},[23],[15,18262,18264],{"className":18263},[27],[29,18265,18266],{"xmlns":31},[34,18267,18268,18276],{},[37,18269,18270,18272,18274],{},[43,18271,10376],{},[50,18273,2682],{},[308,18275,318],{},[109,18277,10880],{"encoding":111},[15,18279,18281,18299],{"className":18280,"ariaHidden":72},[116],[15,18282,18284,18287,18290,18293,18296],{"className":18283},[120],[15,18285],{"className":18286,"style":10414},[124],[15,18288,10376],{"className":18289},[129,133],[15,18291],{"className":18292,"style":3277},[93],[15,18294,2682],{"className":18295},[2436],[15,18297],{"className":18298,"style":3277},[93],[15,18300,18302,18305],{"className":18301},[120],[15,18303],{"className":18304,"style":4886},[124],[15,18306,318],{"className":18307},[129],[18046,18309,18310,18313,18373],{},[18062,18311,18312],{},"轴对齐矩形",[18062,18314,18315],{},[15,18316,18318,18335],{"className":18317,"translate":19},[23],[15,18319,18321],{"className":18320},[27],[29,18322,18323],{"xmlns":31},[34,18324,18325,18333],{},[37,18326,18327],{},[305,18328,18329,18331],{},[43,18330,104],{"mathvariant":103},[43,18332,10376],{},[109,18334,11570],{"encoding":111},[15,18336,18338],{"className":18337,"ariaHidden":72},[116],[15,18339,18341,18344],{"className":18340},[120],[15,18342],{"className":18343,"style":10490},[124],[15,18345,18347,18350],{"className":18346},[129],[15,18348,104],{"className":18349},[129,285],[15,18351,18353],{"className":18352},[137],[15,18354,18356],{"className":18355},[141],[15,18357,18359],{"className":18358},[146],[15,18360,18362],{"className":18361,"style":10490},[150],[15,18363,18364,18367],{"style":383},[15,18365],{"className":18366,"style":159},[158],[15,18368,18370],{"className":18369},[163,164,165,166],[15,18371,10376],{"className":18372},[129,133,166],[18062,18374,18375],{},[15,18376,18378,18393],{"className":18377,"translate":19},[23],[15,18379,18381],{"className":18380},[27],[29,18382,18383],{"xmlns":31},[34,18384,18385,18391],{},[37,18386,18387,18389],{},[308,18388,310],{},[43,18390,10376],{},[109,18392,14280],{"encoding":111},[15,18394,18396],{"className":18395,"ariaHidden":72},[116],[15,18397,18399,18402,18405],{"className":18398},[120],[15,18400],{"className":18401,"style":522},[124],[15,18403,310],{"className":18404},[129],[1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",[15,18531,18533,18546],{"className":18532,"translate":19},[23],[15,18534,18536],{"className":18535},[27],[29,18537,18538],{"xmlns":31},[34,18539,18540,18544],{},[37,18541,18542],{},[43,18543,819],{"mathvariant":596},[109,18545,822],{"encoding":111},[15,18547,18549],{"className":18548,"ariaHidden":72},[116],[15,18550,18552,18555],{"className":18551},[120],[15,18553],{"className":18554,"style":610},[124],[15,18556,819],{"className":18557,"style":835},[129,614],[18062,18559,18560],{},"任意",[18062,18562,18563],{},[15,18564,18566,18596],{"className":18565,"translate":19},[23],[15,18567,18569],{"className":18568},[27],[29,18570,18571],{"xmlns":31},[34,18572,18573,18593],{},[37,18574,18575,18577,18587,18589,18591],{},[50,18576,2717],{},[40,18578,18579,18585],{},[37,18580,18581,18583],{},[43,18582,14727],{},[50,18584,69],{},[308,18586,310],{},[50,18588,2945],{"stretchy":52},[43,18590,819],{"mathvariant":596},[50,18592,2945],{"stretchy":52},[109,18594,18595],{"encoding":111},"\\le\\log_2\\lvert\\mathcal H\\rvert",[15,18597,18599,18611],{"className":18598,"ariaHidden":72},[116],[15,18600,18602,18605,18608],{"className":18601},[120],[15,18603],{"className":18604,"style":7757},[124],[15,18606,2717],{"className":18607},[203],[15,18609],{"className":18610,"style":199},[93],[15,18612,18614,18617,18659,18662,18665],{"className":18613},[120],[15,18615],{"className":18616,"style":125},[124],[15,18618,18620,18625],{"className":18619},[217],[15,18621,14777,18623],{"className":18622},[217],[15,18624,2247],{"style":2246},[15,18626,18628],{"className":18627},[137],[15,18629,18631,18651],{"className":18630},[141,142],[15,18632,18634,18648],{"className":18633},[146],[15,18635,18637],{"className":18636,"style":14792},[150],[15,18638,18639,18642],{"style":14795},[15,18640],{"className":18641,"style":159},[158],[15,18643,18645],{"className":18644},[163,164,165,166],[15,18646,310],{"className":18647},[129,166],[15,18649,175],{"className":18650},[174],[15,18652,18654],{"className":18653},[146],[15,18655,18657],{"className":18656,"style":14814},[150],[15,18658],{},[15,18660,2945],{"className":18661},[188],[15,18663,819],{"className":18664,"style":835},[129,614],[15,18666,2945],{"className":18667},[195],[18046,18669,18670,18740,18770],{},[18062,18671,18672],{},[15,18673,18675,18707],{"className":18674,"translate":19},[23],[15,18676,18678],{"className":18677},[27],[29,18679,18680],{"xmlns":31},[34,18681,18682,18704],{},[37,18683,18684,18686,18688,18690,18692,18694,18696,18698,18700,18702],{},[43,18685,66],{"mathvariant":65},[50,18687,69],{},[50,18689,53],{"stretchy":52},[43,18691,76],{},[50,18693,69],{},[50,18695,53],{"stretchy":52},[43,18697,48],{},[43,18699,56],{},[50,18701,59],{"stretchy":52},[50,18703,59],{"stretchy":52},[109,18705,18706],{"encoding":111},"\\operatorname{sign}(\\sin(\\theta x))",[15,18708,18710],{"className":18709,"ariaHidden":72},[116],[15,18711,18713,18716,18722,18725,18728,18731,18734,18737],{"className":18712},[120],[15,18714],{"className":18715,"style":125},[124],[15,18717,18719],{"className":18718},[217],[15,18720,66],{"className":18721},[129,221],[15,18723,53],{"className":18724},[188],[15,18726,76],{"className":18727},[217],[15,18729,53],{"className":18730},[188],[15,18732,48],{"className":18733,"style":170},[129,133],[15,18735,56],{"className":18736},[129,133],[15,18738,6149],{"className":18739},[195],[18062,18741,18742],{},[15,18743,18745,18758],{"className":18744,"translate":19},[23],[15,18746,18748],{"className":18747},[27],[29,18749,18750],{"xmlns":31},[34,18751,18752,18756],{},[37,18753,18754],{},[43,18755,104],{"mathvariant":103},[109,18757,18083],{"encoding":111},[15,18759,18761],{"className":18760,"ariaHidden":72},[116],[15,18762,18764,18767],{"className":18763},[120],[15,18765],{"className":18766,"style":281},[124],[15,18768,104],{"className":18769},[129,285],[18062,18771,18772],{},[15,18773,18775,18789],{"className":18774,"translate":19},[23],[15,18776,18778],{"className":18777},[27],[29,18779,18780],{"xmlns":31},[34,18781,18782,18786],{},[37,18783,18784],{},[43,18785,4909],{"mathvariant":65},[109,18787,18788],{"encoding":111},"\\infty",[15,18790,18792],{"className":18791,"ariaHidden":72},[116],[15,18793,18795,18798],{"className":18794},[120],[15,18796],{"className":18797,"style":4383},[124],[15,18799,4909],{"className":18800},[129],[18046,18802,18803,18806,18866],{},[18062,18804,18805],{},"1-NN",[18062,18807,18808],{},[15,18809,18811,18828],{"className":18810,"translate":19},[23],[15,18812,18814],{"className":18813},[27],[29,18815,18816],{"xmlns":31},[34,18817,18818,18826],{},[37,18819,18820],{},[305,18821,18822,18824],{},[43,18823,104],{"mathvariant":103},[43,18825,10376],{},[109,18827,11570],{"encoding":111},[15,18829,18831],{"className":18830,"ariaHidden":72},[116],[15,18832,18834,18837],{"className":18833},[120],[15,18835],{"className":18836,"style":10490},[124],[15,18838,18840,18843],{"className":18839},[129],[15,18841,104],{"className":18842},[129,285],[15,18844,18846],{"className":18845},[137],[15,18847,18849],{"className":18848},[141],[15,18850,18852],{"className":18851},[146],[15,18853,18855],{"className":18854,"style":10490},[150],[15,18856,18857,18860],{"style":383},[15,18858],{"className":18859,"style":159},[158],[15,18861,18863],{"className":18862},[163,164,165,166],[15,18864,10376],{"className":18865},[129,133,166],[18062,18867,18868],{},[15,18869,18871,18884],{"className":18870,"translate":19},[23],[15,18872,18874],{"className":18873},[27],[29,18875,18876],{"xmlns":31},[34,18877,18878,18882],{},[37,18879,18880],{},[43,18881,4909],{"mathvariant":65},[109,18883,18788],{"encoding":111},[15,18885,18887],{"className":18886,"ariaHidden":72},[116],[15,18888,18890,18893],{"className":18889},[120],[15,18891],{"className":18892,"style":4383},[124],[15,18894,4909],{"className":18895},[129],[6559,18897,18898],{},[11,18899,18900,18903,18904,19076,19077,19173,19174,19226,19227,19419,19420,19448,19449,19508,19509,19537,19538,19566],{},[698,18901,18902],{},"注（怎么算）"," 对线性分类器，固定点集 ",[15,18905,18907,18945],{"className":18906,"translate":19},[23],[15,18908,18910],{"className":18909},[27],[29,18911,18912],{"xmlns":31},[34,18913,18914,18942],{},[37,18915,18916,18918,18920,18922,18928,18930,18932,18934,18940],{},[43,18917,5619],{},[50,18919,62],{},[50,18921,636],{"stretchy":52},[40,18923,18924,18926],{},[43,18925,56],{},[308,18927,318],{},[50,18929,91],{"separator":72},[50,18931,335],{},[50,18933,91],{"separator":72},[40,18935,18936,18938],{},[43,18937,56],{},[43,18939,16562],{},[50,18941,645],{"stretchy":52},[109,18943,18944],{"encoding":111},"C=\\{x_1,\\dots,x_k\\}",[15,18946,18948,18966],{"className":18947,"ariaHidden":72},[116],[15,18949,18951,18954,18957,18960,18963],{"className":18950},[120],[15,18952],{"className":18953,"style":610},[124],[15,18955,5619],{"className":18956,"style":5665},[129,133],[15,18958],{"className":18959,"style":199},[93],[15,18961,62],{"className":18962},[203],[15,18964],{"className":18965,"style":199},[93],[15,18967,18969,18972,18975,19015,19018,19021,19024,19027,19030,19033,19073],{"className":18968},[120],[15,18970],{"className":18971,"style":125},[124],[15,18973,636],{"className":18974},[188],[15,18976,18978,18981],{"className":18977},[129],[15,18979,56],{"className":18980},[129,133],[15,18982,18984],{"className":18983},[137],[15,18985,18987,19007],{"className":18986},[141,142],[15,18988,18990,19004],{"className":18989},[146],[15,18991,18993],{"className":18992,"style":1051},[150],[15,18994,18995,18998],{"style":154},[15,18996],{"className":18997,"style":159},[158],[15,18999,19001],{"className":19000},[163,164,165,166],[15,19002,318],{"className":19003},[129,166],[15,19005,175],{"className":19006},[174],[15,19008,19010],{"className":19009},[146],[15,19011,19013],{"className":19012,"style":182},[150],[15,19014],{},[15,19016,91],{"className":19017},[254],[15,19019],{"className":19020,"style":262},[93],[15,19022,335],{"className":19023},[449],[15,19025],{"className":19026,"style":262},[93],[15,19028,91],{"className":19029},[254],[15,19031],{"className":19032,"style":262},[93],[15,19034,19036,19039],{"className":19035},[129],[15,19037,56],{"className":19038},[129,133],[15,19040,19042],{"className":19041},[137],[15,19043,19045,19065],{"className":19044},[141,142],[15,19046,19048,19062],{"className":19047},[146],[15,19049,19051],{"className":19050,"style":151},[150],[15,19052,19053,19056],{"style":154},[15,19054],{"className":19055,"style":159},[158],[15,19057,19059],{"className":19058},[163,164,165,166],[15,19060,16562],{"className":19061,"style":16595},[129,133,166],[15,19063,175],{"className":19064},[174],[15,19066,19068],{"className":19067},[146],[15,19069,19071],{"className":19070,"style":182},[150],[15,19072],{},[15,19074,645],{"className":19075},[195],"，对每个标注 ",[15,19078,19080,19108],{"className":19079,"translate":19},[23],[15,19081,19083],{"className":19082},[27],[29,19084,19085],{"xmlns":31},[34,19086,19087,19105],{},[37,19088,19089,19091,19093,19095,19097,19099],{},[43,19090,857],{},[50,19092,100],{},[50,19094,636],{"stretchy":52},[50,19096,10932],{},[308,19098,318],{},[305,19100,19101,19103],{},[50,19102,645],{"stretchy":52},[43,19104,16562],{},[109,19106,19107],{"encoding":111},"y\\in\\{\\pm1\\}^k",[15,19109,19111,19129],{"className":19110,"ariaHidden":72},[116],[15,19112,19114,19117,19120,19123,19126],{"className":19113},[120],[15,19115],{"className":19116,"style":10959},[124],[15,19118,857],{"className":19119,"style":887},[129,133],[15,19121],{"className":19122,"style":199},[93],[15,19124,100],{"className":19125},[203],[15,19127],{"className":19128,"style":199},[93],[15,19130,19132,19135,19138,19141,19144],{"className":19131},[120],[15,19133],{"className":19134,"style":10978},[124],[15,19136,636],{"className":19137},[188],[15,19139,10932],{"className":19140},[129],[15,19142,318],{"className":19143},[129],[15,19145,19147,19150],{"className":19146},[195],[15,19148,645],{"className":19149},[195],[15,19151,19153],{"className":19152},[137],[15,19154,19156],{"className":19155},[141],[15,19157,19159],{"className":19158},[146],[15,19160,19162],{"className":19161,"style":10490},[150],[15,19163,19164,19167],{"style":383},[15,19165],{"className":19166,"style":159},[158],[15,19168,19170],{"className":19169},[163,164,165,166],[15,19171,16562],{"className":19172,"style":16595},[129,133,166]," 解可行性问题：是否存在 ",[15,19175,19177,19199],{"className":19176,"translate":19},[23],[15,19178,19180],{"className":19179},[27],[29,19181,19182],{"xmlns":31},[34,19183,19184,19196],{},[37,19185,19186,19188,19190,19192,19194],{},[50,19187,53],{"stretchy":52},[43,19189,10550],{},[50,19191,91],{"separator":72},[43,19193,8699],{},[50,19195,59],{"stretchy":52},[109,19197,19198],{"encoding":111},"(w,b)",[15,19200,19202],{"className":19201,"ariaHidden":72},[116],[15,19203,19205,19208,19211,19214,19217,19220,19223],{"className":19204},[120],[15,19206],{"className":19207,"style":125},[124],[15,19209,53],{"className":19210},[188],[15,19212,10550],{"className":19213,"style":10627},[129,133],[15,19215,91],{"className":19216},[254],[15,19218],{"className":19219,"style":262},[93],[15,19221,8699],{"className":19222},[129,133],[15,19224,59],{"className":19225},[195]," 使 ",[15,19228,19230,19270],{"className":19229,"translate":19},[23],[15,19231,19233],{"className":19232},[27],[29,19234,19235],{"xmlns":31},[34,19236,19237,19267],{},[37,19238,19239,19245,19247,19249,19251,19257,19259,19261,19263,19265],{},[40,19240,19241,19243],{},[43,19242,857],{},[43,19244,1459],{},[50,19246,53],{"stretchy":52},[43,19248,10550],{},[50,19250,10553],{},[40,19252,19253,19255],{},[43,19254,56],{},[43,19256,1459],{},[50,19258,2682],{},[43,19260,8699],{},[50,19262,59],{"stretchy":52},[50,19264,7473],{},[308,19266,318],{},[109,19268,19269],{"encoding":111},"y_i(w\\cdot x_i+b)\\ge 1",[15,19271,19273,19334,19389,19410],{"className":19272,"ariaHidden":72},[116],[15,19274,19276,19279,19319,19322,19325,19328,19331],{"className":19275},[120],[15,19277],{"className":19278,"style":125},[124],[15,19280,19282,19285],{"className":19281},[129],[15,19283,857],{"className":19284,"style":887},[129,133],[15,19286,19288],{"className":19287},[137],[15,19289,19291,19311],{"className":19290},[141,142],[15,19292,19294,19308],{"className":19293},[146],[15,19295,19297],{"className":19296,"style":1981},[150],[15,19298,19299,19302],{"style":1100},[15,19300],{"className":19301,"style":159},[158],[15,19303,19305],{"className":19304},[163,164,165,166],[15,19306,1459],{"className":19307},[129,133,166],[15,19309,175],{"className":19310},[174],[15,19312,19314],{"className":19313},[146],[15,19315,19317],{"className":19316,"style":182},[150],[15,19318],{},[15,19320,53],{"className":19321},[188],[15,19323,10550],{"className":19324,"style":10627},[129,133],[15,19326],{"className":19327,"style":3277},[93],[15,19329,10553],{"className":19330},[2436],[15,19332],{"className":19333,"style":3277},[93],[15,19335,19337,19340,19380,19383,19386],{"className":19336},[120],[15,19338],{"className":19339,"style":11117},[124],[15,19341,19343,19346],{"className":19342},[129],[15,19344,56],{"className":19345},[129,133],[15,19347,19349],{"className":19348},[137],[15,19350,19352,19372],{"className":19351},[141,142],[15,19353,19355,19369],{"className":19354},[146],[15,19356,19358],{"className":19357,"style":1981},[150],[15,19359,19360,19363],{"style":154},[15,19361],{"className":19362,"style":159},[158],[15,19364,19366],{"className":19365},[163,164,165,166],[15,19367,1459],{"className":19368},[129,133,166],[15,19370,175],{"className":19371},[174],[15,19373,19375],{"className":19374},[146],[15,19376,19378],{"className":19377,"style":182},[150],[15,19379],{},[15,19381],{"className":19382,"style":3277},[93],[15,19384,2682],{"className":19385},[2436],[15,19387],{"className":19388,"style":3277},[93],[15,19390,19392,19395,19398,19401,19404,19407],{"className":19391},[120],[15,19393],{"className":19394,"style":125},[124],[15,19396,8699],{"className":19397},[129,133],[15,19399,59],{"className":19400},[195],[15,19402],{"className":19403,"style":199},[93],[15,19405,7473],{"className":19406},[203],[15,19408],{"className":19409,"style":199},[93],[15,19411,19413,19416],{"className":19412},[120],[15,19414],{"className":19415,"style":4886},[124],[15,19417,318],{"className":19418},[129]," 对所有 ",[15,19421,19423,19436],{"className":19422,"translate":19},[23],[15,19424,19426],{"className":19425},[27],[29,19427,19428],{"xmlns":31},[34,19429,19430,19434],{},[37,19431,19432],{},[43,19433,1459],{},[109,19435,1459],{"encoding":111},[15,19437,19439],{"className":19438,"ariaHidden":72},[116],[15,19440,19442,19445],{"className":19441},[120],[15,19443],{"className":19444,"style":16111},[124],[15,19446,1459],{"className":19447},[129,133]," 成立。这是一个线性规划。若 ",[15,19450,19452,19470],{"className":19451,"translate":19},[23],[15,19453,19455],{"className":19454},[27],[29,19456,19457],{"xmlns":31},[34,19458,19459,19467],{},[37,19460,19461],{},[305,19462,19463,19465],{},[308,19464,310],{},[43,19466,16562],{},[109,19468,19469],{"encoding":111},"2^k",[15,19471,19473],{"className":19472,"ariaHidden":72},[116],[15,19474,19476,19479],{"className":19475},[120],[15,19477],{"className":19478,"style":10490},[124],[15,19480,19482,19485],{"className":19481},[129],[15,19483,310],{"className":19484},[129],[15,19486,19488],{"className":19487},[137],[15,19489,19491],{"className":19490},[141],[15,19492,19494],{"className":19493},[146],[15,19495,19497],{"className":19496,"style":10490},[150],[15,19498,19499,19502],{"style":383},[15,19500],{"className":19501,"style":159},[158],[15,19503,19505],{"className":19504},[163,164,165,166],[15,19506,16562],{"className":19507,"style":16595},[129,133,166]," 个标注全部可行，",[15,19510,19512,19525],{"className":19511,"translate":19},[23],[15,19513,19515],{"className":19514},[27],[29,19516,19517],{"xmlns":31},[34,19518,19519,19523],{},[37,19520,19521],{},[43,19522,5619],{},[109,19524,5619],{"encoding":111},[15,19526,19528],{"className":19527,"ariaHidden":72},[116],[15,19529,19531,19534],{"className":19530},[120],[15,19532],{"className":19533,"style":610},[124],[15,19535,5619],{"className":19536,"style":5665},[129,133]," 就被打散；对所有 ",[15,19539,19541,19554],{"className":19540,"translate":19},[23],[15,19542,19544],{"className":19543},[27],[29,19545,19546],{"xmlns":31},[34,19547,19548,19552],{},[37,19549,19550],{},[43,19551,5619],{},[109,19553,5619],{"encoding":111},[15,19555,19557],{"className":19556,"ariaHidden":72},[116],[15,19558,19560,19563],{"className":19559},[120],[15,19561],{"className":19562,"style":610},[124],[15,19564,5619],{"className":19565,"style":5665},[129,133]," 取最大，就得到 VC 维。对一般的假设类，精确判定通常是组合优化问题，实践中只能靠随机采样加优化或可满足性求解来估计。",[575,19568,19570],{"id":19569},"_23-三个误区","2.3 三个误区",[11,19572,19573,19576],{},[698,19574,19575],{},"第一，「存在」不是「所有」。"," 定义只要求存在一个点集被打散。所以 VC 维大，不意味着假设类在任意数据上都很灵活；VC 维小，也不意味着它在某些特殊点集上不灵活。",[11,19578,19579,19582,19583,19586],{},[698,19580,19581],{},"第二，打散不是「能拟合这组数据」。"," 给定一组带标签的数据，几乎任何足够灵活的类都能拟合。打散要求的是：把这组点的标签",[698,19584,19585],{},"换着花样","全部试一遍，每一种都能实现。前者是插值能力，后者是组合丰富度。",[11,19588,19589,19592,19593,19621,19622,19650],{},[698,19590,19591],{},"第三，VC 维与参数化方式无关。"," 同一个函数集合可以有一万种参数化，VC 维不变。反过来，同一个参数数目可以对应 VC 维为 ",[15,19594,19596,19609],{"className":19595,"translate":19},[23],[15,19597,19599],{"className":19598},[27],[29,19600,19601],{"xmlns":31},[34,19602,19603,19607],{},[37,19604,19605],{},[308,19606,318],{},[109,19608,318],{"encoding":111},[15,19610,19612],{"className":19611,"ariaHidden":72},[116],[15,19613,19615,19618],{"className":19614},[120],[15,19616],{"className":19617,"style":4886},[124],[15,19619,318],{"className":19620},[129]," 的类，也可以对应 VC 维为 ",[15,19623,19625,19638],{"className":19624,"translate":19},[23],[15,19626,19628],{"className":19627},[27],[29,19629,19630],{"xmlns":31},[34,19631,19632,19636],{},[37,19633,19634],{},[43,19635,4909],{"mathvariant":65},[109,19637,18788],{"encoding":111},[15,19639,19641],{"className":19640,"ariaHidden":72},[116],[15,19642,19644,19647],{"className":19643},[120],[15,19645],{"className":19646,"style":4383},[124],[15,19648,4909],{"className":19649},[129]," 的类。参数只是描述函数的一种方式，容量是函数集合本身的性质。",[11,19652,19653,19654,19657],{},"还有一点值得提前说明：上面定义的是",[698,19655,19656],{},"二值","函数类的 VC 维。如果假设输出实数（回归、概率、间隔），就需要 pseudo-dimension 或 fat-shattering dimension，后文会再提。",[570,19659,19661],{"id":19660},"三增长函数与-sauershelah-引理","三、增长函数与 Sauer–Shelah 引理",[575,19663,19665],{"id":19664},"_31-增长函数","3.1 增长函数",[11,19667,19668],{},"逐个点集去数二分太麻烦。我们只关心最坏情况下能产生多少种二分：",[15,19670,19672],{"className":19671,"translate":19},[18],[15,19673,19675,19747],{"className":19674,"translate":19},[23],[15,19676,19678],{"className":19677},[27],[29,19679,19680],{"xmlns":31,"display":32},[34,19681,19682,19744],{},[37,19683,19684,19691,19693,19695,19697,19699,19730,19732,19734,19740,19742],{},[40,19685,19686,19689],{},[43,19687,19688],{"mathvariant":65},"Π",[43,19690,819],{"mathvariant":596},[50,19692,53],{"stretchy":52},[43,19694,348],{},[50,19696,59],{"stretchy":52},[50,19698,62],{},[2140,19700,19701,19708],{},[37,19702,19703,19706],{},[43,19704,19705],{},"max",[50,19707,69],{},[37,19709,19710,19712,19714,19716,19718,19720,19722,19724,19726,19728],{},[43,19711,5619],{},[50,19713,5647],{},[43,19715,597],{"mathvariant":596},[50,19717,91],{"separator":72},[6856,19719,6858],{},[43,19721,2945],{"mathvariant":65},[43,19723,5619],{},[43,19725,2945],{"mathvariant":65},[50,19727,62],{},[43,19729,348],{},[50,19731,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,19733,819],{"mathvariant":596},[40,19735,19736,19738],{},[43,19737,2945],{"mathvariant":65},[43,19739,5619],{},[50,19741,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,19743,107],{"mathvariant":65},[109,19745,19746],{"encoding":111},"\\Pi_{\\mathcal H}(m)=\\max_{C\\subseteq\\mathcal X,\\ |C|=m}\\bigl|\\mathcal H|_C\\bigr|.",[15,19748,19750,19817],{"className":19749,"ariaHidden":72},[116],[15,19751,19753,19756,19799,19802,19805,19808,19811,19814],{"className":19752},[120],[15,19754],{"className":19755,"style":125},[124],[15,19757,19759,19762],{"className":19758},[129],[15,19760,19688],{"className":19761},[129],[15,19763,19765],{"className":19764},[137],[15,19766,19768,19791],{"className":19767},[141,142],[15,19769,19771,19788],{"className":19770},[146],[15,19772,19774],{"className":19773,"style":1525},[150],[15,19775,19776,19779],{"style":154},[15,19777],{"className":19778,"style":159},[158],[15,19780,19782],{"className":19781},[163,164,165,166],[15,19783,19785],{"className":19784},[129,166],[15,19786,819],{"className":19787,"style":835},[129,614,166],[15,19789,175],{"className":19790},[174],[15,19792,19794],{"className":19793},[146],[15,19795,19797],{"className":19796,"style":182},[150],[15,19798],{},[15,19800,53],{"className":19801},[188],[15,19803,348],{"className":19804},[129,133],[15,19806,59],{"className":19807},[195],[15,19809],{"className":19810,"style":199},[93],[15,19812,62],{"className":19813},[203],[15,19815],{"className":19816,"style":199},[93],[15,19818,19820,19824,19903,19940,19943,19983,20020],{"className":19819},[120],[15,19821],{"className":19822,"style":19823},[124],"height:1.816em;vertical-align:-0.966em;",[15,19825,19827],{"className":19826},[217,1874],[15,19828,19830,19894],{"className":19829},[141,142],[15,19831,19833,19891],{"className":19832},[146],[15,19834,19836,19881],{"className":19835,"style":4383},[150],[15,19837,19839,19842],{"style":19838},"top:-2.309em;margin-left:0em;",[15,19840],{"className":19841,"style":1823},[158],[15,19843,19845],{"className":19844},[163,164,165,166],[15,19846,19848,19851,19854,19857,19860,19866,19869,19872,19875,19878],{"className":19847},[129,166],[15,19849,5619],{"className":19850,"style":5665},[129,133,166],[15,19852,5647],{"className":19853},[203,166],[15,19855,597],{"className":19856,"style":615},[129,614,166],[15,19858,91],{"className":19859},[254,166],[15,19861,19863],{"className":19862},[93,166],[15,19864,6858],{"className":19865},[166],[15,19867,2945],{"className":19868},[129,166],[15,19870,5619],{"className":19871,"style":5665},[129,133,166],[15,19873,2945],{"className":19874},[129,166],[15,19876,62],{"className":19877},[203,166],[15,19879,348],{"className":19880},[129,133,166],[15,19882,19883,19886],{"style":2287},[15,19884],{"className":19885,"style":1823},[158],[15,19887,19888],{},[15,19889,19705],{"className":19890},[217],[15,19892,175],{"className":19893},[174],[15,19895,19897],{"className":19896},[146],[15,19898,19901],{"className":19899,"style":19900},[150],"height:0.966em;",[15,19902],{},[15,19904,19906],{"className":19905},[188],[15,19907,19909],{"className":19908},[228,4433],[15,19910,19912,19932],{"className":19911},[141,142],[15,19913,19915,19929],{"className":19914},[146],[15,19916,19918],{"className":19917,"style":4443},[150],[15,19919,19920,19923],{"style":4446},[15,19921],{"className":19922,"style":4450},[158],[15,19924,19925],{"style":4453},[4455,19926,19927],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,19928],{"d":4463},[15,19930,175],{"className":19931},[174],[15,19933,19935],{"className":19934},[146],[15,19936,19938],{"className":19937,"style":4473},[150],[15,19939],{},[15,19941,819],{"className":19942,"style":835},[129,614],[15,19944,19946,19949],{"className":19945},[129],[15,19947,2945],{"className":19948},[129],[15,19950,19952],{"className":19951},[137],[15,19953,19955,19975],{"className":19954},[141,142],[15,19956,19958,19972],{"className":19957},[146],[15,19959,19961],{"className":19960,"style":1525},[150],[15,19962,19963,19966],{"style":154},[15,19964],{"className":19965,"style":159},[158],[15,19967,19969],{"className":19968},[163,164,165,166],[15,19970,5619],{"className":19971,"style":5665},[129,133,166],[15,19973,175],{"className":19974},[174],[15,19976,19978],{"className":19977},[146],[15,19979,19981],{"className":19980,"style":182},[150],[15,19982],{},[15,19984,19986],{"className":19985},[195],[15,19987,19989],{"className":19988},[228,4433],[15,19990,19992,20012],{"className":19991},[141,142],[15,19993,19995,20009],{"className":19994},[146],[15,19996,19998],{"className":19997,"style":4443},[150],[15,19999,20000,20003],{"style":4446},[15,20001],{"className":20002,"style":4450},[158],[15,20004,20005],{"style":4453},[4455,20006,20007],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,20008],{"d":4463},[15,20010,175],{"className":20011},[174],[15,20013,20015],{"className":20014},[146],[15,20016,20018],{"className":20017,"style":4473},[150],[15,20019],{},[15,20021,107],{"className":20022},[129],[11,20024,20025,20026,1345,20054,20057,20058,20198,20199,20273],{},"它称为 ",[15,20027,20029,20042],{"className":20028,"translate":19},[23],[15,20030,20032],{"className":20031},[27],[29,20033,20034],{"xmlns":31},[34,20035,20036,20040],{},[37,20037,20038],{},[43,20039,819],{"mathvariant":596},[109,20041,822],{"encoding":111},[15,20043,20045],{"className":20044,"ariaHidden":72},[116],[15,20046,20048,20051],{"className":20047},[120],[15,20049],{"className":20050,"style":610},[124],[15,20052,819],{"className":20053,"style":835},[129,614],[698,20055,20056],{},"增长函数","（growth function）。显然 ",[15,20059,20061,20093],{"className":20060,"translate":19},[23],[15,20062,20064],{"className":20063},[27],[29,20065,20066],{"xmlns":31},[34,20067,20068,20090],{},[37,20069,20070,20076,20078,20080,20082,20084],{},[40,20071,20072,20074],{},[43,20073,19688],{"mathvariant":65},[43,20075,819],{"mathvariant":596},[50,20077,53],{"stretchy":52},[43,20079,348],{},[50,20081,59],{"stretchy":52},[50,20083,2717],{},[305,20085,20086,20088],{},[308,20087,310],{},[43,20089,348],{},[109,20091,20092],{"encoding":111},"\\Pi_{\\mathcal H}(m)\\le 2^m",[15,20094,20096,20163],{"className":20095,"ariaHidden":72},[116],[15,20097,20099,20102,20145,20148,20151,20154,20157,20160],{"className":20098},[120],[15,20100],{"className":20101,"style":125},[124],[15,20103,20105,20108],{"className":20104},[129],[15,20106,19688],{"className":20107},[129],[15,20109,20111],{"className":20110},[137],[15,20112,20114,20137],{"className":20113},[141,142],[15,20115,20117,20134],{"className":20116},[146],[15,20118,20120],{"className":20119,"style":1525},[150],[15,20121,20122,20125],{"style":154},[15,20123],{"className":20124,"style":159},[158],[15,20126,20128],{"className":20127},[163,164,165,166],[15,20129,20131],{"className":20130},[129,166],[15,20132,819],{"className":20133,"style":835},[129,614,166],[15,20135,175],{"className":20136},[174],[15,20138,20140],{"className":20139},[146],[15,20141,20143],{"className":20142,"style":182},[150],[15,20144],{},[15,20146,53],{"className":20147},[188],[15,20149,348],{"className":20150},[129,133],[15,20152,59],{"className":20153},[195],[15,20155],{"className":20156,"style":199},[93],[15,20158,2717],{"className":20159},[203],[15,20161],{"className":20162,"style":199},[93],[15,20164,20166,20169],{"className":20165},[120],[15,20167],{"className":20168,"style":6377},[124],[15,20170,20172,20175],{"className":20171},[129],[15,20173,310],{"className":20174},[129],[15,20176,20178],{"className":20177},[137],[15,20179,20181],{"className":20180},[141],[15,20182,20184],{"className":20183},[146],[15,20185,20187],{"className":20186,"style":6377},[150],[15,20188,20189,20192],{"style":383},[15,20190],{"className":20191,"style":159},[158],[15,20193,20195],{"className":20194},[163,164,165,166],[15,20196,348],{"className":20197},[129,133,166],"，且 ",[15,20200,20202,20220],{"className":20201,"translate":19},[23],[15,20203,20205],{"className":20204},[27],[29,20206,20207],{"xmlns":31},[34,20208,20209,20217],{},[37,20210,20211],{},[40,20212,20213,20215],{},[43,20214,19688],{"mathvariant":65},[43,20216,819],{"mathvariant":596},[109,20218,20219],{"encoding":111},"\\Pi_{\\mathcal H}",[15,20221,20223],{"className":20222,"ariaHidden":72},[116],[15,20224,20226,20230],{"className":20225},[120],[15,20227],{"className":20228,"style":20229},[124],"height:0.8333em;vertical-align:-0.15em;",[15,20231,20233,20236],{"className":20232},[129],[15,20234,19688],{"className":20235},[129],[15,20237,20239],{"className":20238},[137],[15,20240,20242,20265],{"className":20241},[141,142],[15,20243,20245,20262],{"className":20244},[146],[15,20246,20248],{"className":20247,"style":1525},[150],[15,20249,20250,20253],{"style":154},[15,20251],{"className":20252,"style":159},[158],[15,20254,20256],{"className":20255},[163,164,165,166],[15,20257,20259],{"className":20258},[129,166],[15,20260,819],{"className":20261,"style":835},[129,614,166],[15,20263,175],{"className":20264},[174],[15,20266,20268],{"className":20267},[146],[15,20269,20271],{"className":20270,"style":182},[150],[15,20272],{}," 单调不减。",[11,20275,20276,20277,20326,20327,20467,20468,20496,20497,20525,20526,20554,20555,20583,20584,20612,20613,20709,20710,20761,20762,1313],{},"由定义，",[15,20278,20280,20302],{"className":20279,"translate":19},[23],[15,20281,20283],{"className":20282},[27],[29,20284,20285],{"xmlns":31},[34,20286,20287,20299],{},[37,20288,20289,20291,20293,20295,20297],{},[43,20290,6823],{"mathvariant":65},[50,20292,69],{},[50,20294,53],{"stretchy":52},[43,20296,819],{"mathvariant":596},[50,20298,59],{"stretchy":52},[109,20300,20301],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)",[15,20303,20305],{"className":20304,"ariaHidden":72},[116],[15,20306,20308,20311,20317,20320,20323],{"className":20307},[120],[15,20309],{"className":20310,"style":125},[124],[15,20312,20314],{"className":20313},[217],[15,20315,6823],{"className":20316},[129,221],[15,20318,53],{"className":20319},[188],[15,20321,819],{"className":20322,"style":835},[129,614],[15,20324,59],{"className":20325},[195]," 就是使 ",[15,20328,20330,20362],{"className":20329,"translate":19},[23],[15,20331,20333],{"className":20332},[27],[29,20334,20335],{"xmlns":31},[34,20336,20337,20359],{},[37,20338,20339,20345,20347,20349,20351,20353],{},[40,20340,20341,20343],{},[43,20342,19688],{"mathvariant":65},[43,20344,819],{"mathvariant":596},[50,20346,53],{"stretchy":52},[43,20348,348],{},[50,20350,59],{"stretchy":52},[50,20352,62],{},[305,20354,20355,20357],{},[308,20356,310],{},[43,20358,348],{},[109,20360,20361],{"encoding":111},"\\Pi_{\\mathcal H}(m)=2^m",[15,20363,20365,20432],{"className":20364,"ariaHidden":72},[116],[15,20366,20368,20371,20414,20417,20420,20423,20426,20429],{"className":20367},[120],[15,20369],{"className":20370,"style":125},[124],[15,20372,20374,20377],{"className":20373},[129],[15,20375,19688],{"className":20376},[129],[15,20378,20380],{"className":20379},[137],[15,20381,20383,20406],{"className":20382},[141,142],[15,20384,20386,20403],{"className":20385},[146],[15,20387,20389],{"className":20388,"style":1525},[150],[15,20390,20391,20394],{"style":154},[15,20392],{"className":20393,"style":159},[158],[15,20395,20397],{"className":20396},[163,164,165,166],[15,20398,20400],{"className":20399},[129,166],[15,20401,819],{"className":20402,"style":835},[129,614,166],[15,20404,175],{"className":20405},[174],[15,20407,20409],{"className":20408},[146],[15,20410,20412],{"className":20411,"style":182},[150],[15,20413],{},[15,20415,53],{"className":20416},[188],[15,20418,348],{"className":20419},[129,133],[15,20421,59],{"className":20422},[195],[15,20424],{"className":20425,"style":199},[93],[15,20427,62],{"className":20428},[203],[15,20430],{"className":20431,"style":199},[93],[15,20433,20435,20438],{"className":20434},[120],[15,20436],{"className":20437,"style":6377},[124],[15,20439,20441,20444],{"className":20440},[129],[15,20442,310],{"className":20443},[129],[15,20445,20447],{"className":20446},[137],[15,20448,20450],{"className":20449},[141],[15,20451,20453],{"className":20452},[146],[15,20454,20456],{"className":20455,"style":6377},[150],[15,20457,20458,20461],{"style":383},[15,20459],{"className":20460,"style":159},[158],[15,20462,20464],{"className":20463},[163,164,165,166],[15,20465,348],{"className":20466},[129,133,166]," 成立的最大的 ",[15,20469,20471,20484],{"className":20470,"translate":19},[23],[15,20472,20474],{"className":20473},[27],[29,20475,20476],{"xmlns":31},[34,20477,20478,20482],{},[37,20479,20480],{},[43,20481,348],{},[109,20483,348],{"encoding":111},[15,20485,20487],{"className":20486,"ariaHidden":72},[116],[15,20488,20490,20493],{"className":20489},[120],[15,20491],{"className":20492,"style":4383},[124],[15,20494,348],{"className":20495},[129,133],"（若对所有 ",[15,20498,20500,20513],{"className":20499,"translate":19},[23],[15,20501,20503],{"className":20502},[27],[29,20504,20505],{"xmlns":31},[34,20506,20507,20511],{},[37,20508,20509],{},[43,20510,348],{},[109,20512,348],{"encoding":111},[15,20514,20516],{"className":20515,"ariaHidden":72},[116],[15,20517,20519,20522],{"className":20518},[120],[15,20520],{"className":20521,"style":4383},[124],[15,20523,348],{"className":20524},[129,133]," 都成立，则为 ",[15,20527,20529,20542],{"className":20528,"translate":19},[23],[15,20530,20532],{"className":20531},[27],[29,20533,20534],{"xmlns":31},[34,20535,20536,20540],{},[37,20537,20538],{},[43,20539,4909],{"mathvariant":65},[109,20541,18788],{"encoding":111},[15,20543,20545],{"className":20544,"ariaHidden":72},[116],[15,20546,20548,20551],{"className":20547},[120],[15,20549],{"className":20550,"style":4383},[124],[15,20552,4909],{"className":20553},[129],"）。注意这个性质是「向下封闭」的：若 ",[15,20556,20558,20571],{"className":20557,"translate":19},[23],[15,20559,20561],{"className":20560},[27],[29,20562,20563],{"xmlns":31},[34,20564,20565,20569],{},[37,20566,20567],{},[43,20568,5619],{},[109,20570,5619],{"encoding":111},[15,20572,20574],{"className":20573,"ariaHidden":72},[116],[15,20575,20577,20580],{"className":20576},[120],[15,20578],{"className":20579,"style":610},[124],[15,20581,5619],{"className":20582,"style":5665},[129,133]," 被打散，",[15,20585,20587,20600],{"className":20586,"translate":19},[23],[15,20588,20590],{"className":20589},[27],[29,20591,20592],{"xmlns":31},[34,20593,20594,20598],{},[37,20595,20596],{},[43,20597,5619],{},[109,20599,5619],{"encoding":111},[15,20601,20603],{"className":20602,"ariaHidden":72},[116],[15,20604,20606,20609],{"className":20605},[120],[15,20607],{"className":20608,"style":610},[124],[15,20610,5619],{"className":20611,"style":5665},[129,133]," 的任何子集也被打散，所以一旦 ",[15,20614,20616,20644],{"className":20615,"translate":19},[23],[15,20617,20619],{"className":20618},[27],[29,20620,20621],{"xmlns":31},[34,20622,20623,20641],{},[37,20624,20625,20627,20629,20631,20633,20635],{},[43,20626,19688],{"mathvariant":65},[50,20628,53],{"stretchy":52},[43,20630,348],{},[50,20632,59],{"stretchy":52},[50,20634,62],{},[305,20636,20637,20639],{},[308,20638,310],{},[43,20640,348],{},[109,20642,20643],{"encoding":111},"\\Pi(m)=2^m",[15,20645,20647,20674],{"className":20646,"ariaHidden":72},[116],[15,20648,20650,20653,20656,20659,20662,20665,20668,20671],{"className":20649},[120],[15,20651],{"className":20652,"style":125},[124],[15,20654,19688],{"className":20655},[129],[15,20657,53],{"className":20658},[188],[15,20660,348],{"className":20661},[129,133],[15,20663,59],{"className":20664},[195],[15,20666],{"className":20667,"style":199},[93],[15,20669,62],{"className":20670},[203],[15,20672],{"className":20673,"style":199},[93],[15,20675,20677,20680],{"className":20676},[120],[15,20678],{"className":20679,"style":6377},[124],[15,20681,20683,20686],{"className":20682},[129],[15,20684,310],{"className":20685},[129],[15,20687,20689],{"className":20688},[137],[15,20690,20692],{"className":20691},[141],[15,20693,20695],{"className":20694},[146],[15,20696,20698],{"className":20697,"style":6377},[150],[15,20699,20700,20703],{"style":383},[15,20701],{"className":20702,"style":159},[158],[15,20704,20706],{"className":20705},[163,164,165,166],[15,20707,348],{"className":20708},[129,133,166],"，对所有 ",[15,20711,20713,20731],{"className":20712,"translate":19},[23],[15,20714,20716],{"className":20715},[27],[29,20717,20718],{"xmlns":31},[34,20719,20720,20728],{},[37,20721,20722,20724,20726],{},[43,20723,16562],{},[50,20725,2717],{},[43,20727,348],{},[109,20729,20730],{"encoding":111},"k\\le m",[15,20732,20734,20752],{"className":20733,"ariaHidden":72},[116],[15,20735,20737,20740,20743,20746,20749],{"className":20736},[120],[15,20738],{"className":20739,"style":6966},[124],[15,20741,16562],{"className":20742,"style":16595},[129,133],[15,20744],{"className":20745,"style":199},[93],[15,20747,2717],{"className":20748},[203],[15,20750],{"className":20751,"style":199},[93],[15,20753,20755,20758],{"className":20754},[120],[15,20756],{"className":20757,"style":4383},[124],[15,20759,348],{"className":20760},[129,133]," 都有 ",[15,20763,20765,20793],{"className":20764,"translate":19},[23],[15,20766,20768],{"className":20767},[27],[29,20769,20770],{"xmlns":31},[34,20771,20772,20790],{},[37,20773,20774,20776,20778,20780,20782,20784],{},[43,20775,19688],{"mathvariant":65},[50,20777,53],{"stretchy":52},[43,20779,16562],{},[50,20781,59],{"stretchy":52},[50,20783,62],{},[305,20785,20786,20788],{},[308,20787,310],{},[43,20789,16562],{},[109,20791,20792],{"encoding":111},"\\Pi(k)=2^k",[15,20794,20796,20823],{"className":20795,"ariaHidden":72},[116],[15,20797,20799,20802,20805,20808,20811,20814,20817,20820],{"className":20798},[120],[15,20800],{"className":20801,"style":125},[124],[15,20803,19688],{"className":20804},[129],[15,20806,53],{"className":20807},[188],[15,20809,16562],{"className":20810,"style":16595},[129,133],[15,20812,59],{"className":20813},[195],[15,20815],{"className":20816,"style":199},[93],[15,20818,62],{"className":20819},[203],[15,20821],{"className":20822,"style":199},[93],[15,20824,20826,20829],{"className":20825},[120],[15,20827],{"className":20828,"style":10490},[124],[15,20830,20832,20835],{"className":20831},[129],[15,20833,310],{"className":20834},[129],[15,20836,20838],{"className":20837},[137],[15,20839,20841],{"className":20840},[141],[15,20842,20844],{"className":20843},[146],[15,20845,20847],{"className":20846,"style":10490},[150],[15,20848,20849,20852],{"style":383},[15,20850],{"className":20851,"style":159},[158],[15,20853,20855],{"className":20854},[163,164,165,166],[15,20856,16562],{"className":20857,"style":16595},[129,133,166],[575,20859,20861],{"id":20860},"_32-sauershelah-引理","3.2 Sauer–Shelah 引理",[6559,20863,20864,20991],{},[11,20865,20866,6566,20869,20962,20963,7610],{},[698,20867,20868],{},"定理 3.1（Sauer–Shelah）",[15,20870,20872,20902],{"className":20871,"translate":19},[23],[15,20873,20875],{"className":20874},[27],[29,20876,20877],{"xmlns":31},[34,20878,20879,20899],{},[37,20880,20881,20883,20885,20887,20889,20891,20893,20895,20897],{},[43,20882,6823],{"mathvariant":65},[50,20884,69],{},[50,20886,53],{"stretchy":52},[43,20888,819],{"mathvariant":596},[50,20890,59],{"stretchy":52},[50,20892,62],{},[43,20894,10376],{},[50,20896,7910],{},[43,20898,4909],{"mathvariant":65},[109,20900,20901],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)=d\u003C\\infty",[15,20903,20905,20935,20953],{"className":20904,"ariaHidden":72},[116],[15,20906,20908,20911,20917,20920,20923,20926,20929,20932],{"className":20907},[120],[15,20909],{"className":20910,"style":125},[124],[15,20912,20914],{"className":20913},[217],[15,20915,6823],{"className":20916},[129,221],[15,20918,53],{"className":20919},[188],[15,20921,819],{"className":20922,"style":835},[129,614],[15,20924,59],{"className":20925},[195],[15,20927],{"className":20928,"style":199},[93],[15,20930,62],{"className":20931},[203],[15,20933],{"className":20934,"style":199},[93],[15,20936,20938,20941,20944,20947,20950],{"className":20937},[120],[15,20939],{"className":20940,"style":2405},[124],[15,20942,10376],{"className":20943},[129,133],[15,20945],{"className":20946,"style":199},[93],[15,20948,7910],{"className":20949},[203],[15,20951],{"className":20952,"style":199},[93],[15,20954,20956,20959],{"className":20955},[120],[15,20957],{"className":20958,"style":4383},[124],[15,20960,4909],{"className":20961},[129],"，则对任意 ",[15,20964,20966,20979],{"className":20965,"translate":19},[23],[15,20967,20969],{"className":20968},[27],[29,20970,20971],{"xmlns":31},[34,20972,20973,20977],{},[37,20974,20975],{},[43,20976,348],{},[109,20978,348],{"encoding":111},[15,20980,20982],{"className":20981,"ariaHidden":72},[116],[15,20983,20985,20988],{"className":20984},[120],[15,20986],{"className":20987,"style":4383},[124],[15,20989,348],{"className":20990},[129,133],[15,20992,20994],{"className":20993,"translate":19},[18],[15,20995,20997,21052],{"className":20996,"translate":19},[23],[15,20998,21000],{"className":20999},[27],[29,21001,21002],{"xmlns":31,"display":32},[34,21003,21004,21049],{},[37,21005,21006,21012,21014,21016,21018,21020,21034,21047],{},[40,21007,21008,21010],{},[43,21009,19688],{"mathvariant":65},[43,21011,819],{"mathvariant":596},[50,21013,53],{"stretchy":52},[43,21015,348],{},[50,21017,59],{"stretchy":52},[50,21019,2717],{},[1450,21021,21022,21024,21032],{},[50,21023,1454],{},[37,21025,21026,21028,21030],{},[43,21027,1459],{},[50,21029,62],{},[308,21031,541],{},[43,21033,10376],{},[37,21035,21036,21038,21045],{},[50,21037,53],{"fence":72},[1443,21039,21041,21043],{"linethickness":21040},"0px",[43,21042,348],{},[43,21044,1459],{},[50,21046,59],{"fence":72},[43,21048,107],{"mathvariant":65},[109,21050,21051],{"encoding":111},"\\Pi_{\\mathcal H}(m)\\le\\sum_{i=0}^{d}\\binom{m}{i}.",[15,21053,21055,21122],{"className":21054,"ariaHidden":72},[116],[15,21056,21058,21061,21104,21107,21110,21113,21116,21119],{"className":21057},[120],[15,21059],{"className":21060,"style":125},[124],[15,21062,21064,21067],{"className":21063},[129],[15,21065,19688],{"className":21066},[129],[15,21068,21070],{"className":21069},[137],[15,21071,21073,21096],{"className":21072},[141,142],[15,21074,21076,21093],{"className":21075},[146],[15,21077,21079],{"className":21078,"style":1525},[150],[15,21080,21081,21084],{"style":154},[15,21082],{"className":21083,"style":159},[158],[15,21085,21087],{"className":21086},[163,164,165,166],[15,21088,21090],{"className":21089},[129,166],[15,21091,819],{"className":21092,"style":835},[129,614,166],[15,21094,175],{"className":21095},[174],[15,21097,21099],{"className":21098},[146],[15,21100,21102],{"className":21101,"style":182},[150],[15,21103],{},[15,21105,53],{"className":21106},[188],[15,21108,348],{"className":21109},[129,133],[15,21111,59],{"className":21112},[195],[15,21114],{"className":21115,"style":199},[93],[15,21117,2717],{"className":21118},[203],[15,21120],{"className":21121,"style":199},[93],[15,21123,21125,21129,21197,21200,21261],{"className":21124},[120],[15,21126],{"className":21127,"style":21128},[124],"height:3.1138em;vertical-align:-1.2777em;",[15,21130,21132],{"className":21131},[217,1874],[15,21133,21135,21189],{"className":21134},[141,142],[15,21136,21138,21186],{"className":21137},[146],[15,21139,21142,21162,21172],{"className":21140,"style":21141},[150],"height:1.8361em;",[15,21143,21144,21147],{"style":1887},[15,21145],{"className":21146,"style":1891},[158],[15,21148,21150],{"className":21149},[163,164,165,166],[15,21151,21153,21156,21159],{"className":21152},[129,166],[15,21154,1459],{"className":21155},[129,133,166],[15,21157,62],{"className":21158},[203,166],[15,21160,541],{"className":21161},[129,166],[15,21163,21164,21167],{"style":1909},[15,21165],{"className":21166,"style":1891},[158],[15,21168,21169],{},[15,21170,1454],{"className":21171},[217,1918,1919],[15,21173,21174,21177],{"style":1922},[15,21175],{"className":21176,"style":1891},[158],[15,21178,21180],{"className":21179},[163,164,165,166],[15,21181,21183],{"className":21182},[129,166],[15,21184,10376],{"className":21185},[129,133,166],[15,21187,175],{"className":21188},[174],[15,21190,21192],{"className":21191},[146],[15,21193,21195],{"className":21194,"style":1941},[150],[15,21196],{},[15,21198],{"className":21199,"style":262},[93],[15,21201,21203,21209,21255],{"className":21202},[129],[15,21204,21206],{"className":21205,"style":5052},[188,5051],[15,21207,53],{"className":21208},[228,165],[15,21210,21212],{"className":21211},[1443],[15,21213,21215,21247],{"className":21214},[141,142],[15,21216,21218,21244],{"className":21217},[146],[15,21219,21222,21233],{"className":21220,"style":21221},[150],"height:1.1076em;",[15,21223,21224,21227],{"style":1819},[15,21225],{"className":21226,"style":1823},[158],[15,21228,21230],{"className":21229},[129],[15,21231,1459],{"className":21232},[129,133],[15,21234,21235,21238],{"style":1843},[15,21236],{"className":21237,"style":1823},[158],[15,21239,21241],{"className":21240},[129],[15,21242,348],{"className":21243},[129,133],[15,21245,175],{"className":21246},[174],[15,21248,21250],{"className":21249},[146],[15,21251,21253],{"className":21252,"style":1862},[150],[15,21254],{},[15,21256,21258],{"className":21257,"style":5052},[195,5051],[15,21259,59],{"className":21260},[228,165],[15,21262,107],{"className":21263},[129],[11,21265,21266,21267,21318,21319,21491,21492,21652,21653,21728],{},"证明用归纳法，对 ",[15,21268,21270,21288],{"className":21269,"translate":19},[23],[15,21271,21273],{"className":21272},[27],[29,21274,21275],{"xmlns":31},[34,21276,21277,21285],{},[37,21278,21279,21281,21283],{},[43,21280,348],{},[50,21282,2682],{},[43,21284,10376],{},[109,21286,21287],{"encoding":111},"m+d",[15,21289,21291,21309],{"className":21290,"ariaHidden":72},[116],[15,21292,21294,21297,21300,21303,21306],{"className":21293},[120],[15,21295],{"className":21296,"style":8518},[124],[15,21298,348],{"className":21299},[129,133],[15,21301],{"className":21302,"style":3277},[93],[15,21304,2682],{"className":21305},[2436],[15,21307],{"className":21308,"style":3277},[93],[15,21310,21312,21315],{"className":21311},[120],[15,21313],{"className":21314,"style":522},[124],[15,21316,10376],{"className":21317},[129,133]," 归纳。取任意 ",[15,21320,21322,21360],{"className":21321,"translate":19},[23],[15,21323,21325],{"className":21324},[27],[29,21326,21327],{"xmlns":31},[34,21328,21329,21357],{},[37,21330,21331,21333,21335,21337,21343,21345,21347,21349,21355],{},[43,21332,5619],{},[50,21334,62],{},[50,21336,636],{"stretchy":52},[40,21338,21339,21341],{},[43,21340,5628],{},[308,21342,318],{},[50,21344,91],{"separator":72},[50,21346,335],{},[50,21348,91],{"separator":72},[40,21350,21351,21353],{},[43,21352,5628],{},[43,21354,348],{},[50,21356,645],{"stretchy":52},[109,21358,21359],{"encoding":111},"C=\\{c_1,\\dots,c_m\\}",[15,21361,21363,21381],{"className":21362,"ariaHidden":72},[116],[15,21364,21366,21369,21372,21375,21378],{"className":21365},[120],[15,21367],{"className":21368,"style":610},[124],[15,21370,5619],{"className":21371,"style":5665},[129,133],[15,21373],{"className":21374,"style":199},[93],[15,21376,62],{"className":21377},[203],[15,21379],{"className":21380,"style":199},[93],[15,21382,21384,21387,21390,21430,21433,21436,21439,21442,21445,21448,21488],{"className":21383},[120],[15,21385],{"className":21386,"style":125},[124],[15,21388,636],{"className":21389},[188],[15,21391,21393,21396],{"className":21392},[129],[15,21394,5628],{"className":21395},[129,133],[15,21397,21399],{"className":21398},[137],[15,21400,21402,21422],{"className":21401},[141,142],[15,21403,21405,21419],{"className":21404},[146],[15,21406,21408],{"className":21407,"style":1051},[150],[15,21409,21410,21413],{"style":154},[15,21411],{"className":21412,"style":159},[158],[15,21414,21416],{"className":21415},[163,164,165,166],[15,21417,318],{"className":21418},[129,166],[15,21420,175],{"className":21421},[174],[15,21423,21425],{"className":21424},[146],[15,21426,21428],{"className":21427,"style":182},[150],[15,21429],{},[15,21431,91],{"className":21432},[254],[15,21434],{"className":21435,"style":262},[93],[15,21437,335],{"className":21438},[449],[15,21440],{"className":21441,"style":262},[93],[15,21443,91],{"className":21444},[254],[15,21446],{"className":21447,"style":262},[93],[15,21449,21451,21454],{"className":21450},[129],[15,21452,5628],{"className":21453},[129,133],[15,21455,21457],{"className":21456},[137],[15,21458,21460,21480],{"className":21459},[141,142],[15,21461,21463,21477],{"className":21462},[146],[15,21464,21466],{"className":21465,"style":1163},[150],[15,21467,21468,21471],{"style":154},[15,21469],{"className":21470,"style":159},[158],[15,21472,21474],{"className":21473},[163,164,165,166],[15,21475,348],{"className":21476},[129,133,166],[15,21478,175],{"className":21479},[174],[15,21481,21483],{"className":21482},[146],[15,21484,21486],{"className":21485,"style":182},[150],[15,21487],{},[15,21489,645],{"className":21490},[195],"，记 ",[15,21493,21495,21531],{"className":21494,"translate":19},[23],[15,21496,21498],{"className":21497},[27],[29,21499,21500],{"xmlns":31},[34,21501,21502,21528],{},[37,21503,21504,21511,21513,21515,21518,21520,21526],{},[305,21505,21506,21508],{},[43,21507,5619],{},[50,21509,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},"′",[50,21512,62],{},[43,21514,5619],{},[50,21516,21517],{},"∖",[50,21519,636],{"stretchy":52},[40,21521,21522,21524],{},[43,21523,5628],{},[43,21525,348],{},[50,21527,645],{"stretchy":52},[109,21529,21530],{"encoding":111},"C'=C\\setminus\\{c_m\\}",[15,21532,21534,21582,21600],{"className":21533,"ariaHidden":72},[116],[15,21535,21537,21541,21573,21576,21579],{"className":21536},[120],[15,21538],{"className":21539,"style":21540},[124],"height:0.7519em;",[15,21542,21544,21547],{"className":21543},[129],[15,21545,5619],{"className":21546,"style":5665},[129,133],[15,21548,21550],{"className":21549},[137],[15,21551,21553],{"className":21552},[141],[15,21554,21556],{"className":21555},[146],[15,21557,21559],{"className":21558,"style":21540},[150],[15,21560,21561,21564],{"style":383},[15,21562],{"className":21563,"style":159},[158],[15,21565,21567],{"className":21566},[163,164,165,166],[15,21568,21570],{"className":21569},[129,166],[15,21571,21510],{"className":21572},[129,166],[15,21574],{"className":21575,"style":199},[93],[15,21577,62],{"className":21578},[203],[15,21580],{"className":21581,"style":199},[93],[15,21583,21585,21588,21591,21594,21597],{"className":21584},[120],[15,21586],{"className":21587,"style":125},[124],[15,21589,5619],{"className":21590,"style":5665},[129,133],[15,21592],{"className":21593,"style":3277},[93],[15,21595,21517],{"className":21596},[2436],[15,21598],{"className":21599,"style":3277},[93],[15,21601,21603,21606,21609,21649],{"className":21602},[120],[15,21604],{"className":21605,"style":125},[124],[15,21607,636],{"className":21608},[188],[15,21610,21612,21615],{"className":21611},[129],[15,21613,5628],{"className":21614},[129,133],[15,21616,21618],{"className":21617},[137],[15,21619,21621,21641],{"className":21620},[141,142],[15,21622,21624,21638],{"className":21623},[146],[15,21625,21627],{"className":21626,"style":1163},[150],[15,21628,21629,21632],{"style":154},[15,21630],{"className":21631,"style":159},[158],[15,21633,21635],{"className":21634},[163,164,165,166],[15,21636,348],{"className":21637},[129,133,166],[15,21639,175],{"className":21640},[174],[15,21642,21644],{"className":21643},[146],[15,21645,21647],{"className":21646,"style":182},[150],[15,21648],{},[15,21650,645],{"className":21651},[195],"。把 ",[15,21654,21656,21676],{"className":21655,"translate":19},[23],[15,21657,21659],{"className":21658},[27],[29,21660,21661],{"xmlns":31},[34,21662,21663,21673],{},[37,21664,21665,21667],{},[43,21666,819],{"mathvariant":596},[40,21668,21669,21671],{},[43,21670,2945],{"mathvariant":65},[43,21672,5619],{},[109,21674,21675],{"encoding":111},"\\mathcal H|_C",[15,21677,21679],{"className":21678,"ariaHidden":72},[116],[15,21680,21682,21685,21688],{"className":21681},[120],[15,21683],{"className":21684,"style":125},[124],[15,21686,819],{"className":21687,"style":835},[129,614],[15,21689,21691,21694],{"className":21690},[129],[15,21692,2945],{"className":21693},[129],[15,21695,21697],{"className":21696},[137],[15,21698,21700,21720],{"className":21699},[141,142],[15,21701,21703,21717],{"className":21702},[146],[15,21704,21706],{"className":21705,"style":1525},[150],[15,21707,21708,21711],{"style":154},[15,21709],{"className":21710,"style":159},[158],[15,21712,21714],{"className":21713},[163,164,165,166],[15,21715,5619],{"className":21716,"style":5665},[129,133,166],[15,21718,175],{"className":21719},[174],[15,21721,21723],{"className":21722},[146],[15,21724,21726],{"className":21725,"style":182},[150],[15,21727],{}," 按最后一个坐标分成两类：",[21730,21731,21732,21835],"ul",{},[21733,21734,21735,21805,21806,21834],"li",{},[15,21736,21738,21756],{"className":21737,"translate":19},[23],[15,21739,21741],{"className":21740},[27],[29,21742,21743],{"xmlns":31},[34,21744,21745,21753],{},[37,21746,21747],{},[40,21748,21749,21751],{},[43,21750,13097],{},[308,21752,541],{},[109,21754,21755],{"encoding":111},"A_0",[15,21757,21759],{"className":21758,"ariaHidden":72},[116],[15,21760,21762,21765],{"className":21761},[120],[15,21763],{"className":21764,"style":20229},[124],[15,21766,21768,21771],{"className":21767},[129],[15,21769,13097],{"className":21770},[129,133],[15,21772,21774],{"className":21773},[137],[15,21775,21777,21797],{"className":21776},[141,142],[15,21778,21780,21794],{"className":21779},[146],[15,21781,21783],{"className":21782,"style":1051},[150],[15,21784,21785,21788],{"style":154},[15,21786],{"className":21787,"style":159},[158],[15,21789,21791],{"className":21790},[163,164,165,166],[15,21792,541],{"className":21793},[129,166],[15,21795,175],{"className":21796},[174],[15,21798,21800],{"className":21799},[146],[15,21801,21803],{"className":21802,"style":182},[150],[15,21804],{},"：最后一个坐标为 ",[15,21807,21809,21822],{"className":21808,"translate":19},[23],[15,21810,21812],{"className":21811},[27],[29,21813,21814],{"xmlns":31},[34,21815,21816,21820],{},[37,21817,21818],{},[308,21819,541],{},[109,21821,541],{"encoding":111},[15,21823,21825],{"className":21824,"ariaHidden":72},[116],[15,21826,21828,21831],{"className":21827},[120],[15,21829],{"className":21830,"style":4886},[124],[15,21832,541],{"className":21833},[129]," 的二分；",[21733,21836,21837,21805,21907,21935],{},[15,21838,21840,21858],{"className":21839,"translate":19},[23],[15,21841,21843],{"className":21842},[27],[29,21844,21845],{"xmlns":31},[34,21846,21847,21855],{},[37,21848,21849],{},[40,21850,21851,21853],{},[43,21852,13097],{},[308,21854,318],{},[109,21856,21857],{"encoding":111},"A_1",[15,21859,21861],{"className":21860,"ariaHidden":72},[116],[15,21862,21864,21867],{"className":21863},[120],[15,21865],{"className":21866,"style":20229},[124],[15,21868,21870,21873],{"className":21869},[129],[15,21871,13097],{"className":21872},[129,133],[15,21874,21876],{"className":21875},[137],[15,21877,21879,21899],{"className":21878},[141,142],[15,21880,21882,21896],{"className":21881},[146],[15,21883,21885],{"className":21884,"style":1051},[150],[15,21886,21887,21890],{"style":154},[15,21888],{"className":21889,"style":159},[158],[15,21891,21893],{"className":21892},[163,164,165,166],[15,21894,318],{"className":21895},[129,166],[15,21897,175],{"className":21898},[174],[15,21900,21902],{"className":21901},[146],[15,21903,21905],{"className":21904,"style":182},[150],[15,21906],{},[15,21908,21910,21923],{"className":21909,"translate":19},[23],[15,21911,21913],{"className":21912},[27],[29,21914,21915],{"xmlns":31},[34,21916,21917,21921],{},[37,21918,21919],{},[308,21920,318],{},[109,21922,318],{"encoding":111},[15,21924,21926],{"className":21925,"ariaHidden":72},[116],[15,21927,21929,21932],{"className":21928},[120],[15,21930],{"className":21931,"style":4886},[124],[15,21933,318],{"className":21934},[129]," 的二分。",[11,21937,21938,21939,10179,22008,5831,22077,22139,22140,22169],{},"设 ",[15,21940,21942,21959],{"className":21941,"translate":19},[23],[15,21943,21945],{"className":21944},[27],[29,21946,21947],{"xmlns":31},[34,21948,21949,21957],{},[37,21950,21951],{},[40,21952,21953,21955],{},[43,21954,13097],{},[308,21956,541],{},[109,21958,21755],{"encoding":111},[15,21960,21962],{"className":21961,"ariaHidden":72},[116],[15,21963,21965,21968],{"className":21964},[120],[15,21966],{"className":21967,"style":20229},[124],[15,21969,21971,21974],{"className":21970},[129],[15,21972,13097],{"className":21973},[129,133],[15,21975,21977],{"className":21976},[137],[15,21978,21980,22000],{"className":21979},[141,142],[15,21981,21983,21997],{"className":21982},[146],[15,21984,21986],{"className":21985,"style":1051},[150],[15,21987,21988,21991],{"style":154},[15,21989],{"className":21990,"style":159},[158],[15,21992,21994],{"className":21993},[163,164,165,166],[15,21995,541],{"className":21996},[129,166],[15,21998,175],{"className":21999},[174],[15,22001,22003],{"className":22002},[146],[15,22004,22006],{"className":22005,"style":182},[150],[15,22007],{},[15,22009,22011,22028],{"className":22010,"translate":19},[23],[15,22012,22014],{"className":22013},[27],[29,22015,22016],{"xmlns":31},[34,22017,22018,22026],{},[37,22019,22020],{},[40,22021,22022,22024],{},[43,22023,13097],{},[308,22025,318],{},[109,22027,21857],{"encoding":111},[15,22029,22031],{"className":22030,"ariaHidden":72},[116],[15,22032,22034,22037],{"className":22033},[120],[15,22035],{"className":22036,"style":20229},[124],[15,22038,22040,22043],{"className":22039},[129],[15,22041,13097],{"className":22042},[129,133],[15,22044,22046],{"className":22045},[137],[15,22047,22049,22069],{"className":22048},[141,142],[15,22050,22052,22066],{"className":22051},[146],[15,22053,22055],{"className":22054,"style":1051},[150],[15,22056,22057,22060],{"style":154},[15,22058],{"className":22059,"style":159},[158],[15,22061,22063],{"className":22062},[163,164,165,166],[15,22064,318],{"className":22065},[129,166],[15,22067,175],{"className":22068},[174],[15,22070,22072],{"className":22071},[146],[15,22073,22075],{"className":22074,"style":182},[150],[15,22076],{},[15,22078,22080,22098],{"className":22079,"translate":19},[23],[15,22081,22083],{"className":22082},[27],[29,22084,22085],{"xmlns":31},[34,22086,22087,22095],{},[37,22088,22089],{},[305,22090,22091,22093],{},[43,22092,5619],{},[50,22094,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,22096,22097],{"encoding":111},"C'",[15,22099,22101],{"className":22100,"ariaHidden":72},[116],[15,22102,22104,22107],{"className":22103},[120],[15,22105],{"className":22106,"style":21540},[124],[15,22108,22110,22113],{"className":22109},[129],[15,22111,5619],{"className":22112,"style":5665},[129,133],[15,22114,22116],{"className":22115},[137],[15,22117,22119],{"className":22118},[141],[15,22120,22122],{"className":22121},[146],[15,22123,22125],{"className":22124,"style":21540},[150],[15,22126,22127,22130],{"style":383},[15,22128],{"className":22129,"style":159},[158],[15,22131,22133],{"className":22132},[163,164,165,166],[15,22134,22136],{"className":22135},[129,166],[15,22137,21510],{"className":22138},[129,166]," 上的投影的交集大小为 ",[15,22141,22143,22157],{"className":22142,"translate":19},[23],[15,22144,22146],{"className":22145},[27],[29,22147,22148],{"xmlns":31},[34,22149,22150,22155],{},[37,22151,22152],{},[43,22153,22154],{},"s",[109,22156,22154],{"encoding":111},[15,22158,22160],{"className":22159,"ariaHidden":72},[116],[15,22161,22163,22166],{"className":22162},[120],[15,22164],{"className":22165,"style":4383},[124],[15,22167,22154],{"className":22168},[129,133],"，则",[15,22171,22173],{"className":22172,"translate":19},[18],[15,22174,22176,22248],{"className":22175,"translate":19},[23],[15,22177,22179],{"className":22178},[27],[29,22180,22181],{"xmlns":31,"display":32},[34,22182,22183,22245],{},[37,22184,22185,22187,22189,22195,22197,22199,22201,22207,22209,22211,22213,22219,22221,22223,22229,22231,22233,22235,22237,22239,22241,22243],{},[43,22186,2945],{"mathvariant":65},[43,22188,819],{"mathvariant":596},[40,22190,22191,22193],{},[43,22192,2945],{"mathvariant":65},[43,22194,5619],{},[43,22196,2945],{"mathvariant":65},[50,22198,62],{},[43,22200,2945],{"mathvariant":65},[40,22202,22203,22205],{},[43,22204,13097],{},[308,22206,541],{},[43,22208,2945],{"mathvariant":65},[50,22210,2682],{},[43,22212,2945],{"mathvariant":65},[40,22214,22215,22217],{},[43,22216,13097],{},[308,22218,318],{},[43,22220,2945],{"mathvariant":65},[50,22222,2717],{},[40,22224,22225,22227],{},[43,22226,19688],{"mathvariant":65},[43,22228,819],{"mathvariant":596},[50,22230,53],{"stretchy":52},[43,22232,348],{},[50,22234,315],{},[308,22236,318],{},[50,22238,59],{"stretchy":52},[50,22240,2682],{},[43,22242,22154],{},[43,22244,107],{"mathvariant":65},[109,22246,22247],{"encoding":111},"|\\mathcal H|_C|=|A_0|+|A_1|\\le \\Pi_{\\mathcal H}(m-1)+s.",[15,22249,22251,22315,22376,22437,22501,22522],{"className":22250,"ariaHidden":72},[116],[15,22252,22254,22257,22260,22263,22303,22306,22309,22312],{"className":22253},[120],[15,22255],{"className":22256,"style":125},[124],[15,22258,2945],{"className":22259},[129],[15,22261,819],{"className":22262,"style":835},[129,614],[15,22264,22266,22269],{"className":22265},[129],[15,22267,2945],{"className":22268},[129],[15,22270,22272],{"className":22271},[137],[15,22273,22275,22295],{"className":22274},[141,142],[15,22276,22278,22292],{"className":22277},[146],[15,22279,22281],{"className":22280,"style":1525},[150],[15,22282,22283,22286],{"style":154},[15,22284],{"className":22285,"style":159},[158],[15,22287,22289],{"className":22288},[163,164,165,166],[15,22290,5619],{"className":22291,"style":5665},[129,133,166],[15,22293,175],{"className":22294},[174],[15,22296,22298],{"className":22297},[146],[15,22299,22301],{"className":22300,"style":182},[150],[15,22302],{},[15,22304,2945],{"className":22305},[129],[15,22307],{"className":22308,"style":199},[93],[15,22310,62],{"className":22311},[203],[15,22313],{"className":22314,"style":199},[93],[15,22316,22318,22321,22324,22364,22367,22370,22373],{"className":22317},[120],[15,22319],{"className":22320,"style":125},[124],[15,22322,2945],{"className":22323},[129],[15,22325,22327,22330],{"className":22326},[129],[15,22328,13097],{"className":22329},[129,133],[15,22331,22333],{"className":22332},[137],[15,22334,22336,22356],{"className":22335},[141,142],[15,22337,22339,22353],{"className":22338},[146],[15,22340,22342],{"className":22341,"style":1051},[150],[15,22343,22344,22347],{"style":154},[15,22345],{"className":22346,"style":159},[158],[15,22348,22350],{"className":22349},[163,164,165,166],[15,22351,541],{"className":22352},[129,166],[15,22354,175],{"className":22355},[174],[15,22357,22359],{"className":22358},[146],[15,22360,22362],{"className":22361,"style":182},[150],[15,22363],{},[15,22365,2945],{"className":22366},[129],[15,22368],{"className":22369,"style":3277},[93],[15,22371,2682],{"className":22372},[2436],[15,22374],{"className":22375,"style":3277},[93],[15,22377,22379,22382,22385,22425,22428,22431,22434],{"className":22378},[120],[15,22380],{"className":22381,"style":125},[124],[15,22383,2945],{"className":22384},[129],[15,22386,22388,22391],{"className":22387},[129],[15,22389,13097],{"className":22390},[129,133],[15,22392,22394],{"className":22393},[137],[15,22395,22397,22417],{"className":22396},[141,142],[15,22398,22400,22414],{"className":22399},[146],[15,22401,22403],{"className":22402,"style":1051},[150],[15,22404,22405,22408],{"style":154},[15,22406],{"className":22407,"style":159},[158],[15,22409,22411],{"className":22410},[163,164,165,166],[15,22412,318],{"className":22413},[129,166],[15,22415,175],{"className":22416},[174],[15,22418,22420],{"className":22419},[146],[15,22421,22423],{"className":22422,"style":182},[150],[15,22424],{},[15,22426,2945],{"className":22427},[129],[15,22429],{"className":22430,"style":199},[93],[15,22432,2717],{"className":22433},[203],[15,22435],{"className":22436,"style":199},[93],[15,22438,22440,22443,22486,22489,22492,22495,22498],{"className":22439},[120],[15,22441],{"className":22442,"style":125},[124],[15,22444,22446,22449],{"className":22445},[129],[15,22447,19688],{"className":22448},[129],[15,22450,22452],{"className":22451},[137],[15,22453,22455,22478],{"className":22454},[141,142],[15,22456,22458,22475],{"className":22457},[146],[15,22459,22461],{"className":22460,"style":1525},[150],[15,22462,22463,22466],{"style":154},[15,22464],{"className":22465,"style":159},[158],[15,22467,22469],{"className":22468},[163,164,165,166],[15,22470,22472],{"className":22471},[129,166],[15,22473,819],{"className":22474,"style":835},[129,614,166],[15,22476,175],{"className":22477},[174],[15,22479,22481],{"className":22480},[146],[15,22482,22484],{"className":22483,"style":182},[150],[15,22485],{},[15,22487,53],{"className":22488},[188],[15,22490,348],{"className":22491},[129,133],[15,22493],{"className":22494,"style":3277},[93],[15,22496,315],{"className":22497},[2436],[15,22499],{"className":22500,"style":3277},[93],[15,22502,22504,22507,22510,22513,22516,22519],{"className":22503},[120],[15,22505],{"className":22506,"style":125},[124],[15,22508,318],{"className":22509},[129],[15,22511,59],{"className":22512},[195],[15,22514],{"className":22515,"style":3277},[93],[15,22517,2682],{"className":22518},[2436],[15,22520],{"className":22521,"style":3277},[93],[15,22523,22525,22528,22531],{"className":22524},[120],[15,22526],{"className":22527,"style":4383},[124],[15,22529,22154],{"className":22530},[129,133],[15,22532,107],{"className":22533},[129],[11,22535,22536,22537,22598,22599,22666,22667,22732,22733,22794,22795,22823,22824,22852,22853,22933,22934,22984,22985,23116,23117,1313],{},"交集里的每个二分，都对应 ",[15,22538,22540,22557],{"className":22539,"translate":19},[23],[15,22541,22543],{"className":22542},[27],[29,22544,22545],{"xmlns":31},[34,22546,22547,22555],{},[37,22548,22549],{},[305,22550,22551,22553],{},[43,22552,5619],{},[50,22554,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,22556,22097],{"encoding":111},[15,22558,22560],{"className":22559,"ariaHidden":72},[116],[15,22561,22563,22566],{"className":22562},[120],[15,22564],{"className":22565,"style":21540},[124],[15,22567,22569,22572],{"className":22568},[129],[15,22570,5619],{"className":22571,"style":5665},[129,133],[15,22573,22575],{"className":22574},[137],[15,22576,22578],{"className":22577},[141],[15,22579,22581],{"className":22580},[146],[15,22582,22584],{"className":22583,"style":21540},[150],[15,22585,22586,22589],{"style":383},[15,22587],{"className":22588,"style":159},[158],[15,22590,22592],{"className":22591},[163,164,165,166],[15,22593,22595],{"className":22594},[129,166],[15,22596,21510],{"className":22597},[129,166]," 上一种「最后一位可以任意翻转」的模式。把这些模式组成的集合记为 ",[15,22600,22602,22624],{"className":22601,"translate":19},[23],[15,22603,22605],{"className":22604},[27],[29,22606,22607],{"xmlns":31},[34,22608,22609,22621],{},[37,22610,22611],{},[305,22612,22613,22615],{},[43,22614,819],{"mathvariant":596},[37,22616,22617,22619],{},[50,22618,21510],{"mathvariant":65},[50,22620,21510],{"mathvariant":65},[109,22622,22623],{"encoding":111},"\\mathcal H''",[15,22625,22627],{"className":22626,"ariaHidden":72},[116],[15,22628,22630,22633],{"className":22629},[120],[15,22631],{"className":22632,"style":21540},[124],[15,22634,22636,22639],{"className":22635},[129],[15,22637,819],{"className":22638,"style":835},[129,614],[15,22640,22642],{"className":22641},[137],[15,22643,22645],{"className":22644},[141],[15,22646,22648],{"className":22647},[146],[15,22649,22651],{"className":22650,"style":21540},[150],[15,22652,22653,22656],{"style":383},[15,22654],{"className":22655,"style":159},[158],[15,22657,22659],{"className":22658},[163,164,165,166],[15,22660,22662],{"className":22661},[129,166],[15,22663,22665],{"className":22664},[129,166],"′′","。若 ",[15,22668,22670,22691],{"className":22669,"translate":19},[23],[15,22671,22673],{"className":22672},[27],[29,22674,22675],{"xmlns":31},[34,22676,22677,22689],{},[37,22678,22679],{},[305,22680,22681,22683],{},[43,22682,819],{"mathvariant":596},[37,22684,22685,22687],{},[50,22686,21510],{"mathvariant":65},[50,22688,21510],{"mathvariant":65},[109,22690,22623],{"encoding":111},[15,22692,22694],{"className":22693,"ariaHidden":72},[116],[15,22695,22697,22700],{"className":22696},[120],[15,22698],{"className":22699,"style":21540},[124],[15,22701,22703,22706],{"className":22702},[129],[15,22704,819],{"className":22705,"style":835},[129,614],[15,22707,22709],{"className":22708},[137],[15,22710,22712],{"className":22711},[141],[15,22713,22715],{"className":22714},[146],[15,22716,22718],{"className":22717,"style":21540},[150],[15,22719,22720,22723],{"style":383},[15,22721],{"className":22722,"style":159},[158],[15,22724,22726],{"className":22725},[163,164,165,166],[15,22727,22729],{"className":22728},[129,166],[15,22730,22665],{"className":22731},[129,166]," 打散了 ",[15,22734,22736,22753],{"className":22735,"translate":19},[23],[15,22737,22739],{"className":22738},[27],[29,22740,22741],{"xmlns":31},[34,22742,22743,22751],{},[37,22744,22745],{},[305,22746,22747,22749],{},[43,22748,5619],{},[50,22750,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,22752,22097],{"encoding":111},[15,22754,22756],{"className":22755,"ariaHidden":72},[116],[15,22757,22759,22762],{"className":22758},[120],[15,22760],{"className":22761,"style":21540},[124],[15,22763,22765,22768],{"className":22764},[129],[15,22766,5619],{"className":22767,"style":5665},[129,133],[15,22769,22771],{"className":22770},[137],[15,22772,22774],{"className":22773},[141],[15,22775,22777],{"className":22776},[146],[15,22778,22780],{"className":22779,"style":21540},[150],[15,22781,22782,22785],{"style":383},[15,22783],{"className":22784,"style":159},[158],[15,22786,22788],{"className":22787},[163,164,165,166],[15,22789,22791],{"className":22790},[129,166],[15,22792,21510],{"className":22793},[129,166]," 的某个 ",[15,22796,22798,22811],{"className":22797,"translate":19},[23],[15,22799,22801],{"className":22800},[27],[29,22802,22803],{"xmlns":31},[34,22804,22805,22809],{},[37,22806,22807],{},[43,22808,10376],{},[109,22810,10376],{"encoding":111},[15,22812,22814],{"className":22813,"ariaHidden":72},[116],[15,22815,22817,22820],{"className":22816},[120],[15,22818],{"className":22819,"style":522},[124],[15,22821,10376],{"className":22822},[129,133]," 元子集，那么 ",[15,22825,22827,22840],{"className":22826,"translate":19},[23],[15,22828,22830],{"className":22829},[27],[29,22831,22832],{"xmlns":31},[34,22833,22834,22838],{},[37,22835,22836],{},[43,22837,819],{"mathvariant":596},[109,22839,822],{"encoding":111},[15,22841,22843],{"className":22842,"ariaHidden":72},[116],[15,22844,22846,22849],{"className":22845},[120],[15,22847],{"className":22848,"style":610},[124],[15,22850,819],{"className":22851,"style":835},[129,614]," 就打散了该子集并上 ",[15,22854,22856,22878],{"className":22855,"translate":19},[23],[15,22857,22859],{"className":22858},[27],[29,22860,22861],{"xmlns":31},[34,22862,22863,22875],{},[37,22864,22865,22867,22873],{},[50,22866,636],{"stretchy":52},[40,22868,22869,22871],{},[43,22870,5628],{},[43,22872,348],{},[50,22874,645],{"stretchy":52},[109,22876,22877],{"encoding":111},"\\{c_m\\}",[15,22879,22881],{"className":22880,"ariaHidden":72},[116],[15,22882,22884,22887,22890,22930],{"className":22883},[120],[15,22885],{"className":22886,"style":125},[124],[15,22888,636],{"className":22889},[188],[15,22891,22893,22896],{"className":22892},[129],[15,22894,5628],{"className":22895},[129,133],[15,22897,22899],{"className":22898},[137],[15,22900,22902,22922],{"className":22901},[141,142],[15,22903,22905,22919],{"className":22904},[146],[15,22906,22908],{"className":22907,"style":1163},[150],[15,22909,22910,22913],{"style":154},[15,22911],{"className":22912,"style":159},[158],[15,22914,22916],{"className":22915},[163,164,165,166],[15,22917,348],{"className":22918},[129,133,166],[15,22920,175],{"className":22921},[174],[15,22923,22925],{"className":22924},[146],[15,22926,22928],{"className":22927,"style":182},[150],[15,22929],{},[15,22931,645],{"className":22932},[195],"，共 ",[15,22935,22937,22954],{"className":22936,"translate":19},[23],[15,22938,22940],{"className":22939},[27],[29,22941,22942],{"xmlns":31},[34,22943,22944,22952],{},[37,22945,22946,22948,22950],{},[43,22947,10376],{},[50,22949,2682],{},[308,22951,318],{},[109,22953,10880],{"encoding":111},[15,22955,22957,22975],{"className":22956,"ariaHidden":72},[116],[15,22958,22960,22963,22966,22969,22972],{"className":22959},[120],[15,22961],{"className":22962,"style":10414},[124],[15,22964,10376],{"className":22965},[129,133],[15,22967],{"className":22968,"style":3277},[93],[15,22970,2682],{"className":22971},[2436],[15,22973],{"className":22974,"style":3277},[93],[15,22976,22978,22981],{"className":22977},[120],[15,22979],{"className":22980,"style":4886},[124],[15,22982,318],{"className":22983},[129]," 个点，矛盾。所以 ",[15,22986,22988,23026],{"className":22987,"translate":19},[23],[15,22989,22991],{"className":22990},[27],[29,22992,22993],{"xmlns":31},[34,22994,22995,23023],{},[37,22996,22997,22999,23001,23003,23013,23015,23017,23019,23021],{},[43,22998,6823],{"mathvariant":65},[50,23000,69],{},[50,23002,53],{"stretchy":52},[305,23004,23005,23007],{},[43,23006,819],{"mathvariant":596},[37,23008,23009,23011],{},[50,23010,21510],{"mathvariant":65},[50,23012,21510],{"mathvariant":65},[50,23014,59],{"stretchy":52},[50,23016,2717],{},[43,23018,10376],{},[50,23020,315],{},[308,23022,318],{},[109,23024,23025],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H'')\\le d-1",[15,23027,23029,23089,23107],{"className":23028,"ariaHidden":72},[116],[15,23030,23032,23036,23042,23045,23077,23080,23083,23086],{"className":23031},[120],[15,23033],{"className":23034,"style":23035},[124],"height:1.0019em;vertical-align:-0.25em;",[15,23037,23039],{"className":23038},[217],[15,23040,6823],{"className":23041},[129,221],[15,23043,53],{"className":23044},[188],[15,23046,23048,23051],{"className":23047},[129],[15,23049,819],{"className":23050,"style":835},[129,614],[15,23052,23054],{"className":23053},[137],[15,23055,23057],{"className":23056},[141],[15,23058,23060],{"className":23059},[146],[15,23061,23063],{"className":23062,"style":21540},[150],[15,23064,23065,23068],{"style":383},[15,23066],{"className":23067,"style":159},[158],[15,23069,23071],{"className":23070},[163,164,165,166],[15,23072,23074],{"className":23073},[129,166],[15,23075,22665],{"className":23076},[129,166],[15,23078,59],{"className":23079},[195],[15,23081],{"className":23082,"style":199},[93],[15,23084,2717],{"className":23085},[203],[15,23087],{"className":23088,"style":199},[93],[15,23090,23092,23095,23098,23101,23104],{"className":23091},[120],[15,23093],{"className":23094,"style":10414},[124],[15,23096,10376],{"className":23097},[129,133],[15,23099],{"className":23100,"style":3277},[93],[15,23102,315],{"className":23103},[2436],[15,23105],{"className":23106,"style":3277},[93],[15,23108,23110,23113],{"className":23109},[120],[15,23111],{"className":23112,"style":4886},[124],[15,23114,318],{"className":23115},[129],"，从而 ",[15,23118,23120,23201],{"className":23119,"translate":19},[23],[15,23121,23123],{"className":23122},[27],[29,23124,23125],{"xmlns":31},[34,23126,23127,23198],{},[37,23128,23129,23131,23133,23147,23149,23151,23153,23155,23157,23159,23180],{},[43,23130,22154],{},[50,23132,2717],{},[40,23134,23135,23137],{},[43,23136,19688],{"mathvariant":65},[305,23138,23139,23141],{},[43,23140,819],{"mathvariant":596},[37,23142,23143,23145],{},[50,23144,21510],{"mathvariant":65},[50,23146,21510],{"mathvariant":65},[50,23148,53],{"stretchy":52},[43,23150,348],{},[50,23152,315],{},[308,23154,318],{},[50,23156,59],{"stretchy":52},[50,23158,2717],{},[23160,23161,23162,23164,23172],"msubsup",{},[50,23163,1454],{},[37,23165,23166,23168,23170],{},[43,23167,1459],{},[50,23169,62],{},[308,23171,541],{},[37,23173,23174,23176,23178],{},[43,23175,10376],{},[50,23177,315],{},[308,23179,318],{},[37,23181,23182,23184,23196],{},[50,23183,53],{"fence":72},[1443,23185,23186,23194],{"linethickness":21040},[37,23187,23188,23190,23192],{},[43,23189,348],{},[50,23191,315],{},[308,23193,318],{},[43,23195,1459],{},[50,23197,59],{"fence":72},[109,23199,23200],{"encoding":111},"s\\le\\Pi_{\\mathcal H''}(m-1)\\le\\sum_{i=0}^{d-1}\\binom{m-1}{i}",[15,23202,23204,23222,23317,23338],{"className":23203,"ariaHidden":72},[116],[15,23205,23207,23210,23213,23216,23219],{"className":23206},[120],[15,23208],{"className":23209,"style":7757},[124],[15,23211,22154],{"className":23212},[129,133],[15,23214],{"className":23215,"style":199},[93],[15,23217,2717],{"className":23218},[203],[15,23220],{"className":23221,"style":199},[93],[15,23223,23225,23228,23302,23305,23308,23311,23314],{"className":23224},[120],[15,23226],{"className":23227,"style":125},[124],[15,23229,23231,23234],{"className":23230},[129],[15,23232,19688],{"className":23233},[129],[15,23235,23237],{"className":23236},[137],[15,23238,23240,23294],{"className":23239},[141,142],[15,23241,23243,23291],{"className":23242},[146],[15,23244,23246],{"className":23245,"style":1525},[150],[15,23247,23248,23251],{"style":154},[15,23249],{"className":23250,"style":159},[158],[15,23252,23254],{"className":23253},[163,164,165,166],[15,23255,23257],{"className":23256},[129,166],[15,23258,23260,23263],{"className":23259},[129,166],[15,23261,819],{"className":23262,"style":835},[129,614,166],[15,23264,23266],{"className":23265},[137],[15,23267,23269],{"className":23268},[141],[15,23270,23272],{"className":23271},[146],[15,23273,23276],{"className":23274,"style":23275},[150],"height:0.6828em;",[15,23277,23279,23282],{"style":23278},"top:-2.786em;margin-right:0.0714em;",[15,23280],{"className":23281,"style":15987},[158],[15,23283,23285],{"className":23284},[163,15991,229,166],[15,23286,23288],{"className":23287},[129,166],[15,23289,22665],{"className":23290},[129,166],[15,23292,175],{"className":23293},[174],[15,23295,23297],{"className":23296},[146],[15,23298,23300],{"className":23299,"style":182},[150],[15,23301],{},[15,23303,53],{"className":23304},[188],[15,23306,348],{"className":23307},[129,133],[15,23309],{"className":23310,"style":3277},[93],[15,23312,315],{"className":23313},[2436],[15,23315],{"className":23316,"style":3277},[93],[15,23318,23320,23323,23326,23329,23332,23335],{"className":23319},[120],[15,23321],{"className":23322,"style":125},[124],[15,23324,318],{"className":23325},[129],[15,23327,59],{"className":23328},[195],[15,23330],{"className":23331,"style":199},[93],[15,23333,2717],{"className":23334},[203],[15,23336],{"className":23337,"style":199},[93],[15,23339,23341,23345,23420,23423],{"className":23340},[120],[15,23342],{"className":23343,"style":23344},[124],"height:1.339em;vertical-align:-0.35em;",[15,23346,23348,23353],{"className":23347},[217],[15,23349,1454],{"className":23350,"style":23352},[217,1918,23351],"small-op","position:relative;top:0em;",[15,23354,23356],{"className":23355},[137],[15,23357,23359,23411],{"className":23358},[141,142],[15,23360,23362,23408],{"className":23361},[146],[15,23363,23366,23387],{"className":23364,"style":23365},[150],"height:0.989em;",[15,23367,23369,23372],{"style":23368},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[15,23370],{"className":23371,"style":159},[158],[15,23373,23375],{"className":23374},[163,164,165,166],[15,23376,23378,23381,23384],{"className":23377},[129,166],[15,23379,1459],{"className":23380},[129,133,166],[15,23382,62],{"className":23383},[203,166],[15,23385,541],{"className":23386},[129,166],[15,23388,23390,23393],{"style":23389},"top:-3.2029em;margin-right:0.05em;",[15,23391],{"className":23392,"style":159},[158],[15,23394,23396],{"className":23395},[163,164,165,166],[15,23397,23399,23402,23405],{"className":23398},[129,166],[15,23400,10376],{"className":23401},[129,133,166],[15,23403,315],{"className":23404},[2436,166],[15,23406,318],{"className":23407},[129,166],[15,23409,175],{"className":23410},[174],[15,23412,23414],{"className":23413},[146],[15,23415,23418],{"className":23416,"style":23417},[150],"height:0.2997em;",[15,23419],{},[15,23421],{"className":23422,"style":262},[93],[15,23424,23426,23432,23493],{"className":23425},[129],[15,23427,23429],{"className":23428,"style":5052},[188,5051],[15,23430,53],{"className":23431},[228,229],[15,23433,23435],{"className":23434},[1443],[15,23436,23438,23484],{"className":23437},[141,142],[15,23439,23441,23481],{"className":23440},[146],[15,23442,23445,23460],{"className":23443,"style":23444},[150],"height:0.8951em;",[15,23446,23448,23451],{"style":23447},"top:-2.355em;",[15,23449],{"className":23450,"style":159},[158],[15,23452,23454],{"className":23453},[163,164,165,166],[15,23455,23457],{"className":23456},[129,166],[15,23458,1459],{"className":23459},[129,133,166],[15,23461,23463,23466],{"style":23462},"top:-3.144em;",[15,23464],{"className":23465,"style":159},[158],[15,23467,23469],{"className":23468},[163,164,165,166],[15,23470,23472,23475,23478],{"className":23471},[129,166],[15,23473,348],{"className":23474},[129,133,166],[15,23476,315],{"className":23477},[2436,166],[15,23479,318],{"className":23480},[129,166],[15,23482,175],{"className":23483},[174],[15,23485,23487],{"className":23486},[146],[15,23488,23491],{"className":23489,"style":23490},[150],"height:0.345em;",[15,23492],{},[15,23494,23496],{"className":23495,"style":5052},[195,5051],[15,23497,59],{"className":23498},[228,229],[11,23500,23501],{},"代回并用 Pascal 公式：",[15,23503,23505],{"className":23504,"translate":19},[18],[15,23506,23508,23636],{"className":23507,"translate":19},[23],[15,23509,23511],{"className":23510},[27],[29,23512,23513],{"xmlns":31,"display":32},[34,23514,23515,23633],{},[37,23516,23517,23523,23525,23527,23529,23531,23545,23563,23565,23585,23603,23605,23619,23631],{},[40,23518,23519,23521],{},[43,23520,19688],{"mathvariant":65},[43,23522,819],{"mathvariant":596},[50,23524,53],{"stretchy":52},[43,23526,348],{},[50,23528,59],{"stretchy":52},[50,23530,2717],{},[1450,23532,23533,23535,23543],{},[50,23534,1454],{},[37,23536,23537,23539,23541],{},[43,23538,1459],{},[50,23540,62],{},[308,23542,541],{},[43,23544,10376],{},[37,23546,23547,23549,23561],{},[50,23548,53],{"fence":72},[1443,23550,23551,23559],{"linethickness":21040},[37,23552,23553,23555,23557],{},[43,23554,348],{},[50,23556,315],{},[308,23558,318],{},[43,23560,1459],{},[50,23562,59],{"fence":72},[50,23564,2682],{},[1450,23566,23567,23569,23577],{},[50,23568,1454],{},[37,23570,23571,23573,23575],{},[43,23572,1459],{},[50,23574,62],{},[308,23576,541],{},[37,23578,23579,23581,23583],{},[43,23580,10376],{},[50,23582,315],{},[308,23584,318],{},[37,23586,23587,23589,23601],{},[50,23588,53],{"fence":72},[1443,23590,23591,23599],{"linethickness":21040},[37,23592,23593,23595,23597],{},[43,23594,348],{},[50,23596,315],{},[308,23598,318],{},[43,23600,1459],{},[50,23602,59],{"fence":72},[50,23604,62],{},[1450,23606,23607,23609,23617],{},[50,23608,1454],{},[37,23610,23611,23613,23615],{},[43,23612,1459],{},[50,23614,62],{},[308,23616,541],{},[43,23618,10376],{},[37,23620,23621,23623,23629],{},[50,23622,53],{"fence":72},[1443,23624,23625,23627],{"linethickness":21040},[43,23626,348],{},[43,23628,1459],{},[50,23630,59],{"fence":72},[43,23632,107],{"mathvariant":65},[109,23634,23635],{"encoding":111},"\\Pi_{\\mathcal H}(m)\\le\\sum_{i=0}^{d}\\binom{m-1}{i}+\\sum_{i=0}^{d-1}\\binom{m-1}{i}\n=\\sum_{i=0}^{d}\\binom{m}{i}.",[15,23637,23639,23706,23863,24026],{"className":23638,"ariaHidden":72},[116],[15,23640,23642,23645,23688,23691,23694,23697,23700,23703],{"className":23641},[120],[15,23643],{"className":23644,"style":125},[124],[15,23646,23648,23651],{"className":23647},[129],[15,23649,19688],{"className":23650},[129],[15,23652,23654],{"className":23653},[137],[15,23655,23657,23680],{"className":23656},[141,142],[15,23658,23660,23677],{"className":23659},[146],[15,23661,23663],{"className":23662,"style":1525},[150],[15,23664,23665,23668],{"style":154},[15,23666],{"className":23667,"style":159},[158],[15,23669,23671],{"className":23670},[163,164,165,166],[15,23672,23674],{"className":23673},[129,166],[15,23675,819],{"className":23676,"style":835},[129,614,166],[15,23678,175],{"className":23679},[174],[15,23681,23683],{"className":23682},[146],[15,23684,23686],{"className":23685,"style":182},[150],[15,23687],{},[15,23689,53],{"className":23690},[188],[15,23692,348],{"className":23693},[129,133],[15,23695,59],{"className":23696},[195],[15,23698],{"className":23699,"style":199},[93],[15,23701,2717],{"className":23702},[203],[15,23704],{"className":23705,"style":199},[93],[15,23707,23709,23712,23779,23782,23854,23857,23860],{"className":23708},[120],[15,23710],{"className":23711,"style":21128},[124],[15,23713,23715],{"className":23714},[217,1874],[15,23716,23718,23771],{"className":23717},[141,142],[15,23719,23721,23768],{"className":23720},[146],[15,23722,23724,23744,23754],{"className":23723,"style":21141},[150],[15,23725,23726,23729],{"style":1887},[15,23727],{"className":23728,"style":1891},[158],[15,23730,23732],{"className":23731},[163,164,165,166],[15,23733,23735,23738,23741],{"className":23734},[129,166],[15,23736,1459],{"className":23737},[129,133,166],[15,23739,62],{"className":23740},[203,166],[15,23742,541],{"className":23743},[129,166],[15,23745,23746,23749],{"style":1909},[15,23747],{"className":23748,"style":1891},[158],[15,23750,23751],{},[15,23752,1454],{"className":23753},[217,1918,1919],[15,23755,23756,23759],{"style":1922},[15,23757],{"className":23758,"style":1891},[158],[15,23760,23762],{"className":23761},[163,164,165,166],[15,23763,23765],{"className":23764},[129,166],[15,23766,10376],{"className":23767},[129,133,166],[15,23769,175],{"className":23770},[174],[15,23772,23774],{"className":23773},[146],[15,23775,23777],{"className":23776,"style":1941},[150],[15,23778],{},[15,23780],{"className":23781,"style":262},[93],[15,23783,23785,23791,23848],{"className":23784},[129],[15,23786,23788],{"className":23787,"style":5052},[188,5051],[15,23789,53],{"className":23790},[228,165],[15,23792,23794],{"className":23793},[1443],[15,23795,23797,23840],{"className":23796},[141,142],[15,23798,23800,23837],{"className":23799},[146],[15,23801,23803,23814],{"className":23802,"style":1816},[150],[15,23804,23805,23808],{"style":1819},[15,23806],{"className":23807,"style":1823},[158],[15,23809,23811],{"className":23810},[129],[15,23812,1459],{"className":23813},[129,133],[15,23815,23816,23819],{"style":1843},[15,23817],{"className":23818,"style":1823},[158],[15,23820,23822,23825,23828,23831,23834],{"className":23821},[129],[15,23823,348],{"className":23824},[129,133],[15,23826],{"className":23827,"style":3277},[93],[15,23829,315],{"className":23830},[2436],[15,23832],{"className":23833,"style":3277},[93],[15,23835,318],{"className":23836},[129],[15,23838,175],{"className":23839},[174],[15,23841,23843],{"className":23842},[146],[15,23844,23846],{"className":23845,"style":1862},[150],[15,23847],{},[15,23849,23851],{"className":23850,"style":5052},[195,5051],[15,23852,59],{"className":23853},[228,165],[15,23855],{"className":23856,"style":3277},[93],[15,23858,2682],{"className":23859},[2436],[15,23861],{"className":23862,"style":3277},[93],[15,23864,23866,23869,23942,23945,24017,24020,24023],{"className":23865},[120],[15,23867],{"className":23868,"style":21128},[124],[15,23870,23872],{"className":23871},[217,1874],[15,23873,23875,23934],{"className":23874},[141,142],[15,23876,23878,23931],{"className":23877},[146],[15,23879,23881,23901,23911],{"className":23880,"style":21141},[150],[15,23882,23883,23886],{"style":1887},[15,23884],{"className":23885,"style":1891},[158],[15,23887,23889],{"className":23888},[163,164,165,166],[15,23890,23892,23895,23898],{"className":23891},[129,166],[15,23893,1459],{"className":23894},[129,133,166],[15,23896,62],{"className":23897},[203,166],[15,23899,541],{"className":23900},[129,166],[15,23902,23903,23906],{"style":1909},[15,23904],{"className":23905,"style":1891},[158],[15,23907,23908],{},[15,23909,1454],{"className":23910},[217,1918,1919],[15,23912,23913,23916],{"style":1922},[15,23914],{"className":23915,"style":1891},[158],[15,23917,23919],{"className":23918},[163,164,165,166],[15,23920,23922,23925,23928],{"className":23921},[129,166],[15,23923,10376],{"className":23924},[129,133,166],[15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多项式增长",[11,24173,17335,24174,24225],{},[15,24175,24177,24195],{"className":24176,"translate":19},[23],[15,24178,24180],{"className":24179},[27],[29,24181,24182],{"xmlns":31},[34,24183,24184,24192],{},[37,24185,24186,24188,24190],{},[43,24187,348],{},[50,24189,7473],{},[43,24191,10376],{},[109,24193,24194],{"encoding":111},"m\\ge d",[15,24196,24198,24216],{"className":24197,"ariaHidden":72},[116],[15,24199,24201,24204,24207,24210,24213],{"className":24200},[120],[15,24202],{"className":24203,"style":7757},[124],[15,24205,348],{"className":24206},[129,133],[15,24208],{"className":24209,"style":199},[93],[15,24211,7473],{"className":24212},[203],[15,24214],{"className":24215,"style":199},[93],[15,24217,24219,24222],{"className":24218},[120],[15,24220],{"className":24221,"style":522},[124],[15,24223,10376],{"className":24224},[129,133]," 时，上述和式可以被一个更粗但更好用的界控制：",[15,24227,24229],{"className":24228,"translate":19},[18],[15,24230,24232,24294],{"className":24231,"translate":19},[23],[15,24233,24235],{"className":24234},[27],[29,24236,24237],{"xmlns":31,"display":32},[34,24238,24239,24291],{},[37,24240,24241,24255,24267,24269,24289],{},[1450,24242,24243,24245,24253],{},[50,24244,1454],{},[37,24246,24247,24249,24251],{},[43,24248,1459],{},[50,24250,62],{},[308,24252,541],{},[43,24254,10376],{},[37,24256,24257,24259,24265],{},[50,24258,53],{"fence":72},[1443,24260,24261,24263],{"linethickness":21040},[43,24262,348],{},[43,24264,1459],{},[50,24266,59],{"fence":72},[50,24268,2717],{},[305,24270,24271,24287],{},[37,24272,24273,24275,24285],{},[50,24274,53],{"fence":72},[1443,24276,24277,24283],{},[37,24278,24279,24281],{},[43,24280,10699],{},[43,24282,348],{},[43,24284,10376],{},[50,24286,59],{"fence":72},[43,24288,10376],{},[43,24290,107],{"mathvariant":65},[109,24292,24293],{"encoding":111},"\\sum_{i=0}^{d}\\binom{m}{i}\\le\\left(\\frac{em}{d}\\right)^d.",[15,24295,24297,24442],{"className":24296,"ariaHidden":72},[116],[15,24298,24300,24303,24370,24373,24433,24436,24439],{"className":24299},[120],[15,24301],{"className":24302,"style":21128},[124],[15,24304,24306],{"className":24305},[217,1874],[15,24307,24309,24362],{"className":24308},[141,142],[15,24310,24312,24359],{"className":24311},[146],[15,24313,24315,24335,24345],{"className":24314,"style":21141},[150],[15,24316,24317,24320],{"style":1887},[15,24318],{"className":24319,"style":1891},[158],[15,24321,24323],{"className":24322},[163,164,165,166],[15,24324,24326,24329,24332],{"className":24325},[129,166],[15,24327,1459],{"className":24328},[129,133,166],[15,24330,62],{"className":24331},[203,166],[15,24333,541],{"className":24334},[129,166],[15,24336,24337,24340],{"style":1909},[15,24338],{"className":24339,"style":1891},[158],[15,24341,24342],{},[15,24343,1454],{"className":24344},[217,1918,1919],[15,24346,24347,24350],{"style":1922},[15,24348],{"className":24349,"style":1891},[158],[15,24351,24353],{"className":24352},[163,164,165,166],[15,24354,24356],{"className":24355},[129,166],[15,24357,10376],{"className":24358},[129,133,166],[15,24360,175],{"className":24361},[174],[15,24363,24365],{"className":24364},[146],[15,24366,24368],{"className":24367,"style":1941},[150],[15,24369],{},[15,24371],{"className":24372,"style":262},[93],[15,24374,24376,24382,24427],{"className":24375},[129],[15,24377,24379],{"className":24378,"style":5052},[188,5051],[15,24380,53],{"className":24381},[228,165],[15,24383,24385],{"className":24384},[1443],[15,24386,24388,24419],{"className":24387},[141,142],[15,24389,24391,24416],{"className":24390},[146],[15,24392,24394,24405],{"className":24393,"style":21221},[150],[15,24395,24396,24399],{"style":1819},[15,24397],{"className":24398,"style":1823},[158],[15,24400,24402],{"className":24401},[129],[15,24403,1459],{"className":24404},[129,133],[15,24406,24407,24410],{"style":1843},[15,24408],{"className":24409,"style":1823},[158],[15,24411,24413],{"className":24412},[129],[15,24414,348],{"className":24415},[129,133],[15,24417,175],{"className":24418},[174],[15,24420,24422],{"className":24421},[146],[15,24423,24425],{"className":24424,"style":1862},[150],[15,24426],{},[15,24428,24430],{"className":24429,"style":5052},[195,5051],[15,24431,59],{"className":24432},[228,165],[15,24434],{"className":24435,"style":199},[93],[15,24437,2717],{"className":24438},[203],[15,24440],{"className":24441,"style":199},[93],[15,24443,24445,24449,24558,24561],{"className":24444},[120],[15,24446],{"className":24447,"style":24448},[124],"height:2.075em;vertical-align:-0.686em;",[15,24450,24452,24533],{"className":24451},[449],[15,24453,24455,24462,24527],{"className":24454},[449],[15,24456,24458],{"className":24457,"style":5052},[188,5051],[15,24459,53],{"className":24460},[228,24461],"size2",[15,24463,24465,24468,24524],{"className":24464},[129],[15,24466],{"className":24467},[188,1803],[15,24469,24471],{"className":24470},[1443],[15,24472,24474,24516],{"className":24473},[141,142],[15,24475,24477,24513],{"className":24476},[146],[15,24478,24480,24491,24499],{"className":24479,"style":21221},[150],[15,24481,24482,24485],{"style":1819},[15,24483],{"className":24484,"style":1823},[158],[15,24486,24488],{"className":24487},[129],[15,24489,10376],{"className":24490},[129,133],[15,24492,24493,24496],{"style":1832},[15,24494],{"className":24495,"style":1823},[158],[15,24497],{"className":24498,"style":1840},[1839],[15,24500,24501,24504],{"style":1843},[15,24502],{"className":24503,"style":1823},[158],[15,24505,24507,24510],{"className":24506},[129],[15,24508,10699],{"className":24509},[129,133],[15,24511,348],{"className":24512},[129,133],[15,24514,175],{"className":24515},[174],[15,24517,24519],{"className":24518},[146],[15,24520,24522],{"className":24521,"style":1862},[150],[15,24523],{},[15,24525],{"className":24526},[195,1803],[15,24528,24530],{"className":24529,"style":5052},[195,5051],[15,2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维有限的假设类，在 ",[15,25984,25986,25999],{"className":25985,"translate":19},[23],[15,25987,25989],{"className":25988},[27],[29,25990,25991],{"xmlns":31},[34,25992,25993,25997],{},[37,25994,25995],{},[43,25996,348],{},[109,25998,348],{"encoding":111},[15,26000,26002],{"className":26001,"ariaHidden":72},[116],[15,26003,26005,26008],{"className":26004},[120],[15,26006],{"className":26007,"style":4383},[124],[15,26009,348],{"className":26010},[129,133]," 个点上最多只有 ",[15,26013,26015,26033],{"className":26014,"translate":19},[23],[15,26016,26018],{"className":26017},[27],[29,26019,26020],{"xmlns":31},[34,26021,26022,26030],{},[37,26023,26024],{},[305,26025,26026,26028],{},[43,26027,348],{},[43,26029,10376],{},[109,26031,26032],{"encoding":111},"m^d",[15,26034,26036],{"className":26035,"ariaHidden":72},[116],[15,26037,26039,26042],{"className":26038},[120],[15,26040],{"className":26041,"style":10490},[124],[15,26043,26045,26048],{"className":26044},[129],[15,26046,348],{"className":26047},[129,133],[15,26049,26051],{"className":26050},[137],[15,26052,26054],{"className":26053},[141],[15,26055,26057],{"className":26056},[146],[15,26058,26060],{"className":26059,"style":10490},[150],[15,26061,26062,26065],{"style":383},[15,26063],{"className":26064,"style":159},[158],[15,26066,26068],{"className":26067},[163,164,165,166],[15,26069,10376],{"className":26070},[129,133,166]," 种行为，而不是 ",[15,26073,26075,26092],{"className":26074,"translate":19},[23],[15,26076,26078],{"className":26077},[27],[29,26079,26080],{"xmlns":31},[34,26081,26082,26090],{},[37,26083,26084],{},[305,26085,26086,26088],{},[308,26087,310],{},[43,26089,348],{},[109,26091,6367],{"encoding":111},[15,26093,26095],{"className":26094,"ariaHidden":72},[116],[15,26096,26098,26101],{"className":26097},[120],[15,26099],{"className":26100,"style":6377},[124],[15,26102,26104,26107],{"className":26103},[129],[15,26105,310],{"className":26106},[129],[15,26108,26110],{"className":26109},[137],[15,26111,26113],{"className":26112},[141],[15,26114,26116],{"className":26115},[146],[15,26117,26119],{"className":26118,"style":6377},[150],[15,26120,26121,26124],{"style":383},[15,26122],{"className":26123,"style":159},[158],[15,26125,26127],{"className":26126},[163,164,165,166],[15,26128,348],{"className":26129},[129,133,166]," 种。当 ",[15,26132,26134,26147],{"className":26133,"translate":19},[23],[15,26135,26137],{"className":26136},[27],[29,26138,26139],{"xmlns":31},[34,26140,26141,26145],{},[37,26142,26143],{},[43,26144,348],{},[109,26146,348],{"encoding":111},[15,26148,26150],{"className":26149,"ariaHidden":72},[116],[15,26151,26153,26156],{"className":26152},[120],[15,26154],{"className":26155,"style":4383},[124],[15,26157,348],{"className":26158},[129,133]," 增大时，这两种增长速度的差距是压倒性的——",[15,26161,26163,26180],{"className":26162,"translate":19},[23],[15,26164,26166],{"className":26165},[27],[29,26167,26168],{"xmlns":31},[34,26169,26170,26178],{},[37,26171,26172],{},[305,26173,26174,26176],{},[43,26175,348],{},[43,26177,10376],{},[109,26179,26032],{"encoding":111},[15,26181,26183],{"className":26182,"ariaHidden":72},[116],[15,26184,26186,26189],{"className":26185},[120],[15,26187],{"className":26188,"style":10490},[124],[15,26190,26192,26195],{"className":26191},[129],[15,26193,348],{"className":26194},[129,133],[15,26196,26198],{"className":26197},[137],[15,26199,26201],{"className":26200},[141],[15,26202,26204],{"className":26203},[146],[15,26205,26207],{"className":26206,"style":10490},[150],[15,26208,26209,26212],{"style":383},[15,26210],{"className":26211,"style":159},[158],[15,26213,26215],{"className":26214},[163,164,165,166],[15,26216,10376],{"className":26217},[129,133,166]," 取对数后只有 ",[15,26220,26222,26242],{"className":26221,"translate":19},[23],[15,26223,26225],{"className":26224},[27],[29,26226,26227],{"xmlns":31},[34,26228,26229,26239],{},[37,26230,26231,26233,26235,26237],{},[43,26232,10376],{},[43,26234,14727],{},[50,26236,69],{},[43,26238,348],{},[109,26240,26241],{"encoding":111},"d\\log m",[15,26243,26245],{"className":26244,"ariaHidden":72},[116],[15,26246,26248,26251,26254,26257,26262,26265],{"className":26247},[120],[15,26249],{"className":26250,"style":1682},[124],[15,26252,10376],{"className":26253},[129,133],[15,26255],{"className":26256,"style":262},[93],[15,26258,14777,26260],{"className":26259},[217],[15,26261,2247],{"style":2246},[15,26263],{"className":26264,"style":262},[93],[15,26266,348],{"className":26267},[129,133],"，而 ",[15,26270,26272,26289],{"className":26271,"translate":19},[23],[15,26273,26275],{"className":26274},[27],[29,26276,26277],{"xmlns":31},[34,26278,26279,26287],{},[37,26280,26281],{},[305,26282,26283,26285],{},[308,26284,310],{},[43,26286,348],{},[109,26288,6367],{"encoding":111},[15,26290,26292],{"className":26291,"ariaHidden":72},[116],[15,26293,26295,26298],{"className":26294},[120],[15,26296],{"className":26297,"style":6377},[124],[15,26299,26301,26304],{"className":26300},[129],[15,26302,310],{"className":26303},[129],[15,26305,26307],{"className":26306},[137],[15,26308,26310],{"className":26309},[141],[15,26311,26313],{"className":26312},[146],[15,26314,26316],{"className":26315,"style":6377},[150],[15,26317,26318,26321],{"style":383},[15,26319],{"className":26320,"style":159},[158],[15,26322,26324],{"className":26323},[163,164,165,166],[15,26325,348],{"className":26326},[129,133,166]," 取对数后是 ",[15,26329,26331,26344],{"className":26330,"translate":19},[23],[15,26332,26334],{"className":26333},[27],[29,26335,26336],{"xmlns":31},[34,26337,26338,26342],{},[37,26339,26340],{},[43,26341,348],{},[109,26343,348],{"encoding":111},[15,26345,26347],{"className":26346,"ariaHidden":72},[116],[15,26348,26350,26353],{"className":26349},[120],[15,26351],{"className":26352,"style":4383},[124],[15,26354,348],{"className":26355},[129,133],[11,26357,26358,26359,26362],{},"换句话说，",[698,26360,26361],{},"有限 VC 维等价于增长函数是多项式级的","。这就是「容量」一词的数学来源：容量不是参数个数，而是假设类能产生的二分数量随样本量的增长速度。",[570,26364,26366],{"id":26365},"四一致收敛定理","四、一致收敛定理",[575,26368,26370],{"id":26369},"_41-对称化","4.1 对称化",[11,26372,26373,26374,26377],{},"增长函数是组合量，要把它变成泛化界，需要一步概率论证，称为",[698,26375,26376],{},"对称化","（symmetrization）。",[11,26379,21938,26380,26442,26443,26471,26472,26500],{},[15,26381,26383,26401],{"className":26382,"translate":19},[23],[15,26384,26386],{"className":26385},[27],[29,26387,26388],{"xmlns":31},[34,26389,26390,26398],{},[37,26391,26392],{},[305,26393,26394,26396],{},[43,26395,940],{},[50,26397,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,26399,26400],{"encoding":111},"S'",[15,26402,26404],{"className":26403,"ariaHidden":72},[116],[15,26405,26407,26410],{"className":26406},[120],[15,26408],{"className":26409,"style":21540},[124],[15,26411,26413,26416],{"className":26412},[129],[15,26414,940],{"className":26415,"style":1013},[129,133],[15,26417,26419],{"className":26418},[137],[15,26420,26422],{"className":26421},[141],[15,26423,26425],{"className":26424},[146],[15,26426,26428],{"className":26427,"style":21540},[150],[15,26429,26430,26433],{"style":383},[15,26431],{"className":26432,"style":159},[158],[15,26434,26436],{"className":26435},[163,164,165,166],[15,26437,26439],{"className":26438},[129,166],[15,26440,21510],{"className":26441},[129,166]," 是与 ",[15,26444,26446,26459],{"className":26445,"translate":19},[23],[15,26447,26449],{"className":26448},[27],[29,26450,26451],{"xmlns":31},[34,26452,26453,26457],{},[37,26454,26455],{},[43,26456,940],{},[109,26458,940],{"encoding":111},[15,26460,26462],{"className":26461,"ariaHidden":72},[116],[15,26463,26465,26468],{"className":26464},[120],[15,26466],{"className":26467,"style":610},[124],[15,26469,940],{"className":26470,"style":1013},[129,133]," 独立同分布的另一组 ",[15,26473,26475,26488],{"className":26474,"translate":19},[23],[15,26476,26478],{"className":26477},[27],[29,26479,26480],{"xmlns":31},[34,26481,26482,26486],{},[37,26483,26484],{},[43,26485,348],{},[109,26487,348],{"encoding":111},[15,26489,26491],{"className":26490,"ariaHidden":72},[116],[15,26492,26494,26497],{"className":26493},[120],[15,26495],{"className":26496,"style":4383},[124],[15,26498,348],{"className":26499},[129,133]," 个样本。核心不等式是",[15,26502,26504],{"className":26503,"translate":19},[18],[15,26505,26507,26642],{"className":26506,"translate":19},[23],[15,26508,26510],{"className":26509},[27],[29,26511,26512],{"xmlns":31,"display":32},[34,26513,26514,26639],{},[37,26515,26516,26532,26534,26540,26542,26544,26546,26548,26554,26556,26558,26560,26562,26565,26568,26570,26587,26603,26605,26611,26613,26615,26617,26619,26629,26631,26633,26635,26637],{},[2140,26517,26518,26524],{},[37,26519,26520,26522],{},[43,26521,2932],{},[50,26523,69],{},[37,26525,26526,26528,26530],{},[43,26527,45],{},[50,26529,100],{},[43,26531,819],{"mathvariant":596},[50,26533,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,26535,26536,26538],{},[43,26537,1373],{},[43,26539,906],{"mathvariant":596},[50,26541,53],{"stretchy":52},[43,26543,45],{},[50,26545,59],{"stretchy":52},[50,26547,315],{},[40,26549,26550,26552],{},[43,26551,1373],{},[43,26553,940],{},[50,26555,53],{"stretchy":52},[43,26557,45],{},[50,26559,59],{"stretchy":52},[50,26561,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[6856,26563,26564],{},"  ",[50,26566,26567],{},"≲",[6856,26569,26564],{},[40,26571,26572,26575],{},[43,26573,26574],{"mathvariant":103},"E",[37,26576,26577,26579,26581],{},[43,26578,940],{},[50,26580,91],{"separator":72},[305,26582,26583,26585],{},[43,26584,940],{},[50,26586,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[2140,26588,26589,26595],{},[37,26590,26591,26593],{},[43,26592,2932],{},[50,26594,69],{},[37,26596,26597,26599,26601],{},[43,26598,45],{},[50,26600,100],{},[43,26602,819],{"mathvariant":596},[50,26604,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,26606,26607,26609],{},[43,26608,1373],{},[43,26610,940],{},[50,26612,53],{"stretchy":52},[43,26614,45],{},[50,26616,59],{"stretchy":52},[50,26618,315],{},[40,26620,26621,26623],{},[43,26622,1373],{},[305,26624,26625,26627],{},[43,26626,940],{},[50,26628,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[50,26630,53],{"stretchy":52},[43,26632,45],{},[50,26634,59],{"stretchy":52},[50,26636,2945],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,26638,91],{"separator":72},[109,26640,26641],{"encoding":111},"\\sup_{h\\in\\mathcal H}\\bigl|L_{\\mathcal D}(h)-L_S(h)\\bigr|\n\\;\\lesssim\\;\n\\mathbb E_{S,S'}\\sup_{h\\in\\mathcal H}\\bigl|L_S(h)-L_{S'}(h)\\bigr|,",[15,26643,26645,26803,26910,27145],{"className":26644,"ariaHidden":72},[116],[15,26646,26648,26652,26705,26742,26785,26788,26791,26794,26797,26800],{"className":26647},[120],[15,26649],{"className":26650,"style":26651},[124],"height:1.8239em;vertical-align:-0.9739em;",[15,26653,26655],{"className":26654},[217,1874],[15,26656,26658,26697],{"className":26657},[141,142],[15,26659,26661,26694],{"className":26660},[146],[15,26662,26664,26684],{"className":26663,"style":4383},[150],[15,26665,26666,26669],{"style":4386},[15,26667],{"className":26668,"style":1823},[158],[15,26670,26672],{"className":26671},[163,164,165,166],[15,26673,26675,26678,26681],{"className":26674},[129,166],[15,26676,45],{"className":26677},[129,133,166],[15,26679,100],{"className":26680},[203,166],[15,26682,819],{"className":26683,"style":835},[129,614,166],[15,26685,26686,26689],{"style":2287},[15,26687],{"className":26688,"style":1823},[158],[15,26690,26691],{},[15,26692,2932],{"className":26693},[217],[15,26695,175],{"className":26696},[174],[15,26698,26700],{"className":26699},[146],[15,26701,26703],{"className":26702,"style":4424},[150],[15,26704],{},[15,26706,26708],{"className":26707},[188],[15,26709,26711],{"className":26710},[228,4433],[15,26712,26714,26734],{"className":26713},[141,142],[15,26715,26717,26731],{"className":26716},[146],[15,26718,26720],{"className":26719,"style":4443},[150],[15,26721,26722,26725],{"style":4446},[15,26723],{"className":26724,"style":4450},[158],[15,26726,26727],{"style":4453},[4455,26728,26729],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,26730],{"d":4463},[15,26732,175],{"className":26733},[174],[15,26735,26737],{"className":26736},[146],[15,26738,26740],{"className":26739,"style":4473},[150],[15,26741],{},[15,26743,26745,26748],{"className":26744},[129],[15,26746,1373],{"className":26747},[129,133],[15,26749,26751],{"className":26750},[137],[15,26752,26754,26777],{"className":26753},[141,142],[15,26755,26757,26774],{"className":26756},[146],[15,26758,26760],{"className":26759,"style":1525},[150],[15,26761,26762,26765],{"style":154},[15,26763],{"className":26764,"style":159},[158],[15,26766,26768],{"className":26767},[163,164,165,166],[15,26769,26771],{"className":26770},[129,166],[15,26772,906],{"className":26773,"style":170},[129,614,166],[15,26775,175],{"className":26776},[174],[15,26778,26780],{"className":26779},[146],[15,26781,26783],{"className":26782,"style":182},[150],[15,26784],{},[15,26786,53],{"className":26787},[188],[15,26789,45],{"className":26790},[129,133],[15,26792,59],{"className":26793},[195],[15,26795],{"className":26796,"style":3277},[93],[15,26798,315],{"className":26799},[2436],[15,26801],{"className":26802,"style":3277},[93],[15,26804,26806,26809,26849,26852,26855,26858,26895,26898,26901,26904,26907],{"className":26805},[120],[15,26807],{"className":26808,"style":213},[124],[15,26810,26812,26815],{"className":26811},[129],[15,26813,1373],{"className":26814},[129,133],[15,26816,26818],{"className":26817},[137],[15,26819,26821,26841],{"className":26820},[141,142],[15,26822,26824,26838],{"className":26823},[146],[15,26825,26827],{"className":26826,"style":1525},[150],[15,26828,26829,26832],{"style":154},[15,26830],{"className":26831,"style":159},[158],[15,26833,26835],{"className":26834},[163,164,165,166],[15,26836,940],{"className":26837,"style":1013},[129,133,166],[15,26839,175],{"className":26840},[174],[15,26842,26844],{"className":26843},[146],[15,26845,26847],{"className":26846,"style":182},[150],[15,26848],{},[15,26850,53],{"className":26851},[188],[15,26853,45],{"className":26854},[129,133],[15,26856,59],{"className":26857},[195],[15,26859,26861],{"className":26860},[195],[15,26862,26864],{"className":26863},[228,4433],[15,26865,26867,26887],{"className":26866},[141,142],[15,26868,26870,26884],{"className":26869},[146],[15,26871,26873],{"className":26872,"style":4443},[150],[15,26874,26875,26878],{"style":4446},[15,26876],{"className":26877,"style":4450},[158],[15,26879,26880],{"style":4453},[4455,26881,26882],{"xmlns":4457,"width":4458,"height":73,"viewBox":4459},[4461,26883],{"d":4463},[15,26885,175],{"className":26886},[174],[15,26888,26890],{"className":26889},[146],[15,26891,26893],{"className":26892,"style":4473},[150],[15,26894],{},[15,26896],{"className":26897,"style":199},[93],[15,26899],{"className":26900,"style":199},[93],[15,26902,26567],{"className":26903},[203,13316],[15,26905],{"className":26906,"style":199},[93],[15,26908],{"className":26909,"style":199},[93],[15,26911,26913,26916,26994,26997,27050,27087,27127,27130,27133,27136,27139,27142],{"className":26912},[120],[15,26914],{"className":26915,"style":26651},[124],[15,26917,26919,26922],{"className":26918},[129],[15,26920,26574],{"className":26921},[129,285],[15,26923,26925],{"className":26924},[137],[15,26926,26928,26986],{"className":26927},[141,142],[15,26929,26931,26983],{"className":26930},[146],[15,26932,26934],{"className":26933,"style":1525},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不等式），代价是概率意义上的常数因子。",[11,27276,27277,27278,27386,27387,27457],{},"接着引入独立同分布的 Rademacher 变量 ",[15,27279,27281,27310],{"className":27280,"translate":19},[23],[15,27282,27284],{"className":27283},[27],[29,27285,27286],{"xmlns":31},[34,27287,27288,27307],{},[37,27289,27290,27297,27299,27301,27303,27305],{},[40,27291,27292,27295],{},[43,27293,27294],{},"σ",[43,27296,1459],{},[50,27298,100],{},[50,27300,636],{"stretchy":52},[50,27302,10932],{},[308,27304,318],{},[50,27306,645],{"stretchy":52},[109,27308,27309],{"encoding":111},"\\sigma_i\\in\\{\\pm1\\}",[15,27311,27313,27368],{"className":27312,"ariaHidden":72},[116],[15,27314,27316,27319,27359,27362,27365],{"className":27315},[120],[15,27317],{"className":27318,"style":7929},[124],[15,27320,27322,27325],{"className":27321},[129],[15,27323,27294],{"className":27324,"style":887},[129,133],[15,27326,27328],{"className":27327},[137],[15,27329,27331,27351],{"className":27330},[141,142],[15,27332,27334,27348],{"className":27333},[146],[15,27335,27337],{"className":27336,"style":1981},[150],[15,27338,27339,27342],{"style":1100},[15,27340],{"className":27341,"style":159},[158],[15,27343,27345],{"className":27344},[163,164,165,166],[15,27346,1459],{"className":27347},[129,133,166],[15,27349,175],{"className":27350},[174],[15,27352,27354],{"className":27353},[146],[15,27355,27357],{"className":27356,"style":182},[150],[15,27358],{},[15,27360],{"className":27361,"style":199},[93],[15,27363,100],{"className":27364},[203],[15,27366],{"className":27367,"style":199},[93],[15,27369,27371,27374,27377,27380,27383],{"className":27370},[120],[15,27372],{"className":27373,"style":125},[124],[15,27375,636],{"className":27376},[188],[15,27378,10932],{"className":27379},[129],[15,27381,318],{"className":27382},[129],[15,27384,645],{"className":27385},[195],"。由于 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",[15,28699,28701,28745],{"className":28700,"translate":19},[23],[15,28702,28704],{"className":28703},[27],[29,28705,28706],{"xmlns":31},[34,28707,28708,28742],{},[37,28709,28710,28716,28718,28720,28722,28724,28726,28728,28730,28732,28734,28736,28738,28740],{},[40,28711,28712,28714],{},[43,28713,27529],{"mathvariant":65},[43,28715,45],{},[50,28717,53],{"stretchy":52},[43,28719,27538],{},[50,28721,59],{"stretchy":52},[50,28723,62],{},[308,28725,318],{"mathvariant":1468},[50,28727,1407],{"stretchy":52},[43,28729,45],{},[50,28731,53],{"stretchy":52},[43,28733,56],{},[50,28735,59],{"stretchy":52},[50,28737,1418],{"mathvariant":65},[43,28739,857],{},[50,28741,1423],{"stretchy":52},[109,28743,28744],{"encoding":111},"\\ell_h(z)=\\mathbf 1[h(x)\\neq y]",[15,28746,28748,28812,28878],{"className":28747,"ariaHidden":72},[116],[15,28749,28751,28754,28794,28797,28800,28803,28806,28809],{"className":28750},[120],[15,28752],{"className":28753,"style":125},[124],[15,28755,28757,28760],{"className":28756},[129],[15,28758,27529],{"className":28759},[129],[15,28761,28763],{"className":28762},[137],[15,28764,28766,28786],{"className":28765},[141,142],[15,28767,28769,28783],{"className":28768},[146],[15,28770,28772],{"className":28771,"style":151},[150],[15,28773,28774,28777],{"style":154},[15,28775],{"className":28776,"style":159},[158],[15,28778,28780],{"className":28779},[163,164,165,166],[15,28781,45],{"className":28782},[129,133,166],[15,28784,175],{"className":28785},[174],[15,28787,28789],{"className":28788},[146],[15,28790,28792],{"className":28791,"style":182},[150],[15,28793],{},[15,28795,53],{"className":28796},[188],[15,28798,27538],{"className":28799,"style":28028},[129,133],[15,28801,59],{"className":28802},[195],[15,28804],{"className":28805,"style":199},[93],[15,28807,62],{"className":28808},[203],[15,28810],{"className":28811,"style":199},[93],[15,28813,28815,28818,28821,28824,28827,28830,28833,28836,28839,28875],{"className":28814},[120],[15,28816],{"className":28817,"style":125},[124],[15,28819,318],{"className":28820},[129,1950],[15,28822,1407],{"className":28823},[188],[15,28825,45],{"className":28826},[129,133],[15,28828,53],{"className":28829},[188],[15,28831,56],{"className":28832},[129,133],[15,28834,59],{"className":28835},[195],[15,28837],{"className":28838,"style":199},[93],[15,28840,28842,28869,28872],{"className":28841},[203],[15,28843,28845],{"className":28844},[203],[15,28846,28848],{"className":28847},[129,1670],[15,28849,28851],{"className":28850},[1674],[15,28852,28854,28857,28866],{"className":28853},[1678],[15,28855],{"className":28856,"style":1682},[124],[15,28858,28860],{"className":28859},[1686],[15,28861,28863],{"className":28862},[129],[15,28864,1693],{"className":28865},[203],[15,28867],{"className":28868},[1697],[15,28870],{"className":28871},[93,1701],[15,28873,62],{"className":28874},[203],[15,28876],{"className":28877,"style":199},[93],[15,28879,28881,28884,28887],{"className":28880},[120],[15,28882],{"className":28883,"style":125},[124],[15,28885,857],{"className":28886,"style":887},[129,133],[15,28888,1423],{"className":28889},[195],"。右侧就是 ",[698,28892,28893],{},"Rademacher 复杂度",[15,28895,28897,28922],{"className":28896,"translate":19},[23],[15,28898,28900],{"className":28899},[27],[29,28901,28902],{"xmlns":31},[34,28903,28904,28919],{},[37,28905,28906,28913,28915,28917],{},[40,28907,28908,28911],{},[43,28909,104],{"mathvariant":28910},"fraktur",[43,28912,348],{},[50,28914,53],{"stretchy":52},[43,28916,819],{"mathvariant":596},[50,28918,59],{"stretchy":52},[109,28920,28921],{"encoding":111},"\\mathfrak R_m(\\mathcal H)",[15,28923,28925],{"className":28924,"ariaHidden":72},[116],[15,28926,28928,28931,28972,28975,28978],{"className":28927},[120],[15,28929],{"className":28930,"style":125},[124],[15,28932,28934,28938],{"className":28933},[129],[15,28935,104],{"className":28936},[129,28937],"mathfrak",[15,28939,28941],{"className":28940},[137],[15,28942,28944,28964],{"className":28943},[141,142],[15,28945,28947,28961],{"className":28946},[146],[15,28948,28950],{"className":28949,"style":1163},[150],[15,28951,28952,28955],{"style":154},[15,28953],{"className":28954,"style":159},[158],[15,28956,28958],{"className":28957},[163,164,165,166],[15,28959,348],{"className":28960},[129,133,166],[15,28962,175],{"className":28963},[174],[15,28965,28967],{"className":28966},[146],[15,28968,28970],{"className":28969,"style":182},[150],[15,28971],{},[15,28973,53],{"className":28974},[188],[15,28976,819],{"className":28977,"style":835},[129,614],[15,28979,59],{"className":28980},[195],"，它度量的是：给每个样本随机翻一个符号，假设类还能拟合得多好。随机符号是纯噪声，能拟合噪声就说明容量大。",[11,28983,28984],{},"对二值函数类，Rademacher 复杂度可以被增长函数控制：",[15,28986,28988],{"className":28987,"translate":19},[18],[15,28989,28991,29046],{"className":28990,"translate":19},[23],[15,28992,28994],{"className":28993},[27],[29,28995,28996],{"xmlns":31,"display":32},[34,28997,28998,29043],{},[37,28999,29000,29006,29008,29010,29012,29014,29041],{},[40,29001,29002,29004],{},[43,29003,104],{"mathvariant":28910},[43,29005,348],{},[50,29007,53],{"stretchy":52},[43,29009,819],{"mathvariant":596},[50,29011,59],{"stretchy":52},[50,29013,2717],{},[29015,29016,29017],"msqrt",{},[1443,29018,29019,29039],{},[37,29020,29021,29023,29025,29027,29033,29035,29037],{},[308,29022,310],{},[43,29024,14727],{},[50,29026,69],{},[40,29028,29029,29031],{},[43,29030,19688],{"mathvariant":65},[43,29032,819],{"mathvariant":596},[50,29034,53],{"stretchy":52},[43,29036,348],{},[50,29038,59],{"stretchy":52},[43,29040,348],{},[43,29042,107],{"mathvariant":65},[109,29044,29045],{"encoding":111},"\\mathfrak R_m(\\mathcal H)\\le\\sqrt{\\frac{2\\log\\Pi_{\\mathcal H}(m)}{m}}.",[15,29047,29049,29113],{"className":29048,"ariaHidden":72},[116],[15,29050,29052,29055,29095,29098,29101,29104,29107,29110],{"className":29051},[120],[15,29053],{"className":29054,"style":125},[124],[15,29056,29058,29061],{"className":29057},[129],[15,29059,104],{"className":29060},[129,28937],[15,29062,29064],{"className":29063},[137],[15,29065,29067,29087],{"className":29066},[141,142],[15,29068,29070,29084],{"className":29069},[146],[15,29071,29073],{"className":29072,"style":1163},[150],[15,29074,29075,29078],{"style":154},[15,29076],{"className":29077,"style":159},[158],[15,29079,29081],{"className":29080},[163,164,165,166],[15,29082,348],{"className":29083},[129,133,166],[15,29085,175],{"className":29086},[174],[15,29088,29090],{"className":29089},[146],[15,29091,29093],{"className":29092,"style":182},[150],[15,29094],{},[15,29096,53],{"className":29097},[188],[15,29099,819],{"className":29100,"style":835},[129,614],[15,29102,59],{"className":29103},[195],[15,29105],{"className":29106,"style":199},[93],[15,29108,2717],{"className":29109},[203],[15,29111],{"className":29112,"style":199},[93],[15,29114,29116,29120,29305],{"className":29115},[120],[15,29117],{"className":29118,"style":29119},[124],"height:2.44em;vertical-align:-0.7356em;",[15,29121,29124],{"className":29122},[129,29123],"sqrt",[15,29125,29127,29296],{"className":29126},[141,142],[15,29128,29130,29293],{"className":29129},[146],[15,29131,29134,29273],{"className":29132,"style":29133},[150],"height:1.7044em;",[15,29135,29139,29143],{"className":29136,"style":29138},[29137],"svg-align","top:-4.4em;",[15,29140],{"className":29141,"style":29142},[158],"height:4.4em;",[15,29144,29147],{"className":29145,"style":29146},[129],"padding-left:1em;",[15,29148,29150,29153,29270],{"className":29149},[129],[15,29151],{"className":29152},[188,1803],[15,29154,29156],{"className":29155},[1443],[15,29157,29159,29262],{"className":29158},[141,142],[15,29160,29162,29259],{"className":29161},[146],[15,29163,29166,29177,29185],{"className":29164,"style":29165},[150],"height:1.427em;",[15,29167,29168,29171],{"style":1819},[15,29169],{"className":29170,"style":1823},[158],[15,29172,29174],{"className":29173},[129],[15,29175,348],{"className":29176},[129,133],[15,29178,29179,29182],{"style":1832},[15,29180],{"className":29181,"style":1823},[158],[15,29183],{"className":29184,"style":1840},[1839],[15,29186,29187,29190],{"style":1843},[15,29188],{"className":29189,"style":1823},[158],[15,29191,29193,29196,29199,29204,29207,29250,29253,29256],{"className":29192},[129],[15,29194,310],{"className":29195},[129],[15,29197],{"className":29198,"style":262},[93],[15,29200,14777,29202],{"className":29201},[217],[15,29203,2247],{"style":2246},[15,29205],{"className":29206,"style":262},[93],[15,29208,29210,29213],{"className":29209},[129],[15,29211,19688],{"className":29212},[129],[15,29214,29216],{"className":29215},[137],[15,29217,29219,29242],{"className":29218},[141,142],[15,29220,29222,29239],{"className":29221},[146],[15,29223,29225],{"className":29224,"style":1525},[150],[15,29226,29227,29230],{"style":154},[15,29228],{"className":29229,"style":159},[158],[15,29231,29233],{"className":29232},[163,164,165,166],[15,29234,29236],{"className":29235},[129,166],[15,29237,819],{"className":29238,"style":835},[129,614,166],[15,29240,175],{"className":29241},[174],[15,29243,29245],{"className":29244},[146],[15,29246,29248],{"className":29247,"style":182},[150],[15,29249],{},[15,29251,53],{"className":29252},[188],[15,29254,348],{"className":29255},[129,133],[15,29257,59],{"className":29258},[195],[15,29260,175],{"className":29261},[174],[15,29263,29265],{"className":29264},[146],[15,29266,29268],{"className":29267,"style":1862},[150],[15,29269],{},[15,29271],{"className":29272},[195,1803],[15,29274,29276,29279],{"style":29275},"top:-3.6644em;",[15,29277],{"className":29278,"style":29142},[158],[15,29280,29284],{"className":29281,"style":29283},[29282],"hide-tail","min-width:1.02em;height:2.48em;",[4455,29285,29290],{"xmlns":4457,"width":29286,"height":29287,"viewBox":29288,"preserveAspectRatio":29289},"400em","2.48em","0 0 400000 2592","xMinYMin slice",[4461,29291],{"d":29292},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[15,29294,175],{"className":29295},[174],[15,29297,29299],{"className":29298},[146],[15,29300,29303],{"className":29301,"style":29302},[150],"height:0.7356em;",[15,29304],{},[15,29306,107],{"className":29307},[129],[11,29309,29310,29311,29339,29340,29428],{},"这一步用的是有限类的 Massart 引理：把 ",[15,29312,29314,29327],{"className":29313,"translate":19},[23],[15,29315,29317],{"className":29316},[27],[29,29318,29319],{"xmlns":31},[34,29320,29321,29325],{},[37,29322,29323],{},[43,29324,819],{"mathvariant":596},[109,29326,822],{"encoding":111},[15,29328,29330],{"className":29329,"ariaHidden":72},[116],[15,29331,29333,29336],{"className":29332},[120],[15,29334],{"className":29335,"style":610},[124],[15,29337,819],{"className":29338,"style":835},[129,614]," 限制到样本上，最多只有 ",[15,29341,29343,29367],{"className":29342,"translate":19},[23],[15,29344,29346],{"className":29345},[27],[29,29347,29348],{"xmlns":31},[34,29349,29350,29364],{},[37,29351,29352,29358,29360,29362],{},[40,29353,29354,29356],{},[43,29355,19688],{"mathvariant":65},[43,29357,819],{"mathvariant":596},[50,29359,53],{"stretchy":52},[43,29361,348],{},[50,29363,59],{"stretchy":52},[109,29365,29366],{"encoding":111},"\\Pi_{\\mathcal H}(m)",[15,29368,29370],{"className":29369,"ariaHidden":72},[116],[15,29371,29373,29376,29419,29422,29425],{"className":29372},[120],[15,29374],{"className":29375,"style":125},[124],[15,29377,29379,29382],{"className":29378},[129],[15,29380,19688],{"className":29381},[129],[15,29383,29385],{"className":29384},[137],[15,29386,29388,29411],{"className":29387},[141,142],[15,29389,29391,29408],{"className":29390},[146],[15,29392,29394],{"className":29393,"style":1525},[150],[15,29395,29396,29399],{"style":154},[15,29397],{"className":29398,"style":159},[158],[15,29400,29402],{"className":29401},[163,164,165,166],[15,29403,29405],{"className":29404},[129,166],[15,29406,819],{"className":29407,"style":835},[129,614,166],[15,29409,175],{"className":29410},[174],[15,29412,29414],{"className":29413},[146],[15,29415,29417],{"className":29416,"style":182},[150],[15,29418],{},[15,29420,53],{"className":29421},[188],[15,29423,348],{"className":29424},[129,133],[15,29426,59],{"className":29427},[195]," 种不同的二分，再用 union bound 或鞅不等式。",[575,29430,29432],{"id":29431},"_42-泛化界","4.2 泛化界",[11,29434,29435,29436,14475],{},"把上面的估计串起来，得到本文唯一需要记住的公式。当 ",[15,29437,29439,29456],{"className":29438,"translate":19},[23],[15,29440,29442],{"className":29441},[27],[29,29443,29444],{"xmlns":31},[34,29445,29446,29454],{},[37,29447,29448,29450,29452],{},[43,29449,348],{},[50,29451,7473],{},[43,29453,10376],{},[109,29455,24194],{"encoding":111},[15,29457,29459,29477],{"className":29458,"ariaHidden":72},[116],[15,29460,29462,29465,29468,29471,29474],{"className":29461},[120],[15,29463],{"className":29464,"style":7757},[124],[15,29466,348],{"className":29467},[129,133],[15,29469],{"className":29470,"style":199},[93],[15,29472,7473],{"className":29473},[203],[15,29475],{"className":29476,"style":199},[93],[15,29478,29480,29483],{"className":29479},[120],[15,29481],{"className":29482,"style":522},[124],[15,29484,10376],{"className":29485},[129,133],[15,29487,29489],{"className":29488,"translate":19},[18],[15,29490,29492,29585],{"className":29491,"translate":19},[23],[15,29493,29495],{"className":29494},[27],[29,29496,29497],{"xmlns":31,"display":32},[34,29498,29499,29582],{},[37,29500,29501,29507,29509,29511,29513,29515,29517,29519,29525,29527,29529,29531,29533,29535,29537,29539,29580],{},[40,29502,29503,29505],{},[43,29504,1373],{},[43,29506,906],{"mathvariant":596},[50,29508,53],{"stretchy":52},[43,29510,45],{},[50,29512,59],{"stretchy":52},[6856,29514,26564],{},[50,29516,2717],{},[6856,29518,26564],{},[40,29520,29521,29523],{},[43,29522,1373],{},[43,29524,940],{},[50,29526,53],{"stretchy":52},[43,29528,45],{},[50,29530,59],{"stretchy":52},[6856,29532,26564],{},[50,29534,2682],{},[6856,29536,26564],{},[43,29538,5619],{},[29015,29540,29541],{},[1443,29542,29543,29578],{},[37,29544,29545,29547,29549,29551,29553,29555,29557,29559,29561,29563,29565,29567,29569,29571,29573,29576],{},[43,29546,10376],{},[43,29548,14727],{},[50,29550,69],{},[50,29552,53],{"stretchy":52},[43,29554,348],{},[43,29556,544],{"mathvariant":65},[43,29558,10376],{},[50,29560,59],{"stretchy":52},[50,29562,2682],{},[43,29564,14727],{},[50,29566,69],{},[50,29568,53],{"stretchy":52},[308,29570,318],{},[43,29572,544],{"mathvariant":65},[43,29574,29575],{},"δ",[50,29577,59],{"stretchy":52},[43,29579,348],{},[50,29581,91],{"separator":72},[109,29583,29584],{"encoding":111},"L_{\\mathcal D}(h)\\;\\le\\;L_S(h)\\;+\\;C\\sqrt{\\frac{d\\log(m\u002Fd)+\\log(1\u002F\\delta)}{m}},",[15,29586,29588,29661,29731],{"className":29587,"ariaHidden":72},[116],[15,29589,29591,29594,29637,29640,29643,29646,29649,29652,29655,29658],{"className":29590},[120],[15,29592],{"className":29593,"style":125},[124],[15,29595,29597,29600],{"className":29596},[129],[15,29598,1373],{"className":29599},[129,133],[15,29601,29603],{"className":29602},[137],[15,29604,29606,29629],{"className":29605},[141,142],[15,29607,29609,29626],{"className":29608},[146],[15,29610,29612],{"className":29611,"style":1525},[150],[15,29613,29614,29617],{"style":154},[15,29615],{"className":29616,"style":159},[158],[15,29618,29620],{"className":29619},[163,164,165,166],[15,29621,29623],{"className":29622},[129,166],[15,29624,906],{"className":29625,"style":170},[129,614,166],[15,29627,175],{"className":29628},[174],[15,29630,29632],{"className":29631},[146],[15,29633,29635],{"className":29634,"style":182},[150],[15,29636],{},[15,29638,53],{"className":29639},[188],[15,29641,45],{"className":29642},[129,133],[15,29644,59],{"className":29645},[195],[15,29647],{"className":29648,"style":199},[93],[15,29650],{"className":29651,"style":199},[93],[15,29653,2717],{"className":29654},[203],[15,29656],{"className":29657,"style":199},[93],[15,29659],{"className":29660,"style":199},[93],[15,29662,29664,29667,29707,29710,29713,29716,29719,29722,29725,29728],{"className":29663},[120],[15,29665],{"className":29666,"style":125},[124],[15,29668,29670,29673],{"className":29669},[129],[15,29671,1373],{"className":29672},[129,133],[15,29674,29676],{"className":29675},[137],[15,29677,29679,29699],{"className":29678},[141,142],[15,29680,29682,29696],{"className":29681},[146],[15,29683,29685],{"className":29684,"style":1525},[150],[15,29686,29687,29690],{"style":154},[15,29688],{"className":29689,"style":159},[158],[15,29691,29693],{"className":29692},[163,164,165,166],[15,29694,940],{"className":29695,"style":1013},[129,133,166],[15,29697,175],{"className":29698},[174],[15,29700,29702],{"className":29701},[146],[15,29703,29705],{"className":29704,"style":182},[150],[15,29706],{},[15,29708,53],{"className":29709},[188],[15,29711,45],{"className":29712},[129,133],[15,29714,59],{"className":29715},[195],[15,29717],{"className":29718,"style":199},[93],[15,29720],{"className":29721,"style":3277},[93],[15,29723,2682],{"className":29724},[2436],[15,29726],{"className":29727,"style":199},[93],[15,29729],{"className":29730,"style":3277},[93],[15,29732,29734,29737,29740,29897],{"className":29733},[120],[15,29735],{"className":29736,"style":29119},[124],[15,29738,5619],{"className":29739,"style":5665},[129,133],[15,29741,29743],{"className":29742},[129,29123],[15,29744,29746,29889],{"className":29745},[141,142],[15,29747,29749,29886],{"className":29748},[146],[15,29750,29752,29874],{"className":29751,"style":29133},[150],[15,29753,29755,29758],{"className":29754,"style":29138},[29137],[15,29756],{"className":29757,"style":29142},[158],[15,29759,29761],{"className":29760,"style":29146},[129],[15,29762,29764,29767,29871],{"className":29763},[129],[15,29765],{"className":29766},[188,1803],[15,29768,29770],{"className":29769},[1443],[15,29771,29773,29863],{"className":29772},[141,142],[15,29774,29776,29860],{"className":29775},[146],[15,29777,29779,29790,29798],{"className":29778,"style":29165},[150],[15,29780,29781,29784],{"style":1819},[15,29782],{"className":29783,"style":1823},[158],[15,29785,29787],{"className":29786},[129],[15,29788,348],{"className":29789},[129,133],[15,29791,29792,29795],{"style":1832},[15,29793],{"className":29794,"style":1823},[158],[15,29796],{"className":29797,"style":1840},[1839],[15,29799,29800,29803],{"style":1843},[15,29801],{"className":29802,"style":1823},[158],[15,29804,29806,29809,29812,29817,29820,29823,29826,29829,29832,29835,29838,29841,29846,29849,29853,29857],{"className":29805},[129],[15,29807,10376],{"className":29808},[129,133],[15,29810],{"className":29811,"style":262},[93],[15,29813,14777,29815],{"className":29814},[217],[15,29816,2247],{"style":2246},[15,29818,53],{"className":29819},[188],[15,29821,348],{"className":29822},[129,133],[15,29824,544],{"className":29825},[129],[15,29827,10376],{"className":29828},[129,133],[15,29830,59],{"className":29831},[195],[15,29833],{"className":29834,"style":3277},[93],[15,29836,2682],{"className":29837},[2436],[15,29839],{"className":29840,"style":3277},[93],[15,29842,14777,29844],{"className":29843},[217],[15,29845,2247],{"style":2246},[15,29847,53],{"className":29848},[188],[15,29850,29852],{"className":29851},[129],"1\u002F",[15,29854,29575],{"className":29855,"style":29856},[129,133],"margin-right:0.0379em;",[15,29858,59],{"className":29859},[195],[15,29861,175],{"className":29862},[174],[15,29864,29866],{"className":29865},[146],[15,29867,29869],{"className":29868,"style":1862},[150],[15,29870],{},[15,29872],{"className":29873},[195,1803],[15,29875,29876,29879],{"style":29275},[15,29877],{"className":29878,"style":29142},[158],[15,29880,29882],{"className":29881,"style":29283},[29282],[4455,29883,29884],{"xmlns":4457,"width":29286,"height":29287,"viewBox":29288,"preserveAspectRatio":29289},[4461,29885],{"d":29292},[15,29887,175],{"className":29888},[174],[15,29890,29892],{"className":29891},[146],[15,29893,29895],{"className":29894,"style":29302},[150],[15,29896],{},[15,29898,91],{"className":29899},[254],[11,29901,29902,29903,29954,29955,804,29957,30008,30009,7610,30080,30108],{},"以至少 ",[15,29904,29906,29924],{"className":29905,"translate":19},[23],[15,29907,29909],{"className":29908},[27],[29,29910,29911],{"xmlns":31},[34,29912,29913,29921],{},[37,29914,29915,29917,29919],{},[308,29916,318],{},[50,29918,315],{},[43,29920,29575],{},[109,29922,29923],{"encoding":111},"1-\\delta",[15,29925,29927,29945],{"className":29926,"ariaHidden":72},[116],[15,29928,29930,29933,29936,29939,29942],{"className":29929},[120],[15,29931],{"className":29932,"style":16484},[124],[15,29934,318],{"className":29935},[129],[15,29937],{"className":29938,"style":3277},[93],[15,29940,315],{"className":29941},[2436],[15,29943],{"className":29944,"style":3277},[93],[15,29946,29948,29951],{"className":29947},[120],[15,29949],{"className":29950,"style":522},[124],[15,29952,29575],{"className":29953,"style":29856},[129,133]," 的概率对",[698,29956,7250],{},[15,29958,29960,29978],{"className":29959,"translate":19},[23],[15,29961,29963],{"className":29962},[27],[29,29964,29965],{"xmlns":31},[34,29966,29967,29975],{},[37,29968,29969,29971,29973],{},[43,29970,45],{},[50,29972,100],{},[43,29974,819],{"mathvariant":596},[109,29976,29977],{"encoding":111},"h\\in\\mathcal H",[15,29979,29981,29999],{"className":29980,"ariaHidden":72},[116],[15,29982,29984,29987,29990,29993,29996],{"className":29983},[120],[15,29985],{"className":29986,"style":2405},[124],[15,29988,45],{"className":29989},[129,133],[15,29991],{"className":29992,"style":199},[93],[15,29994,100],{"className":29995},[203],[15,29997],{"className":29998,"style":199},[93],[15,30000,30002,30005],{"className":30001},[120],[15,30003],{"className":30004,"style":610},[124],[15,30006,819],{"className":30007,"style":835},[129,614]," 同时成立，其中 ",[15,30010,30012,30038],{"className":30011,"translate":19},[23],[15,30013,30015],{"className":30014},[27],[29,30016,30017],{"xmlns":31},[34,30018,30019,30035],{},[37,30020,30021,30023,30025,30027,30029,30031,30033],{},[43,30022,10376],{},[50,30024,62],{},[43,30026,6823],{"mathvariant":65},[50,30028,69],{},[50,30030,53],{"stretchy":52},[43,30032,819],{"mathvariant":596},[50,30034,59],{"stretchy":52},[109,30036,30037],{"encoding":111},"d=\\operatorname{VCdim}(\\mathcal H)",[15,30039,30041,30059],{"className":30040,"ariaHidden":72},[116],[15,30042,30044,30047,30050,30053,30056],{"className":30043},[120],[15,30045],{"className":30046,"style":522},[124],[15,30048,10376],{"className":30049},[129,133],[15,30051],{"className":30052,"style":199},[93],[15,30054,62],{"className":30055},[203],[15,30057],{"className":30058,"style":199},[93],[15,30060,30062,30065,30071,30074,30077],{"className":30061},[120],[15,30063],{"className":30064,"style":125},[124],[15,30066,30068],{"className":30067},[217],[15,30069,6823],{"className":30070},[129,221],[15,30072,53],{"className":30073},[188],[15,30075,819],{"className":30076,"style":835},[129,614],[15,30078,59],{"className":30079},[195],[15,30081,30083,30096],{"className":30082,"translate":19},[23],[15,30084,30086],{"className":30085},[27],[29,30087,30088],{"xmlns":31},[34,30089,30090,30094],{},[37,30091,30092],{},[43,30093,5619],{},[109,30095,5619],{"encoding":111},[15,30097,30099],{"className":30098,"ariaHidden":72},[116],[15,30100,30102,30105],{"className":30101},[120],[15,30103],{"className":30104,"style":610},[124],[15,30106,5619],{"className":30107,"style":5665},[129,133]," 是某个绝对常数。",[11,30110,30111,30112,30145,30146,30212,30213,30289],{},"不同教材的常数和 ",[15,30113,30115,30131],{"className":30114,"translate":19},[23],[15,30116,30118],{"className":30117},[27],[29,30119,30120],{"xmlns":31},[34,30121,30122,30128],{},[37,30123,30124,30126],{},[43,30125,14727],{},[50,30127,69],{},[109,30129,30130],{"encoding":111},"\\log",[15,30132,30134],{"className":30133,"ariaHidden":72},[116],[15,30135,30137,30140],{"className":30136},[120],[15,30138],{"className":30139,"style":1682},[124],[15,30141,14777,30143],{"className":30142},[217],[15,30144,2247],{"style":2246}," 里的具体形式略有出入（常见的一种写法是把 ",[15,30147,30149,30177],{"className":30148,"translate":19},[23],[15,30150,30152],{"className":30151},[27],[29,30153,30154],{"xmlns":31},[34,30155,30156,30174],{},[37,30157,30158,30160,30162,30164,30166,30168,30170,30172],{},[43,30159,10376],{},[43,30161,14727],{},[50,30163,69],{},[50,30165,53],{"stretchy":52},[43,30167,348],{},[43,30169,544],{"mathvariant":65},[43,30171,10376],{},[50,30173,59],{"stretchy":52},[109,30175,30176],{"encoding":111},"d\\log(m\u002Fd)",[15,30178,30180],{"className":30179,"ariaHidden":72},[116],[15,30181,30183,30186,30189,30192,30197,30200,30203,30206,30209],{"className":30182},[120],[15,30184],{"className":30185,"style":125},[124],[15,30187,10376],{"className":30188},[129,133],[15,30190],{"className":30191,"style":262},[93],[15,30193,14777,30195],{"className":30194},[217],[15,30196,2247],{"style":2246},[15,30198,53],{"className":30199},[188],[15,30201,348],{"className":30202},[129,133],[15,30204,544],{"className":30205},[129],[15,30207,10376],{"className":30208},[129,133],[15,30210,59],{"className":30211},[195]," 换成 ",[15,30214,30216,30248],{"className":30215,"translate":19},[23],[15,30217,30219],{"className":30218},[27],[29,30220,30221],{"xmlns":31},[34,30222,30223,30245],{},[37,30224,30225,30227,30229,30231,30233,30235,30237,30239,30241,30243],{},[43,30226,10376],{},[43,30228,14727],{},[50,30230,69],{},[50,30232,53],{"stretchy":52},[308,30234,310],{},[43,30236,10699],{},[43,30238,348],{},[43,30240,544],{"mathvariant":65},[43,30242,10376],{},[50,30244,59],{"stretchy":52},[109,30246,30247],{"encoding":111},"d\\log(2em\u002Fd)",[15,30249,30251],{"className":30250,"ariaHidden":72},[116],[15,30252,30254,30257,30260,30263,30268,30271,30274,30277,30280,30283,30286],{"className":30253},[120],[15,30255],{"className":30256,"style":125},[124],[15,30258,10376],{"className":30259},[129,133],[15,30261],{"className":30262,"style":262},[93],[15,30264,14777,30266],{"className":30265},[217],[15,30267,2247],{"style":2246},[15,30269,53],{"className":30270},[188],[15,30272,310],{"className":30273},[129],[15,30275,10699],{"className":30276},[129,133],[15,30278,348],{"className":30279},[129,133],[15,30281,544],{"className":30282},[129],[15,30284,10376],{"className":30285},[129,133],[15,30287,59],{"className":30288},[195],"），但量级是一致的。重要的是它的结构：",[21730,30291,30292,30380,30412],{},[21733,30293,30294,30295,30323,30324,30379],{},"分子里的 ",[15,30296,30298,30311],{"className":30297,"translate":19},[23],[15,30299,30301],{"className":30300},[27],[29,30302,30303],{"xmlns":31},[34,30304,30305,30309],{},[37,30306,30307],{},[43,30308,10376],{},[109,30310,10376],{"encoding":111},[15,30312,30314],{"className":30313,"ariaHidden":72},[116],[15,30315,30317,30320],{"className":30316},[120],[15,30318],{"className":30319,"style":522},[124],[15,30321,10376],{"className":30322},[129,133]," 是容量，",[15,30325,30327,30353],{"className":30326,"translate":19},[23],[15,30328,30330],{"className":30329},[27],[29,30331,30332],{"xmlns":31},[34,30333,30334,30350],{},[37,30335,30336,30338,30340,30342,30344,30346,30348],{},[43,30337,14727],{},[50,30339,69],{},[50,30341,53],{"stretchy":52},[308,30343,318],{},[43,30345,544],{"mathvariant":65},[43,30347,29575],{},[50,30349,59],{"stretchy":52},[109,30351,30352],{"encoding":111},"\\log(1\u002F\\delta)",[15,30354,30356],{"className":30355,"ariaHidden":72},[116],[15,30357,30359,30362,30367,30370,30373,30376],{"className":30358},[120],[15,30360],{"className":30361,"style":125},[124],[15,30363,14777,30365],{"className":30364},[217],[15,30366,2247],{"style":2246},[15,30368,53],{"className":30369},[188],[15,30371,29852],{"className":30372},[129],[15,30374,29575],{"className":30375,"style":29856},[129,133],[15,30377,59],{"className":30378},[195]," 是置信度；",[21733,30381,30382,30383,30411],{},"分母里的 ",[15,30384,30386,30399],{"className":30385,"translate":19},[23],[15,30387,30389],{"className":30388},[27],[29,30390,30391],{"xmlns":31},[34,30392,30393,30397],{},[37,30394,30395],{},[43,30396,348],{},[109,30398,348],{"encoding":111},[15,30400,30402],{"className":30401,"ariaHidden":72},[116],[15,30403,30405,30408],{"className":30404},[120],[15,30406],{"className":30407,"style":4383},[124],[15,30409,348],{"className":30410},[129,133]," 是样本量；",[21733,30413,30414,30415,30503],{},"整个界以 ",[15,30416,30418,30438],{"className":30417,"translate":19},[23],[15,30419,30421],{"className":30420},[27],[29,30422,30423],{"xmlns":31},[34,30424,30425,30435],{},[37,30426,30427,30429,30431],{},[308,30428,318],{},[43,30430,544],{"mathvariant":65},[29015,30432,30433],{},[43,30434,348],{},[109,30436,30437],{"encoding":111},"1\u002F\\sqrt m",[15,30439,30441],{"className":30440,"ariaHidden":72},[116],[15,30442,30444,30448,30451],{"className":30443},[120],[15,30445],{"className":30446,"style":30447},[124],"height:1.0503em;vertical-align:-0.25em;",[15,30449,29852],{"className":30450},[129],[15,30452,30454],{"className":30453},[129,29123],[15,30455,30457,30494],{"className":30456},[141,142],[15,30458,30460,30491],{"className":30459},[146],[15,30461,30464,30474],{"className":30462,"style":30463},[150],"height:0.8003em;",[15,30465,30467,30470],{"className":30466,"style":2287},[29137],[15,30468],{"className":30469,"style":1823},[158],[15,30471,348],{"className":30472,"style":30473},[129,133],"padding-left:0.833em;",[15,30475,30477,30480],{"style":30476},"top:-2.7603em;",[15,30478],{"className":30479,"style":1823},[158],[15,30481,30484],{"className":30482,"style":30483},[29282],"min-width:0.853em;height:1.08em;",[4455,30485,30488],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},"1.08em","0 0 400000 1080",[4461,30489],{"d":30490},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[15,30492,175],{"className":30493},[174],[15,30495,30497],{"className":30496},[146],[15,30498,30501],{"className":30499,"style":30500},[150],"height:0.2397em;",[15,30502],{}," 的速度下降。在 agnostic 设定下，这个速度是最优的：存在匹配的下界，样本量再少就不可能保证同样的一致收敛。",[11,30505,30506,30507,30600],{},"把 ",[15,30508,30510,30536],{"className":30509,"translate":19},[23],[15,30511,30513],{"className":30512},[27],[29,30514,30515],{"xmlns":31},[34,30516,30517,30533],{},[37,30518,30519,30525,30527,30529,30531],{},[40,30520,30521,30523],{},[43,30522,19688],{"mathvariant":65},[43,30524,819],{"mathvariant":596},[50,30526,53],{"stretchy":52},[308,30528,310],{},[43,30530,348],{},[50,30532,59],{"stretchy":52},[109,30534,30535],{"encoding":111},"\\Pi_{\\mathcal H}(2m)",[15,30537,30539],{"className":30538,"ariaHidden":72},[116],[15,30540,30542,30545,30588,30591,30594,30597],{"className":30541},[120],[15,30543],{"className":30544,"style":125},[124],[15,30546,30548,30551],{"className":30547},[129],[15,30549,19688],{"className":30550},[129],[15,30552,30554],{"className":30553},[137],[15,30555,30557,30580],{"className":30556},[141,142],[15,30558,30560,30577],{"className":30559},[146],[15,30561,30563],{"className":30562,"style":1525},[150],[15,30564,30565,30568],{"style":154},[15,30566],{"className":30567,"style":159},[158],[15,30569,30571],{"className":30570},[163,164,165,166],[15,30572,30574],{"className":30573},[129,166],[15,30575,819],{"className":30576,"style":835},[129,614,166],[15,30578,175],{"className":30579},[174],[15,30581,30583],{"className":30582},[146],[15,30584,30586],{"className":30585,"style":182},[150],[15,30587],{},[15,30589,53],{"className":30590},[188],[15,30592,310],{"className":30593},[129],[15,30595,348],{"className":30596},[129,133],[15,30598,59],{"className":30599},[195]," 用 Sauer 界代入，就得到只依赖 VC 维的形式。再把它代回误差分解，就得到 ERM 的泛化保证。",[575,30602,30604],{"id":30603},"_43-基本定理","4.3 基本定理",[6559,30606,30607,30671],{},[11,30608,30609,30612,30613,30641,30642,30670],{},[698,30610,30611],{},"定理 4.1（统计学习基本定理）"," 对二分类、",[15,30614,30616,30629],{"className":30615,"translate":19},[23],[15,30617,30619],{"className":30618},[27],[29,30620,30621],{"xmlns":31},[34,30622,30623,30627],{},[37,30624,30625],{},[308,30626,541],{},[109,30628,541],{"encoding":111},[15,30630,30632],{"className":30631,"ariaHidden":72},[116],[15,30633,30635,30638],{"className":30634},[120],[15,30636],{"className":30637,"style":4886},[124],[15,30639,541],{"className":30640},[129],"-",[15,30643,30645,30658],{"className":30644,"translate":19},[23],[15,30646,30648],{"className":30647},[27],[29,30649,30650],{"xmlns":31},[34,30651,30652,30656],{},[37,30653,30654],{},[308,30655,318],{},[109,30657,318],{"encoding":111},[15,30659,30661],{"className":30660,"ariaHidden":72},[116],[15,30662,30664,30667],{"className":30663},[120],[15,30665],{"className":30666,"style":4886},[124],[15,30668,318],{"className":30669},[129]," 损失、分布无关（distribution-free）的设定，以下四条等价：",[30672,30673,30674,30747,30778,30781],"ol",{},[21733,30675,30676,9211],{},[15,30677,30679,30705],{"className":30678,"translate":19},[23],[15,30680,30682],{"className":30681},[27],[29,30683,30684],{"xmlns":31},[34,30685,30686,30702],{},[37,30687,30688,30690,30692,30694,30696,30698,30700],{},[43,30689,6823],{"mathvariant":65},[50,30691,69],{},[50,30693,53],{"stretchy":52},[43,30695,819],{"mathvariant":596},[50,30697,59],{"stretchy":52},[50,30699,7910],{},[43,30701,4909],{"mathvariant":65},[109,30703,30704],{"encoding":111},"\\operatorname{VCdim}(\\mathcal H)\u003C\\infty",[15,30706,30708,30738],{"className":30707,"ariaHidden":72},[116],[15,30709,30711,30714,30720,30723,30726,30729,30732,30735],{"className":30710},[120],[15,30712],{"className":30713,"style":125},[124],[15,30715,30717],{"className":30716},[217],[15,30718,6823],{"className":30719},[129,221],[15,30721,53],{"className":30722},[188],[15,30724,819],{"className":30725,"style":835},[129,614],[15,30727,59],{"className":30728},[195],[15,30730],{"className":30731,"style":199},[93],[15,30733,7910],{"className":30734},[203],[15,30736],{"className":30737,"style":199},[93],[15,30739,30741,30744],{"className":30740},[120],[15,30742],{"className":30743,"style":4383},[124],[15,30745,4909],{"className":30746},[129],[21733,30748,30749,30777],{},[15,30750,30752,30765],{"className":30751,"translate":19},[23],[15,30753,30755],{"className":30754},[27],[29,30756,30757],{"xmlns":31},[34,30758,30759,30763],{},[37,30760,30761],{},[43,30762,819],{"mathvariant":596},[109,30764,822],{"encoding":111},[15,30766,30768],{"className":30767,"ariaHidden":72},[116],[15,30769,30771,30774],{"className":30770},[120],[15,30772],{"className":30773,"style":610},[124],[15,30775,819],{"className":30776,"style":835},[129,614]," 具有一致收敛性；",[21733,30779,30780],{},"ERM 是 agnostic PAC 一致的；",[21733,30782,30783,30811],{},[15,30784,30786,30799],{"className":30785,"translate":19},[23],[15,30787,30789],{"className":30788},[27],[29,30790,30791],{"xmlns":31},[34,30792,30793,30797],{},[37,30794,30795],{},[43,30796,819],{"mathvariant":596},[109,30798,822],{"encoding":111},[15,30800,30802],{"className":30801,"ariaHidden":72},[116],[15,30803,30805,30808],{"className":30804},[120],[15,30806],{"className":30807,"style":610},[124],[15,30809,819],{"className":30810,"style":835},[129,614]," 是 agnostic PAC 可学习的。",[11,30813,30814,30815],{},"在可实现（realizable）情形下，有限 VC 维同样等价于 PAC 可学习。这个定理把「组合上的有限性」和「统计上的可学习性」严格地焊在了一起：",[698,30816,30817],{},"在分布无关的框架里，有限 VC 维不只是充分条件，而是充要条件。",[575,30819,30821],{"id":30820},"_44-样本复杂度","4.4 样本复杂度",[11,30823,30824,30825,30853],{},"由上面的界解出 ",[15,30826,30828,30841],{"className":30827,"translate":19},[23],[15,30829,30831],{"className":30830},[27],[29,30832,30833],{"xmlns":31},[34,30834,30835,30839],{},[37,30836,30837],{},[43,30838,348],{},[109,30840,348],{"encoding":111},[15,30842,30844],{"className":30843,"ariaHidden":72},[116],[15,30845,30847,30850],{"className":30846},[120],[15,30848],{"className":30849,"style":4383},[124],[15,30851,348],{"className":30852},[129,133],"，得到 agnostic 情形下的样本复杂度：",[15,30855,30857],{"className":30856,"translate":19},[18],[15,30858,30860,30915],{"className":30859,"translate":19},[23],[15,30861,30863],{"className":30862},[27],[29,30864,30865],{"xmlns":31,"display":32},[34,30866,30867,30912],{},[37,30868,30869,30871,30873,30876,30910],{},[43,30870,348],{},[50,30872,62],{},[43,30874,30875],{"mathvariant":65},"Θ",[37,30877,30878,30880,30908],{},[50,30879,53],{"fence":72},[1443,30881,30882,30902],{},[37,30883,30884,30886,30888,30890,30892,30894,30896,30898,30900],{},[43,30885,10376],{},[50,30887,2682],{},[43,30889,14727],{},[50,30891,69],{},[50,30893,53],{"stretchy":52},[308,30895,318],{},[43,30897,544],{"mathvariant":65},[43,30899,29575],{},[50,30901,59],{"stretchy":52},[305,30903,30904,30906],{},[43,30905,4849],{},[308,30907,310],{},[50,30909,59],{"fence":72},[50,30911,91],{"separator":72},[109,30913,30914],{"encoding":111},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon^2}\\right),",[15,30916,30918,30936],{"className":30917,"ariaHidden":72},[116],[15,30919,30921,30924,30927,30930,30933],{"className":30920},[120],[15,30922],{"className":30923,"style":4383},[124],[15,30925,348],{"className":30926},[129,133],[15,30928],{"className":30929,"style":199},[93],[15,30931,62],{"className":30932},[203],[15,30934],{"className":30935,"style":199},[93],[15,30937,30939,30942,30945,30948,31078,31081],{"className":30938},[120],[15,30940],{"className":30941,"style":17014},[124],[15,30943,30875],{"className":30944},[129],[15,30946],{"className":30947,"style":262},[93],[15,30949,30951,30957,31072],{"className":30950},[449],[15,30952,30954],{"className":30953,"style":5052},[188,5051],[15,30955,53],{"className":30956},[228,165],[15,30958,30960,30963,31069],{"className":30959},[129],[15,30961],{"className":30962},[188,1803],[15,30964,30966],{"className":30965},[1443],[15,30967,30969,31061],{"className":30968},[141,142],[15,30970,30972,31058],{"className":30971},[146],[15,30973,30975,31013,31021],{"className":30974,"style":29165},[150],[15,30976,30977,30980],{"style":1819},[15,30978],{"className":30979,"style":1823},[158],[15,30981,30983],{"className":30982},[129],[15,30984,30986,30989],{"className":30985},[129],[15,30987,4849],{"className":30988},[129,133],[15,30990,30992],{"className":30991},[137],[15,30993,30995],{"className":30994},[141],[15,30996,30998],{"className":30997},[146],[15,30999,31002],{"className":31000,"style":31001},[150],"height:0.7401em;",[15,31003,31004,31007],{"style":17201},[15,31005],{"className":31006,"style":159},[158],[15,31008,31010],{"className":31009},[163,164,165,166],[15,31011,310],{"className":31012},[129,166],[15,31014,31015,31018],{"style":1832},[15,31016],{"className":31017,"style":1823},[158],[15,31019],{"className":31020,"style":1840},[1839],[15,31022,31023,31026],{"style":1843},[15,31024],{"className":31025,"style":1823},[158],[15,31027,31029,31032,31035,31038,31041,31046,31049,31052,31055],{"className":31028},[129],[15,31030,10376],{"className":31031},[129,133],[15,31033],{"className":31034,"style":3277},[93],[15,31036,2682],{"className":31037},[2436],[15,31039],{"className":31040,"style":3277},[93],[15,31042,14777,31044],{"className":31043},[217],[15,31045,2247],{"style":2246},[15,31047,53],{"className":31048},[188],[15,31050,29852],{"className":31051},[129],[15,31053,29575],{"className":31054,"style":29856},[129,133],[15,31056,59],{"className":31057},[195],[15,31059,175],{"className":31060},[174],[15,31062,31064],{"className":31063},[146],[15,31065,31067],{"className":31066,"style":1862},[150],[15,31068],{},[15,31070],{"className":31071},[195,1803],[15,31073,31075],{"className":31074,"style":5052},[195,5051],[15,31076,59],{"className":31077},[228,165],[15,31079],{"className":31080,"style":262},[93],[15,31082,91],{"className":31083},[254],[11,31085,31086],{},"可实现情形下则是",[15,31088,31090],{"className":31089,"translate":19},[18],[15,31091,31093,31143],{"className":31092,"translate":19},[23],[15,31094,31096],{"className":31095},[27],[29,31097,31098],{"xmlns":31,"display":32},[34,31099,31100,31140],{},[37,31101,31102,31104,31106,31108,31138],{},[43,31103,348],{},[50,31105,62],{},[43,31107,30875],{"mathvariant":65},[37,31109,31110,31112,31136],{},[50,31111,53],{"fence":72},[1443,31113,31114,31134],{},[37,31115,31116,31118,31120,31122,31124,31126,31128,31130,31132],{},[43,31117,10376],{},[50,31119,2682],{},[43,31121,14727],{},[50,31123,69],{},[50,31125,53],{"stretchy":52},[308,31127,318],{},[43,31129,544],{"mathvariant":65},[43,31131,29575],{},[50,31133,59],{"stretchy":52},[43,31135,4849],{},[50,31137,59],{"fence":72},[43,31139,107],{"mathvariant":65},[109,31141,31142],{"encoding":111},"m=\\Theta\\left(\\frac{d+\\log(1\u002F\\delta)}{\\varepsilon}\\right).",[15,31144,31146,31164],{"className":31145,"ariaHidden":72},[116],[15,31147,31149,31152,31155,31158,31161],{"className":31148},[120],[15,31150],{"className":31151,"style":4383},[124],[15,31153,348],{"className":31154},[129,133],[15,31156],{"className":31157,"style":199},[93],[15,31159,62],{"className":31160},[203],[15,31162],{"className":31163,"style":199},[93],[15,31165,31167,31170,31173,31176,31279,31282],{"className":31166},[120],[15,31168],{"className":31169,"style":17014},[124],[15,31171,30875],{"className":31172},[129],[15,31174],{"className":31175,"style":262},[93],[15,31177,31179,31185,31273],{"className":31178},[449],[15,31180,31182],{"className":31181,"style":5052},[188,5051],[15,31183,53],{"className":31184},[228,165],[15,31186,31188,31191,31270],{"className":31187},[129],[15,31189],{"className":31190},[188,1803],[15,31192,31194],{"className":31193},[1443],[15,31195,31197,31262],{"className":31196},[141,142],[15,31198,31200,31259],{"className":31199},[146],[15,31201,31203,31214,31222],{"className":31202,"style":29165},[150],[15,31204,31205,31208],{"style":1819},[15,31206],{"className":31207,"style":1823},[158],[15,31209,31211],{"className":31210},[129],[15,31212,4849],{"className":31213},[129,133],[15,31215,31216,31219],{"style":1832},[15,31217],{"className":31218,"style":1823},[158],[15,31220],{"className":31221,"style":1840},[1839],[15,31223,31224,31227],{"style":1843},[15,31225],{"className":31226,"style":1823},[158],[15,31228,31230,31233,31236,31239,31242,31247,31250,31253,31256],{"className":31229},[129],[15,31231,10376],{"className":31232},[129,133],[15,31234],{"className":31235,"style":3277},[93],[15,31237,2682],{"className":31238},[2436],[15,31240],{"className":31241,"style":3277},[93],[15,31243,14777,31245],{"className":31244},[217],[15,31246,2247],{"style":2246},[15,31248,53],{"className":31249},[188],[15,31251,29852],{"className":31252},[129],[15,31254,29575],{"className":31255,"style":29856},[129,133],[15,31257,59],{"className":31258},[195],[15,31260,175],{"className":31261},[174],[15,31263,31265],{"className":31264},[146],[15,31266,31268],{"className":31267,"style":1862},[150],[15,31269],{},[15,31271],{"className":31272},[195,1803],[15,31274,31276],{"className":31275,"style":5052},[195,5051],[15,31277,59],{"className":31278},[228,165],[15,31280],{"className":31281,"style":262},[93],[15,31283,107],{"className":31284},[129],[11,31286,31287,31288,31317,31318,31321],{},"这两个 ",[15,31289,31291,31305],{"className":31290,"translate":19},[23],[15,31292,31294],{"className":31293},[27],[29,31295,31296],{"xmlns":31},[34,31297,31298,31302],{},[37,31299,31300],{},[43,31301,30875],{"mathvariant":65},[109,31303,31304],{"encoding":111},"\\Theta",[15,31306,31308],{"className":31307,"ariaHidden":72},[116],[15,31309,31311,31314],{"className":31310},[120],[15,31312],{"className":31313,"style":610},[124],[15,31315,30875],{"className":31316},[129]," 是双向的：上界由 ERM 达到，下界则是信息论下界——存在一个分布，使任何算法都需要这么多样本。所以 VC 维不仅给出了一个可用的界，还在 minimax 意义下给出了",[698,31319,31320],{},"最优","的样本复杂度量级。",[11,31323,31324],{},"这一点很重要。很多人批评 VC 界「太松」，但在分布无关、最坏情况的设定里，它并不松：你不可能用更少的样本保证同样的一致收敛。问题不在于界本身，而在于这个设定是否匹配你关心的场景。",[570,31326,31328],{"id":31327},"五vc-维的含义","五、VC 维的含义",[11,31330,31331],{},"到这里，数学已经完整了。但公式正确不等于理解。这一节把 VC 维的语义拆成五句话。",[11,31333,31334,31337,31338,31414,31415,31492,31493,1313],{},[698,31335,31336],{},"第一，它是 min-max 的。"," 定义里的 ",[15,31339,31341,31363],{"className":31340,"translate":19},[23],[15,31342,31344],{"className":31343},[27],[29,31345,31346],{"xmlns":31},[34,31347,31348,31360],{},[37,31349,31350],{},[40,31351,31352,31358],{},[37,31353,31354,31356],{},[43,31355,2932],{},[50,31357,69],{},[43,31359,5619],{},[109,31361,31362],{"encoding":111},"\\sup_C",[15,31364,31366],{"className":31365,"ariaHidden":72},[116],[15,31367,31369,31373],{"className":31368},[120],[15,31370],{"className":31371,"style":31372},[124],"height:0.6747em;vertical-align:-0.2441em;",[15,31374,31376,31379],{"className":31375},[217],[15,31377,2932],{"className":31378},[217],[15,31380,31382],{"className":31381},[137],[15,31383,31385,31406],{"className":31384},[141,142],[15,31386,31388,31403],{"className":31387},[146],[15,31389,31392],{"className":31390,"style":31391},[150],"height:0.2342em;",[15,31393,31394,31397],{"style":14795},[15,31395],{"className":31396,"style":159},[158],[15,31398,31400],{"className":31399},[163,164,165,166],[15,31401,5619],{"className":31402,"style":5665},[129,133,166],[15,31404,175],{"className":31405},[174],[15,31407,31409],{"className":31408},[146],[15,31410,31412],{"className":31411,"style":14814},[150],[15,31413],{}," 对应界里的 ",[15,31416,31418,31440],{"className":31417,"translate":19},[23],[15,31419,31421],{"className":31420},[27],[29,31422,31423],{"xmlns":31},[34,31424,31425,31437],{},[37,31426,31427],{},[40,31428,31429,31435],{},[37,31430,31431,31433],{},[43,31432,2932],{},[50,31434,69],{},[43,31436,906],{"mathvariant":596},[109,31438,31439],{"encoding":111},"\\sup_{\\mathcal D}",[15,31441,31443],{"className":31442,"ariaHidden":72},[116],[15,31444,31446,31449],{"className":31445},[120],[15,31447],{"className":31448,"style":31372},[124],[15,31450,31452,31455],{"className":31451},[217],[15,31453,2932],{"className":31454},[217],[15,31456,31458],{"className":31457},[137],[15,31459,31461,31484],{"className":31460},[141,142],[15,31462,31464,31481],{"className":31463},[146],[15,31465,31467],{"className":31466,"style":31391},[150],[15,31468,31469,31472],{"style":14795},[15,31470],{"className":31471,"style":159},[158],[15,31473,31475],{"className":31474},[163,164,165,166],[15,31476,31478],{"className":31477},[129,166],[15,31479,906],{"className":31480,"style":170},[129,614,166],[15,31482,175],{"className":31483},[174],[15,31485,31487],{"className":31486},[146],[15,31488,31490],{"className":31489,"style":14814},[150],[15,31491],{},"。VC 理论先让对手选一个最坏的分布，再让对手选一个最坏的点集，然后问：你还剩什么保证？所以它天生悲观。它给出的不是一个典型情形下的估计，而是一个",[698,31494,31495],{},"任何情形下都不被打破的承诺",[11,31497,31498,5831,31501,30641,31529,31557],{},[698,31499,31500],{},"第二，它是充要的。",[15,31502,31504,31517],{"className":31503,"translate":19},[23],[15,31505,31507],{"className":31506},[27],[29,31508,31509],{"xmlns":31},[34,31510,31511,31515],{},[37,31512,31513],{},[308,31514,541],{},[109,31516,541],{"encoding":111},[15,31518,31520],{"className":31519,"ariaHidden":72},[116],[15,31521,31523,31526],{"className":31522},[120],[15,31524],{"className":31525,"style":4886},[124],[15,31527,541],{"className":31528},[129],[15,31530,31532,31545],{"className":31531,"translate":19},[23],[15,31533,31535],{"className":31534},[27],[29,31536,31537],{"xmlns":31},[34,31538,31539,31543],{},[37,31540,31541],{},[308,31542,318],{},[109,31544,318],{"encoding":111},[15,31546,31548],{"className":31547,"ariaHidden":72},[116],[15,31549,31551,31554],{"className":31550},[120],[15,31552],{"className":31553,"style":4886},[124],[15,31555,318],{"className":31556},[129]," 损失、二分类、要求对所有分布一致的框架里，有限 VC 维精确刻画了可学习性。它不是「一个充分但过强的条件」；它就是那个条件。问题不在定理，而在框架。",[11,31559,31560,31563],{},[698,31561,31562],{},"第三，它是组合量，概率只出现在最后一步。"," 打散、增长函数、Sauer 引理都是纯组合的，与分布、与概率无关；对称化把组合量变成概率界。这也解释了为什么 VC 维对数据长什么样一无所知：它的输入只有假设类的集合结构。",[11,31565,31566,31569,31570,31634],{},[698,31567,31568],{},"第四，它衡量的是「能实现多少种二分」，不是「能拟合多复杂的函数」。"," 一个类可以 VC 维无穷却在某个分布上表现很好（1-NN），也可以 VC 维很小却拟合得很差（区间无法实现 ",[15,31571,31573,31598],{"className":31572,"translate":19},[23],[15,31574,31576],{"className":31575},[27],[29,31577,31578],{"xmlns":31},[34,31579,31580,31596],{},[37,31581,31582,31584,31586,31588,31590,31592,31594],{},[50,31583,53],{"stretchy":52},[308,31585,318],{},[50,31587,91],{"separator":72},[308,31589,541],{},[50,31591,91],{"separator":72},[308,31593,318],{},[50,31595,59],{"stretchy":52},[109,31597,10071],{"encoding":111},[15,31599,31601],{"className":31600,"ariaHidden":72},[116],[15,31602,31604,31607,31610,31613,31616,31619,31622,31625,31628,31631],{"className":31603},[120],[15,31605],{"className":31606,"style":125},[124],[15,31608,53],{"className":31609},[188],[15,31611,318],{"className":31612},[129],[15,31614,91],{"className":31615},[254],[15,31617],{"className":31618,"style":262},[93],[15,31620,541],{"className":31621},[129],[15,31623,91],{"className":31624},[254],[15,31626],{"className":31627,"style":262},[93],[15,31629,318],{"className":31630},[129],[15,31632,59],{"className":31633},[195]," 这种标注）。VC 维是容量，不是质量。",[11,31636,31637,804,31640,31800,31801,31829],{},[698,31638,31639],{},"第五，它是上界，不是预测。",[15,31641,31643,31674],{"className":31642,"translate":19},[23],[15,31644,31646],{"className":31645},[27],[29,31647,31648],{"xmlns":31},[34,31649,31650,31671],{},[37,31651,31652,31658,31660,31666,31668],{},[40,31653,31654,31656],{},[43,31655,1373],{},[43,31657,906],{"mathvariant":596},[50,31659,2717],{},[40,31661,31662,31664],{},[43,31663,1373],{},[43,31665,940],{},[50,31667,2682],{},[50,31669,31670],{},"⋯",[109,31672,31673],{"encoding":111},"L_{\\mathcal D}\\le L_S+\\cdots",[15,31675,31677,31735,31790],{"className":31676,"ariaHidden":72},[116],[15,31678,31680,31683,31726,31729,31732],{"className":31679},[120],[15,31681],{"className":31682,"style":20229},[124],[15,31684,31686,31689],{"className":31685},[129],[15,31687,1373],{"className":31688},[129,133],[15,31690,31692],{"className":31691},[137],[15,31693,31695,31718],{"className":31694},[141,142],[15,31696,31698,31715],{"className":31697},[146],[15,31699,31701],{"className":31700,"style":1525},[150],[15,31702,31703,31706],{"style":154},[15,31704],{"className":31705,"style":159},[158],[15,31707,31709],{"className":31708},[163,164,165,166],[15,31710,31712],{"className":31711},[129,166],[15,31713,906],{"className":31714,"style":170},[129,614,166],[15,31716,175],{"className":31717},[174],[15,31719,31721],{"className":31720},[146],[15,31722,31724],{"className":31723,"style":182},[150],[15,31725],{},[15,31727],{"className":31728,"style":199},[93],[15,31730,2717],{"className":31731},[203],[15,31733],{"className":31734,"style":199},[93],[15,31736,31738,31741,31781,31784,31787],{"className":31737},[120],[15,31739],{"className":31740,"style":20229},[124],[15,31742,31744,31747],{"className":31743},[129],[15,31745,1373],{"className":31746},[129,133],[15,31748,31750],{"className":31749},[137],[15,31751,31753,31773],{"className":31752},[141,142],[15,31754,31756,31770],{"className":31755},[146],[15,31757,31759],{"className":31758,"style":1525},[150],[15,31760,31761,31764],{"style":154},[15,31762],{"className":31763,"style":159},[158],[15,31765,31767],{"className":31766},[163,164,165,166],[15,31768,940],{"className":31769,"style":1013},[129,133,166],[15,31771,175],{"className":31772},[174],[15,31774,31776],{"className":31775},[146],[15,31777,31779],{"className":31778,"style":182},[150],[15,31780],{},[15,31782],{"className":31783,"style":3277},[93],[15,31785,2682],{"className":31786},[2436],[15,31788],{"className":31789,"style":3277},[93],[15,31791,31793,31797],{"className":31792},[120],[15,31794],{"className":31795,"style":31796},[124],"height:0.313em;",[15,31798,31670],{"className":31799},[449]," 是一个保证（guarantee），不是一个估计（estimate）。当界大于 ",[15,31802,31804,31817],{"className":31803,"translate":19},[23],[15,31805,31807],{"className":31806},[27],[29,31808,31809],{"xmlns":31},[34,31810,31811,31815],{},[37,31812,31813],{},[308,31814,318],{},[109,31816,318],{"encoding":111},[15,31818,31820],{"className":31819,"ariaHidden":72},[116],[15,31821,31823,31826],{"className":31822},[120],[15,31824],{"className":31825,"style":4886},[124],[15,31827,318],{"className":31828},[129]," 时，它没有说「误差很大」，它只是没有信息。把泛化界当成测试误差的预测器，是误用的常见来源。",[11,31831,31832,31833,31836],{},"这五句话合起来，解释了为什么 VC 理论在深度学习时代显得「失灵」：它回答的是一个比深度学习实际面对的问题",[698,31834,31835],{},"更难","的问题。深度网络不是在一个任意的分布上学习，而是在图像、文本、语音这些有强结构的分布上学习；不是用任意算法，而是用 SGD 及其变体；不是在整个假设类里搜索，而是在一个由初始化、架构和训练动力学共同决定的子集里搜索。VC 维对这三件事都不敏感。",[31838,31839,31840],"div",{},[31841,31842],"img",{"height":31843,"width":31844,"style":31845,"data-d2-theme":31846,"alt":31847,"src":31848},68,760,"max-width:100%;height:auto","default","VC 理论的逻辑链：从打散到泛化界","data:image\u002Fsvg+xml;base64,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\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",[570,31850,31852],{"id":31851},"六深度学习中的-vc-维","六、深度学习中的 VC 维",[575,31854,31856],{"id":31855},"_61-参数量估算","6.1 参数量估算",[11,31858,31859,31860,31956,31957,31985,31986,32015,32016,32048],{},"把上一节的界套到实际模型上，只需估算 ",[15,31861,31863,31885],{"className":31862,"translate":19},[23],[15,31864,31866],{"className":31865},[27],[29,31867,31868],{"xmlns":31},[34,31869,31870,31882],{},[37,31871,31872],{},[29015,31873,31874],{},[37,31875,31876,31878,31880],{},[43,31877,10376],{},[43,31879,544],{"mathvariant":65},[43,31881,348],{},[109,31883,31884],{"encoding":111},"\\sqrt{d\u002Fm}",[15,31886,31888],{"className":31887,"ariaHidden":72},[116],[15,31889,31891,31895],{"className":31890},[120],[15,31892],{"className":31893,"style":31894},[124],"height:1.24em;vertical-align:-0.305em;",[15,31896,31898],{"className":31897},[129,29123],[15,31899,31901,31947],{"className":31900},[141,142],[15,31902,31904,31944],{"className":31903},[146],[15,31905,31908,31927],{"className":31906,"style":31907},[150],"height:0.935em;",[15,31909,31912,31915],{"className":31910,"style":31911},[29137],"top:-3.2em;",[15,31913],{"className":31914,"style":4450},[158],[15,31916,31918,31921,31924],{"className":31917,"style":29146},[129],[15,31919,10376],{"className":31920},[129,133],[15,31922,544],{"className":31923},[129],[15,31925,348],{"className":31926},[129,133],[15,31928,31930,31933],{"style":31929},"top:-2.895em;",[15,31931],{"className":31932,"style":4450},[158],[15,31934,31937],{"className":31935,"style":31936},[29282],"min-width:1.02em;height:1.28em;",[4455,31938,31941],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},"1.28em","0 0 400000 1296",[4461,31942],{"d":31943},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[15,31945,175],{"className":31946},[174],[15,31948,31950],{"className":31949},[146],[15,31951,31954],{"className":31952,"style":31953},[150],"height:0.305em;",[15,31955],{}," 的量级（暂时把 ",[15,31958,31960,31973],{"className":31959,"translate":19},[23],[15,31961,31963],{"className":31962},[27],[29,31964,31965],{"xmlns":31},[34,31966,31967,31971],{},[37,31968,31969],{},[43,31970,10376],{},[109,31972,10376],{"encoding":111},[15,31974,31976],{"className":31975,"ariaHidden":72},[116],[15,31977,31979,31982],{"className":31978},[120],[15,31980],{"className":31981,"style":522},[124],[15,31983,10376],{"className":31984},[129,133]," 低估成参数量 ",[15,31987,31989,32003],{"className":31988,"translate":19},[23],[15,31990,31992],{"className":31991},[27],[29,31993,31994],{"xmlns":31},[34,31995,31996,32001],{},[37,31997,31998],{},[43,31999,32000],{},"W",[109,32002,32000],{"encoding":111},[15,32004,32006],{"className":32005,"ariaHidden":72},[116],[15,32007,32009,32012],{"className":32008},[120],[15,32010],{"className":32011,"style":610},[124],[15,32013,32000],{"className":32014,"style":12449},[129,133],"，忽略 ",[15,32017,32019,32034],{"className":32018,"translate":19},[23],[15,32020,32022],{"className":32021},[27],[29,32023,32024],{"xmlns":31},[34,32025,32026,32032],{},[37,32027,32028,32030],{},[43,32029,14727],{},[50,32031,69],{},[109,32033,30130],{"encoding":111},[15,32035,32037],{"className":32036,"ariaHidden":72},[116],[15,32038,32040,32043],{"className":32039},[120],[15,32041],{"className":32042,"style":1682},[124],[15,32044,14777,32046],{"className":32045},[217],[15,32047,2247],{"style":2246}," 因子）：",[18040,32050,32051,32209],{},[18043,32052,32053],{},[18046,32054,32055,32058,32089,32120],{},[18049,32056,32057],{},"模型",[18049,32059,32060,32061],{},"参数量 ",[15,32062,32064,32077],{"className":32063,"translate":19},[23],[15,32065,32067],{"className":32066},[27],[29,32068,32069],{"xmlns":31},[34,32070,32071,32075],{},[37,32072,32073],{},[43,32074,32000],{},[109,32076,32000],{"encoding":111},[15,32078,32080],{"className":32079,"ariaHidden":72},[116],[15,32081,32083,32086],{"className":32082},[120],[15,32084],{"className":32085,"style":610},[124],[15,32087,32000],{"className":32088,"style":12449},[129,133],[18049,32090,32091,32092],{},"训练样本 ",[15,32093,32095,32108],{"className":32094,"translate":19},[23],[15,32096,32098],{"className":32097},[27],[29,32099,32100],{"xmlns":31},[34,32101,32102,32106],{},[37,32103,32104],{},[43,32105,348],{},[109,32107,348],{"encoding":111},[15,32109,32111],{"className":32110,"ariaHidden":72},[116],[15,32112,32114,32117],{"className":32113},[120],[15,32115],{"className":32116,"style":4383},[124],[15,32118,348],{"className":32119},[129,133],[18049,32121,32122],{},[15,32123,32125,32147],{"className":32124,"translate":19},[23],[15,32126,32128],{"className":32127},[27],[29,32129,32130],{"xmlns":31},[34,32131,32132,32144],{},[37,32133,32134],{},[29015,32135,32136],{},[37,32137,32138,32140,32142],{},[43,32139,32000],{},[43,32141,544],{"mathvariant":65},[43,32143,348],{},[109,32145,32146],{"encoding":111},"\\sqrt{W\u002Fm}",[15,32148,32150],{"className":32149,"ariaHidden":72},[116],[15,32151,32153,32156],{"className":32152},[120],[15,32154],{"className":32155,"style":31894},[124],[15,32157,32159],{"className":32158},[129,29123],[15,32160,32162,32201],{"className":32161},[141,142],[15,32163,32165,32198],{"className":32164},[146],[15,32166,32168,32186],{"className":32167,"style":31907},[150],[15,32169,32171,32174],{"className":32170,"style":31911},[29137],[15,32172],{"className":32173,"style":4450},[158],[15,32175,32177,32180,32183],{"className":32176,"style":29146},[129],[15,32178,32000],{"className":32179,"style":12449},[129,133],[15,32181,544],{"className":32182},[129],[15,32184,348],{"className":32185},[129,133],[15,32187,32188,32191],{"style":31929},[15,32189],{"className":32190,"style":4450},[158],[15,32192,32194],{"className":32193,"style":31936},[29282],[4455,32195,32196],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},[4461,32197],{"d":31943},[15,32199,175],{"className":32200},[174],[15,32202,32204],{"className":32203},[146],[15,32205,32207],{"className":32206,"style":31953},[150],[15,32208],{},[18057,32210,32211,32436,32662,32885],{},[18046,32212,32213,32216,32304,32389],{},[18062,32214,32215],{},"LeNet-5（MNIST）",[18062,32217,32218],{},[15,32219,32221,32245],{"className":32220,"translate":19},[23],[15,32222,32224],{"className":32223},[27],[29,32225,32226],{"xmlns":31},[34,32227,32228,32242],{},[37,32229,32230,32233,32235],{},[308,32231,32232],{},"6",[50,32234,11422],{},[305,32236,32237,32240],{},[308,32238,32239],{},"10",[308,32241,14508],{},[109,32243,32244],{"encoding":111},"6\\times10^4",[15,32246,32248,32266],{"className":32247,"ariaHidden":72},[116],[15,32249,32251,32254,32257,32260,32263],{"className":32250},[120],[15,32252],{"className":32253,"style":16484},[124],[15,32255,32232],{"className":32256},[129],[15,32258],{"className":32259,"style":3277},[93],[15,32261,11422],{"className":32262},[2436],[15,32264],{"className":32265,"style":3277},[93],[15,32267,32269,32272,32275],{"className":32268},[120],[15,32270],{"className":32271,"style":380},[124],[15,32273,318],{"className":32274},[129],[15,32276,32278,32281],{"className":32277},[129],[15,32279,541],{"className":32280},[129],[15,32282,32284],{"className":32283},[137],[15,32285,32287],{"className":32286},[141],[15,32288,32290],{"className":32289},[146],[15,32291,32293],{"className":32292,"style":380},[150],[15,32294,32295,32298],{"style":383},[15,32296],{"className":32297,"style":159},[158],[15,32299,32301],{"className":32300},[163,164,165,166],[15,32302,14508],{"className":32303},[129,166],[18062,32305,32306],{},[15,32307,32309,32330],{"className":32308,"translate":19},[23],[15,32310,32312],{"className":32311},[27],[29,32313,32314],{"xmlns":31},[34,32315,32316,32328],{},[37,32317,32318,32320,32322],{},[308,32319,32232],{},[50,32321,11422],{},[305,32323,32324,32326],{},[308,32325,32239],{},[308,32327,14508],{},[109,32329,32244],{"encoding":111},[15,32331,32333,32351],{"className":32332,"ariaHidden":72},[116],[15,32334,32336,32339,32342,32345,32348],{"className":32335},[120],[15,32337],{"className":32338,"style":16484},[124],[15,32340,32232],{"className":32341},[129],[15,32343],{"className":32344,"style":3277},[93],[15,32346,11422],{"className":32347},[2436],[15,32349],{"className":32350,"style":3277},[93],[15,32352,32354,32357,32360],{"className":32353},[120],[15,32355],{"className":32356,"style":380},[124],[15,32358,318],{"className":32359},[129],[15,32361,32363,32366],{"className":32362},[129],[15,32364,541],{"className":32365},[129],[15,32367,32369],{"className":32368},[137],[15,32370,32372],{"className":32371},[141],[15,32373,32375],{"className":32374},[146],[15,32376,32378],{"className":32377,"style":380},[150],[15,32379,32380,32383],{"style":383},[15,32381],{"className":32382,"style":159},[158],[15,32384,32386],{"className":32385},[163,164,165,166],[15,32387,14508],{"className":32388},[129,166],[18062,32390,32391],{},[15,32392,32394,32411],{"className":32393,"translate":19},[23],[15,32395,32397],{"className":32396},[27],[29,32398,32399],{"xmlns":31},[34,32400,32401,32408],{},[37,32402,32403,32406],{},[50,32404,32405],{},"≈",[308,32407,318],{},[109,32409,32410],{"encoding":111},"\\approx 1",[15,32412,32414,32427],{"className":32413,"ariaHidden":72},[116],[15,32415,32417,32421,32424],{"className":32416},[120],[15,32418],{"className":32419,"style":32420},[124],"height:0.4831em;",[15,32422,32405],{"className":32423},[203],[15,32425],{"className":32426,"style":199},[93],[15,32428,32430,32433],{"className":32429},[120],[15,32431],{"className":32432,"style":4886},[124],[15,32434,318],{"className":32435},[129],[18046,32437,32438,32441,32529,32616],{},[18062,32439,32440],{},"ResNet-18（ImageNet）",[18062,32442,32443],{},[15,32444,32446,32470],{"className":32445,"translate":19},[23],[15,32447,32449],{"className":32448},[27],[29,32450,32451],{"xmlns":31},[34,32452,32453,32467],{},[37,32454,32455,32458,32460],{},[308,32456,32457],{},"1.17",[50,32459,11422],{},[305,32461,32462,32464],{},[308,32463,32239],{},[308,32465,32466],{},"7",[109,32468,32469],{"encoding":111},"1.17\\times10^7",[15,32471,32473,32491],{"className":32472,"ariaHidden":72},[116],[15,32474,32476,32479,32482,32485,32488],{"className":32475},[120],[15,32477],{"className":32478,"style":16484},[124],[15,32480,32457],{"className":32481},[129],[15,32483],{"className":32484,"style":3277},[93],[15,32486,11422],{"className":32487},[2436],[15,32489],{"className":32490,"style":3277},[93],[15,32492,32494,32497,32500],{"className":32493},[120],[15,32495],{"className":32496,"style":380},[124],[15,32498,318],{"className":32499},[129],[15,32501,32503,32506],{"className":32502},[129],[15,32504,541],{"className":32505},[129],[15,32507,32509],{"className":32508},[137],[15,32510,32512],{"className":32511},[141],[15,32513,32515],{"className":32514},[146],[15,32516,32518],{"className":32517,"style":380},[150],[15,32519,32520,32523],{"style":383},[15,32521],{"className":32522,"style":159},[158],[15,32524,32526],{"className":32525},[163,164,165,166],[15,32527,32466],{"className":32528},[129,166],[18062,32530,32531],{},[15,32532,32534,32557],{"className":32533,"translate":19},[23],[15,32535,32537],{"className":32536},[27],[29,32538,32539],{"xmlns":31},[34,32540,32541,32554],{},[37,32542,32543,32546,32548],{},[308,32544,32545],{},"1.28",[50,32547,11422],{},[305,32549,32550,32552],{},[308,32551,32239],{},[308,32553,32232],{},[109,32555,32556],{"encoding":111},"1.28\\times10^6",[15,32558,32560,32578],{"className":32559,"ariaHidden":72},[116],[15,32561,32563,32566,32569,32572,32575],{"className":32562},[120],[15,32564],{"className":32565,"style":16484},[124],[15,32567,32545],{"className":32568},[129],[15,32570],{"className":32571,"style":3277},[93],[15,32573,11422],{"className":32574},[2436],[15,32576],{"className":32577,"style":3277},[93],[15,32579,32581,32584,32587],{"className":32580},[120],[15,32582],{"className":32583,"style":380},[124],[15,32585,318],{"className":32586},[129],[15,32588,32590,32593],{"className":32589},[129],[15,32591,541],{"className":32592},[129],[15,32594,32596],{"className":32595},[137],[15,32597,32599],{"className":32598},[141],[15,32600,32602],{"className":32601},[146],[15,32603,32605],{"className":32604,"style":380},[150],[15,32606,32607,32610],{"style":383},[15,32608],{"className":32609,"style":159},[158],[15,32611,32613],{"className":32612},[163,164,165,166],[15,32614,32232],{"className":32615},[129,166],[18062,32617,32618],{},[15,32619,32621,32638],{"className":32620,"translate":19},[23],[15,32622,32624],{"className":32623},[27],[29,32625,32626],{"xmlns":31},[34,32627,32628,32635],{},[37,32629,32630,32632],{},[50,32631,32405],{},[308,32633,32634],{},"3.0",[109,32636,32637],{"encoding":111},"\\approx 3.0",[15,32639,32641,32653],{"className":32640,"ariaHidden":72},[116],[15,32642,32644,32647,32650],{"className":32643},[120],[15,32645],{"className":32646,"style":32420},[124],[15,32648,32405],{"className":32649},[203],[15,32651],{"className":32652,"style":199},[93],[15,32654,32656,32659],{"className":32655},[120],[15,32657],{"className":32658,"style":4886},[124],[15,32660,32634],{"className":32661},[129],[18046,32663,32664,32667,32754,32839],{},[18062,32665,32666],{},"ResNet-50（ImageNet）",[18062,32668,32669],{},[15,32670,32672,32695],{"className":32671,"translate":19},[23],[15,32673,32675],{"className":32674},[27],[29,32676,32677],{"xmlns":31},[34,32678,32679,32692],{},[37,32680,32681,32684,32686],{},[308,32682,32683],{},"2.56",[50,32685,11422],{},[305,32687,32688,32690],{},[308,32689,32239],{},[308,32691,32466],{},[109,32693,32694],{"encoding":111},"2.56\\times10^7",[15,32696,32698,32716],{"className":32697,"ariaHidden":72},[116],[15,32699,32701,32704,32707,32710,32713],{"className":32700},[120],[15,32702],{"className":32703,"style":16484},[124],[15,32705,32683],{"className":32706},[129],[15,32708],{"className":32709,"style":3277},[93],[15,32711,11422],{"className":32712},[2436],[15,32714],{"className":32715,"style":3277},[93],[15,32717,32719,32722,32725],{"className":32718},[120],[15,32720],{"className":32721,"style":380},[124],[15,32723,318],{"className":32724},[129],[15,32726,32728,32731],{"className":32727},[129],[15,32729,541],{"className":32730},[129],[15,32732,32734],{"className":32733},[137],[15,32735,32737],{"className":32736},[141],[15,32738,32740],{"className":32739},[146],[15,32741,32743],{"className":32742,"style":380},[150],[15,32744,32745,32748],{"style":383},[15,32746],{"className":32747,"style":159},[158],[15,32749,32751],{"className":32750},[163,164,165,166],[15,32752,32466],{"className":32753},[129,166],[18062,32755,32756],{},[15,32757,32759,32780],{"className":32758,"translate":19},[23],[15,32760,32762],{"className":32761},[27],[29,32763,32764],{"xmlns":31},[34,32765,32766,32778],{},[37,32767,32768,32770,32772],{},[308,32769,32545],{},[50,32771,11422],{},[305,32773,32774,32776],{},[308,32775,32239],{},[308,32777,32232],{},[109,32779,32556],{"encoding":111},[15,32781,32783,32801],{"className":32782,"ariaHidden":72},[116],[15,32784,32786,32789,32792,32795,32798],{"className":32785},[120],[15,32787],{"className":32788,"style":16484},[124],[15,32790,32545],{"className":32791},[129],[15,32793],{"className":32794,"style":3277},[93],[15,32796,11422],{"className":32797},[2436],[15,32799],{"className":32800,"style":3277},[93],[15,32802,32804,32807,32810],{"className":32803},[120],[15,32805],{"className":32806,"style":380},[124],[15,32808,318],{"className":32809},[129],[15,32811,32813,32816],{"className":32812},[129],[15,32814,541],{"className":32815},[129],[15,32817,32819],{"className":32818},[137],[15,32820,32822],{"className":32821},[141],[15,32823,32825],{"className":32824},[146],[15,32826,32828],{"className":32827,"style":380},[150],[15,32829,32830,32833],{"style":383},[15,32831],{"className":32832,"style":159},[158],[15,32834,32836],{"className":32835},[163,164,165,166],[15,32837,32232],{"className":32838},[129,166],[18062,32840,32841],{},[15,32842,32844,32861],{"className":32843,"translate":19},[23],[15,32845,32847],{"className":32846},[27],[29,32848,32849],{"xmlns":31},[34,32850,32851,32858],{},[37,32852,32853,32855],{},[50,32854,32405],{},[308,32856,32857],{},"4.5",[109,32859,32860],{"encoding":111},"\\approx 4.5",[15,32862,32864,32876],{"className":32863,"ariaHidden":72},[116],[15,32865,32867,32870,32873],{"className":32866},[120],[15,32868],{"className":32869,"style":32420},[124],[15,32871,32405],{"className":32872},[203],[15,32874],{"className":32875,"style":199},[93],[15,32877,32879,32882],{"className":32878},[120],[15,32880],{"className":32881,"style":4886},[124],[15,32883,32857],{"className":32884},[129],[18046,32886,32887,32890,32978,33063],{},[18062,32888,32889],{},"ViT-L\u002F16（ImageNet）",[18062,32891,32892],{},[15,32893,32895,32919],{"className":32894,"translate":19},[23],[15,32896,32898],{"className":32897},[27],[29,32899,32900],{"xmlns":31},[34,32901,32902,32916],{},[37,32903,32904,32907,32909],{},[308,32905,32906],{},"3.07",[50,32908,11422],{},[305,32910,32911,32913],{},[308,32912,32239],{},[308,32914,32915],{},"8",[109,32917,32918],{"encoding":111},"3.07\\times10^8",[15,32920,32922,32940],{"className":32921,"ariaHidden":72},[116],[15,32923,32925,32928,32931,32934,32937],{"className":32924},[120],[15,32926],{"className":32927,"style":16484},[124],[15,32929,32906],{"className":32930},[129],[15,32932],{"className":32933,"style":3277},[93],[15,32935,11422],{"className":32936},[2436],[15,32938],{"className":32939,"style":3277},[93],[15,32941,32943,32946,32949],{"className":32942},[120],[15,32944],{"className":32945,"style":380},[124],[15,32947,318],{"className":32948},[129],[15,32950,32952,32955],{"className":32951},[129],[15,32953,541],{"className":32954},[129],[15,32956,32958],{"className":32957},[137],[15,32959,32961],{"className":32960},[141],[15,32962,32964],{"className":32963},[146],[15,32965,32967],{"className":32966,"style":380},[150],[15,32968,32969,32972],{"style":383},[15,32970],{"className":32971,"style":159},[158],[15,32973,32975],{"className":32974},[163,164,165,166],[15,32976,32915],{"className":32977},[129,166],[18062,32979,32980],{},[15,32981,32983,33004],{"className":32982,"translate":19},[23],[15,32984,32986],{"className":32985},[27],[29,32987,32988],{"xmlns":31},[34,32989,32990,33002],{},[37,32991,32992,32994,32996],{},[308,32993,32545],{},[50,32995,11422],{},[305,32997,32998,33000],{},[308,32999,32239],{},[308,33001,32232],{},[109,33003,32556],{"encoding":111},[15,33005,33007,33025],{"className":33006,"ariaHidden":72},[116],[15,33008,33010,33013,33016,33019,33022],{"className":33009},[120],[15,33011],{"className":33012,"style":16484},[124],[15,33014,32545],{"className":33015},[129],[15,33017],{"className":33018,"style":3277},[93],[15,33020,11422],{"className":33021},[2436],[15,33023],{"className":33024,"style":3277},[93],[15,33026,33028,33031,33034],{"className":33027},[120],[15,33029],{"className":33030,"style":380},[124],[15,33032,318],{"className":33033},[129],[15,33035,33037,33040],{"className":33036},[129],[15,33038,541],{"className":33039},[129],[15,33041,33043],{"className":33042},[137],[15,33044,33046],{"className":33045},[141],[15,33047,33049],{"className":33048},[146],[15,33050,33052],{"className":33051,"style":380},[150],[15,33053,33054,33057],{"style":383},[15,33055],{"className":33056,"style":159},[158],[15,33058,33060],{"className":33059},[163,164,165,166],[15,33061,32232],{"className":33062},[129,166],[18062,33064,33065],{},[15,33066,33068,33085],{"className":33067,"translate":19},[23],[15,33069,33071],{"className":33070},[27],[29,33072,33073],{"xmlns":31},[34,33074,33075,33082],{},[37,33076,33077,33079],{},[50,33078,32405],{},[308,33080,33081],{},"15.5",[109,33083,33084],{"encoding":111},"\\approx 15.5",[15,33086,33088,33100],{"className":33087,"ariaHidden":72},[116],[15,33089,33091,33094,33097],{"className":33090},[120],[15,33092],{"className":33093,"style":32420},[124],[15,33095,32405],{"className":33096},[203],[15,33098],{"className":33099,"style":199},[93],[15,33101,33103,33106],{"className":33102},[120],[15,33104],{"className":33105,"style":4886},[124],[15,33107,33081],{"className":33108},[129],[11,33110,33111,33112,33140,33141,33169,33170,33198,33199,33231],{},"每一个都在 ",[15,33113,33115,33128],{"className":33114,"translate":19},[23],[15,33116,33118],{"className":33117},[27],[29,33119,33120],{"xmlns":31},[34,33121,33122,33126],{},[37,33123,33124],{},[308,33125,318],{},[109,33127,318],{"encoding":111},[15,33129,33131],{"className":33130,"ariaHidden":72},[116],[15,33132,33134,33137],{"className":33133},[120],[15,33135],{"className":33136,"style":4886},[124],[15,33138,318],{"className":33139},[129]," 附近或远大于 ",[15,33142,33144,33157],{"className":33143,"translate":19},[23],[15,33145,33147],{"className":33146},[27],[29,33148,33149],{"xmlns":31},[34,33150,33151,33155],{},[37,33152,33153],{},[308,33154,318],{},[109,33156,318],{"encoding":111},[15,33158,33160],{"className":33159,"ariaHidden":72},[116],[15,33161,33163,33166],{"className":33162},[120],[15,33164],{"className":33165,"style":4886},[124],[15,33167,318],{"className":33168},[129],"。一个「错误率上界」一旦达到或超过 ",[15,33171,33173,33186],{"className":33172,"translate":19},[23],[15,33174,33176],{"className":33175},[27],[29,33177,33178],{"xmlns":31},[34,33179,33180,33184],{},[37,33181,33182],{},[308,33183,318],{},[109,33185,318],{"encoding":111},[15,33187,33189],{"className":33188,"ariaHidden":72},[116],[15,33190,33192,33195],{"className":33191},[120],[15,33193],{"className":33194,"style":4886},[124],[15,33196,318],{"className":33197},[129],"，就等于什么都没说——它连「模型会犯错」这种同义反复都不如，因为它没有排除任何东西。而且这还是把 VC 维低估成参数量的结果；下一小节会看到，深度网络的 VC 维还要乘上 ",[15,33200,33202,33217],{"className":33201,"translate":19},[23],[15,33203,33205],{"className":33204},[27],[29,33206,33207],{"xmlns":31},[34,33208,33209,33215],{},[37,33210,33211,33213],{},[43,33212,14727],{},[50,33214,69],{},[109,33216,30130],{"encoding":111},[15,33218,33220],{"className":33219,"ariaHidden":72},[116],[15,33221,33223,33226],{"className":33222},[120],[15,33224],{"className":33225,"style":1682},[124],[15,33227,14777,33229],{"className":33228},[217],[15,33230,2247],{"style":2246}," 因子。",[11,33233,33234,33235,33263,33264,33292,33293,33354],{},"有人会说：那就把界改紧一点。但问题不是常数，是量级。ImageNet 有 128 万张训练图，ResNet-50 有 2560 万参数，",[15,33236,33238,33251],{"className":33237,"translate":19},[23],[15,33239,33241],{"className":33240},[27],[29,33242,33243],{"xmlns":31},[34,33244,33245,33249],{},[37,33246,33247],{},[43,33248,348],{},[109,33250,348],{"encoding":111},[15,33252,33254],{"className":33253,"ariaHidden":72},[116],[15,33255,33257,33260],{"className":33256},[120],[15,33258],{"className":33259,"style":4383},[124],[15,33261,348],{"className":33262},[129,133]," 比 ",[15,33265,33267,33280],{"className":33266,"translate":19},[23],[15,33268,33270],{"className":33269},[27],[29,33271,33272],{"xmlns":31},[34,33273,33274,33278],{},[37,33275,33276],{},[43,33277,10376],{},[109,33279,10376],{"encoding":111},[15,33281,33283],{"className":33282,"ariaHidden":72},[116],[15,33284,33286,33289],{"className":33285},[120],[15,33287],{"className":33288,"style":522},[124],[15,33290,10376],{"className":33291},[129,133]," 小二十倍。",[698,33294,33295,33296,33324,33325,33353],{},"任何以 ",[15,33297,33299,33312],{"className":33298,"translate":19},[23],[15,33300,33302],{"className":33301},[27],[29,33303,33304],{"xmlns":31},[34,33305,33306,33310],{},[37,33307,33308],{},[43,33309,10376],{},[109,33311,10376],{"encoding":111},[15,33313,33315],{"className":33314,"ariaHidden":72},[116],[15,33316,33318,33321],{"className":33317},[120],[15,33319],{"className":33320,"style":522},[124],[15,33322,10376],{"className":33323},[129,133]," 为主项的界，在这个区间里都只能给出与 ",[15,33326,33328,33341],{"className":33327,"translate":19},[23],[15,33329,33331],{"className":33330},[27],[29,33332,33333],{"xmlns":31},[34,33334,33335,33339],{},[37,33336,33337],{},[308,33338,318],{},[109,33340,318],{"encoding":111},[15,33342,33344],{"className":33343,"ariaHidden":72},[116],[15,33345,33347,33350],{"className":33346},[120],[15,33348],{"className":33349,"style":4886},[124],[15,33351,318],{"className":33352},[129]," 同阶或更大的量级。"," 这是样本量与容量之间的量级失配，不是分析技巧的问题。",[575,33356,33358],{"id":33357},"_62-深度网络的-vc-维","6.2 深度网络的 VC 维",[11,33360,33361],{},"先确认一件事：深度网络的 VC 维是有限的。只要参数个数有限、激活函数是分段多项式（ReLU、Leaky ReLU、MaxOut 都是），VC 维就有限。深度网络并不是一个无限容量的对象。",[11,33363,33364],{},"那么它有多大？",[21730,33366,33367,33529],{},[21733,33368,33369,33372,33373,16663,33438,33466,33467,33499,33500,33528],{},[698,33370,33371],{},"阈值网络。"," Baum 与 Haussler（1989）证明，由阈值门组成的多层网络，VC 维是 ",[15,33374,33376,33403],{"className":33375,"translate":19},[23],[15,33377,33379],{"className":33378},[27],[29,33380,33381],{"xmlns":31},[34,33382,33383,33400],{},[37,33384,33385,33388,33390,33392,33394,33396,33398],{},[43,33386,33387],{},"O",[50,33389,53],{"stretchy":52},[43,33391,32000],{},[43,33393,14727],{},[50,33395,69],{},[43,33397,32000],{},[50,33399,59],{"stretchy":52},[109,33401,33402],{"encoding":111},"O(W\\log W)",[15,33404,33406],{"className":33405,"ariaHidden":72},[116],[15,33407,33409,33412,33415,33418,33421,33424,33429,33432,33435],{"className":33408},[120],[15,33410],{"className":33411,"style":125},[124],[15,33413,33387],{"className":33414,"style":170},[129,133],[15,33416,53],{"className":33417},[188],[15,33419,32000],{"className":33420,"style":12449},[129,133],[15,33422],{"className":33423,"style":262},[93],[15,33425,14777,33427],{"className":33426},[217],[15,33428,2247],{"style":2246},[15,33430],{"className":33431,"style":262},[93],[15,33433,32000],{"className":33434,"style":12449},[129,133],[15,33436,59],{"className":33437},[195],[15,33439,33441,33454],{"className":33440,"translate":19},[23],[15,33442,33444],{"className":33443},[27],[29,33445,33446],{"xmlns":31},[34,33447,33448,33452],{},[37,33449,33450],{},[43,33451,32000],{},[109,33453,32000],{"encoding":111},[15,33455,33457],{"className":33456,"ariaHidden":72},[116],[15,33458,33460,33463],{"className":33459},[120],[15,33461],{"className":33462,"style":610},[124],[15,33464,32000],{"className":33465,"style":12449},[129,133]," 是权重个数。",[15,33468,33470,33485],{"className":33469,"translate":19},[23],[15,33471,33473],{"className":33472},[27],[29,33474,33475],{"xmlns":31},[34,33476,33477,33483],{},[37,33478,33479,33481],{},[43,33480,14727],{},[50,33482,69],{},[109,33484,30130],{"encoding":111},[15,33486,33488],{"className":33487,"ariaHidden":72},[116],[15,33489,33491,33494],{"className":33490},[120],[15,33492],{"className":33493,"style":1682},[124],[15,33495,14777,33497],{"className":33496},[217],[15,33498,2247],{"style":2246}," 因子来自网络内部的分段线性结构：不同的权重符号组合对应不同的线性区域，区域数量随 ",[15,33501,33503,33516],{"className":33502,"translate":19},[23],[15,33504,33506],{"className":33505},[27],[29,33507,33508],{"xmlns":31},[34,33509,33510,33514],{},[37,33511,33512],{},[43,33513,32000],{},[109,33515,32000],{"encoding":111},[15,33517,33519],{"className":33518,"ariaHidden":72},[116],[15,33520,33522,33525],{"className":33521},[120],[15,33523],{"className":33524,"style":610},[124],[15,33526,32000],{"className":33527,"style":12449},[129,133]," 组合增长。",[21733,33530,33531,33534,33535,33604,33605,33633,33634,33718],{},[698,33532,33533],{},"ReLU 网络。"," Bartlett、Harvey、Liaw 与 Mehrabian（2019）给出 ",[15,33536,33538,33566],{"className":33537,"translate":19},[23],[15,33539,33541],{"className":33540},[27],[29,33542,33543],{"xmlns":31},[34,33544,33545,33563],{},[37,33546,33547,33549,33551,33553,33555,33557,33559,33561],{},[43,33548,33387],{},[50,33550,53],{"stretchy":52},[43,33552,32000],{},[43,33554,1373],{},[43,33556,14727],{},[50,33558,69],{},[43,33560,32000],{},[50,33562,59],{"stretchy":52},[109,33564,33565],{"encoding":111},"O(WL\\log W)",[15,33567,33569],{"className":33568,"ariaHidden":72},[116],[15,33570,33572,33575,33578,33581,33584,33587,33590,33595,33598,33601],{"className":33571},[120],[15,33573],{"className":33574,"style":125},[124],[15,33576,33387],{"className":33577,"style":170},[129,133],[15,33579,53],{"className":33580},[188],[15,33582,32000],{"className":33583,"style":12449},[129,133],[15,33585,1373],{"className":33586},[129,133],[15,33588],{"className":33589,"style":262},[93],[15,33591,14777,33593],{"className":33592},[217],[15,33594,2247],{"style":2246},[15,33596],{"className":33597,"style":262},[93],[15,33599,32000],{"className":33600,"style":12449},[129,133],[15,33602,59],{"className":33603},[195]," 的上界（",[15,33606,33608,33621],{"className":33607,"translate":19},[23],[15,33609,33611],{"className":33610},[27],[29,33612,33613],{"xmlns":31},[34,33614,33615,33619],{},[37,33616,33617],{},[43,33618,1373],{},[109,33620,1373],{"encoding":111},[15,33622,33624],{"className":33623,"ariaHidden":72},[116],[15,33625,33627,33630],{"className":33626},[120],[15,33628],{"className":33629,"style":610},[124],[15,33631,1373],{"className":33632},[129,133]," 为层数），并给出几乎匹配的下界 ",[15,33635,33637,33674],{"className":33636,"translate":19},[23],[15,33638,33640],{"className":33639},[27],[29,33641,33642],{"xmlns":31},[34,33643,33644,33671],{},[37,33645,33646,33649,33651,33653,33655,33657,33659,33661,33663,33665,33667,33669],{},[43,33647,33648],{"mathvariant":65},"Ω",[50,33650,53],{"stretchy":52},[43,33652,32000],{},[43,33654,1373],{},[43,33656,14727],{},[50,33658,69],{},[50,33660,53],{"stretchy":52},[43,33662,32000],{},[43,33664,544],{"mathvariant":65},[43,33666,1373],{},[50,33668,59],{"stretchy":52},[50,33670,59],{"stretchy":52},[109,33672,33673],{"encoding":111},"\\Omega(WL\\log(W\u002FL))",[15,33675,33677],{"className":33676,"ariaHidden":72},[116],[15,33678,33680,33683,33686,33689,33692,33695,33698,33703,33706,33709,33712,33715],{"className":33679},[120],[15,33681],{"className":33682,"style":125},[124],[15,33684,33648],{"className":33685},[129],[15,33687,53],{"className":33688},[188],[15,33690,32000],{"className":33691,"style":12449},[129,133],[15,33693,1373],{"className":33694},[129,133],[15,33696],{"className":33697,"style":262},[93],[15,33699,14777,33701],{"className":33700},[217],[15,33702,2247],{"style":2246},[15,33704,53],{"className":33705},[188],[15,33707,32000],{"className":33708,"style":12449},[129,133],[15,33710,544],{"className":33711},[129],[15,33713,1373],{"className":33714},[129,133],[15,33716,6149],{"className":33717},[195],"；伪维（pseudo-dimension）同阶。",[11,33720,33721,33722,33758],{},"结论很清楚：",[698,33723,33724,33725,33757],{},"深度网络的 VC 维随参数线性增长（乘一个 ",[15,33726,33728,33743],{"className":33727,"translate":19},[23],[15,33729,33731],{"className":33730},[27],[29,33732,33733],{"xmlns":31},[34,33734,33735,33741],{},[37,33736,33737,33739],{},[43,33738,14727],{},[50,33740,69],{},[109,33742,30130],{"encoding":111},[15,33744,33746],{"className":33745,"ariaHidden":72},[116],[15,33747,33749,33752],{"className":33748},[120],[15,33750],{"className":33751,"style":1682},[124],[15,33753,14777,33755],{"className":33754},[217],[15,33756,2247],{"style":2246},"）。"," 「深度学习之所以泛化，是因为它的 VC 维很小」这个流行说法是错的。深度网络和线性模型一样，容量随参数增长；区别在于，深度网络的实际泛化误差可以远小于这个容量所暗示的上界。",[11,33760,33761],{},"另一个方向的反例是 1-NN：VC 维无穷，但在很多分布上表现并不差。这说明 VC 维无穷并不等于「不能用」；它只意味着不存在一个对所有分布一致的保证。",[575,33763,33765],{"id":33764},"_63-三条出路","6.3 三条出路",[11,33767,33768],{},"既然「整个假设类的 VC 维」给不出有用的话，理论就往三个方向走。",[11,33770,33771],{},[698,33772,33773],{},"（a）把容量做得更细：从假设类到算法输出",[11,33775,33776,33777,33805],{},"与其用整个 ",[15,33778,33780,33793],{"className":33779,"translate":19},[23],[15,33781,33783],{"className":33782},[27],[29,33784,33785],{"xmlns":31},[34,33786,33787,33791],{},[37,33788,33789],{},[43,33790,819],{"mathvariant":596},[109,33792,822],{"encoding":111},[15,33794,33796],{"className":33795,"ariaHidden":72},[116],[15,33797,33799,33802],{"className":33798},[120],[15,33800],{"className":33801,"style":610},[124],[15,33803,819],{"className":33804,"style":835},[129,614]," 的 VC 维，不如用「训练解所在的那个小子类」的容量。",[11,33807,33808,33809,33872,33873,33936,33937,33968,33969,34054,34055,34083],{},"线性情形是最好的例子。若 ",[15,33810,33812,33835],{"className":33811,"translate":19},[23],[15,33813,33815],{"className":33814},[27],[29,33816,33817],{"xmlns":31},[34,33818,33819,33832],{},[37,33820,33821,33824,33826,33828,33830],{},[50,33822,33823],{"stretchy":52},"∥",[43,33825,56],{},[50,33827,33823],{"stretchy":52},[50,33829,2717],{},[43,33831,104],{},[109,33833,33834],{"encoding":111},"\\lVert x\\rVert\\le R",[15,33836,33838,33862],{"className":33837,"ariaHidden":72},[116],[15,33839,33841,33844,33847,33850,33853,33856,33859],{"className":33840},[120],[15,33842],{"className":33843,"style":125},[124],[15,33845,33823],{"className":33846},[188],[15,33848,56],{"className":33849},[129,133],[15,33851,33823],{"className":33852},[195],[15,33854],{"className":33855,"style":199},[93],[15,33857,2717],{"className":33858},[203],[15,33860],{"className":33861,"style":199},[93],[15,33863,33865,33868],{"className":33864},[120],[15,33866],{"className":33867,"style":610},[124],[15,33869,104],{"className":33870,"style":33871},[129,133],"margin-right:0.0077em;","、",[15,33874,33876,33899],{"className":33875,"translate":19},[23],[15,33877,33879],{"className":33878},[27],[29,33880,33881],{"xmlns":31},[34,33882,33883,33896],{},[37,33884,33885,33887,33889,33891,33893],{},[50,33886,33823],{"stretchy":52},[43,33888,10550],{},[50,33890,33823],{"stretchy":52},[50,33892,2717],{},[43,33894,33895],{},"B",[109,33897,33898],{"encoding":111},"\\lVert w\\rVert\\le B",[15,33900,33902,33926],{"className":33901,"ariaHidden":72},[116],[15,33903,33905,33908,33911,33914,33917,33920,33923],{"className":33904},[120],[15,33906],{"className":33907,"style":125},[124],[15,33909,33823],{"className":33910},[188],[15,33912,10550],{"className":33913,"style":10627},[129,133],[15,33915,33823],{"className":33916},[195],[15,33918],{"className":33919,"style":199},[93],[15,33921,2717],{"className":33922},[203],[15,33924],{"className":33925,"style":199},[93],[15,33927,33929,33932],{"className":33928},[120],[15,33930],{"className":33931,"style":610},[124],[15,33933,33895],{"className":33934,"style":33935},[129,133],"margin-right:0.0502em;","，且分类间隔至少为 ",[15,33938,33940,33955],{"className":33939,"translate":19},[23],[15,33941,33943],{"className":33942},[27],[29,33944,33945],{"xmlns":31},[34,33946,33947,33952],{},[37,33948,33949],{},[43,33950,33951],{},"γ",[109,33953,33954],{"encoding":111},"\\gamma",[15,33956,33958],{"className":33957,"ariaHidden":72},[116],[15,33959,33961,33964],{"className":33960},[120],[15,33962],{"className":33963,"style":5399},[124],[15,33965,33951],{"className":33966,"style":33967},[129,133],"margin-right:0.0556em;","，则有效容量约为 ",[15,33970,33972,34000],{"className":33971,"translate":19},[23],[15,33973,33975],{"className":33974},[27],[29,33976,33977],{"xmlns":31},[34,33978,33979,33997],{},[37,33980,33981,33983,33985,33987,33989,33991],{},[50,33982,53],{"stretchy":52},[43,33984,104],{},[43,33986,33895],{},[43,33988,544],{"mathvariant":65},[43,33990,33951],{},[305,33992,33993,33995],{},[50,33994,59],{"stretchy":52},[308,33996,310],{},[109,33998,33999],{"encoding":111},"(RB\u002F\\gamma)^2",[15,34001,34003],{"className":34002,"ariaHidden":72},[116],[15,34004,34006,34010,34013,34016,34019,34022,34025],{"className":34005},[120],[15,34007],{"className":34008,"style":34009},[124],"height:1.0641em;vertical-align:-0.25em;",[15,34011,53],{"className":34012},[188],[15,34014,104],{"className":34015,"style":33871},[129,133],[15,34017,33895],{"className":34018,"style":33935},[129,133],[15,34020,544],{"className":34021},[129],[15,34023,33951],{"className":34024,"style":33967},[129,133],[15,34026,34028,34031],{"className":34027},[195],[15,34029,59],{"className":34030},[195],[15,34032,34034],{"className":34033},[137],[15,34035,34037],{"className":34036},[141],[15,34038,34040],{"className":34039},[146],[15,34041,34043],{"className":34042,"style":380},[150],[15,34044,34045,34048],{"style":383},[15,34046],{"className":34047,"style":159},[158],[15,34049,34051],{"className":34050},[163,164,165,166],[15,34052,310],{"className":34053},[129,166],"，与输入维度 ",[15,34056,34058,34071],{"className":34057,"translate":19},[23],[15,34059,34061],{"className":34060},[27],[29,34062,34063],{"xmlns":31},[34,34064,34065,34069],{},[37,34066,34067],{},[43,34068,10376],{},[109,34070,10376],{"encoding":111},[15,34072,34074],{"className":34073,"ariaHidden":72},[116],[15,34075,34077,34080],{"className":34076},[120],[15,34078],{"className":34079,"style":522},[124],[15,34081,10376],{"className":34082},[129,133]," 无关。这正是 SVM 的理论基础：在高维甚至无限维的特征空间里，只要间隔大，容量就小。",[11,34085,34086,34087,34126,34127,34300],{},"深度网络有类似的谱范数版本。把 ",[15,34088,34090,34108],{"className":34089,"translate":19},[23],[15,34091,34093],{"className":34092},[27],[29,34094,34095],{"xmlns":31},[34,34096,34097,34105],{},[37,34098,34099,34101,34103],{},[50,34100,33823],{"stretchy":52},[43,34102,10550],{},[50,34104,33823],{"stretchy":52},[109,34106,34107],{"encoding":111},"\\lVert w\\rVert",[15,34109,34111],{"className":34110,"ariaHidden":72},[116],[15,34112,34114,34117,34120,34123],{"className":34113},[120],[15,34115],{"className":34116,"style":125},[124],[15,34118,33823],{"className":34119},[188],[15,34121,10550],{"className":34122,"style":10627},[129,133],[15,34124,33823],{"className":34125},[195]," 换成各层谱范数的乘积 ",[15,34128,34130,34164],{"className":34129,"translate":19},[23],[15,34131,34133],{"className":34132},[27],[29,34134,34135],{"xmlns":31},[34,34136,34137,34161],{},[37,34138,34139,34147,34149,34155],{},[40,34140,34141,34144],{},[50,34142,34143],{},"∏",[43,34145,34146],{},"l",[50,34148,33823],{"stretchy":52},[40,34150,34151,34153],{},[43,34152,32000],{},[43,34154,34146],{},[40,34156,34157,34159],{},[50,34158,33823],{"stretchy":52},[43,34160,27294],{},[109,34162,34163],{"encoding":111},"\\prod_l\\lVert W_l\\rVert_\\sigma",[15,34165,34167],{"className":34166,"ariaHidden":72},[116],[15,34168,34170,34174,34216,34219,34260],{"className":34169},[120],[15,34171],{"className":34172,"style":34173},[124],"height:1.0497em;vertical-align:-0.2997em;",[15,34175,34177,34180],{"className":34176},[217],[15,34178,34143],{"className":34179,"style":23352},[217,1918,23351],[15,34181,34183],{"className":34182},[137],[15,34184,34186,34208],{"className":34185},[141,142],[15,34187,34189,34205],{"className":34188},[146],[15,34190,34193],{"className":34191,"style":34192},[150],"height:0.1864em;",[15,34194,34195,34198],{"style":23368},[15,34196],{"className":34197,"style":159},[158],[15,34199,34201],{"className":34200},[163,164,165,166],[15,34202,34146],{"className":34203,"style":34204},[129,133,166],"margin-right:0.0197em;",[15,34206,175],{"className":34207},[174],[15,34209,34211],{"className":34210},[146],[15,34212,34214],{"className":34213,"style":23417},[150],[15,34215],{},[15,34217,33823],{"className":34218},[188],[15,34220,34222,34225],{"className":34221},[129],[15,34223,32000],{"className":34224,"style":12449},[129,133],[15,34226,34228],{"className":34227},[137],[15,34229,34231,34252],{"className":34230},[141,142],[15,34232,34234,34249],{"className":34233},[146],[15,34235,34237],{"className":34236,"style":151},[150],[15,34238,34240,34243],{"style":34239},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[15,34241],{"className":34242,"style":159},[158],[15,34244,34246],{"className":34245},[163,164,165,166],[15,34247,34146],{"className":34248,"style":34204},[129,133,166],[15,34250,175],{"className":34251},[174],[15,34253,34255],{"className":34254},[146],[15,34256,34258],{"className":34257,"style":182},[150],[15,34259],{},[15,34261,34263,34266],{"className":34262},[195],[15,34264,33823],{"className":34265},[195],[15,34267,34269],{"className":34268},[137],[15,34270,34272,34292],{"className":34271},[141,142],[15,34273,34275,34289],{"className":34274},[146],[15,34276,34278],{"className":34277,"style":1163},[150],[15,34279,34280,34283],{"style":154},[15,34281],{"className":34282,"style":159},[158],[15,34284,34286],{"className":34285},[163,164,165,166],[15,34287,27294],{"className":34288,"style":887},[129,133,166],[15,34290,175],{"className":34291},[174],[15,34293,34295],{"className":34294},[146],[15,34296,34298],{"className":34297,"style":182},[150],[15,34299],{},"，再乘上深度和宽度带来的多项式因子、除以间隔，就得到 Bartlett–Foster–Telgarsky（2017）与 Neyshabur 等人（2018）的界。这类界的形式大致是「分子为谱范数乘积的平方，分母为间隔与样本量」；它们在经过特殊正则的小网络上可以非空，但在标准训练的 ImageNet 网络上仍然空泛。",[11,34302,34303,34304,34333],{},"覆盖数（covering number）是同一思想的度量几何版本：VC 维回答「能打散多少点」，覆盖数回答「用多少个半径为 ",[15,34305,34307,34321],{"className":34306,"translate":19},[23],[15,34308,34310],{"className":34309},[27],[29,34311,34312],{"xmlns":31},[34,34313,34314,34318],{},[37,34315,34316],{},[43,34317,4849],{},[109,34319,34320],{"encoding":111},"\\varepsilon",[15,34322,34324],{"className":34323,"ariaHidden":72},[116],[15,34325,34327,34330],{"className":34326},[120],[15,34328],{"className":34329,"style":4383},[124],[15,34331,4849],{"className":34332},[129,133]," 的球能把假设类盖住」。二者通过 Dudley 熵积分相连。",[11,34335,34336,34339,34340,34370,34371,34421],{},[698,34337,34338],{},"PAC-Bayes"," 则把容量变成了「后验与先验的距离」。对任意后验分布 ",[15,34341,34343,34357],{"className":34342,"translate":19},[23],[15,34344,34346],{"className":34345},[27],[29,34347,34348],{"xmlns":31},[34,34349,34350,34355],{},[37,34351,34352],{},[43,34353,34354],{},"Q",[109,34356,34354],{"encoding":111},[15,34358,34360],{"className":34359,"ariaHidden":72},[116],[15,34361,34363,34367],{"className":34362},[120],[15,34364],{"className":34365,"style":34366},[124],"height:0.8778em;vertical-align:-0.1944em;",[15,34368,34354],{"className":34369},[129,133],"（通常是训练后权重的分布），以 ",[15,34372,34374,34391],{"className":34373,"translate":19},[23],[15,34375,34377],{"className":34376},[27],[29,34378,34379],{"xmlns":31},[34,34380,34381,34389],{},[37,34382,34383,34385,34387],{},[308,34384,318],{},[50,34386,315],{},[43,34388,29575],{},[109,34390,29923],{"encoding":111},[15,34392,34394,34412],{"className":34393,"ariaHidden":72},[116],[15,34395,34397,34400,34403,34406,34409],{"className":34396},[120],[15,34398],{"className":34399,"style":16484},[124],[15,34401,318],{"className":34402},[129],[15,34404],{"className":34405,"style":3277},[93],[15,34407,315],{"className":34408},[2436],[15,34410],{"className":34411,"style":3277},[93],[15,34413,34415,34418],{"className":34414},[120],[15,34416],{"className":34417,"style":522},[124],[15,34419,29575],{"className":34420,"style":29856},[129,133]," 的概率，",[15,34423,34425],{"className":34424,"translate":19},[18],[15,34426,34428,34562],{"className":34427,"translate":19},[23],[15,34429,34431],{"className":34430},[27],[29,34432,34433],{"xmlns":31,"display":32},[34,34434,34435,34559],{},[37,34436,34437,34449,34451,34457,34459,34461,34463,34465,34467,34469,34471,34483,34485,34491,34493,34495,34497,34499,34501,34503,34505,34507,34510,34557],{},[40,34438,34439,34441],{},[43,34440,26574],{"mathvariant":103},[37,34442,34443,34445,34447],{},[43,34444,45],{},[50,34446,991],{},[43,34448,34354],{},[50,34450,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,34452,34453,34455],{},[43,34454,1373],{},[43,34456,906],{"mathvariant":596},[50,34458,53],{"stretchy":52},[43,34460,45],{},[50,34462,59],{"stretchy":52},[50,34464,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[6856,34466,26564],{},[50,34468,2717],{},[6856,34470,26564],{},[40,34472,34473,34475],{},[43,34474,26574],{"mathvariant":103},[37,34476,34477,34479,34481],{},[43,34478,45],{},[50,34480,991],{},[43,34482,34354],{},[50,34484,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,34486,34487,34489],{},[43,34488,1373],{},[43,34490,940],{},[50,34492,53],{"stretchy":52},[43,34494,45],{},[50,34496,59],{"stretchy":52},[50,34498,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[6856,34500,26564],{},[50,34502,2682],{},[6856,34504,26564],{},[43,34506,33387],{},[6856,34508,34509],{}," ⁣",[37,34511,34512,34514,34555],{},[50,34513,53],{"fence":72},[29015,34515,34516],{},[1443,34517,34518,34553],{},[37,34519,34520,34527,34529,34531,34533,34535,34537,34539,34541,34543,34545,34547,34549,34551],{},[37,34521,34522,34525],{},[43,34523,34524],{"mathvariant":65},"K",[43,34526,1373],{"mathvariant":65},[50,34528,53],{"stretchy":52},[43,34530,34354],{},[43,34532,33823],{"mathvariant":65},[43,34534,1388],{},[50,34536,59],{"stretchy":52},[50,34538,2682],{},[43,34540,14727],{},[50,34542,69],{},[50,34544,53],{"stretchy":52},[308,34546,318],{},[43,34548,544],{"mathvariant":65},[43,34550,29575],{},[50,34552,59],{"stretchy":52},[43,34554,348],{},[50,34556,59],{"fence":72},[43,34558,107],{"mathvariant":65},[109,34560,34561],{"encoding":111},"\\mathbb E_{h\\sim Q}\\bigl[L_{\\mathcal D}(h)\\bigr]\n\\;\\le\\;\n\\mathbb E_{h\\sim Q}\\bigl[L_S(h)\\bigr]\n\\;+\\;O\\!\\left(\\sqrt{\\frac{\\mathrm{KL}(Q\\Vert P)+\\log(1\u002F\\delta)}{m}}\\right).",[15,34563,34565,34699,34830],{"className":34564,"ariaHidden":72},[116],[15,34566,34568,34571,34620,34626,34669,34672,34675,34678,34684,34687,34690,34693,34696],{"className":34567},[120],[15,34569],{"className":34570,"style":213},[124],[15,34572,34574,34577],{"className":34573},[129],[15,34575,26574],{"className":34576},[129,285],[15,34578,34580],{"className":34579},[137],[15,34581,34583,34612],{"className":34582},[141,142],[15,34584,34586,34609],{"className":34585},[146],[15,34587,34589],{"className":34588,"style":151},[150],[15,34590,34591,34594],{"style":154},[15,34592],{"className":34593,"style":159},[158],[15,34595,34597],{"className":34596},[163,164,165,166],[15,34598,34600,34603,34606],{"className":34599},[129,166],[15,34601,45],{"className":34602},[129,133,166],[15,34604,991],{"className":34605},[203,166],[15,34607,34354],{"className":34608},[129,133,166],[15,34610,175],{"className":34611},[174],[15,34613,34615],{"className":34614},[146],[15,34616,34618],{"className":34617,"style":14041},[150],[15,34619],{},[15,34621,34623],{"className":34622},[188],[15,34624,1407],{"className":34625},[228,229],[15,34627,34629,34632],{"className":34628},[129],[15,34630,1373],{"className":34631},[129,133],[15,34633,34635],{"className":34634},[137],[15,34636,34638,34661],{"className":34637},[141,142],[15,34639,34641,34658],{"className":34640},[146],[15,34642,34644],{"className":34643,"style":1525},[150],[15,34645,34646,34649],{"style":154},[15,34647],{"className":34648,"style":159},[158],[15,34650,34652],{"className":34651},[163,164,165,166],[15,34653,34655],{"className":34654},[129,166],[15,34656,906],{"className":34657,"style":170},[129,614,166],[15,34659,175],{"className":34660},[174],[15,34662,34664],{"className":34663},[146],[15,34665,34667],{"className":34666,"style":182},[150],[15,34668],{},[15,34670,53],{"className":34671},[188],[15,34673,45],{"className":34674},[129,133],[15,34676,59],{"className":34677},[195],[15,34679,34681],{"className":34680},[195],[15,34682,1423],{"className":34683},[228,229],[15,34685],{"className":34686,"style":199},[93],[15,34688],{"className":34689,"style":199},[93],[15,34691,2717],{"className":34692},[203],[15,34694],{"className":34695,"style":199},[93],[15,34697],{"className":34698,"style":199},[93],[15,34700,34702,34705,34754,34760,34800,34803,34806,34809,34815,34818,34821,34824,34827],{"className":34701},[120],[15,34703],{"className":34704,"style":213},[124],[15,34706,34708,34711],{"className":34707},[129],[15,34709,26574],{"className":34710},[129,285],[15,34712,34714],{"className":34713},[137],[15,34715,34717,34746],{"className":34716},[141,142],[15,34718,34720,34743],{"className":34719},[146],[15,34721,34723],{"className":34722,"style":151},[150],[15,34724,34725,34728],{"style":154},[15,34726],{"className":34727,"style":159},[158],[15,34729,34731],{"className":34730},[163,164,165,166],[15,34732,34734,34737,34740],{"className":34733},[129,166],[15,34735,45],{"className":34736},[129,133,166],[15,34738,991],{"className":34739},[203,166],[15,34741,34354],{"className":34742},[129,133,166],[15,34744,175],{"className":34745},[174],[15,34747,34749],{"className":34748},[146],[15,34750,34752],{"className":34751,"style":14041},[150],[15,34753],{},[15,34755,34757],{"className":34756},[188],[15,34758,1407],{"className":34759},[228,229],[15,34761,34763,34766],{"className":34762},[129],[15,34764,1373],{"className":34765},[129,133],[15,34767,34769],{"className":34768},[137],[15,34770,34772,34792],{"className":34771},[141,142],[15,34773,34775,34789],{"className":34774},[146],[15,34776,34778],{"className":34777,"style":1525},[150],[15,34779,34780,34783],{"style":154},[15,34781],{"className":34782,"style":159},[158],[15,34784,34786],{"className":34785},[163,164,165,166],[15,34787,940],{"className":34788,"style":1013},[129,133,166],[15,34790,175],{"className":34791},[174],[15,34793,34795],{"className":34794},[146],[15,34796,34798],{"className":34797,"style":182},[150],[15,34799],{},[15,34801,53],{"className":34802},[188],[15,34804,45],{"className":34805},[129,133],[15,34807,59],{"className":34808},[195],[15,34810,34812],{"className":34811},[195],[15,34813,1423],{"className":34814},[228,229],[15,34816],{"className":34817,"style":199},[93],[15,34819],{"className":34820,"style":3277},[93],[15,34822,2682],{"className":34823},[2436],[15,34825],{"className":34826,"style":199},[93],[15,34828],{"className":34829,"style":3277},[93],[15,34831,34833,34837,34840,34844,34847,35013,35016],{"className":34832},[120],[15,34834],{"className":34835,"style":34836},[124],"height:3em;vertical-align:-1.25em;",[15,34838,33387],{"className":34839,"style":170},[129,133],[15,34841],{"className":34842,"style":34843},[93],"margin-right:-0.1667em;",[15,34845],{"className":34846,"style":262},[93],[15,34848,34850,34856,35007],{"className":34849},[449],[15,34851,34853],{"className":34852,"style":5052},[188,5051],[15,34854,53],{"className":34855},[228,15857],[15,34857,34859],{"className":3485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VC 维，而是「训练后的分布 ",[15,35023,35025,35038],{"className":35024,"translate":19},[23],[15,35026,35028],{"className":35027},[27],[29,35029,35030],{"xmlns":31},[34,35031,35032,35036],{},[37,35033,35034],{},[43,35035,34354],{},[109,35037,34354],{"encoding":111},[15,35039,35041],{"className":35040,"ariaHidden":72},[116],[15,35042,35044,35047],{"className":35043},[120],[15,35045],{"className":35046,"style":34366},[124],[15,35048,34354],{"className":35049},[129,133]," 离先验 ",[15,35052,35054,35067],{"className":35053,"translate":19},[23],[15,35055,35057],{"className":35056},[27],[29,35058,35059],{"xmlns":31},[34,35060,35061,35065],{},[37,35062,35063],{},[43,35064,1388],{},[109,35066,1388],{"encoding":111},[15,35068,35070],{"className":35069,"ariaHidden":72},[116],[15,35071,35073,35076],{"className":35072},[120],[15,35074],{"className":35075,"style":610},[124],[15,35077,1388],{"className":35078,"style":12449},[129,133]," 有多远」。Dziugaite 与 Roy（2017）用它在 MNIST 规模的网络上算出了非空的界，这是 PAC-Bayes 在深度网络上最早显示出实际内容的结果之一。",[11,35081,35082,35085,35086,35114,35115,35202],{},[698,35083,35084],{},"压缩"," 是另一条路：若训练好的网络可以被压缩成 ",[15,35087,35089,35102],{"className":35088,"translate":19},[23],[15,35090,35092],{"className":35091},[27],[29,35093,35094],{"xmlns":31},[34,35095,35096,35100],{},[37,35097,35098],{},[43,35099,16562],{},[109,35101,16562],{"encoding":111},[15,35103,35105],{"className":35104,"ariaHidden":72},[116],[15,35106,35108,35111],{"className":35107},[120],[15,35109],{"className":35110,"style":522},[124],[15,35112,16562],{"className":35113,"style":16595},[129,133]," 比特而不改变其在训练集上的预测，则泛化误差约为 ",[15,35116,35118,35140],{"className":35117,"translate":19},[23],[15,35119,35121],{"className":35120},[27],[29,35122,35123],{"xmlns":31},[34,35124,35125,35137],{},[37,35126,35127],{},[29015,35128,35129],{},[37,35130,35131,35133,35135],{},[43,35132,16562],{},[43,35134,544],{"mathvariant":65},[43,35136,348],{},[109,35138,35139],{"encoding":111},"\\sqrt{k\u002Fm}",[15,35141,35143],{"className":35142,"ariaHidden":72},[116],[15,35144,35146,35149],{"className":35145},[120],[15,35147],{"className":35148,"style":31894},[124],[15,35150,35152],{"className":35151},[129,29123],[15,35153,35155,35194],{"className":35154},[141,142],[15,35156,35158,35191],{"className":35157},[146],[15,35159,35161,35179],{"className":35160,"style":31907},[150],[15,35162,35164,35167],{"className":35163,"style":31911},[29137],[15,35165],{"className":35166,"style":4450},[158],[15,35168,35170,35173,35176],{"className":35169,"style":29146},[129],[15,35171,16562],{"className":35172,"style":16595},[129,133],[15,35174,544],{"className":35175},[129],[15,35177,348],{"className":35178},[129,133],[15,35180,35181,35184],{"style":31929},[15,35182],{"className":35183,"style":4450},[158],[15,35185,35187],{"className":35186,"style":31936},[29282],[4455,35188,35189],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},[4461,35190],{"d":31943},[15,35192,175],{"className":35193},[174],[15,35195,35197],{"className":35196},[146],[15,35198,35200],{"className":35199,"style":31953},[150],[15,35201],{},"。Zhou 等人（2019）用剪枝加量化的压缩方案，在 ImageNet 规模上给出了非空（但仍然很松）的界。「非空」是里程碑，但离「解释」还很远。",[11,35204,35205,35208],{},[698,35206,35207],{},"稳定性"," 则把容量挂在算法上：若算法对替换一个样本不敏感，泛化误差就可以被稳定性系数控制（Bousquet–Elisseeff，2002）。Hardt、Recht 与 Singer（2016）把这一框架用到了 SGD 上。",[11,35210,35211,35212,35215,35216,35244],{},"这些界的共同点很重要：",[698,35213,35214],{},"它们中的大多数不再是纯粹「假设类」的属性，而是「算法输出」或「数据」的属性。"," 容量从单一的 ",[15,35217,35219,35232],{"className":35218,"translate":19},[23],[15,35220,35222],{"className":35221},[27],[29,35223,35224],{"xmlns":31},[34,35225,35226,35230],{},[37,35227,35228],{},[43,35229,819],{"mathvariant":596},[109,35231,822],{"encoding":111},[15,35233,35235],{"className":35234,"ariaHidden":72},[116],[15,35236,35238,35241],{"className":35237},[120],[15,35239],{"className":35240,"style":610},[124],[15,35242,819],{"className":35243,"style":835},[129,614]," 转移到了假设类、算法、数据三者共同决定的整体上。",[11,35246,35247],{},[698,35248,35249],{},"（b）算法偏置：SGD 只在一个很小的子集里搜索",[11,35251,35252,35280,35281,35309,35310,35338],{},[15,35253,35255,35268],{"className":35254,"translate":19},[23],[15,35256,35258],{"className":35257},[27],[29,35259,35260],{"xmlns":31},[34,35261,35262,35266],{},[37,35263,35264],{},[43,35265,819],{"mathvariant":596},[109,35267,822],{"encoding":111},[15,35269,35271],{"className":35270,"ariaHidden":72},[116],[15,35272,35274,35277],{"className":35273},[120],[15,35275],{"className":35276,"style":610},[124],[15,35278,819],{"className":35279,"style":835},[129,614]," 很大，但 SGD 不是均匀地在 ",[15,35282,35284,35297],{"className":35283,"translate":19},[23],[15,35285,35287],{"className":35286},[27],[29,35288,35289],{"xmlns":31},[34,35290,35291,35295],{},[37,35292,35293],{},[43,35294,819],{"mathvariant":596},[109,35296,822],{"encoding":111},[15,35298,35300],{"className":35299,"ariaHidden":72},[116],[15,35301,35303,35306],{"className":35302},[120],[15,35304],{"className":35305,"style":610},[124],[15,35307,819],{"className":35308,"style":835},[129,614]," 里搜索。从初始化出发、沿着梯度的方向、在有限步内，它只能到达 ",[15,35311,35313,35326],{"className":35312,"translate":19},[23],[15,35314,35316],{"className":35315},[27],[29,35317,35318],{"xmlns":31},[34,35319,35320,35324],{},[37,35321,35322],{},[43,35323,819],{"mathvariant":596},[109,35325,822],{"encoding":111},[15,35327,35329],{"className":35328,"ariaHidden":72},[116],[15,35330,35332,35335],{"className":35331},[120],[15,35333],{"className":35334,"style":610},[124],[15,35336,819],{"className":35337,"style":835},[129,614]," 的一个很小的子集。",[11,35340,35341],{},"线性可分的情形已经被严格刻画：梯度下降收敛到最大间隔解（Soudry 等，2018），即使假设类包含所有线性分类器。深度网络里也有大量关于隐式正则的工作，试图说明梯度方法偏向于某种「最小范数」或「最大间隔」的解。",[11,35343,35344],{},"神经正切核（NTK）提供了一个更清晰的图景：当网络宽度趋于无穷时，训练动力学等价于核回归，泛化由核的谱和 RKHS 范数控制。这把容量从参数空间搬到了函数空间：不是问「有多少个参数」，而是问「在核的度量下，解有多大的范数」。",[11,35346,35347],{},"但要小心：NTK 只描述「懒惰训练」（lazy regime），即特征几乎不变的情形。真实网络会学特征，这时 NTK 不再适用，隐式偏置的理论也远未完成。这是当前深度学习理论最核心的开放问题之一。",[11,35349,35350],{},[698,35351,35352],{},"（c）数据分布：最坏情况太坏",[11,35354,35355],{},"VC 界对每个分布都成立，因此必须按最坏的分布来定价。真实数据不是最坏的：图像有低维结构，类别之间有间隔，长尾分布有层次。",[11,35357,35358,35361,35362,35390,35391,35419],{},[698,35359,35360],{},"双下降","（double descent）是一个典型现象：当模型容量、训练时间或样本量越过插值阈值时，测试误差可以出现第二次下降，而不是经典 U 形曲线预言的那样一路变差（Belkin 等，2019；Nakkiran 等，2020）。需要注意，经典理论从来没有证明过 U 形曲线——它只是小模型时代的经验总结；双下降也不「推翻」VC 理论，它只是说明在 ",[15,35363,35365,35378],{"className":35364,"translate":19},[23],[15,35366,35368],{"className":35367},[27],[29,35369,35370],{"xmlns":31},[34,35371,35372,35376],{},[37,35373,35374],{},[43,35375,10376],{},[109,35377,10376],{"encoding":111},[15,35379,35381],{"className":35380,"ariaHidden":72},[116],[15,35382,35384,35387],{"className":35383},[120],[15,35385],{"className":35386,"style":522},[124],[15,35388,10376],{"className":35389},[129,133]," 随 ",[15,35392,35394,35407],{"className":35393,"translate":19},[23],[15,35395,35397],{"className":35396},[27],[29,35398,35399],{"xmlns":31},[34,35400,35401,35405],{},[37,35402,35403],{},[43,35404,348],{},[109,35406,348],{"encoding":111},[15,35408,35410],{"className":35409,"ariaHidden":72},[116],[15,35411,35413,35416],{"className":35412},[120],[15,35414],{"className":35415,"style":4383},[124],[15,35417,348],{"className":35418},[129,133]," 变化的区域，偏差 - 方差分解的直觉不再适用。",[11,35421,35422,35425],{},[698,35423,35424],{},"良性过拟合","（benign overfitting）则更反直觉：在高维线性回归中，最小范数插值器可以是一致的，只要数据协方差的谱衰减得足够快（Bartlett、Long、Lugosi、Tsigler，2020）。「插值」不等于「过拟合」——这是一个已经被严格证明的现象，而它的成立强烈依赖数据分布的具体性质。",[11,35427,35428],{},"这些结果都指向同一件事：泛化不是假设类单独的性质，而是假设类、算法、分布三者共同的性质。VC 维只看到了第一个。",[575,35430,35432],{"id":35431},"_64-一致收敛的局限","6.4 一致收敛的局限",[11,35434,35435,35436,35439],{},"还有一条更根本的批评。Nagarajan 与 Kolter（2019）构造了一类学习问题，使得真实泛化误差可以很小，但",[698,35437,35438],{},"任何","基于一致收敛的界都必然很大。换句话说，「把 VC 界改紧」这条路存在原理性的上限。",[11,35441,35442],{},"这不是说他们的构造代表了真实网络，而是说：要解释深度学习，必须放弃「对所有假设一致」的要求，转而利用算法或分布的结构。一致收敛是一个充分条件，不是必要条件。",[11,35444,35445,35446,30641,35474,35502,35503],{},"这里有一个容易被忽略的逻辑细节：对二分类 ",[15,35447,35449,35462],{"className":35448,"translate":19},[23],[15,35450,35452],{"className":35451},[27],[29,35453,35454],{"xmlns":31},[34,35455,35456,35460],{},[37,35457,35458],{},[308,35459,541],{},[109,35461,541],{"encoding":111},[15,35463,35465],{"className":35464,"ariaHidden":72},[116],[15,35466,35468,35471],{"className":35467},[120],[15,35469],{"className":35470,"style":4886},[124],[15,35472,541],{"className":35473},[129],[15,35475,35477,35490],{"className":35476,"translate":19},[23],[15,35478,35480],{"className":35479},[27],[29,35481,35482],{"xmlns":31},[34,35483,35484,35488],{},[37,35485,35486],{},[308,35487,318],{},[109,35489,318],{"encoding":111},[15,35491,35493],{"className":35492,"ariaHidden":72},[116],[15,35494,35496,35499],{"className":35495},[120],[15,35497],{"className":35498,"style":4886},[124],[15,35500,318],{"className":35501},[129]," 损失，有限 VC 维确实是 PAC 可学习的充要条件。所以如果深度网络在分布无关的意义下可学，它的 VC 维必须有限——而它确实有限。逻辑上没有矛盾；矛盾只在「界」与「实际」的巨大差距。",[698,35504,35505],{},"VC 理论没有被证伪，它只是没有提供我们想要的那种解释。",[570,35507,35509],{"id":35508},"七容量度量谱系","七、容量度量谱系",[11,35511,35512,35513,35516,35517,35545,35546,35549,35550,35578],{},"在前面的讨论里，VC 维一直默认假设输出是二值的。对实值函数类（回归、概率输出、间隔），它不能直接使用。标准的替代有两种：把实值函数按阈值二值化后取 VC 维，得到 ",[698,35514,35515],{},"pseudo-dimension","；或者要求在某个尺度 ",[15,35518,35520,35533],{"className":35519,"translate":19},[23],[15,35521,35523],{"className":35522},[27],[29,35524,35525],{"xmlns":31},[34,35526,35527,35531],{},[37,35528,35529],{},[43,35530,33951],{},[109,35532,33954],{"encoding":111},[15,35534,35536],{"className":35535,"ariaHidden":72},[116],[15,35537,35539,35542],{"className":35538},[120],[15,35540],{"className":35541,"style":5399},[124],[15,35543,33951],{"className":35544,"style":33967},[129,133]," 下仍能打散，得到 ",[698,35547,35548],{},"fat-shattering dimension","。后者是间隔理论的正式语言：一个类完全可以在 VC 维意义下无穷，却在每个固定的 ",[15,35551,35553,35566],{"className":35552,"translate":19},[23],[15,35554,35556],{"className":35555},[27],[29,35557,35558],{"xmlns":31},[34,35559,35560,35564],{},[37,35561,35562],{},[43,35563,33951],{},[109,35565,33954],{"encoding":111},[15,35567,35569],{"className":35568,"ariaHidden":72},[116],[15,35570,35572,35575],{"className":35571},[120],[15,35573],{"className":35574,"style":5399},[124],[15,35576,33951],{"className":35577,"style":33967},[129,133]," 下只有有限的 fat-shattering 维数——这正是「间隔大则容量小」的精确表述。",[11,35580,35581],{},"把上面提到的各种度量放在一张表里，可以看到它们各自回答的是不同的问题。",[18040,35583,35584,35603],{},[18043,35585,35586],{},[18046,35587,35588,35591,35594,35597,35600],{},[18049,35589,35590],{},"度量",[18049,35592,35593],{},"度量对象",[18049,35595,35596],{},"核心思想",[18049,35598,35599],{},"典型界",[18049,35601,35602],{},"主要局限",[18057,35604,35605,35813,35828,35845,35890,35936,36092,36250,36297,36398],{},[18046,35606,35607,35609,35612,35615,35753],{},[18062,35608,6804],{},[18062,35610,35611],{},"二值函数类",[18062,35613,35614],{},"最大打散集",[18062,35616,35617],{},[15,35618,35620,35662],{"className":35619,"translate":19},[23],[15,35621,35623],{"className":35622},[27],[29,35624,35625],{"xmlns":31},[34,35626,35627,35659],{},[37,35628,35629],{},[29015,35630,35631],{},[37,35632,35633,35635,35637,35639,35641,35643,35645,35647,35649,35651,35653,35655,35657],{},[50,35634,53],{"stretchy":52},[43,35636,10376],{},[50,35638,2682],{},[43,35640,14727],{},[50,35642,69],{},[50,35644,53],{"stretchy":52},[308,35646,318],{},[43,35648,544],{"mathvariant":65},[43,35650,29575],{},[50,35652,59],{"stretchy":52},[50,35654,59],{"stretchy":52},[43,35656,544],{"mathvariant":65},[43,35658,348],{},[109,35660,35661],{"encoding":111},"\\sqrt{(d+\\log(1\u002F\\delta))\u002Fm}",[15,35663,35665],{"className":35664,"ariaHidden":72},[116],[15,35666,35668,35671],{"className":35667},[120],[15,35669],{"className":35670,"style":31894},[124],[15,35672,35674],{"className":35673},[129,29123],[15,35675,35677,35745],{"className":35676},[141,142],[15,35678,35680,35742],{"className":35679},[146],[15,35681,35683,35730],{"className":35682,"style":31907},[150],[15,35684,35686,35689],{"className":35685,"style":31911},[29137],[15,35687],{"className":35688,"style":4450},[158],[15,35690,35692,35695,35698,35701,35704,35707,35712,35715,35718,35721,35724,35727],{"className":35691,"style":29146},[129],[15,35693,53],{"className":35694},[188],[15,35696,10376],{"className":35697},[129,133],[15,35699],{"className":35700,"style":3277},[93],[15,35702,2682],{"className":35703},[2436],[15,35705],{"className":35706,"style":3277},[93],[15,35708,14777,35710],{"className":35709},[217],[15,35711,2247],{"style":2246},[15,35713,53],{"className":35714},[188],[15,35716,29852],{"className":35717},[129],[15,35719,29575],{"className":35720,"style":29856},[129,133],[15,35722,6149],{"className":35723},[195],[15,35725,544],{"className":35726},[129],[15,35728,348],{"className":35729},[129,133],[15,35731,35732,35735],{"style":31929},[15,35733],{"className":35734,"style":4450},[158],[15,35736,35738],{"className":35737,"style":31936},[29282],[4455,35739,35740],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},[4461,35741],{"d":31943},[15,35743,175],{"className":35744},[174],[15,35746,35748],{"className":35747},[146],[15,35749,35751],{"className":35750,"style":31953},[150],[15,35752],{},[18062,35754,35755,35756,30641,35784,35812],{},"分布无关、最坏情况、只适用 ",[15,35757,35759,35772],{"className":35758,"translate":19},[23],[15,35760,35762],{"className":35761},[27],[29,35763,35764],{"xmlns":31},[34,35765,35766,35770],{},[37,35767,35768],{},[308,35769,541],{},[109,35771,541],{"encoding":111},[15,35773,35775],{"className":35774,"ariaHidden":72},[116],[15,35776,35778,35781],{"className":35777},[120],[15,35779],{"className":35780,"style":4886},[124],[15,35782,541],{"className":35783},[129],[15,35785,35787,35800],{"className":35786,"translate":19},[23],[15,35788,35790],{"className":35789},[27],[29,35791,35792],{"xmlns":31},[34,35793,35794,35798],{},[37,35795,35796],{},[308,35797,318],{},[109,35799,318],{"encoding":111},[15,35801,35803],{"className":35802,"ariaHidden":72},[116],[15,35804,35806,35809],{"className":35805},[120],[15,35807],{"className":35808,"style":4886},[124],[15,35810,318],{"className":35811},[129]," 损失",[18046,35814,35815,35817,35819,35822,35825],{},[18062,35816,20056],{},[18062,35818,35611],{},[18062,35820,35821],{},"限制类的最大规模",[18062,35823,35824],{},"Sauer 多项式界",[18062,35826,35827],{},"组合量，需要再转概率",[18046,35829,35830,35833,35836,35839,35842],{},[18062,35831,35832],{},"Pseudo-dimension",[18062,35834,35835],{},"实值函数类",[18062,35837,35838],{},"阈值化后的 VC 维",[18062,35840,35841],{},"与 VC 同阶",[18062,35843,35844],{},"仍是最坏情况",[18046,35846,35847,35850,35852,35884,35887],{},[18062,35848,35849],{},"Fat-shattering",[18062,35851,35835],{},[18062,35853,35854,35855,35883],{},"尺度 ",[15,35856,35858,35871],{"className":35857,"translate":19},[23],[15,35859,35861],{"className":35860},[27],[29,35862,35863],{"xmlns":31},[34,35864,35865,35869],{},[37,35866,35867],{},[43,35868,33951],{},[109,35870,33954],{"encoding":111},[15,35872,35874],{"className":35873,"ariaHidden":72},[116],[15,35875,35877,35880],{"className":35876},[120],[15,35878],{"className":35879,"style":5399},[124],[15,35881,33951],{"className":35882,"style":33967},[129,133]," 下的打散",[18062,35885,35886],{},"间隔界",[18062,35888,35889],{},"依赖尺度选择",[18046,35891,35892,35895,35898,35930,35933],{},[18062,35893,35894],{},"覆盖数",[18062,35896,35897],{},"度量空间",[18062,35899,35900,35901,35929],{},"用 ",[15,35902,35904,35917],{"className":35903,"translate":19},[23],[15,35905,35907],{"className":35906},[27],[29,35908,35909],{"xmlns":31},[34,35910,35911,35915],{},[37,35912,35913],{},[43,35914,4849],{},[109,35916,34320],{"encoding":111},[15,35918,35920],{"className":35919,"ariaHidden":72},[116],[15,35921,35923,35926],{"className":35922},[120],[15,35924],{"className":35925,"style":4383},[124],[15,35927,4849],{"className":35928},[129,133],"-球覆盖",[18062,35931,35932],{},"Dudley 熵积分",[18062,35934,35935],{},"计算困难",[18046,35937,35938,35940,35943,35946,36089],{},[18062,35939,28893],{},[18062,35941,35942],{},"数据相关",[18062,35944,35945],{},"随机符号下的拟合能力",[18062,35947,35948],{},[15,35949,35951,35991],{"className":35950,"translate":19},[23],[15,35952,35954],{"className":35953},[27],[29,35955,35956],{"xmlns":31},[34,35957,35958,35988],{},[37,35959,35960,35962,35964,35966],{},[308,35961,310],{},[43,35963,104],{"mathvariant":28910},[50,35965,2682],{},[29015,35967,35968],{},[37,35969,35970,35972,35974,35976,35978,35980,35982,35984,35986],{},[43,35971,14727],{},[50,35973,69],{},[50,35975,53],{"stretchy":52},[308,35977,318],{},[43,35979,544],{"mathvariant":65},[43,35981,29575],{},[50,35983,59],{"stretchy":52},[43,35985,544],{"mathvariant":65},[43,35987,348],{},[109,35989,35990],{"encoding":111},"2\\mathfrak R+\\sqrt{\\log(1\u002F\\delta)\u002Fm}",[15,35992,35994,36016],{"className":35993,"ariaHidden":72},[116],[15,35995,35997,36001,36004,36007,36010,36013],{"className":35996},[120],[15,35998],{"className":35999,"style":36000},[124],"height:0.7747em;vertical-align:-0.0833em;",[15,36002,310],{"className":36003},[129],[15,36005,104],{"className":36006},[129,28937],[15,36008],{"className":36009,"style":3277},[93],[15,36011,2682],{"className":36012},[2436],[15,36014],{"className":36015,"style":3277},[93],[15,36017,36019,36022],{"className":36018},[120],[15,36020],{"className":36021,"style":31894},[124],[15,36023,36025],{"className":36024},[129,29123],[15,36026,36028,36081],{"className":36027},[141,142],[15,36029,36031,36078],{"className":36030},[146],[15,36032,36034,36066],{"className":36033,"style":31907},[150],[15,36035,36037,36040],{"className":36036,"style":31911},[29137],[15,36038],{"className":36039,"style":4450},[158],[15,36041,36043,36048,36051,36054,36057,36060,36063],{"className":36042,"style":29146},[129],[15,36044,14777,36046],{"className":36045},[217],[15,36047,2247],{"style":2246},[15,36049,53],{"className":36050},[188],[15,36052,29852],{"className":36053},[129],[15,36055,29575],{"className":36056,"style":29856},[129,133],[15,36058,59],{"className":36059},[195],[15,36061,544],{"className":36062},[129],[15,36064,348],{"className":36065},[129,133],[15,36067,36068,36071],{"style":31929},[15,36069],{"className":36070,"style":4450},[158],[15,36072,36074],{"className":36073,"style":31936},[29282],[4455,36075,36076],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},[4461,36077],{"d":31943},[15,36079,175],{"className":36080},[174],[15,36082,36084],{"className":36083},[146],[15,36085,36087],{"className":36086,"style":31953},[150],[15,36088],{},[18062,36090,36091],{},"需要估计，通常偏松",[18046,36093,36094,36096,36099,36102,36247],{},[18062,36095,34338],{},[18062,36097,36098],{},"后验与先验",[18062,36100,36101],{},"KL 复杂度",[18062,36103,36104],{},[15,36105,36107,36153],{"className":36106,"translate":19},[23],[15,36108,36110],{"className":36109},[27],[29,36111,36112],{"xmlns":31},[34,36113,36114,36150],{},[37,36115,36116],{},[29015,36117,36118],{},[37,36119,36120,36122,36128,36130,36132,36134,36136,36138,36140,36142,36144,36146,36148],{},[50,36121,53],{"stretchy":52},[37,36123,36124,36126],{},[43,36125,34524],{"mathvariant":65},[43,36127,1373],{"mathvariant":65},[50,36129,2682],{},[43,36131,14727],{},[50,36133,69],{},[50,36135,53],{"stretchy":52},[308,36137,318],{},[43,36139,544],{"mathvariant":65},[43,36141,29575],{},[50,36143,59],{"stretchy":52},[50,36145,59],{"stretchy":52},[43,36147,544],{"mathvariant":65},[43,36149,348],{},[109,36151,36152],{"encoding":111},"\\sqrt{(\\mathrm{KL}+\\log(1\u002F\\delta))\u002Fm}",[15,36154,36156],{"className":36155,"ariaHidden":72},[116],[15,36157,36159,36162],{"className":36158},[120],[15,36160],{"className":36161,"style":31894},[124],[15,36163,36165],{"className":36164},[129,29123],[15,36166,36168,36239],{"className":36167},[141,142],[15,36169,36171,36236],{"className":36170},[146],[15,36172,36174,36224],{"className":36173,"style":31907},[150],[15,36175,36177,36180],{"className":36176,"style":31911},[29137],[15,36178],{"className":36179,"style":4450},[158],[15,36181,36183,36186,36192,36195,36198,36201,36206,36209,36212,36215,36218,36221],{"className":36182,"style":29146},[129],[15,36184,53],{"className":36185},[188],[15,36187,36189],{"className":36188},[129],[15,36190,34928],{"className":36191},[129,221],[15,36193],{"className":36194,"style":3277},[93],[15,36196,2682],{"className":36197},[2436],[15,36199],{"className":36200,"style":3277},[93],[15,36202,14777,36204],{"className":36203},[217],[15,36205,2247],{"style":2246},[15,36207,53],{"className":36208},[188],[15,36210,29852],{"className":36211},[129],[15,36213,29575],{"className":36214,"style":29856},[129,133],[15,36216,6149],{"className":36217},[195],[15,36219,544],{"className":36220},[129],[15,36222,348],{"className":36223},[129,133],[15,36225,36226,36229],{"style":31929},[15,36227],{"className":36228,"style":4450},[158],[15,36230,36232],{"className":36231,"style":31936},[29282],[4455,36233,36234],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},[4461,36235],{"d":31943},[15,36237,175],{"className":36238},[174],[15,36240,36242],{"className":36241},[146],[15,36243,36245],{"className":36244,"style":31953},[150],[15,36246],{},[18062,36248,36249],{},"先验的选择",[18046,36251,36252,36254,36257,36260,36294],{},[18062,36253,35207],{},[18062,36255,36256],{},"算法",[18062,36258,36259],{},"替换一个样本的影响",[18062,36261,36262,36263],{},"稳定性系数 ",[15,36264,36266,36281],{"className":36265,"translate":19},[23],[15,36267,36269],{"className":36268},[27],[29,36270,36271],{"xmlns":31},[34,36272,36273,36278],{},[37,36274,36275],{},[43,36276,36277],{},"β",[109,36279,36280],{"encoding":111},"\\beta",[15,36282,36284],{"className":36283,"ariaHidden":72},[116],[15,36285,36287,36290],{"className":36286},[120],[15,36288],{"className":36289,"style":1682},[124],[15,36291,36277],{"className":36292,"style":36293},[129,133],"margin-right:0.0528em;",[18062,36295,36296],{},"对 SGD 难以验证",[18046,36298,36299,36301,36304,36307,36395],{},[18062,36300,35084],{},[18062,36302,36303],{},"算法输出",[18062,36305,36306],{},"描述长度",[18062,36308,36309],{},[15,36310,36312,36333],{"className":36311,"translate":19},[23],[15,36313,36315],{"className":36314},[27],[29,36316,36317],{"xmlns":31},[34,36318,36319,36331],{},[37,36320,36321],{},[29015,36322,36323],{},[37,36324,36325,36327,36329],{},[43,36326,16562],{},[43,36328,544],{"mathvariant":65},[43,36330,348],{},[109,36332,35139],{"encoding":111},[15,36334,36336],{"className":36335,"ariaHidden":72},[116],[15,36337,36339,36342],{"className":36338},[120],[15,36340],{"className":36341,"style":31894},[124],[15,36343,36345],{"className":36344},[129,29123],[15,36346,36348,36387],{"className":36347},[141,142],[15,36349,36351,36384],{"className":36350},[146],[15,36352,36354,36372],{"className":36353,"style":31907},[150],[15,36355,36357,36360],{"className":36356,"style":31911},[29137],[15,36358],{"className":36359,"style":4450},[158],[15,36361,36363,36366,36369],{"className":36362,"style":29146},[129],[15,36364,16562],{"className":36365,"style":16595},[129,133],[15,36367,544],{"className":36368},[129],[15,36370,348],{"className":36371},[129,133],[15,36373,36374,36377],{"style":31929},[15,36375],{"className":36376,"style":4450},[158],[15,36378,36380],{"className":36379,"style":31936},[29282],[4455,36381,36382],{"xmlns":4457,"width":29286,"height":31939,"viewBox":31940,"preserveAspectRatio":29289},[4461,36383],{"d":31943},[15,36385,175],{"className":36386},[174],[15,36388,36390],{"className":36389},[146],[15,36391,36393],{"className":36392,"style":31953},[150],[15,36394],{},[18062,36396,36397],{},"需要构造压缩方案",[18046,36399,36400,36403,36406,36409,36412],{},[18062,36401,36402],{},"NTK \u002F RKHS 范数",[18062,36404,36405],{},"核函数",[18062,36407,36408],{},"函数空间中的范数",[18062,36410,36411],{},"谱条件",[18062,36413,36414],{},"只适用于懒惰训练",[11,36416,36417,36418,36421],{},"这张表想说明一件事：",[698,36419,36420],{},"「容量」不是一个数，而是一个依设定而变的函数。"," 你选择哪种度量，等于选择了对「什么算简单」的回答。VC 维的回答是「组合上简单」，间隔界的回答是「几何上简单」，PAC-Bayes 的回答是「相对先验简单」，压缩的回答是「描述上简单」。它们不是互相替代的关系，而是从不同角度逼近同一个问题。",[570,36423,36424],{"id":36424},"结语",[11,36426,36427,36428],{},"回到开头那两个数字。一个只有 1 个参数的正弦分类器可以有无限 VC 维；一个 2500 万参数的 ResNet 却只能在 ImageNet 上得到一个大于 1 的界。这两件事并不矛盾，它们从两个方向说明了同一件事：",[698,36429,36430],{},"容量不是参数的函数，而是假设类结构的函数；泛化不是容量的函数，而是容量、算法与分布三者共同的函数。",{"title":36432,"searchDepth":36433,"depth":36433,"links":36434},"",2,[36435,36441,36446,36451,36457,36458,36464,36465],{"id":572,"depth":36433,"text":573,"children":36436},[36437,36439,36440],{"id":577,"depth":36438,"text":578},3,{"id":2365,"depth":36438,"text":2366},{"id":4801,"depth":36438,"text":4802},{"id":5596,"depth":36433,"text":5597,"children":36442},[36443,36444,36445],{"id":5600,"depth":36438,"text":5601},{"id":7375,"depth":36438,"text":7376},{"id":19569,"depth":36438,"text":19570},{"id":19660,"depth":36433,"text":19661,"children":36447},[36448,36449,36450],{"id":19664,"depth":36438,"text":19665},{"id":20860,"depth":36438,"text":20861},{"id":24170,"depth":36438,"text":24171},{"id":26365,"depth":36433,"text":26366,"children":36452},[36453,36454,36455,36456],{"id":26369,"depth":36438,"text":26370},{"id":29431,"depth":36438,"text":29432},{"id":30603,"depth":36438,"text":30604},{"id":30820,"depth":36438,"text":30821},{"id":31327,"depth":36433,"text":31328},{"id":31851,"depth":36433,"text":31852,"children":36459},[36460,36461,36462,36463],{"id":31855,"depth":36438,"text":31856},{"id":33357,"depth":36438,"text":33358},{"id":33764,"depth":36438,"text":33765},{"id":35431,"depth":36438,"text":35432},{"id":35508,"depth":36433,"text":35509},{"id":36424,"depth":36433,"text":36424},false,true,"md",null,{},"\u002Fblog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension",{"title":6,"description":13},"blog\u002F2025\u002F2025-10-05-vapnik-chervonenkis-dimension","Vapnik-Chervonenkis Dimension 是统计学习理论中刻画假设类容量的核心概念。本文从打散与增长函数出发，完整推导 Sauer–Shelah 引理与一致收敛泛化界，再讨论它为何在深度网络上失效，以及间隔界、PAC-Bayes、稳定性等替代方案如何回应这一困境。",[36476,36477],"mathematics","machine-learning","2025-10-05T18:09:22+08:00","pm0qsEbi4FVab4f8OGbmKFSlwV0EdZ0Hzz_xiuH0fJU",{"id":36481,"title":36482,"body":36483,"description":36432,"draft":36466,"enableComment":36467,"extension":36468,"image":36432,"important":36466,"location":36469,"meta":47890,"navigation":36467,"ogImage":36469,"path":47891,"seo":47892,"stem":47893,"summary":47894,"tags":47895,"time":47896,"__hash__":47897},"blog\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations.md","非线性与激活函数",{"type":8,"value":36484,"toc":47876},[36485,36488,36491,36494,36585,36588,36591,36734,37530,37533,37536,37695,38462,38465,38468,38528,38761,38764,38801,39295,39298,39301,39439,39536,39596,39599,39852,40071,40074,40798,40801,40804,41137,41403,41463,41466,41469,41472,41650,41653,41813,41816,42050,42053,42056,42274,42277,42425,42428,42431,42527,42695,42881,42913,42916,43155,43229,43262,43299,43302,43305,43308,43311,43314,43319,43322,43324,43544,43633,43947,43950,43953,44650,44709,44712,44919,44922,44925,45447,45450,45669,45672,46314,46317,46320,46520,46523,46582,46585,46823,46826,46829,47568,47571,47645,47648,47651,47654,47657,47660,47663,47855,47858,47861,47864,47867,47870,47873],[570,36486,36487],{"id":36487},"线性组合",[11,36489,36490],{},"「线性」（linearity）这一概念贯穿数学、物理、工程学以及许多其他领域。简单来说，线性描述了两种基本的运算性质：可加性与齐次性。",[11,36492,36493],{},"如果一个关系、函数或系统同时满足以下两个条件，则称其为线性的。",[11,36495,36496,36497,36525,36526,36554,36555,1313],{},"考虑作用在输入 ",[15,36498,36500,36513],{"className":36499,"translate":19},[23],[15,36501,36503],{"className":36502},[27],[29,36504,36505],{"xmlns":31},[34,36506,36507,36511],{},[37,36508,36509],{},[43,36510,56],{},[109,36512,56],{"encoding":111},[15,36514,36516],{"className":36515,"ariaHidden":72},[116],[15,36517,36519,36522],{"className":36518},[120],[15,36520],{"className":36521,"style":4383},[124],[15,36523,56],{"className":36524},[129,133]," 和 ",[15,36527,36529,36542],{"className":36528,"translate":19},[23],[15,36530,36532],{"className":36531},[27],[29,36533,36534],{"xmlns":31},[34,36535,36536,36540],{},[37,36537,36538],{},[43,36539,857],{},[109,36541,857],{"encoding":111},[15,36543,36545],{"className":36544,"ariaHidden":72},[116],[15,36546,36548,36551],{"className":36547},[120],[15,36549],{"className":36550,"style":5399},[124],[15,36552,857],{"className":36553,"style":887},[129,133]," 上的变换或函数 ",[15,36556,36558,36572],{"className":36557,"translate":19},[23],[15,36559,36561],{"className":36560},[27],[29,36562,36563],{"xmlns":31},[34,36564,36565,36570],{},[37,36566,36567],{},[43,36568,36569],{},"f",[109,36571,36569],{"encoding":111},[15,36573,36575],{"className":36574,"ariaHidden":72},[116],[15,36576,36578,36581],{"className":36577},[120],[15,36579],{"className":36580,"style":1682},[124],[15,36582,36569],{"className":36583,"style":36584},[129,133],"margin-right:0.1076em;",[575,36586,36587],{"id":36587},"可加性",[11,36589,36590],{},"「部分之和等于整体」。也就是说，分别处理两个输入再相加结果，与先将输入相加再处理其和，是等价的。",[15,36592,36594],{"className":36593,"translate":19},[18],[15,36595,36597,36641],{"className":36596,"translate":19},[23],[15,36598,36600],{"className":36599},[27],[29,36601,36602],{"xmlns":31,"display":32},[34,36603,36604,36638],{},[37,36605,36606,36608,36610,36612,36614,36616,36618,36620,36622,36624,36626,36628,36630,36632,36634,36636],{},[43,36607,36569],{},[50,36609,53],{"stretchy":52},[43,36611,56],{},[50,36613,2682],{},[43,36615,857],{},[50,36617,59],{"stretchy":52},[50,36619,62],{},[43,36621,36569],{},[50,36623,53],{"stretchy":52},[43,36625,56],{},[50,36627,59],{"stretchy":52},[50,36629,2682],{},[43,36631,36569],{},[50,36633,53],{"stretchy":52},[43,36635,857],{},[50,36637,59],{"stretchy":52},[109,36639,36640],{"encoding":111},"f(x + y) = f(x) + f(y)",[15,36642,36644,36668,36689,36716],{"className":36643,"ariaHidden":72},[116],[15,36645,36647,36650,36653,36656,36659,36662,36665],{"className":36646},[120],[15,36648],{"className":36649,"style":125},[124],[15,36651,36569],{"className":36652,"style":36584},[129,133],[15,36654,53],{"className":36655},[188],[15,36657,56],{"className":36658},[129,133],[15,36660],{"className":36661,"style":3277},[93],[15,36663,2682],{"className":36664},[2436],[15,36666],{"className":36667,"style":3277},[93],[15,36669,36671,36674,36677,36680,36683,36686],{"className":36670},[120],[15,36672],{"className":36673,"style":125},[124],[15,36675,857],{"className":36676,"style":887},[129,133],[15,36678,59],{"className":36679},[195],[15,36681],{"className":36682,"style":199},[93],[15,36684,62],{"className":36685},[203],[15,36687],{"className":36688,"style":199},[93],[15,36690,36692,36695,36698,36701,36704,36707,36710,36713],{"className":36691},[120],[15,36693],{"className":36694,"style":125},[124],[15,36696,36569],{"className":36697,"style":36584},[129,133],[15,36699,53],{"className":36700},[188],[15,36702,56],{"className":36703},[129,133],[15,36705,59],{"className":36706},[195],[15,36708],{"className":36709,"style":3277},[93],[15,36711,2682],{"className":36712},[2436],[15,36714],{"className":36715,"style":3277},[93],[15,36717,36719,36722,36725,36728,36731],{"className":36718},[120],[15,36720],{"className":36721,"style":125},[124],[15,36723,36569],{"className":36724,"style":36584},[129,133],[15,36726,53],{"className":36727},[188],[15,36729,857],{"className":36730,"style":887},[129,133],[15,36732,59],{"className":36733},[195],[6559,36735,36736,36812,37132,37232],{},[11,36737,36738,36739,36811],{},"示例：乘法 ",[15,36740,36742,36769],{"className":36741,"translate":19},[23],[15,36743,36745],{"className":36744},[27],[29,36746,36747],{"xmlns":31},[34,36748,36749,36766],{},[37,36750,36751,36753,36755,36757,36759,36761,36764],{},[43,36752,36569],{},[50,36754,53],{"stretchy":52},[43,36756,56],{},[50,36758,59],{"stretchy":52},[50,36760,62],{},[308,36762,36763],{},"5",[43,36765,56],{},[109,36767,36768],{"encoding":111},"f(x) = 5x",[15,36770,36772,36799],{"className":36771,"ariaHidden":72},[116],[15,36773,36775,36778,36781,36784,36787,36790,36793,36796],{"className":36774},[120],[15,36776],{"className":36777,"style":125},[124],[15,36779,36569],{"className":36780,"style":36584},[129,133],[15,36782,53],{"className":36783},[188],[15,36785,56],{"className":36786},[129,133],[15,36788,59],{"className":36789},[195],[15,36791],{"className":36792,"style":199},[93],[15,36794,62],{"className":36795},[203],[15,36797],{"className":36798,"style":199},[93],[15,36800,36802,36805,36808],{"className":36801},[120],[15,36803],{"className":36804,"style":4886},[124],[15,36806,36763],{"className":36807},[129],[15,36809,56],{"className":36810},[129,133]," 满足可加性。",[21730,36813,36814],{},[21733,36815,36816,36817,36982,36983,37131],{},"左边：",[15,36818,36820,36865],{"className":36819,"translate":19},[23],[15,36821,36823],{"className":36822},[27],[29,36824,36825],{"xmlns":31},[34,36826,36827,36862],{},[37,36828,36829,36831,36833,36835,36837,36839,36841,36843,36845,36847,36849,36851,36853,36855,36857,36859],{},[43,36830,36569],{},[50,36832,53],{"stretchy":52},[308,36834,310],{},[50,36836,2682],{},[308,36838,9879],{},[50,36840,59],{"stretchy":52},[50,36842,62],{},[308,36844,36763],{},[50,36846,11422],{},[50,36848,53],{"stretchy":52},[308,36850,310],{},[50,36852,2682],{},[308,36854,9879],{},[50,36856,59],{"stretchy":52},[50,36858,62],{},[308,36860,36861],{},"25",[109,36863,36864],{"encoding":111},"f(2+3) = 5 \\times (2+3) = 25",[15,36866,36868,36892,36913,36931,36952,36973],{"className":36867,"ariaHidden":72},[116],[15,36869,36871,36874,36877,36880,36883,36886,36889],{"className":36870},[120],[15,36872],{"className":36873,"style":125},[124],[15,36875,36569],{"className":36876,"style":36584},[129,133],[15,36878,53],{"className":36879},[188],[15,36881,310],{"className":36882},[129],[15,36884],{"className":36885,"style":3277},[93],[15,36887,2682],{"className":36888},[2436],[15,36890],{"className":36891,"style":3277},[93],[15,36893,36895,36898,36901,36904,36907,36910],{"className":36894},[120],[15,36896],{"className":36897,"style":125},[124],[15,36899,9879],{"className":36900},[129],[15,36902,59],{"className":36903},[195],[15,36905],{"className":36906,"style":199},[93],[15,36908,62],{"className":36909},[203],[15,36911],{"className":36912,"style":199},[93],[15,36914,36916,36919,36922,36925,36928],{"className":36915},[120],[15,36917],{"className":36918,"style":16484},[124],[15,36920,36763],{"className":36921},[129],[15,36923],{"className":36924,"style":3277},[93],[15,36926,11422],{"className":36927},[2436],[15,36929],{"className":36930,"style":3277},[93],[15,36932,36934,36937,36940,36943,36946,36949],{"className":36933},[120],[15,36935],{"className":36936,"style":125},[124],[15,36938,53],{"className":36939},[188],[15,36941,310],{"className":36942},[129],[15,36944],{"className":36945,"style":3277},[93],[15,36947,2682],{"className":36948},[2436],[15,36950],{"className":36951,"style":3277},[93],[15,36953,36955,36958,36961,36964,36967,36970],{"className":36954},[120],[15,36956],{"className":36957,"style":125},[124],[15,36959,9879],{"className":36960},[129],[15,36962,59],{"className":36963},[195],[15,36965],{"className":36966,"style":199},[93],[15,36968,62],{"className":36969},[203],[15,36971],{"className":36972,"style":199},[93],[15,36974,36976,36979],{"className":36975},[120],[15,36977],{"className":36978,"style":4886},[124],[15,36980,36861],{"className":36981},[129],"，右边：",[15,36984,36986,37029],{"className":36985,"translate":19},[23],[15,36987,36989],{"className":36988},[27],[29,36990,36991],{"xmlns":31},[34,36992,36993,37026],{},[37,36994,36995,36997,36999,37001,37003,37005,37007,37009,37011,37013,37015,37017,37019,37022,37024],{},[43,36996,36569],{},[50,36998,53],{"stretchy":52},[308,37000,310],{},[50,37002,59],{"stretchy":52},[50,37004,2682],{},[43,37006,36569],{},[50,37008,53],{"stretchy":52},[308,37010,9879],{},[50,37012,59],{"stretchy":52},[50,37014,62],{},[308,37016,32239],{},[50,37018,2682],{},[308,37020,37021],{},"15",[50,37023,62],{},[308,37025,36861],{},[109,37027,37028],{"encoding":111},"f(2) + f(3) = 10 + 15 = 25",[15,37030,37032,37059,37086,37104,37122],{"className":37031,"ariaHidden":72},[116],[15,37033,37035,37038,37041,37044,37047,37050,37053,37056],{"className":37034},[120],[15,37036],{"className":37037,"style":125},[124],[15,37039,36569],{"className":37040,"style":36584},[129,133],[15,37042,53],{"className":37043},[188],[15,37045,310],{"className":37046},[129],[15,37048,59],{"className":37049},[195],[15,37051],{"className":37052,"style":3277},[93],[15,37054,2682],{"className":37055},[2436],[15,37057],{"className":37058,"style":3277},[93],[15,37060,37062,37065,37068,37071,37074,37077,37080,37083],{"className":37061},[120],[15,37063],{"className":37064,"style":125},[124],[15,37066,36569],{"className":37067,"style":36584},[129,133],[15,37069,53],{"className":37070},[188],[15,37072,9879],{"className":37073},[129],[15,37075,59],{"className":37076},[195],[15,37078],{"className":37079,"style":199},[93],[15,37081,62],{"className":37082},[203],[15,37084],{"className":37085,"style":199},[93],[15,37087,37089,37092,37095,37098,37101],{"className":37088},[120],[15,37090],{"className":37091,"style":16484},[124],[15,37093,32239],{"className":37094},[129],[15,37096],{"className":37097,"style":3277},[93],[15,37099,2682],{"className":37100},[2436],[15,37102],{"className":37103,"style":3277},[93],[15,37105,37107,37110,37113,37116,37119],{"className":37106},[120],[15,37108],{"className":37109,"style":4886},[124],[15,37111,37021],{"className":37112},[129],[15,37114],{"className":37115,"style":199},[93],[15,37117,62],{"className":37118},[203],[15,37120],{"className":37121,"style":199},[93],[15,37123,37125,37128],{"className":37124},[120],[15,37126],{"className":37127,"style":4886},[124],[15,37129,36861],{"className":37130},[129],"，相等。",[11,37133,37134,37135,37231],{},"反例：平方函数 ",[15,37136,37138,37166],{"className":37137,"translate":19},[23],[15,37139,37141],{"className":37140},[27],[29,37142,37143],{"xmlns":31},[34,37144,37145,37163],{},[37,37146,37147,37149,37151,37153,37155,37157],{},[43,37148,36569],{},[50,37150,53],{"stretchy":52},[43,37152,56],{},[50,37154,59],{"stretchy":52},[50,37156,62],{},[305,37158,37159,37161],{},[43,37160,56],{},[308,37162,310],{},[109,37164,37165],{"encoding":111},"f(x) = x^2",[15,37167,37169,37196],{"className":37168,"ariaHidden":72},[116],[15,37170,37172,37175,37178,37181,37184,37187,37190,37193],{"className":37171},[120],[15,37173],{"className":37174,"style":125},[124],[15,37176,36569],{"className":37177,"style":36584},[129,133],[15,37179,53],{"className":37180},[188],[15,37182,56],{"className":37183},[129,133],[15,37185,59],{"className":37186},[195],[15,37188],{"className":37189,"style":199},[93],[15,37191,62],{"className":37192},[203],[15,37194],{"className":37195,"style":199},[93],[15,37197,37199,37202],{"className":37198},[120],[15,37200],{"className":37201,"style":380},[124],[15,37203,37205,37208],{"className":37204},[129],[15,37206,56],{"className":37207},[129,133],[15,37209,37211],{"className":37210},[137],[15,37212,37214],{"className":37213},[141],[15,37215,37217],{"className":37216},[146],[15,37218,37220],{"className":37219,"style":380},[150],[15,37221,37222,37225],{"style":383},[15,37223],{"className":37224,"style":159},[158],[15,37226,37228],{"className":37227},[163,164,165,166],[15,37229,310],{"className":37230},[129,166]," 不满足可加性。",[21730,37233,37234],{},[21733,37235,36816,37236,36982,37349,37529],{},[15,37237,37239,37269],{"className":37238,"translate":19},[23],[15,37240,37242],{"className":37241},[27],[29,37243,37244],{"xmlns":31},[34,37245,37246,37266],{},[37,37247,37248,37250,37252,37254,37256,37262,37264],{},[50,37249,53],{"stretchy":52},[308,37251,310],{},[50,37253,2682],{},[308,37255,9879],{},[305,37257,37258,37260],{},[50,37259,59],{"stretchy":52},[308,37261,310],{},[50,37263,62],{},[308,37265,36861],{},[109,37267,37268],{"encoding":111},"(2+3)^2 = 25",[15,37270,37272,37293,37340],{"className":37271,"ariaHidden":72},[116],[15,37273,37275,37278,37281,37284,37287,37290],{"className":37274},[120],[15,37276],{"className":37277,"style":125},[124],[15,37279,53],{"className":37280},[188],[15,37282,310],{"className":37283},[129],[15,37285],{"className":37286,"style":3277},[93],[15,37288,2682],{"className":37289},[2436],[15,37291],{"className":37292,"style":3277},[93],[15,37294,37296,37299,37302,37331,37334,37337],{"className":37295},[120],[15,37297],{"className":37298,"style":34009},[124],[15,37300,9879],{"className":37301},[129],[15,37303,37305,37308],{"className":37304},[195],[15,37306,59],{"className":37307},[195],[15,37309,37311],{"className":37310},[137],[15,37312,37314],{"className":37313},[141],[15,37315,37317],{"className":37316},[146],[15,37318,37320],{"className":37319,"style":380},[150],[15,37321,37322,37325],{"style":383},[15,37323],{"className":37324,"style":159},[158],[15,37326,37328],{"className":37327},[163,164,165,166],[15,37329,310],{"className":37330},[129,166],[15,37332],{"className":37333,"style":199},[93],[15,37335,62],{"className":37336},[203],[15,37338],{"className":37339,"style":199},[93],[15,37341,37343,37346],{"className":37342},[120],[15,37344],{"className":37345,"style":4886},[124],[15,37347,36861],{"className":37348},[129],[15,37350,37352,37392],{"className":37351,"translate":19},[23],[15,37353,37355],{"className":37354},[27],[29,37356,37357],{"xmlns":31},[34,37358,37359,37389],{},[37,37360,37361,37367,37369,37375,37377,37379,37381,37384,37386],{},[305,37362,37363,37365],{},[308,37364,310],{},[308,37366,310],{},[50,37368,2682],{},[305,37370,37371,37373],{},[308,37372,9879],{},[308,37374,310],{},[50,37376,62],{},[308,37378,14508],{},[50,37380,2682],{},[308,37382,37383],{},"9",[50,37385,62],{},[308,37387,37388],{},"13",[109,37390,37391],{"encoding":111},"2^2 + 3^2 = 4+9=13",[15,37393,37395,37440,37484,37502,37520],{"className":37394,"ariaHidden":72},[116],[15,37396,37398,37402,37431,37434,37437],{"className":37397},[120],[15,37399],{"className":37400,"style":37401},[124],"height:0.8974em;vertical-align:-0.0833em;",[15,37403,37405,37408],{"className":37404},[129],[15,37406,310],{"className":37407},[129],[15,37409,37411],{"className":37410},[137],[15,37412,37414],{"className":37413},[141],[15,37415,37417],{"className":37416},[146],[15,37418,37420],{"className":37419,"style":380},[150],[15,37421,37422,37425],{"style":383},[15,37423],{"className":37424,"style":159},[158],[15,37426,37428],{"className":37427},[163,164,165,166],[15,37429,310],{"className":37430},[129,166],[15,37432],{"className":37433,"style":3277},[93],[15,37435,2682],{"className":37436},[2436],[15,37438],{"className":37439,"style":3277},[93],[15,37441,37443,37446,37475,37478,37481],{"className":37442},[120],[15,37444],{"className":37445,"style":380},[124],[15,37447,37449,37452],{"className":37448},[129],[15,37450,9879],{"className":37451},[129],[15,37453,37455],{"className":37454},[137],[15,37456,37458],{"className":37457},[141],[15,37459,37461],{"className":37460},[146],[15,37462,37464],{"className":37463,"style":380},[150],[15,37465,37466,37469],{"style":383},[15,37467],{"className":37468,"style":159},[158],[15,37470,37472],{"className":37471},[163,164,165,166],[15,37473,310],{"className":37474},[129,166],[15,37476],{"className":37477,"style":199},[93],[15,37479,62],{"className":37480},[203],[15,37482],{"className":37483,"style":199},[93],[15,37485,37487,37490,37493,37496,37499],{"className":37486},[120],[15,37488],{"className":37489,"style":16484},[124],[15,37491,14508],{"className":37492},[129],[15,37494],{"className":37495,"style":3277},[93],[15,37497,2682],{"className":37498},[2436],[15,37500],{"className":37501,"style":3277},[93],[15,37503,37505,37508,37511,37514,37517],{"className":37504},[120],[15,37506],{"className":37507,"style":4886},[124],[15,37509,37383],{"className":37510},[129],[15,37512],{"className":37513,"style":199},[93],[15,37515,62],{"className":37516},[203],[15,37518],{"className":37519,"style":199},[93],[15,37521,37523,37526],{"className":37522},[120],[15,37524],{"className":37525,"style":4886},[124],[15,37527,37388],{"className":37528},[129],"，不相等。",[575,37531,37532],{"id":37532},"齐次性",[11,37534,37535],{},"「输入的缩放会使输出按相同比例缩放」。也就是说，将输入乘以一个常数（标量），得到的结果等于原输出乘以同一个常数。",[15,37537,37539],{"className":37538,"translate":19},[18],[15,37540,37542,37589],{"className":37541,"translate":19},[23],[15,37543,37545],{"className":37544},[27],[29,37546,37547],{"xmlns":31,"display":32},[34,37548,37549,37586],{},[37,37550,37551,37553,37555,37557,37559,37561,37563,37565,37567,37569,37571,37573,37575,37577,37579,37581,37584],{},[43,37552,36569],{},[50,37554,53],{"stretchy":52},[43,37556,5628],{},[50,37558,10553],{},[43,37560,56],{},[50,37562,59],{"stretchy":52},[50,37564,62],{},[43,37566,5628],{},[50,37568,10553],{},[43,37570,36569],{},[50,37572,53],{"stretchy":52},[43,37574,56],{},[50,37576,59],{"stretchy":52},[93,37578],{"width":16921},[50,37580,53],{"stretchy":52},[6856,37582,37583],{},"c 为任意常数",[50,37585,59],{"stretchy":52},[109,37587,37588],{"encoding":111},"f(c \\cdot x) = c \\cdot f(x) \\quad (\\text{c 为任意常数})",[15,37590,37592,37616,37637,37655],{"className":37591,"ariaHidden":72},[116],[15,37593,37595,37598,37601,37604,37607,37610,37613],{"className":37594},[120],[15,37596],{"className":37597,"style":125},[124],[15,37599,36569],{"className":37600,"style":36584},[129,133],[15,37602,53],{"className":37603},[188],[15,37605,5628],{"className":37606},[129,133],[15,37608],{"className":37609,"style":3277},[93],[15,37611,10553],{"className":37612},[2436],[15,37614],{"className":37615,"style":3277},[93],[15,37617,37619,37622,37625,37628,37631,37634],{"className":37618},[120],[15,37620],{"className":37621,"style":125},[124],[15,37623,56],{"className":37624},[129,133],[15,37626,59],{"className":37627},[195],[15,37629],{"className":37630,"style":199},[93],[15,37632,62],{"className":37633},[203],[15,37635],{"className":37636,"style":199},[93],[15,37638,37640,37643,37646,37649,37652],{"className":37639},[120],[15,37641],{"className":37642,"style":11098},[124],[15,37644,5628],{"className":37645},[129,133],[15,37647],{"className":37648,"style":3277},[93],[15,37650,10553],{"className":37651},[2436],[15,37653],{"className":37654,"style":3277},[93],[15,37656,37658,37661,37664,37667,37670,37673,37676,37679,37692],{"className":37657},[120],[15,37659],{"className":37660,"style":125},[124],[15,37662,36569],{"className":37663,"style":36584},[129,133],[15,37665,53],{"className":37666},[188],[15,37668,56],{"className":37669},[129,133],[15,37671,59],{"className":37672},[195],[15,37674],{"className":37675,"style":17294},[93],[15,37677,53],{"className":37678},[188],[15,37680,37683,37687],{"className":37681},[129,37682],"text",[15,37684,37686],{"className":37685},[129],"c ",[15,37688,37691],{"className":37689},[129,37690],"cjk_fallback","为任意常数",[15,37693,59],{"className":37694},[195],[6559,37696,37697,37771,38041,38138],{},[11,37698,37699,37700,37770],{},"示例：",[15,37701,37703,37728],{"className":37702,"translate":19},[23],[15,37704,37706],{"className":37705},[27],[29,37707,37708],{"xmlns":31},[34,37709,37710,37726],{},[37,37711,37712,37714,37716,37718,37720,37722,37724],{},[43,37713,36569],{},[50,37715,53],{"stretchy":52},[43,37717,56],{},[50,37719,59],{"stretchy":52},[50,37721,62],{},[308,37723,36763],{},[43,37725,56],{},[109,37727,36768],{"encoding":111},[15,37729,37731,37758],{"className":37730,"ariaHidden":72},[116],[15,37732,37734,37737,37740,37743,37746,37749,37752,37755],{"className":37733},[120],[15,37735],{"className":37736,"style":125},[124],[15,37738,36569],{"className":37739,"style":36584},[129,133],[15,37741,53],{"className":37742},[188],[15,37744,56],{"className":37745},[129,133],[15,37747,59],{"className":37748},[195],[15,37750],{"className":37751,"style":199},[93],[15,37753,62],{"className":37754},[203],[15,37756],{"className":37757,"style":199},[93],[15,37759,37761,37764,37767],{"className":37760},[120],[15,37762],{"className":37763,"style":4886},[124],[15,37765,36763],{"className":37766},[129],[15,37768,56],{"className":37769},[129,133]," 满足齐次性。",[21730,37772,37773],{},[21733,37774,36816,37775,36982,37909,37131],{},[15,37776,37778,37816],{"className":37777,"translate":19},[23],[15,37779,37781],{"className":37780},[27],[29,37782,37783],{"xmlns":31},[34,37784,37785,37813],{},[37,37786,37787,37789,37791,37793,37795,37797,37799,37801,37803,37805,37808,37810],{},[43,37788,36569],{},[50,37790,53],{"stretchy":52},[308,37792,14508],{},[50,37794,11422],{},[308,37796,9879],{},[50,37798,59],{"stretchy":52},[50,37800,62],{},[308,37802,36763],{},[50,37804,11422],{},[308,37806,37807],{},"12",[50,37809,62],{},[308,37811,37812],{},"60",[109,37814,37815],{"encoding":111},"f(4 \\times 3) = 5 \\times 12 = 60",[15,37817,37819,37843,37864,37882,37900],{"className":37818,"ariaHidden":72},[116],[15,37820,37822,37825,37828,37831,37834,37837,37840],{"className":37821},[120],[15,37823],{"className":37824,"style":125},[124],[15,37826,36569],{"className":37827,"style":36584},[129,133],[15,37829,53],{"className":37830},[188],[15,37832,14508],{"className":37833},[129],[15,37835],{"className":37836,"style":3277},[93],[15,37838,11422],{"className":37839},[2436],[15,37841],{"className":37842,"style":3277},[93],[15,37844,37846,37849,37852,37855,37858,37861],{"className":37845},[120],[15,37847],{"className":37848,"style":125},[124],[15,37850,9879],{"className":37851},[129],[15,37853,59],{"className":37854},[195],[15,37856],{"className":37857,"style":199},[93],[15,37859,62],{"className":37860},[203],[15,37862],{"className":37863,"style":199},[93],[15,37865,37867,37870,37873,37876,37879],{"className":37866},[120],[15,37868],{"className":37869,"style":16484},[124],[15,37871,36763],{"className":37872},[129],[15,37874],{"className":37875,"style":3277},[93],[15,37877,11422],{"className":37878},[2436],[15,37880],{"className":37881,"style":3277},[93],[15,37883,37885,37888,37891,37894,37897],{"className":37884},[120],[15,37886],{"className":37887,"style":4886},[124],[15,37889,37807],{"className":37890},[129],[15,37892],{"className":37893,"style":199},[93],[15,37895,62],{"className":37896},[203],[15,37898],{"className":37899,"style":199},[93],[15,37901,37903,37906],{"className":37902},[120],[15,37904],{"className":37905,"style":4886},[124],[15,37907,37812],{"className":37908},[129],[15,37910,37912,37948],{"className":37911,"translate":19},[23],[15,37913,37915],{"className":37914},[27],[29,37916,37917],{"xmlns":31},[34,37918,37919,37945],{},[37,37920,37921,37923,37925,37927,37929,37931,37933,37935,37937,37939,37941,37943],{},[308,37922,14508],{},[50,37924,11422],{},[43,37926,36569],{},[50,37928,53],{"stretchy":52},[308,37930,9879],{},[50,37932,59],{"stretchy":52},[50,37934,62],{},[308,37936,14508],{},[50,37938,11422],{},[308,37940,37021],{},[50,37942,62],{},[308,37944,37812],{},[109,37946,37947],{"encoding":111},"4 \\times f(3) = 4 \\times 15 = 60",[15,37949,37951,37969,37996,38014,38032],{"className":37950,"ariaHidden":72},[116],[15,37952,37954,37957,37960,37963,37966],{"className":37953},[120],[15,37955],{"className":37956,"style":16484},[124],[15,37958,14508],{"className":37959},[129],[15,37961],{"className":37962,"style":3277},[93],[15,37964,11422],{"className":37965},[2436],[15,37967],{"className":37968,"style":3277},[93],[15,37970,37972,37975,37978,37981,37984,37987,37990,37993],{"className":37971},[120],[15,37973],{"className":37974,"style":125},[124],[15,37976,36569],{"className":37977,"style":36584},[129,133],[15,37979,53],{"className":37980},[188],[15,37982,9879],{"className":37983},[129],[15,37985,59],{"className":37986},[195],[15,37988],{"className":37989,"style":199},[93],[15,37991,62],{"className":37992},[203],[15,37994],{"className":37995,"style":199},[93],[15,37997,37999,38002,38005,38008,38011],{"className":37998},[120],[15,38000],{"className":38001,"style":16484},[124],[15,38003,14508],{"className":38004},[129],[15,38006],{"className":38007,"style":3277},[93],[15,38009,11422],{"className":38010},[2436],[15,38012],{"className":38013,"style":3277},[93],[15,38015,38017,38020,38023,38026,38029],{"className":38016},[120],[15,38018],{"className":38019,"style":4886},[124],[15,38021,37021],{"className":38022},[129],[15,38024],{"className":38025,"style":199},[93],[15,38027,62],{"className":38028},[203],[15,38030],{"className":38031,"style":199},[93],[15,38033,38035,38038],{"className":38034},[120],[15,38036],{"className":38037,"style":4886},[124],[15,38039,37812],{"className":38040},[129],[11,38042,38043,38044,38137],{},"反例：加上常数项 ",[15,38045,38047,38077],{"className":38046,"translate":19},[23],[15,38048,38050],{"className":38049},[27],[29,38051,38052],{"xmlns":31},[34,38053,38054,38074],{},[37,38055,38056,38058,38060,38062,38064,38066,38068,38070,38072],{},[43,38057,36569],{},[50,38059,53],{"stretchy":52},[43,38061,56],{},[50,38063,59],{"stretchy":52},[50,38065,62],{},[308,38067,36763],{},[43,38069,56],{},[50,38071,2682],{},[308,38073,310],{},[109,38075,38076],{"encoding":111},"f(x) = 5x + 2",[15,38078,38080,38107,38128],{"className":38079,"ariaHidden":72},[116],[15,38081,38083,38086,38089,38092,38095,38098,38101,38104],{"className":38082},[120],[15,38084],{"className":38085,"style":125},[124],[15,38087,36569],{"className":38088,"style":36584},[129,133],[15,38090,53],{"className":38091},[188],[15,38093,56],{"className":38094},[129,133],[15,38096,59],{"className":38097},[195],[15,38099],{"className":38100,"style":199},[93],[15,38102,62],{"className":38103},[203],[15,38105],{"className":38106,"style":199},[93],[15,38108,38110,38113,38116,38119,38122,38125],{"className":38109},[120],[15,38111],{"className":38112,"style":16484},[124],[15,38114,36763],{"className":38115},[129],[15,38117,56],{"className":38118},[129,133],[15,38120],{"className":38121,"style":3277},[93],[15,38123,2682],{"className":38124},[2436],[15,38126],{"className":38127,"style":3277},[93],[15,38129,38131,38134],{"className":38130},[120],[15,38132],{"className":38133,"style":4886},[124],[15,38135,310],{"className":38136},[129]," 不满足齐次性。",[21730,38139,38140],{},[21733,38141,36816,38142,36982,38297,37529],{},[15,38143,38145,38186],{"className":38144,"translate":19},[23],[15,38146,38148],{"className":38147},[27],[29,38149,38150],{"xmlns":31},[34,38151,38152,38183],{},[37,38153,38154,38156,38158,38160,38162,38164,38166,38168,38170,38172,38174,38176,38178,38180],{},[43,38155,36569],{},[50,38157,53],{"stretchy":52},[308,38159,14508],{},[50,38161,11422],{},[308,38163,9879],{},[50,38165,59],{"stretchy":52},[50,38167,62],{},[308,38169,36763],{},[50,38171,11422],{},[308,38173,37807],{},[50,38175,2682],{},[308,38177,310],{},[50,38179,62],{},[308,38181,38182],{},"62",[109,38184,38185],{"encoding":111},"f(4 \\times 3) = 5 \\times 12 + 2 = 62",[15,38187,38189,38213,38234,38252,38270,38288],{"className":38188,"ariaHidden":72},[116],[15,38190,38192,38195,38198,38201,38204,38207,38210],{"className":38191},[120],[15,38193],{"className":38194,"style":125},[124],[15,38196,36569],{"className":38197,"style":36584},[129,133],[15,38199,53],{"className":38200},[188],[15,38202,14508],{"className":38203},[129],[15,38205],{"className":38206,"style":3277},[93],[15,38208,11422],{"className":38209},[2436],[15,38211],{"className":38212,"style":3277},[93],[15,38214,38216,38219,38222,38225,38228,38231],{"className":38215},[120],[15,38217],{"className":38218,"style":125},[124],[15,38220,9879],{"className":38221},[129],[15,38223,59],{"className":38224},[195],[15,38226],{"className":38227,"style":199},[93],[15,38229,62],{"className":38230},[203],[15,38232],{"className":38233,"style":199},[93],[15,38235,38237,38240,38243,38246,38249],{"className":38236},[120],[15,38238],{"className":38239,"style":16484},[124],[15,38241,36763],{"className":38242},[129],[15,38244],{"className":38245,"style":3277},[93],[15,38247,11422],{"className":38248},[2436],[15,38250],{"className":38251,"style":3277},[93],[15,38253,38255,38258,38261,38264,38267],{"className":38254},[120],[15,38256],{"className":38257,"style":16484},[124],[15,38259,37807],{"className":38260},[129],[15,38262],{"className":38263,"style":3277},[93],[15,38265,2682],{"className":38266},[2436],[15,38268],{"className":38269,"style":3277},[93],[15,38271,38273,38276,38279,38282,38285],{"className":38272},[120],[15,38274],{"className":38275,"style":4886},[124],[15,38277,310],{"className":38278},[129],[15,38280],{"className":38281,"style":199},[93],[15,38283,62],{"className":38284},[203],[15,38286],{"className":38287,"style":199},[93],[15,38289,38291,38294],{"className":38290},[120],[15,38292],{"className":38293,"style":4886},[124],[15,38295,38182],{"className":38296},[129],[15,38298,38300,38345],{"className":38299,"translate":19},[23],[15,38301,38303],{"className":38302},[27],[29,38304,38305],{"xmlns":31},[34,38306,38307,38342],{},[37,38308,38309,38311,38313,38315,38317,38319,38321,38323,38325,38327,38329,38331,38333,38335,38337,38339],{},[308,38310,14508],{},[50,38312,11422],{},[43,38314,36569],{},[50,38316,53],{"stretchy":52},[308,38318,9879],{},[50,38320,59],{"stretchy":52},[50,38322,62],{},[308,38324,14508],{},[50,38326,11422],{},[50,38328,53],{"stretchy":52},[308,38330,37021],{},[50,38332,2682],{},[308,38334,310],{},[50,38336,59],{"stretchy":52},[50,38338,62],{},[308,38340,38341],{},"68",[109,38343,38344],{"encoding":111},"4 \\times f(3) = 4 \\times (15+2) = 68",[15,38346,38348,38366,38393,38411,38432,38453],{"className":38347,"ariaHidden":72},[116],[15,38349,38351,38354,38357,38360,38363],{"className":38350},[120],[15,38352],{"className":38353,"style":16484},[124],[15,38355,14508],{"className":38356},[129],[15,38358],{"className":38359,"style":3277},[93],[15,38361,11422],{"className":38362},[2436],[15,38364],{"className":38365,"style":3277},[93],[15,38367,38369,38372,38375,38378,38381,38384,38387,38390],{"className":38368},[120],[15,38370],{"className":38371,"style":125},[124],[15,38373,36569],{"className":38374,"style":36584},[129,133],[15,38376,53],{"className":38377},[188],[15,38379,9879],{"className":38380},[129],[15,38382,59],{"className":38383},[195],[15,38385],{"className":38386,"style":199},[93],[15,38388,62],{"className":38389},[203],[15,38391],{"className":38392,"style":199},[93],[15,38394,38396,38399,38402,38405,38408],{"className":38395},[120],[15,38397],{"className":38398,"style":16484},[124],[15,38400,14508],{"className":38401},[129],[15,38403],{"className":38404,"style":3277},[93],[15,38406,11422],{"className":38407},[2436],[15,38409],{"className":38410,"style":3277},[93],[15,38412,38414,38417,38420,38423,38426,38429],{"className":38413},[120],[15,38415],{"className":38416,"style":125},[124],[15,38418,53],{"className":38419},[188],[15,38421,37021],{"className":38422},[129],[15,38424],{"className":38425,"style":3277},[93],[15,38427,2682],{"className":38428},[2436],[15,38430],{"className":38431,"style":3277},[93],[15,38433,38435,38438,38441,38444,38447,38450],{"className":38434},[120],[15,38436],{"className":38437,"style":125},[124],[15,38439,310],{"className":38440},[129],[15,38442,59],{"className":38443},[195],[15,38445],{"className":38446,"style":199},[93],[15,38448,62],{"className":38449},[203],[15,38451],{"className":38452,"style":199},[93],[15,38454,38456,38459],{"className":38455},[120],[15,38457],{"className":38458,"style":4886},[124],[15,38460,38341],{"className":38461},[129],[11,38463,38464],{},"在上述定义中，同时满足可加性与齐次性是线性成立的充分必要条件。",[11,38466,38467],{},"许多人错误地认为「线性」就是「图像是一条直线」。这种理解并不准确。",[11,38469,38470,38471,38527],{},"「真正的线性」：一条必须过原点的直线。例如 ",[15,38472,38474,38494],{"className":38473,"translate":19},[23],[15,38475,38477],{"className":38476},[27],[29,38478,38479],{"xmlns":31},[34,38480,38481,38491],{},[37,38482,38483,38485,38487,38489],{},[43,38484,857],{},[50,38486,62],{},[308,38488,310],{},[43,38490,56],{},[109,38492,38493],{"encoding":111},"y = 2x",[15,38495,38497,38515],{"className":38496,"ariaHidden":72},[116],[15,38498,38500,38503,38506,38509,38512],{"className":38499},[120],[15,38501],{"className":38502,"style":5399},[124],[15,38504,857],{"className":38505,"style":887},[129,133],[15,38507],{"className":38508,"style":199},[93],[15,38510,62],{"className":38511},[203],[15,38513],{"className":38514,"style":199},[93],[15,38516,38518,38521,38524],{"className":38517},[120],[15,38519],{"className":38520,"style":4886},[124],[15,38522,310],{"className":38523},[129],[15,38525,56],{"className":38526},[129,133],"。它同时满足可加性与齐次性。如果输入为 0，输出也必为 0。",[11,38529,38530,38531,38559,38560,38638,38639,38760],{},"严格来说，如果加入偏置项 ",[15,38532,38534,38547],{"className":38533,"translate":19},[23],[15,38535,38537],{"className":38536},[27],[29,38538,38539],{"xmlns":31},[34,38540,38541,38545],{},[37,38542,38543],{},[43,38544,8699],{},[109,38546,8699],{"encoding":111},[15,38548,38550],{"className":38549,"ariaHidden":72},[116],[15,38551,38553,38556],{"className":38552},[120],[15,38554],{"className":38555,"style":522},[124],[15,38557,8699],{"className":38558},[129,133],"，它就变成了仿射（affine）。它仍是一条直线，但未必经过原点。例如 ",[15,38561,38563,38587],{"className":38562,"translate":19},[23],[15,38564,38566],{"className":38565},[27],[29,38567,38568],{"xmlns":31},[34,38569,38570,38584],{},[37,38571,38572,38574,38576,38578,38580,38582],{},[43,38573,857],{},[50,38575,62],{},[308,38577,310],{},[43,38579,56],{},[50,38581,2682],{},[308,38583,318],{},[109,38585,38586],{"encoding":111},"y = 2x + 1",[15,38588,38590,38608,38629],{"className":38589,"ariaHidden":72},[116],[15,38591,38593,38596,38599,38602,38605],{"className":38592},[120],[15,38594],{"className":38595,"style":5399},[124],[15,38597,857],{"className":38598,"style":887},[129,133],[15,38600],{"className":38601,"style":199},[93],[15,38603,62],{"className":38604},[203],[15,38606],{"className":38607,"style":199},[93],[15,38609,38611,38614,38617,38620,38623,38626],{"className":38610},[120],[15,38612],{"className":38613,"style":16484},[124],[15,38615,310],{"className":38616},[129],[15,38618,56],{"className":38619},[129,133],[15,38621],{"className":38622,"style":3277},[93],[15,38624,2682],{"className":38625},[2436],[15,38627],{"className":38628,"style":3277},[93],[15,38630,38632,38635],{"className":38631},[120],[15,38633],{"className":38634,"style":4886},[124],[15,38636,318],{"className":38637},[129],"。它不满足齐次性（因为 ",[15,38640,38642,38670],{"className":38641,"translate":19},[23],[15,38643,38645],{"className":38644},[27],[29,38646,38647],{"xmlns":31},[34,38648,38649,38667],{},[37,38650,38651,38653,38655,38657,38659,38661,38663,38665],{},[43,38652,36569],{},[50,38654,53],{"stretchy":52},[308,38656,541],{},[50,38658,59],{"stretchy":52},[50,38660,62],{},[308,38662,318],{},[50,38664,1418],{"mathvariant":65},[308,38666,541],{},[109,38668,38669],{"encoding":111},"f(0)=1 \\neq 0",[15,38671,38673,38700,38751],{"className":38672,"ariaHidden":72},[116],[15,38674,38676,38679,38682,38685,38688,38691,38694,38697],{"className":38675},[120],[15,38677],{"className":38678,"style":125},[124],[15,38680,36569],{"className":38681,"style":36584},[129,133],[15,38683,53],{"className":38684},[188],[15,38686,541],{"className":38687},[129],[15,38689,59],{"className":38690},[195],[15,38692],{"className":38693,"style":199},[93],[15,38695,62],{"className":38696},[203],[15,38698],{"className":38699,"style":199},[93],[15,38701,38703,38706,38709,38712,38748],{"className":38702},[120],[15,38704],{"className":38705,"style":1682},[124],[15,38707,318],{"className":38708},[129],[15,38710],{"className":38711,"style":199},[93],[15,38713,38715,38742,38745],{"className":38714},[203],[15,38716,38718],{"className":38717},[203],[15,38719,38721],{"className":38720},[129,1670],[15,38722,38724],{"className":38723},[1674],[15,38725,38727,38730,38739],{"className":38726},[1678],[15,38728],{"className":38729,"style":1682},[124],[15,38731,38733],{"className":38732},[1686],[15,38734,38736],{"className":38735},[129],[15,38737,1693],{"className":38738},[203],[15,38740],{"className":38741},[1697],[15,38743],{"className":38744},[93,1701],[15,38746,62],{"className":38747},[203],[15,38749],{"className":38750,"style":199},[93],[15,38752,38754,38757],{"className":38753},[120],[15,38755],{"className":38756,"style":4886},[124],[15,38758,541],{"className":38759},[129],"）。尽管它常被称为「线性函数」，但在严格定义下它并非线性，而是「线性 + 平移」，即仿射。",[11,38762,38763],{},"现在我们就能理解为什么称为「线性」组合了。这种组合只允许两类「线性」运算：",[30672,38765,38766,38798],{},[21733,38767,38768,38769,38797],{},"数乘（对应齐次性）：将向量按因子 ",[15,38770,38772,38785],{"className":38771,"translate":19},[23],[15,38773,38775],{"className":38774},[27],[29,38776,38777],{"xmlns":31},[34,38778,38779,38783],{},[37,38780,38781],{},[43,38782,5628],{},[109,38784,5628],{"encoding":111},[15,38786,38788],{"className":38787,"ariaHidden":72},[116],[15,38789,38791,38794],{"className":38790},[120],[15,38792],{"className":38793,"style":4383},[124],[15,38795,5628],{"className":38796},[129,133]," 拉伸或压缩。",[21733,38799,38800],{},"加法（对应可加性）：将处理后的向量相加。",[11,38802,38803,38804,39046,39047,39294],{},"像平方、取绝对值或对数这类非线性运算是不被允许的。因此，形如 ",[15,38805,38807,38848],{"className":38806,"translate":19},[23],[15,38808,38810],{"className":38809},[27],[29,38811,38812],{"xmlns":31},[34,38813,38814,38845],{},[37,38815,38816,38822,38831,38833,38839],{},[40,38817,38818,38820],{},[43,38819,5628],{},[308,38821,318],{},[23160,38823,38824,38827,38829],{},[43,38825,38826],{},"v",[308,38828,318],{},[308,38830,310],{},[50,38832,2682],{},[40,38834,38835,38837],{},[43,38836,5628],{},[308,38838,310],{},[40,38840,38841,38843],{},[43,38842,38826],{},[308,38844,310],{},[109,38846,38847],{"encoding":111},"c_1 v_1^2 + c_2 v_2",[15,38849,38851,38960],{"className":38850,"ariaHidden":72},[116],[15,38852,38854,38858,38898,38951,38954,38957],{"className":38853},[120],[15,38855],{"className":38856,"style":38857},[124],"height:1.0622em;vertical-align:-0.2481em;",[15,38859,38861,38864],{"className":38860},[129],[15,38862,5628],{"className":38863},[129,133],[15,38865,38867],{"className":38866},[137],[15,38868,38870,38890],{"className":38869},[141,142],[15,38871,38873,38887],{"className":38872},[146],[15,38874,38876],{"className":38875,"style":1051},[150],[15,38877,38878,38881],{"style":154},[15,38879],{"className":38880,"style":159},[158],[15,38882,38884],{"className":38883},[163,164,165,166],[15,38885,318],{"className":38886},[129,166],[15,38888,175],{"className":38889},[174],[15,38891,38893],{"className":38892},[146],[15,38894,38896],{"className":38895,"style":182},[150],[15,38897],{},[15,38899,38901,38904],{"className":38900},[129],[15,38902,38826],{"className":38903,"style":887},[129,133],[15,38905,38907],{"className":38906},[137],[15,38908,38910,38942],{"className":38909},[141,142],[15,38911,38913,38939],{"className":38912},[146],[15,38914,38916,38928],{"className":38915,"style":380},[150],[15,38917,38919,38922],{"style":38918},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[15,38920],{"className":38921,"style":159},[158],[15,38923,38925],{"className":38924},[163,164,165,166],[15,38926,318],{"className":38927},[129,166],[15,38929,38930,38933],{"style":383},[15,38931],{"className":38932,"style":159},[158],[15,38934,38936],{"className":38935},[163,164,165,166],[15,38937,310],{"className":38938},[129,166],[15,38940,175],{"className":38941},[174],[15,38943,38945],{"className":38944},[146],[15,38946,38949],{"className":38947,"style":38948},[150],"height:0.2481em;",[15,38950],{},[15,38952],{"className":38953,"style":3277},[93],[15,38955,2682],{"className":38956},[2436],[15,38958],{"className":38959,"style":3277},[93],[15,38961,38963,38966,39006],{"className":38962},[120],[15,38964],{"className":38965,"style":7985},[124],[15,38967,38969,38972],{"className":38968},[129],[15,38970,5628],{"className":38971},[129,133],[15,38973,38975],{"className":38974},[137],[15,38976,38978,38998],{"className":38977},[141,142],[15,38979,38981,38995],{"className":38980},[146],[15,38982,38984],{"className":38983,"style":1051},[150],[15,38985,38986,38989],{"style":154},[15,38987],{"className":38988,"style":159},[158],[15,38990,38992],{"className":38991},[163,164,165,166],[15,38993,310],{"className":38994},[129,166],[15,38996,175],{"className":38997},[174],[15,38999,39001],{"className":39000},[146],[15,39002,39004],{"className":39003,"style":182},[150],[15,39005],{},[15,39007,39009,39012],{"className":39008},[129],[15,39010,38826],{"className":39011,"style":887},[129,133],[15,39013,39015],{"className":39014},[137],[15,39016,39018,39038],{"className":39017},[141,142],[15,39019,39021,39035],{"className":39020},[146],[15,39022,39024],{"className":39023,"style":1051},[150],[15,39025,39026,39029],{"style":1100},[15,39027],{"className":39028,"style":159},[158],[15,39030,39032],{"className":39031},[163,164,165,166],[15,39033,310],{"className":39034},[129,166],[15,39036,175],{"className":39037},[174],[15,39039,39041],{"className":39040},[146],[15,39042,39044],{"className":39043,"style":182},[150],[15,39045],{}," 或 ",[15,39048,39050,39092],{"className":39049,"translate":19},[23],[15,39051,39053],{"className":39052},[27],[29,39054,39055],{"xmlns":31},[34,39056,39057,39089],{},[37,39058,39059,39065,39071,39073,39079,39085,39087],{},[40,39060,39061,39063],{},[43,39062,5628],{},[308,39064,318],{},[40,39066,39067,39069],{},[43,39068,38826],{},[308,39070,318],{},[50,39072,2682],{},[40,39074,39075,39077],{},[43,39076,5628],{},[308,39078,310],{},[40,39080,39081,39083],{},[43,39082,38826],{},[308,39084,310],{},[50,39086,2682],{},[308,39088,9879],{},[109,39090,39091],{"encoding":111},"c_1 v_1 + c_2 v_2 + 3",[15,39093,39095,39190,39285],{"className":39094,"ariaHidden":72},[116],[15,39096,39098,39101,39141,39181,39184,39187],{"className":39097},[120],[15,39099],{"className":39100,"style":11117},[124],[15,39102,39104,39107],{"className":39103},[129],[15,39105,5628],{"className":39106},[129,133],[15,39108,39110],{"className":39109},[137],[15,39111,39113,39133],{"className":39112},[141,142],[15,39114,39116,39130],{"className":39115},[146],[15,39117,39119],{"className":39118,"style":1051},[150],[15,39120,39121,39124],{"style":154},[15,39122],{"className":39123,"style":159},[158],[15,39125,39127],{"className":39126},[163,164,165,166],[15,39128,318],{"className":39129},[129,166],[15,39131,175],{"className":39132},[174],[15,39134,39136],{"className":39135},[146],[15,39137,39139],{"className":39138,"style":182},[150],[15,39140],{},[15,39142,39144,39147],{"className":39143},[129],[15,39145,38826],{"className":39146,"style":887},[129,133],[15,39148,39150],{"className":39149},[137],[15,39151,39153,39173],{"className":39152},[141,142],[15,39154,39156,39170],{"className":39155},[146],[15,39157,39159],{"className":39158,"style":1051},[150],[15,39160,39161,39164],{"style":1100},[15,39162],{"className":39163,"style":159},[158],[15,39165,39167],{"className":39166},[163,164,165,166],[15,39168,318],{"className":39169},[129,166],[15,39171,175],{"className":39172},[174],[15,39174,39176],{"className":39175},[146],[15,39177,39179],{"className":39178,"style":182},[150],[15,39180],{},[15,39182],{"className":39183,"style":3277},[93],[15,39185,2682],{"className":39186},[2436],[15,39188],{"className":39189,"style":3277},[93],[15,39191,39193,39196,39236,39276,39279,39282],{"className":39192},[120],[15,39194],{"className":39195,"style":11117},[124],[15,39197,39199,39202],{"className":39198},[129],[15,39200,5628],{"className":39201},[129,133],[15,39203,39205],{"className":39204},[137],[15,39206,39208,39228],{"className":39207},[141,142],[15,39209,39211,39225],{"className":39210},[146],[15,39212,39214],{"className":39213,"style":1051},[150],[15,39215,39216,39219],{"style":154},[15,39217],{"className":39218,"style":159},[158],[15,39220,39222],{"className":39221},[163,164,165,166],[15,39223,310],{"className":39224},[129,166],[15,39226,175],{"className":39227},[174],[15,39229,39231],{"className":39230},[146],[15,39232,39234],{"className":39233,"style":182},[150],[15,39235],{},[15,39237,39239,39242],{"className":39238},[129],[15,39240,38826],{"className":39241,"style":887},[129,133],[15,39243,39245],{"className":39244},[137],[15,39246,39248,39268],{"className":39247},[141,142],[15,39249,39251,39265],{"className":39250},[146],[15,39252,39254],{"className":39253,"style":1051},[150],[15,39255,39256,39259],{"style":1100},[15,39257],{"className":39258,"style":159},[158],[15,39260,39262],{"className":39261},[163,164,165,166],[15,39263,310],{"className":39264},[129,166],[15,39266,175],{"className":39267},[174],[15,39269,39271],{"className":39270},[146],[15,39272,39274],{"className":39273,"style":182},[150],[15,39275],{},[15,39277],{"className":39278,"style":3277},[93],[15,39280,2682],{"className":39281},[2436],[15,39283],{"className":39284,"style":3277},[93],[15,39286,39288,39291],{"className":39287},[120],[15,39289],{"className":39290,"style":4886},[124],[15,39292,9879],{"className":39293},[129]," 的表达式不能称为线性组合。",[11,39296,39297],{},"可以说「线性」是数学与科学中最简单、最核心的关系，因为它允许我们把复杂问题分解为简单部分的叠加。",[570,39299,39300],{"id":39300},"仿射",[11,39302,39303,39304,39438],{},"仿射函数 ",[15,39305,39307,39338],{"className":39306,"translate":19},[23],[15,39308,39310],{"className":39309},[27],[29,39311,39312],{"xmlns":31},[34,39313,39314,39335],{},[37,39315,39316,39318,39320,39327,39329],{},[43,39317,36569],{},[50,39319,718],{},[305,39321,39322,39324],{},[43,39323,104],{"mathvariant":103},[43,39325,39326],{},"n",[50,39328,723],{},[305,39330,39331,39333],{},[43,39332,104],{"mathvariant":103},[43,39334,348],{},[109,39336,39337],{"encoding":111},"f: \\mathbb{R}^n \\to \\mathbb{R}^m",[15,39339,39341,39359,39403],{"className":39340,"ariaHidden":72},[116],[15,39342,39344,39347,39350,39353,39356],{"className":39343},[120],[15,39345],{"className":39346,"style":1682},[124],[15,39348,36569],{"className":39349,"style":36584},[129,133],[15,39351],{"className":39352,"style":199},[93],[15,39354,718],{"className":39355},[203],[15,39357],{"className":39358,"style":199},[93],[15,39360,39362,39365,39394,39397,39400],{"className":39361},[120],[15,39363],{"className":39364,"style":281},[124],[15,39366,39368,39371],{"className":39367},[129],[15,39369,104],{"className":39370},[129,285],[15,39372,39374],{"className":39373},[137],[15,39375,39377],{"className":39376},[141],[15,39378,39380],{"className":39379},[146],[15,39381,39383],{"className":39382,"style":6377},[150],[15,39384,39385,39388],{"style":383},[15,39386],{"className":39387,"style":159},[158],[15,39389,39391],{"className":39390},[163,164,165,166],[15,39392,39326],{"className":39393},[129,133,166],[15,39395],{"className":39396,"style":199},[93],[15,39398,723],{"className":39399},[203],[15,39401],{"className":39402,"style":199},[93],[15,39404,39406,39409],{"className":39405},[120],[15,39407],{"className":39408,"style":281},[124],[15,39410,39412,39415],{"className":39411},[129],[15,39413,104],{"className":39414},[129,285],[15,39416,39418],{"className":39417},[137],[15,39419,39421],{"className":39420},[141],[15,39422,39424],{"className":39423},[146],[15,39425,39427],{"className":39426,"style":6377},[150],[15,39428,39429,39432],{"style":383},[15,39430],{"className":39431,"style":159},[158],[15,39433,39435],{"className":39434},[163,164,165,166],[15,39436,348],{"className":39437},[129,133,166]," 的定义如下：",[15,39440,39442],{"className":39441,"translate":19},[18],[15,39443,39445,39475],{"className":39444,"translate":19},[23],[15,39446,39448],{"className":39447},[27],[29,39449,39450],{"xmlns":31,"display":32},[34,39451,39452,39472],{},[37,39453,39454,39456,39458,39460,39462,39464,39466,39468,39470],{},[43,39455,36569],{},[50,39457,53],{"stretchy":52},[43,39459,56],{},[50,39461,59],{"stretchy":52},[50,39463,62],{},[43,39465,13097],{},[43,39467,56],{},[50,39469,2682],{},[43,39471,8699],{},[109,39473,39474],{"encoding":111},"f(x) = Ax + b",[15,39476,39478,39505,39527],{"className":39477,"ariaHidden":72},[116],[15,39479,39481,39484,39487,39490,39493,39496,39499,39502],{"className":39480},[120],[15,39482],{"className":39483,"style":125},[124],[15,39485,36569],{"className":39486,"style":36584},[129,133],[15,39488,53],{"className":39489},[188],[15,39491,56],{"className":39492},[129,133],[15,39494,59],{"className":39495},[195],[15,39497],{"className":39498,"style":199},[93],[15,39500,62],{"className":39501},[203],[15,39503],{"className":39504,"style":199},[93],[15,39506,39508,39512,39515,39518,39521,39524],{"className":39507},[120],[15,39509],{"className":39510,"style":39511},[124],"height:0.7667em;vertical-align:-0.0833em;",[15,39513,13097],{"className":39514},[129,133],[15,39516,56],{"className":39517},[129,133],[15,39519],{"className":39520,"style":3277},[93],[15,39522,2682],{"className":39523},[2436],[15,39525],{"className":39526,"style":3277},[93],[15,39528,39530,39533],{"className":39529},[120],[15,39531],{"className":39532,"style":522},[124],[15,39534,8699],{"className":39535},[129,133],[11,39537,28697,39538,39566,39567,39595],{},[15,39539,39541,39554],{"className":39540,"translate":19},[23],[15,39542,39544],{"className":39543},[27],[29,39545,39546],{"xmlns":31},[34,39547,39548,39552],{},[37,39549,39550],{},[43,39551,13097],{},[109,39553,13097],{"encoding":111},[15,39555,39557],{"className":39556,"ariaHidden":72},[116],[15,39558,39560,39563],{"className":39559},[120],[15,39561],{"className":39562,"style":610},[124],[15,39564,13097],{"className":39565},[129,133]," 是矩阵，",[15,39568,39570,39583],{"className":39569,"translate":19},[23],[15,39571,39573],{"className":39572},[27],[29,39574,39575],{"xmlns":31},[34,39576,39577,39581],{},[37,39578,39579],{},[43,39580,8699],{},[109,39582,8699],{"encoding":111},[15,39584,39586],{"className":39585,"ariaHidden":72},[116],[15,39587,39589,39592],{"className":39588},[120],[15,39590],{"className":39591,"style":522},[124],[15,39593,8699],{"className":39594},[129,133]," 是常向量。它就是线性函数加上偏置。",[11,39597,39598],{},"如果取仿射函数的加权线性组合（系数可以是标量）：",[15,39600,39602],{"className":39601,"translate":19},[18],[15,39603,39605,39659],{"className":39604,"translate":19},[23],[15,39606,39608],{"className":39607},[27],[29,39609,39610],{"xmlns":31,"display":32},[34,39611,39612,39656],{},[37,39613,39614,39616,39618,39620,39622,39624,39638,39644,39650,39652,39654],{},[43,39615,2247],{},[50,39617,53],{"stretchy":52},[43,39619,56],{},[50,39621,59],{"stretchy":52},[50,39623,62],{},[1450,39625,39626,39628,39636],{},[50,39627,1454],{},[37,39629,39630,39632,39634],{},[43,39631,1459],{},[50,39633,62],{},[308,39635,318],{},[43,39637,16562],{},[40,39639,39640,39642],{},[43,39641,12008],{},[43,39643,1459],{},[40,39645,39646,39648],{},[43,39647,36569],{},[43,39649,1459],{},[50,39651,53],{"stretchy":52},[43,39653,56],{},[50,39655,59],{"stretchy":52},[109,39657,39658],{"encoding":111},"g(x) = \\sum_{i=1}^k \\alpha_i f_i(x)",[15,39660,39662,39689],{"className":39661,"ariaHidden":72},[116],[15,39663,39665,39668,39671,39674,39677,39680,39683,39686],{"className":39664},[120],[15,39666],{"className":39667,"style":125},[124],[15,39669,2247],{"className":39670,"style":887},[129,133],[15,39672,53],{"className":39673},[188],[15,39675,56],{"className":39676},[129,133],[15,39678,59],{"className":39679},[195],[15,39681],{"className":39682,"style":199},[93],[15,39684,62],{"className":39685},[203],[15,39687],{"className":39688,"style":199},[93],[15,39690,39692,39695,39759,39762,39802,39843,39846,39849],{"className":39691},[120],[15,39693],{"className":39694,"style":21128},[124],[15,39696,39698],{"className":39697},[217,1874],[15,39699,39701,39751],{"className":39700},[141,142],[15,39702,39704,39748],{"className":39703},[146],[15,39705,39707,39727,39737],{"className":39706,"style":21141},[150],[15,39708,39709,39712],{"style":1887},[15,39710],{"className":39711,"style":1891},[158],[15,39713,39715],{"className":39714},[163,164,165,166],[15,39716,39718,39721,39724],{"className":39717},[129,166],[15,39719,1459],{"className":39720},[129,133,166],[15,39722,62],{"className":39723},[203,166],[15,39725,318],{"className":39726},[129,166],[15,39728,39729,39732],{"style":1909},[15,39730],{"className":39731,"style":1891},[158],[15,39733,39734],{},[15,39735,1454],{"className":39736},[217,1918,1919],[15,39738,39739,39742],{"style":1922},[15,39740],{"className":39741,"style":1891},[158],[15,39743,39745],{"className":39744},[163,164,165,166],[15,39746,16562],{"className":39747,"style":16595},[129,133,166],[15,39749,175],{"className":39750},[174],[15,39752,39754],{"className":39753},[146],[15,39755,39757],{"className":39756,"style":1941},[150],[15,39758],{},[15,39760],{"className":39761,"style":262},[93],[15,39763,39765,39768],{"className":39764},[129],[15,39766,12008],{"className":39767,"style":12029},[129,133],[15,39769,39771],{"className":39770},[137],[15,39772,39774,39794],{"className":39773},[141,142],[15,39775,39777,39791],{"className":39776},[146],[15,39778,39780],{"className":39779,"style":1981},[150],[15,39781,39782,39785],{"style":12044},[15,39783],{"className":39784,"style":159},[158],[15,39786,39788],{"className":39787},[163,164,165,166],[15,39789,1459],{"className":39790},[129,133,166],[15,39792,175],{"className":39793},[174],[15,39795,39797],{"className":39796},[146],[15,39798,39800],{"className":39799,"style":182},[150],[15,39801],{},[15,39803,39805,39808],{"className":39804},[129],[15,39806,36569],{"className":39807,"style":36584},[129,133],[15,39809,39811],{"className":39810},[137],[15,39812,39814,39835],{"className":39813},[141,142],[15,39815,39817,39832],{"className":39816},[146],[15,39818,39820],{"className":39819,"style":1981},[150],[15,39821,39823,39826],{"style":39822},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[15,39824],{"className":39825,"style":159},[158],[15,39827,39829],{"className":39828},[163,164,165,166],[15,39830,1459],{"className":39831},[129,133,166],[15,39833,175],{"className":39834},[174],[15,39836,39838],{"className":39837},[146],[15,39839,39841],{"className":39840,"style":182},[150],[15,39842],{},[15,39844,53],{"className":39845},[188],[15,39847,56],{"className":39848},[129,133],[15,39850,59],{"className":39851},[195],[11,39853,39854,39855,1313],{},"其中每个 ",[15,39856,39858,39900],{"className":39857,"translate":19},[23],[15,39859,39861],{"className":39860},[27],[29,39862,39863],{"xmlns":31},[34,39864,39865,39897],{},[37,39866,39867,39873,39875,39877,39879,39881,39887,39889,39891],{},[40,39868,39869,39871],{},[43,39870,36569],{},[43,39872,1459],{},[50,39874,53],{"stretchy":52},[43,39876,56],{},[50,39878,59],{"stretchy":52},[50,39880,62],{},[40,39882,39883,39885],{},[43,39884,13097],{},[43,39886,1459],{},[43,39888,56],{},[50,39890,2682],{},[40,39892,39893,39895],{},[43,39894,8699],{},[43,39896,1459],{},[109,39898,39899],{"encoding":111},"f_i(x) = A_i x + b_i",[15,39901,39903,39967,40025],{"className":39902,"ariaHidden":72},[116],[15,39904,39906,39909,39949,39952,39955,39958,39961,39964],{"className":39905},[120],[15,39907],{"className":39908,"style":125},[124],[15,39910,39912,39915],{"className":39911},[129],[15,39913,36569],{"className":39914,"style":36584},[129,133],[15,39916,39918],{"className":39917},[137],[15,39919,39921,39941],{"className":39920},[141,142],[15,39922,39924,39938],{"className":39923},[146],[15,39925,39927],{"className":39926,"style":1981},[150],[15,39928,39929,39932],{"style":39822},[15,39930],{"className":39931,"style":159},[158],[15,39933,39935],{"className":39934},[163,164,165,166],[15,39936,1459],{"className":39937},[129,133,166],[15,39939,175],{"className":39940},[174],[15,39942,39944],{"className":39943},[146],[15,39945,39947],{"className":39946,"style":182},[150],[15,39948],{},[15,39950,53],{"className":39951},[188],[15,39953,56],{"className":39954},[129,133],[15,39956,59],{"className":39957},[195],[15,39959],{"className":39960,"style":199},[93],[15,39962,62],{"className":39963},[203],[15,39965],{"className":39966,"style":199},[93],[15,39968,39970,39973,40013,40016,40019,40022],{"className":39969},[120],[15,39971],{"className":39972,"style":20229},[124],[15,39974,39976,39979],{"className":39975},[129],[15,39977,13097],{"className":39978},[129,133],[15,39980,39982],{"className":39981},[137],[15,39983,39985,40005],{"className":39984},[141,142],[15,39986,39988,40002],{"className":39987},[146],[15,39989,39991],{"className":39990,"style":1981},[150],[15,39992,39993,39996],{"style":154},[15,39994],{"className":39995,"style":159},[158],[15,39997,39999],{"className":39998},[163,164,165,166],[15,40000,1459],{"className":40001},[129,133,166],[15,40003,175],{"className":40004},[174],[15,40006,40008],{"className":40007},[146],[15,40009,40011],{"className":40010,"style":182},[150],[15,40012],{},[15,40014,56],{"className":40015},[129,133],[15,40017],{"className":40018,"style":3277},[93],[15,40020,2682],{"className":40021},[2436],[15,40023],{"className":40024,"style":3277},[93],[15,40026,40028,40031],{"className":40027},[120],[15,40029],{"className":40030,"style":2183},[124],[15,40032,40034,40037],{"className":40033},[129],[15,40035,8699],{"className":40036},[129,133],[15,40038,40040],{"className":40039},[137],[15,40041,40043,40063],{"className":40042},[141,142],[15,40044,40046,40060],{"className":40045},[146],[15,40047,40049],{"className":40048,"style":1981},[150],[15,40050,40051,40054],{"style":154},[15,40052],{"className":40053,"style":159},[158],[15,40055,40057],{"className":40056},[163,164,165,166],[15,40058,1459],{"className":40059},[129,133,166],[15,40061,175],{"className":40062},[174],[15,40064,40066],{"className":40065},[146],[15,40067,40069],{"className":40068,"style":182},[150],[15,40070],{},[11,40072,40073],{},"展开后：",[15,40075,40077],{"className":40076,"translate":19},[18],[15,40078,40080,40206],{"className":40079,"translate":19},[23],[15,40081,40083],{"className":40082},[27],[29,40084,40085],{"xmlns":31,"display":32},[34,40086,40087,40203],{},[37,40088,40089,40091,40093,40095,40097,40099,40113,40119,40121,40127,40129,40131,40137,40139,40141,40173,40175,40177,40191,40197],{},[43,40090,2247],{},[50,40092,53],{"stretchy":52},[43,40094,56],{},[50,40096,59],{"stretchy":52},[50,40098,62],{},[1450,40100,40101,40103,40111],{},[50,40102,1454],{},[37,40104,40105,40107,40109],{},[43,40106,1459],{},[50,40108,62],{},[308,40110,318],{},[43,40112,16562],{},[40,40114,40115,40117],{},[43,40116,12008],{},[43,40118,1459],{},[50,40120,53],{"stretchy":52},[40,40122,40123,40125],{},[43,40124,13097],{},[43,40126,1459],{},[43,40128,56],{},[50,40130,2682],{},[40,40132,40133,40135],{},[43,40134,8699],{},[43,40136,1459],{},[50,40138,59],{"stretchy":52},[50,40140,62],{},[37,40142,40143,40145,40159,40165,40171],{},[50,40144,53],{"fence":72},[1450,40146,40147,40149,40157],{},[50,40148,1454],{},[37,40150,40151,40153,40155],{},[43,40152,1459],{},[50,40154,62],{},[308,40156,318],{},[43,40158,16562],{},[40,40160,40161,40163],{},[43,40162,12008],{},[43,40164,1459],{},[40,40166,40167,40169],{},[43,40168,13097],{},[43,40170,1459],{},[50,40172,59],{"fence":72},[43,40174,56],{},[50,40176,2682],{},[1450,40178,40179,40181,40189],{},[50,40180,1454],{},[37,40182,40183,40185,40187],{},[43,40184,1459],{},[50,40186,62],{},[308,40188,318],{},[43,40190,16562],{},[40,40192,40193,40195],{},[43,40194,12008],{},[43,40196,1459],{},[40,40198,40199,40201],{},[43,40200,8699],{},[43,40202,1459],{},[109,40204,40205],{"encoding":111},"g(x) = \\sum_{i=1}^k \\alpha_i (A_i x + b_i) = \\left(\\sum_{i=1}^k \\alpha_i A_i \\right) x + \\sum_{i=1}^k \\alpha_i b_i",[15,40207,40209,40236,40404,40462,40645],{"className":40208,"ariaHidden":72},[116],[15,40210,40212,40215,40218,40221,40224,40227,40230,40233],{"className":40211},[120],[15,40213],{"className":40214,"style":125},[124],[15,40216,2247],{"className":40217,"style":887},[129,133],[15,40219,53],{"className":40220},[188],[15,40222,56],{"className":40223},[129,133],[15,40225,59],{"className":40226},[195],[15,40228],{"className":40229,"style":199},[93],[15,40231,62],{"className":40232},[203],[15,40234],{"className":40235,"style":199},[93],[15,40237,40239,40242,40306,40309,40349,40352,40392,40395,40398,40401],{"className":40238},[120],[15,40240],{"className":40241,"style":21128},[124],[15,40243,40245],{"className":40244},[217,1874],[15,40246,40248,40298],{"className":40247},[141,142],[15,40249,40251,40295],{"className":40250},[146],[15,40252,40254,40274,40284],{"className":40253,"style":21141},[150],[15,40255,40256,40259],{"style":1887},[15,40257],{"className":40258,"style":1891},[158],[15,40260,40262],{"className":40261},[163,164,165,166],[15,40263,40265,40268,40271],{"className":40264},[129,166],[15,40266,1459],{"className":40267},[129,133,166],[15,40269,62],{"className":40270},[203,166],[15,40272,318],{"className":40273},[129,166],[15,40275,40276,40279],{"style":1909},[15,40277],{"className":40278,"style":1891},[158],[15,40280,40281],{},[15,40282,1454],{"className":40283},[217,1918,1919],[15,40285,40286,40289],{"style":1922},[15,40287],{"className":40288,"style":1891},[158],[15,40290,40292],{"className":40291},[163,164,165,166],[15,40293,16562],{"className":40294,"style":16595},[129,133,166],[15,40296,175],{"className":40297},[174],[15,40299,40301],{"className":40300},[146],[15,40302,40304],{"className":40303,"style":1941},[150],[15,40305],{},[15,40307],{"className":40308,"style":262},[93],[15,40310,40312,40315],{"className":40311},[129],[15,40313,12008],{"className":40314,"style":12029},[129,133],[15,40316,40318],{"className":40317},[137],[15,40319,40321,40341],{"className":40320},[141,142],[15,40322,40324,40338],{"className":40323},[146],[15,40325,40327],{"className":40326,"style":1981},[150],[15,40328,40329,40332],{"style":12044},[15,40330],{"className":40331,"style":159},[158],[15,40333,40335],{"className":40334},[163,164,165,166],[15,40336,1459],{"className":40337},[129,133,166],[15,40339,175],{"className":40340},[174],[15,40342,40344],{"className":40343},[146],[15,40345,40347],{"className":40346,"style":182},[150],[15,40348],{},[15,40350,53],{"className":40351},[188],[15,40353,40355,40358],{"className":40354},[129],[15,40356,13097],{"className":40357},[129,133],[15,40359,40361],{"className":40360},[137],[15,40362,40364,40384],{"className":40363},[141,142],[15,40365,40367,40381],{"className":40366},[146],[15,40368,40370],{"className":40369,"style":1981},[150],[15,40371,40372,40375],{"style":154},[15,40373],{"className":40374,"style":159},[158],[15,40376,40378],{"className":40377},[163,164,165,166],[15,40379,1459],{"className":40380},[129,133,166],[15,40382,175],{"className":40383},[174],[15,40385,40387],{"className":40386},[146],[15,40388,40390],{"className":40389,"style":182},[150],[15,40391],{},[15,40393,56],{"className":40394},[129,133],[15,40396],{"className":40397,"style":3277},[93],[15,40399,2682],{"className":40400},[2436],[15,40402],{"className":40403,"style":3277},[93],[15,40405,40407,40410,40450,40453,40456,40459],{"className":40406},[120],[15,40408],{"className":40409,"style":125},[124],[15,40411,40413,40416],{"className":40412},[129],[15,40414,8699],{"className":40415},[129,133],[15,40417,40419],{"className":40418},[137],[15,40420,40422,40442],{"className":40421},[141,142],[15,40423,40425,40439],{"className":40424},[146],[15,40426,40428],{"className":40427,"style":1981},[150],[15,40429,40430,40433],{"style":154},[15,40431],{"className":40432,"style":159},[158],[15,40434,40436],{"className":40435},[163,164,165,166],[15,40437,1459],{"className":40438},[129,133,166],[15,40440,175],{"className":40441},[174],[15,40443,40445],{"className":40444},[146],[15,40446,40448],{"className":40447,"style":182},[150],[15,40449],{},[15,40451,59],{"className":40452},[195],[15,40454],{"className":40455,"style":199},[93],[15,40457,62],{"className":40458},[203],[15,40460],{"className":40461,"style":199},[93],[15,40463,40465,40468,40630,40633,40636,40639,40642],{"className":40464},[120],[15,40466],{"className":40467,"style":21128},[124],[15,40469,40471,40477,40541,40544,40584,40624],{"className":40470},[449],[15,40472,40474],{"className":40473,"style":5052},[188,5051],[15,40475,53],{"className":40476},[228,15857],[15,40478,40480],{"className":40479},[217,1874],[15,40481,40483,40533],{"className":40482},[141,142],[15,40484,40486,40530],{"className":40485},[146],[15,40487,40489,40509,40519],{"className":40488,"style":21141},[150],[15,40490,40491,40494],{"style":1887},[15,40492],{"className":40493,"style":1891},[158],[15,40495,40497],{"className":40496},[163,164,165,166],[15,40498,40500,40503,40506],{"className":40499},[129,166],[15,40501,1459],{"className":40502},[129,133,166],[15,40504,62],{"className":40505},[203,166],[15,40507,318],{"className":40508},[129,166],[15,40510,40511,40514],{"style":1909},[15,40512],{"className":40513,"style":1891},[158],[15,40515,40516],{},[15,40517,1454],{"className":40518},[217,1918,1919],[15,40520,40521,40524],{"style":1922},[15,40522],{"className":40523,"style":1891},[158],[15,40525,40527],{"className":40526},[163,164,165,166],[15,40528,16562],{"className":40529,"style":16595},[129,133,166],[15,40531,175],{"className":40532},[174],[15,40534,40536],{"className":40535},[146],[15,40537,40539],{"className":40538,"style":1941},[150],[15,40540],{},[15,40542],{"className":40543,"style":262},[93],[15,40545,40547,40550],{"className":40546},[129],[15,40548,12008],{"className":40549,"style":12029},[129,133],[15,40551,40553],{"className":40552},[137],[15,40554,40556,40576],{"className":40555},[141,142],[15,40557,40559,40573],{"className":40558},[146],[15,40560,40562],{"className":40561,"style":1981},[150],[15,40563,40564,40567],{"style":12044},[15,40565],{"className":40566,"style":159},[158],[15,40568,40570],{"className":40569},[163,164,165,166],[15,40571,1459],{"className":40572},[129,133,166],[15,40574,175],{"className":40575},[174],[15,40577,40579],{"className":40578},[146],[15,40580,40582],{"className":40581,"style":182},[150],[15,40583],{},[15,40585,40587,40590],{"className":40586},[129],[15,40588,13097],{"className":40589},[129,133],[15,40591,40593],{"className":40592},[137],[15,40594,40596,40616],{"className":40595},[141,142],[15,40597,40599,40613],{"className":40598},[146],[15,40600,40602],{"className":40601,"style":1981},[150],[15,40603,40604,40607],{"style":154},[15,40605],{"className":40606,"style":159},[158],[15,40608,40610],{"className":40609},[163,164,165,166],[15,40611,1459],{"className":40612},[129,133,166],[15,40614,175],{"className":40615},[174],[15,40617,40619],{"className":40618},[146],[15,40620,40622],{"className":40621,"style":182},[150],[15,40623],{},[15,40625,40627],{"className":40626,"style":5052},[195,5051],[15,40628,59],{"className":40629},[228,15857],[15,40631],{"className":40632,"style":262},[93],[15,40634,56],{"className":40635},[129,133],[15,40637],{"className":40638,"style":3277},[93],[15,40640,2682],{"className":40641},[2436],[15,40643],{"className":40644,"style":3277},[93],[15,40646,40648,40651,40715,40718,40758],{"className":40647},[120],[15,40649],{"className":40650,"style":21128},[124],[15,40652,40654],{"className":40653},[217,1874],[15,40655,40657,40707],{"className":40656},[141,142],[15,40658,40660,40704],{"className":40659},[146],[15,40661,40663,40683,40693],{"className":40662,"style":21141},[150],[15,40664,40665,40668],{"style":1887},[15,40666],{"className":40667,"style":1891},[158],[15,40669,40671],{"className":40670},[163,164,165,166],[15,40672,40674,40677,40680],{"className":40673},[129,166],[15,40675,1459],{"className":40676},[129,133,166],[15,40678,62],{"className":40679},[203,166],[15,40681,318],{"className":40682},[129,166],[15,40684,40685,40688],{"style":1909},[15,40686],{"className":40687,"style":1891},[158],[15,40689,40690],{},[15,40691,1454],{"className":40692},[217,1918,1919],[15,40694,40695,40698],{"style":1922},[15,40696],{"className":40697,"style":1891},[158],[15,40699,40701],{"className":40700},[163,164,165,166],[15,40702,16562],{"className":40703,"style":16595},[129,133,166],[15,40705,175],{"className":40706},[174],[15,40708,40710],{"className":40709},[146],[15,40711,40713],{"className":40712,"style":1941},[150],[15,40714],{},[15,40716],{"className":40717,"style":262},[93],[15,40719,40721,40724],{"className":40720},[129],[15,40722,12008],{"className":40723,"style":12029},[129,133],[15,40725,40727],{"className":40726},[137],[15,40728,40730,40750],{"className":40729},[141,142],[15,40731,40733,40747],{"className":40732},[146],[15,40734,40736],{"className":40735,"style":1981},[150],[15,40737,40738,40741],{"style":12044},[15,40739],{"className":40740,"style":159},[158],[15,40742,40744],{"className":40743},[163,164,165,166],[15,40745,1459],{"className":40746},[129,133,166],[15,40748,175],{"className":40749},[174],[15,40751,40753],{"className":40752},[146],[15,40754,40756],{"className":40755,"style":182},[150],[15,40757],{},[15,40759,40761,40764],{"className":40760},[129],[15,40762,8699],{"className":40763},[129,133],[15,40765,40767],{"className":40766},[137],[15,40768,40770,40790],{"className":40769},[141,142],[15,40771,40773,40787],{"className":40772},[146],[15,40774,40776],{"className":40775,"style":1981},[150],[15,40777,40778,40781],{"style":154},[15,40779],{"className":40780,"style":159},[158],[15,40782,40784],{"className":40783},[163,164,165,166],[15,40785,1459],{"className":40786},[129,133,166],[15,40788,175],{"className":40789},[174],[15,40791,40793],{"className":40792},[146],[15,40794,40796],{"className":40795,"style":182},[150],[15,40797],{},[11,40799,40800],{},"这仍然是仿射函数。因此，加权组合不会破坏仿射性。",[11,40802,40803],{},"如果用非线性的多项式运算来组合仿射函数，例如平方或更高次幂：",[15,40805,40807],{"className":40806,"translate":19},[18],[15,40808,40810,40884],{"className":40809,"translate":19},[23],[15,40811,40813],{"className":40812},[27],[29,40814,40815],{"xmlns":31,"display":32},[34,40816,40817,40881],{},[37,40818,40819,40821,40823,40825,40827,40829,40831,40837,40839,40841,40843,40849,40851,40853,40859,40861,40863,40865,40871,40873,40875,40877,40879],{},[43,40820,45],{},[50,40822,53],{"stretchy":52},[43,40824,56],{},[50,40826,59],{"stretchy":52},[50,40828,62],{},[50,40830,53],{"stretchy":52},[40,40832,40833,40835],{},[43,40834,36569],{},[308,40836,318],{},[50,40838,53],{"stretchy":52},[43,40840,56],{},[50,40842,59],{"stretchy":52},[305,40844,40845,40847],{},[50,40846,59],{"stretchy":52},[308,40848,310],{},[50,40850,2682],{},[308,40852,9879],{},[40,40854,40855,40857],{},[43,40856,36569],{},[308,40858,310],{},[50,40860,53],{"stretchy":52},[43,40862,56],{},[50,40864,59],{"stretchy":52},[40,40866,40867,40869],{},[43,40868,36569],{},[308,40870,9879],{},[50,40872,53],{"stretchy":52},[43,40874,56],{},[50,40876,59],{"stretchy":52},[50,40878,2682],{},[308,40880,36763],{},[109,40882,40883],{"encoding":111},"h(x) = (f_1(x))^2 + 3 f_2(x) f_3(x) + 5",[15,40885,40887,40914,41012,41128],{"className":40886,"ariaHidden":72},[116],[15,40888,40890,40893,40896,40899,40902,40905,40908,40911],{"className":40889},[120],[15,40891],{"className":40892,"style":125},[124],[15,40894,45],{"className":40895},[129,133],[15,40897,53],{"className":40898},[188],[15,40900,56],{"className":40901},[129,133],[15,40903,59],{"className":40904},[195],[15,40906],{"className":40907,"style":199},[93],[15,40909,62],{"className":40910},[203],[15,40912],{"className":40913,"style":199},[93],[15,40915,40917,40921,40924,40964,40967,40970,40973,41003,41006,41009],{"className":40916},[120],[15,40918],{"className":40919,"style":40920},[124],"height:1.1141em;vertical-align:-0.25em;",[15,40922,53],{"className":40923},[188],[15,40925,40927,40930],{"className":40926},[129],[15,40928,36569],{"className":40929,"style":36584},[129,133],[15,40931,40933],{"className":40932},[137],[15,40934,40936,40956],{"className":40935},[141,142],[15,40937,40939,40953],{"className":40938},[146],[15,40940,40942],{"className":40941,"style":1051},[150],[15,40943,40944,40947],{"style":39822},[15,40945],{"className":40946,"style":159},[158],[15,40948,40950],{"className":40949},[163,164,165,166],[15,40951,318],{"className":40952},[129,166],[15,40954,175],{"className":40955},[174],[15,40957,40959],{"className":40958},[146],[15,40960,40962],{"className":40961,"style":182},[150],[15,40963],{},[15,40965,53],{"className":40966},[188],[15,40968,56],{"className":40969},[129,133],[15,40971,59],{"className":40972},[195],[15,40974,40976,40979],{"className":40975},[195],[15,40977,59],{"className":40978},[195],[15,40980,40982],{"className":40981},[137],[15,40983,40985],{"className":40984},[141],[15,40986,40988],{"className":40987},[146],[15,40989,40992],{"className":40990,"style":40991},[150],"height:0.8641em;",[15,40993,40994,40997],{"style":1272},[15,40995],{"className":40996,"style":159},[158],[15,40998,41000],{"className":40999},[163,164,165,166],[15,41001,310],{"className":41002},[129,166],[15,41004],{"className":41005,"style":3277},[93],[15,41007,2682],{"className":41008},[2436],[15,41010],{"className":41011,"style":3277},[93],[15,41013,41015,41018,41021,41061,41064,41067,41070,41110,41113,41116,41119,41122,41125],{"className":41014},[120],[15,41016],{"className":41017,"style":125},[124],[15,41019,9879],{"className":41020},[129],[15,41022,41024,41027],{"className":41023},[129],[15,41025,36569],{"className":41026,"style":36584},[129,133],[15,41028,41030],{"className":41029},[137],[15,41031,41033,41053],{"className":41032},[141,142],[15,41034,41036,41050],{"className":41035},[146],[15,41037,41039],{"className":41038,"style":1051},[150],[15,41040,41041,41044],{"style":39822},[15,41042],{"className":41043,"style":159},[158],[15,41045,41047],{"className":41046},[163,164,165,166],[15,41048,310],{"className":41049},[129,166],[15,41051,175],{"className":41052},[174],[15,41054,41056],{"className":41055},[146],[15,41057,41059],{"className":41058,"style":182},[150],[15,41060],{},[15,41062,53],{"className":41063},[188],[15,41065,56],{"className":41066},[129,133],[15,41068,59],{"className":41069},[195],[15,41071,41073,41076],{"className":41072},[129],[15,41074,36569],{"className":41075,"style":36584},[129,133],[15,41077,41079],{"className":41078},[137],[15,41080,41082,41102],{"className":41081},[141,142],[15,41083,41085,41099],{"className":41084},[146],[15,41086,41088],{"className":41087,"style":1051},[150],[15,41089,41090,41093],{"style":39822},[15,41091],{"className":41092,"style":159},[158],[15,41094,41096],{"className":41095},[163,164,165,166],[15,41097,9879],{"className":41098},[129,166],[15,41100,175],{"className":41101},[174],[15,41103,41105],{"className":41104},[146],[15,41106,41108],{"className":41107,"style":182},[150],[15,41109],{},[15,41111,53],{"className":41112},[188],[15,41114,56],{"className":41115},[129,133],[15,41117,59],{"className":41118},[195],[15,41120],{"className":41121,"style":3277},[93],[15,41123,2682],{"className":41124},[2436],[15,41126],{"className":41127,"style":3277},[93],[15,41129,41131,41134],{"className":41130},[120],[15,41132],{"className":41133,"style":4886},[124],[15,41135,36763],{"className":41136},[129],[11,41138,41139,41140,41255,41256,41402],{},"注意：",[15,41141,41143,41171],{"className":41142,"translate":19},[23],[15,41144,41146],{"className":41145},[27],[29,41147,41148],{"xmlns":31},[34,41149,41150,41168],{},[37,41151,41152,41158,41160,41162],{},[40,41153,41154,41156],{},[43,41155,36569],{},[308,41157,318],{},[50,41159,53],{"stretchy":52},[43,41161,56],{},[305,41163,41164,41166],{},[50,41165,59],{"stretchy":52},[308,41167,310],{},[109,41169,41170],{"encoding":111},"f_1(x)^2",[15,41172,41174],{"className":41173,"ariaHidden":72},[116],[15,41175,41177,41180,41220,41223,41226],{"className":41176},[120],[15,41178],{"className":41179,"style":34009},[124],[15,41181,41183,41186],{"className":41182},[129],[15,41184,36569],{"className":41185,"style":36584},[129,133],[15,41187,41189],{"className":41188},[137],[15,41190,41192,41212],{"className":41191},[141,142],[15,41193,41195,41209],{"className":41194},[146],[15,41196,41198],{"className":41197,"style":1051},[150],[15,41199,41200,41203],{"style":39822},[15,41201],{"className":41202,"style":159},[158],[15,41204,41206],{"className":41205},[163,164,165,166],[15,41207,318],{"className":41208},[129,166],[15,41210,175],{"className":41211},[174],[15,41213,41215],{"className":41214},[146],[15,41216,41218],{"className":41217,"style":182},[150],[15,41219],{},[15,41221,53],{"className":41222},[188],[15,41224,56],{"className":41225},[129,133],[15,41227,41229,41232],{"className":41228},[195],[15,41230,59],{"className":41231},[195],[15,41233,41235],{"className":41234},[137],[15,41236,41238],{"className":41237},[141],[15,41239,41241],{"className":41240},[146],[15,41242,41244],{"className":41243,"style":380},[150],[15,41245,41246,41249],{"style":383},[15,41247],{"className":41248,"style":159},[158],[15,41250,41252],{"className":41251},[163,164,165,166],[15,41253,310],{"className":41254},[129,166]," 是逐元素平方，",[15,41257,41259,41295],{"className":41258,"translate":19},[23],[15,41260,41262],{"className":41261},[27],[29,41263,41264],{"xmlns":31},[34,41265,41266,41292],{},[37,41267,41268,41274,41276,41278,41280,41286,41288,41290],{},[40,41269,41270,41272],{},[43,41271,36569],{},[308,41273,310],{},[50,41275,53],{"stretchy":52},[43,41277,56],{},[50,41279,59],{"stretchy":52},[40,41281,41282,41284],{},[43,41283,36569],{},[308,41285,9879],{},[50,41287,53],{"stretchy":52},[43,41289,56],{},[50,41291,59],{"stretchy":52},[109,41293,41294],{"encoding":111},"f_2(x) f_3(x)",[15,41296,41298],{"className":41297,"ariaHidden":72},[116],[15,41299,41301,41304,41344,41347,41350,41353,41393,41396,41399],{"className":41300},[120],[15,41302],{"className":41303,"style":125},[124],[15,41305,41307,41310],{"className":41306},[129],[15,41308,36569],{"className":41309,"style":36584},[129,133],[15,41311,41313],{"className":41312},[137],[15,41314,41316,41336],{"className":41315},[141,142],[15,41317,41319,41333],{"className":41318},[146],[15,41320,41322],{"className":41321,"style":1051},[150],[15,41323,41324,41327],{"style":39822},[15,41325],{"className":41326,"style":159},[158],[15,41328,41330],{"className":41329},[163,164,165,166],[15,41331,310],{"className":41332},[129,166],[15,41334,175],{"className":41335},[174],[15,41337,41339],{"className":41338},[146],[15,41340,41342],{"className":41341,"style":182},[150],[15,41343],{},[15,41345,53],{"className":41346},[188],[15,41348,56],{"className":41349},[129,133],[15,41351,59],{"className":41352},[195],[15,41354,41356,41359],{"className":41355},[129],[15,41357,36569],{"className":41358,"style":36584},[129,133],[15,41360,41362],{"className":41361},[137],[15,41363,41365,41385],{"className":41364},[141,142],[15,41366,41368,41382],{"className":41367},[146],[15,41369,41371],{"className":41370,"style":1051},[150],[15,41372,41373,41376],{"style":39822},[15,41374],{"className":41375,"style":159},[158],[15,41377,41379],{"className":41378},[163,164,165,166],[15,41380,9879],{"className":41381},[129,166],[15,41383,175],{"className":41384},[174],[15,41386,41388],{"className":41387},[146],[15,41389,41391],{"className":41390,"style":182},[150],[15,41392],{},[15,41394,53],{"className":41395},[188],[15,41397,56],{"className":41398},[129,133],[15,41400,59],{"className":41401},[195]," 是逐元素相乘或点积。",[11,41404,41405,41406,41462],{},"这些运算会引入二次项或更高阶项，不再保持 ",[15,41407,41409,41429],{"className":41408,"translate":19},[23],[15,41410,41412],{"className":41411},[27],[29,41413,41414],{"xmlns":31},[34,41415,41416,41426],{},[37,41417,41418,41420,41422,41424],{},[43,41419,13097],{},[43,41421,56],{},[50,41423,2682],{},[43,41425,8699],{},[109,41427,41428],{"encoding":111},"Ax+b",[15,41430,41432,41453],{"className":41431,"ariaHidden":72},[116],[15,41433,41435,41438,41441,41444,41447,41450],{"className":41434},[120],[15,41436],{"className":41437,"style":39511},[124],[15,41439,13097],{"className":41440},[129,133],[15,41442,56],{"className":41443},[129,133],[15,41445],{"className":41446,"style":3277},[93],[15,41448,2682],{"className":41449},[2436],[15,41451],{"className":41452,"style":3277},[93],[15,41454,41456,41459],{"className":41455},[120],[15,41457],{"className":41458,"style":522},[124],[15,41460,8699],{"className":41461},[129,133]," 的形式。因此，非线性多项式组合通常不是仿射函数。",[570,41464,41465],{"id":41465},"仿射复合",[11,41467,41468],{},"这个问题非常重要，是理解神经网络本质的关键。让我们仔细分析仿射函数的复合是否仍然是仿射的。",[11,41470,41471],{},"假设有两个仿射函数：",[15,41473,41475],{"className":41474,"translate":19},[18],[15,41476,41478,41530],{"className":41477,"translate":19},[23],[15,41479,41481],{"className":41480},[27],[29,41482,41483],{"xmlns":31,"display":32},[34,41484,41485,41527],{},[37,41486,41487,41489,41491,41493,41495,41497,41499,41501,41503,41505,41507,41509,41511,41513,41515,41517,41519,41521,41523,41525],{},[43,41488,36569],{},[50,41490,53],{"stretchy":52},[43,41492,56],{},[50,41494,59],{"stretchy":52},[50,41496,62],{},[43,41498,13097],{},[43,41500,56],{},[50,41502,2682],{},[43,41504,8699],{},[50,41506,91],{"separator":72},[93,41508],{"width":16921},[43,41510,2247],{},[50,41512,53],{"stretchy":52},[43,41514,56],{},[50,41516,59],{"stretchy":52},[50,41518,62],{},[43,41520,5619],{},[43,41522,56],{},[50,41524,2682],{},[43,41526,10376],{},[109,41528,41529],{"encoding":111},"f(x) = A x + b, \\quad g(x) = C x + d",[15,41531,41533,41560,41581,41620,41641],{"className":41532,"ariaHidden":72},[116],[15,41534,41536,41539,41542,41545,41548,41551,41554,41557],{"className":41535},[120],[15,41537],{"className":41538,"style":125},[124],[15,41540,36569],{"className":41541,"style":36584},[129,133],[15,41543,53],{"className":41544},[188],[15,41546,56],{"className":41547},[129,133],[15,41549,59],{"className":41550},[195],[15,41552],{"className":41553,"style":199},[93],[15,41555,62],{"className":41556},[203],[15,41558],{"className":41559,"style":199},[93],[15,41561,41563,41566,41569,41572,41575,41578],{"className":41562},[120],[15,41564],{"className":41565,"style":39511},[124],[15,41567,13097],{"className":41568},[129,133],[15,41570,56],{"className":41571},[129,133],[15,41573],{"className":41574,"style":3277},[93],[15,41576,2682],{"className":41577},[2436],[15,41579],{"className":41580,"style":3277},[93],[15,41582,41584,41587,41590,41593,41596,41599,41602,41605,41608,41611,41614,41617],{"className":41583},[120],[15,41585],{"className":41586,"style":125},[124],[15,41588,8699],{"className":41589},[129,133],[15,41591,91],{"className":41592},[254],[15,41594],{"className":41595,"style":17294},[93],[15,41597],{"className":41598,"style":262},[93],[15,41600,2247],{"className":41601,"style":887},[129,133],[15,41603,53],{"className":41604},[188],[15,41606,56],{"className":41607},[129,133],[15,41609,59],{"className":41610},[195],[15,41612],{"className":41613,"style":199},[93],[15,41615,62],{"className":41616},[203],[15,41618],{"className":41619,"style":199},[93],[15,41621,41623,41626,41629,41632,41635,41638],{"className":41622},[120],[15,41624],{"className":41625,"style":39511},[124],[15,41627,5619],{"className":41628,"style":5665},[129,133],[15,41630,56],{"className":41631},[129,133],[15,41633],{"className":41634,"style":3277},[93],[15,41636,2682],{"className":41637},[2436],[15,41639],{"className":41640,"style":3277},[93],[15,41642,41644,41647],{"className":41643},[120],[15,41645],{"className":41646,"style":522},[124],[15,41648,10376],{"className":41649},[129,133],[11,41651,41652],{},"其复合为：",[15,41654,41656],{"className":41655,"translate":19},[18],[15,41657,41659,41711],{"className":41658,"translate":19},[23],[15,41660,41662],{"className":41661},[27],[29,41663,41664],{"xmlns":31,"display":32},[34,41665,41666,41708],{},[37,41667,41668,41670,41672,41674,41676,41678,41680,41682,41684,41686,41688,41690,41692,41694,41696,41698,41700,41702,41704,41706],{},[43,41669,45],{},[50,41671,53],{"stretchy":52},[43,41673,56],{},[50,41675,59],{"stretchy":52},[50,41677,62],{},[43,41679,36569],{},[50,41681,53],{"stretchy":52},[43,41683,2247],{},[50,41685,53],{"stretchy":52},[43,41687,56],{},[50,41689,59],{"stretchy":52},[50,41691,59],{"stretchy":52},[50,41693,62],{},[43,41695,36569],{},[50,41697,53],{"stretchy":52},[43,41699,5619],{},[43,41701,56],{},[50,41703,2682],{},[43,41705,10376],{},[50,41707,59],{"stretchy":52},[109,41709,41710],{"encoding":111},"h(x) = f(g(x)) = f(Cx + d)",[15,41712,41714,41741,41774,41801],{"className":41713,"ariaHidden":72},[116],[15,41715,41717,41720,41723,41726,41729,41732,41735,41738],{"className":41716},[120],[15,41718],{"className":41719,"style":125},[124],[15,41721,45],{"className":41722},[129,133],[15,41724,53],{"className":41725},[188],[15,41727,56],{"className":41728},[129,133],[15,41730,59],{"className":41731},[195],[15,41733],{"className":41734,"style":199},[93],[15,41736,62],{"className":41737},[203],[15,41739],{"className":41740,"style":199},[93],[15,41742,41744,41747,41750,41753,41756,41759,41762,41765,41768,41771],{"className":41743},[120],[15,41745],{"className":41746,"style":125},[124],[15,41748,36569],{"className":41749,"style":36584},[129,133],[15,41751,53],{"className":41752},[188],[15,41754,2247],{"className":41755,"style":887},[129,133],[15,41757,53],{"className":41758},[188],[15,41760,56],{"className":41761},[129,133],[15,41763,6149],{"className":41764},[195],[15,41766],{"className":41767,"style":199},[93],[15,41769,62],{"className":41770},[203],[15,41772],{"className":41773,"style":199},[93],[15,41775,41777,41780,41783,41786,41789,41792,41795,41798],{"className":41776},[120],[15,41778],{"className":41779,"style":125},[124],[15,41781,36569],{"className":41782,"style":36584},[129,133],[15,41784,53],{"className":41785},[188],[15,41787,5619],{"className":41788,"style":5665},[129,133],[15,41790,56],{"className":41791},[129,133],[15,41793],{"className":41794,"style":3277},[93],[15,41796,2682],{"className":41797},[2436],[15,41799],{"className":41800,"style":3277},[93],[15,41802,41804,41807,41810],{"className":41803},[120],[15,41805],{"className":41806,"style":125},[124],[15,41808,10376],{"className":41809},[129,133],[15,41811,59],{"className":41812},[195],[11,41814,41815],{},"展开来看其形式：",[15,41817,41819],{"className":41818,"translate":19},[18],[15,41820,41822,41888],{"className":41821,"translate":19},[23],[15,41823,41825],{"className":41824},[27],[29,41826,41827],{"xmlns":31,"display":32},[34,41828,41829,41885],{},[37,41830,41831,41833,41835,41837,41839,41841,41843,41845,41847,41849,41851,41853,41855,41857,41859,41861,41863,41865,41867,41869,41871,41873,41875,41877,41879,41881,41883],{},[43,41832,45],{},[50,41834,53],{"stretchy":52},[43,41836,56],{},[50,41838,59],{"stretchy":52},[50,41840,62],{},[43,41842,13097],{},[50,41844,53],{"stretchy":52},[43,41846,5619],{},[43,41848,56],{},[50,41850,2682],{},[43,41852,10376],{},[50,41854,59],{"stretchy":52},[50,41856,2682],{},[43,41858,8699],{},[50,41860,62],{},[50,41862,53],{"stretchy":52},[43,41864,13097],{},[43,41866,5619],{},[50,41868,59],{"stretchy":52},[43,41870,56],{},[50,41872,2682],{},[50,41874,53],{"stretchy":52},[43,41876,13097],{},[43,41878,10376],{},[50,41880,2682],{},[43,41882,8699],{},[50,41884,59],{"stretchy":52},[109,41886,41887],{"encoding":111},"h(x) = A(Cx + d) + b = (AC)x + (Ad + b)",[15,41889,41891,41918,41945,41966,41984,42014,42038],{"className":41890,"ariaHidden":72},[116],[15,41892,41894,41897,41900,41903,41906,41909,41912,41915],{"className":41893},[120],[15,41895],{"className":41896,"style":125},[124],[15,41898,45],{"className":41899},[129,133],[15,41901,53],{"className":41902},[188],[15,41904,56],{"className":41905},[129,133],[15,41907,59],{"className":41908},[195],[15,41910],{"className":41911,"style":199},[93],[15,41913,62],{"className":41914},[203],[15,41916],{"className":41917,"style":199},[93],[15,41919,41921,41924,41927,41930,41933,41936,41939,41942],{"className":41920},[120],[15,41922],{"className":41923,"style":125},[124],[15,41925,13097],{"className":41926},[129,133],[15,41928,53],{"className":41929},[188],[15,41931,5619],{"className":41932,"style":5665},[129,133],[15,41934,56],{"className":41935},[129,133],[15,41937],{"className":41938,"style":3277},[93],[15,41940,2682],{"className":41941},[2436],[15,41943],{"className":41944,"style":3277},[93],[15,41946,41948,41951,41954,41957,41960,41963],{"className":41947},[120],[15,41949],{"className":41950,"style":125},[124],[15,41952,10376],{"className":41953},[129,133],[15,41955,59],{"className":41956},[195],[15,41958],{"className":41959,"style":3277},[93],[15,41961,2682],{"className":41962},[2436],[15,41964],{"className":41965,"style":3277},[93],[15,41967,41969,41972,41975,41978,41981],{"className":41968},[120],[15,41970],{"className":41971,"style":522},[124],[15,41973,8699],{"className":41974},[129,133],[15,41976],{"className":41977,"style":199},[93],[15,41979,62],{"className":41980},[203],[15,41982],{"className":41983,"style":199},[93],[15,41985,41987,41990,41993,41996,41999,42002,42005,42008,42011],{"className":41986},[120],[15,41988],{"className":41989,"style":125},[124],[15,41991,53],{"className":41992},[188],[15,41994,13097],{"className":41995},[129,133],[15,41997,5619],{"className":41998,"style":5665},[129,133],[15,42000,59],{"className":42001},[195],[15,42003,56],{"className":42004},[129,133],[15,42006],{"className":42007,"style":3277},[93],[15,42009,2682],{"className":42010},[2436],[15,42012],{"className":42013,"style":3277},[93],[15,42015,42017,42020,42023,42026,42029,42032,42035],{"className":42016},[120],[15,42018],{"className":42019,"style":125},[124],[15,42021,53],{"className":42022},[188],[15,42024,13097],{"className":42025},[129,133],[15,42027,10376],{"className":42028},[129,133],[15,42030],{"className":42031,"style":3277},[93],[15,42033,2682],{"className":42034},[2436],[15,42036],{"className":42037,"style":3277},[93],[15,42039,42041,42044,42047],{"className":42040},[120],[15,42042],{"className":42043,"style":125},[124],[15,42045,8699],{"className":42046},[129,133],[15,42048,59],{"className":42049},[195],[11,42051,42052],{},"结果仍然是仿射函数。",[11,42054,42055],{},"由此可见，仿射函数的复合仍然是仿射的。这一点可以推广：复合任意多个仿射函数：",[15,42057,42059],{"className":42058,"translate":19},[18],[15,42060,42062,42112],{"className":42061,"translate":19},[23],[15,42063,42065],{"className":42064},[27],[29,42066,42067],{"xmlns":31,"display":32},[34,42068,42069,42109],{},[37,42070,42071,42077,42079,42081,42087,42089,42095,42097,42099,42101,42103,42105,42107],{},[40,42072,42073,42075],{},[43,42074,36569],{},[43,42076,39326],{},[50,42078,53],{"stretchy":52},[50,42080,335],{},[40,42082,42083,42085],{},[43,42084,36569],{},[308,42086,310],{},[50,42088,53],{"stretchy":52},[40,42090,42091,42093],{},[43,42092,36569],{},[308,42094,318],{},[50,42096,53],{"stretchy":52},[43,42098,56],{},[50,42100,59],{"stretchy":52},[50,42102,59],{"stretchy":52},[50,42104,335],{},[6856,42106,15788],{},[50,42108,59],{"stretchy":52},[109,42110,42111],{"encoding":111},"f_n(\\dots f_2(f_1(x))\\dots)",[15,42113,42115],{"className":42114,"ariaHidden":72},[116],[15,42116,42118,42121,42161,42164,42167,42170,42210,42213,42253,42256,42259,42262,42265,42268,42271],{"className":42117},[120],[15,42119],{"className":42120,"style":125},[124],[15,42122,42124,42127],{"className":42123},[129],[15,42125,36569],{"className":42126,"style":36584},[129,133],[15,42128,42130],{"className":42129},[137],[15,42131,42133,42153],{"className":42132},[141,142],[15,42134,42136,42150],{"className":42135},[146],[15,42137,42139],{"className":42138,"style":1163},[150],[15,42140,42141,42144],{"style":39822},[15,42142],{"className":42143,"style":159},[158],[15,42145,42147],{"className":42146},[163,164,165,166],[15,42148,39326],{"className":42149},[129,133,166],[15,42151,175],{"className":42152},[174],[15,42154,42156],{"className":42155},[146],[15,42157,42159],{"className":42158,"style":182},[150],[15,42160],{},[15,42162,53],{"className":42163},[188],[15,42165,335],{"className":42166},[449],[15,42168],{"className":42169,"style":262},[93],[15,42171,42173,42176],{"className":42172},[129],[15,42174,36569],{"className":42175,"style":36584},[129,133],[15,42177,42179],{"className":42178},[137],[15,42180,42182,42202],{"className":42181},[141,142],[15,42183,42185,42199],{"className":42184},[146],[15,42186,42188],{"className":42187,"style":1051},[150],[15,42189,42190,42193],{"style":39822},[15,42191],{"className":42192,"style":159},[158],[15,42194,42196],{"className":42195},[163,164,165,166],[15,42197,310],{"className":42198},[129,166],[15,42200,175],{"className":42201},[174],[15,42203,42205],{"className":42204},[146],[15,42206,42208],{"className":42207,"style":182},[150],[15,42209],{},[15,42211,53],{"className":42212},[188],[15,42214,42216,42219],{"className":42215},[129],[15,42217,36569],{"className":42218,"style":36584},[129,133],[15,42220,42222],{"className":42221},[137],[15,42223,42225,42245],{"className":42224},[141,142],[15,42226,42228,42242],{"className":42227},[146],[15,42229,42231],{"className":42230,"style":1051},[150],[15,42232,42233,42236],{"style":39822},[15,42234],{"className":42235,"style":159},[158],[15,42237,42239],{"className":42238},[163,164,165,166],[15,42240,318],{"className":42241},[129,166],[15,42243,175],{"className":42244},[174],[15,42246,42248],{"className":42247},[146],[15,42249,42251],{"className":42250,"style":182},[150],[15,42252],{},[15,42254,53],{"className":42255},[188],[15,42257,56],{"className":42258},[129,133],[15,42260,6149],{"className":42261},[195],[15,42263],{"className":42264,"style":262},[93],[15,42266,335],{"className":42267},[449],[15,42269],{"className":42270,"style":262},[93],[15,42272,59],{"className":42273},[195],[11,42275,42276],{},"总能写成：",[15,42278,42280],{"className":42279,"translate":19},[18],[15,42281,42283,42312],{"className":42282,"translate":19},[23],[15,42284,42286],{"className":42285},[27],[29,42287,42288],{"xmlns":31,"display":32},[34,42289,42290,42309],{},[37,42291,42292,42299,42301,42303],{},[40,42293,42294,42296],{},[43,42295,13097],{},[6856,42297,42298],{},"总",[43,42300,56],{},[50,42302,2682],{},[40,42304,42305,42307],{},[43,42306,8699],{},[6856,42308,42298],{},[109,42310,42311],{"encoding":111},"A_\\text{总} x + b_\\text{总}",[15,42313,42315,42376],{"className":42314,"ariaHidden":72},[116],[15,42316,42318,42321,42364,42367,42370,42373],{"className":42317},[120],[15,42319],{"className":42320,"style":20229},[124],[15,42322,42324,42327],{"className":42323},[129],[15,42325,13097],{"className":42326},[129,133],[15,42328,42330],{"className":42329},[137],[15,42331,42333,42356],{"className":42332},[141,142],[15,42334,42336,42353],{"className":42335},[146],[15,42337,42339],{"className":42338,"style":1525},[150],[15,42340,42341,42344],{"style":154},[15,42342],{"className":42343,"style":159},[158],[15,42345,42347],{"className":42346},[163,164,165,166],[15,42348,42350],{"className":42349},[129,37682,166],[15,42351,42298],{"className":42352},[129,37690,166],[15,42354,175],{"className":42355},[174],[15,42357,42359],{"className":42358},[146],[15,42360,42362],{"className":42361,"style":182},[150],[15,42363],{},[15,42365,56],{"className":42366},[129,133],[15,42368],{"className":42369,"style":3277},[93],[15,42371,2682],{"className":42372},[2436],[15,42374],{"className":42375,"style":3277},[93],[15,42377,42379,42382],{"className":42378},[120],[15,42380],{"className":42381,"style":2183},[124],[15,42383,42385,42388],{"className":42384},[129],[15,42386,8699],{"className":42387},[129,133],[15,42389,42391],{"className":42390},[137],[15,42392,42394,42417],{"className":42393},[141,142],[15,42395,42397,42414],{"className":42396},[146],[15,42398,42400],{"className":42399,"style":1525},[150],[15,42401,42402,42405],{"style":154},[15,42403],{"className":42404,"style":159},[158],[15,42406,42408],{"className":42407},[163,164,165,166],[15,42409,42411],{"className":42410},[129,37682,166],[15,42412,42298],{"className":42413},[129,37690,166],[15,42415,175],{"className":42416},[174],[15,42418,42420],{"className":42419},[146],[15,42421,42423],{"className":42422,"style":182},[150],[15,42424],{},[11,42426,42427],{},"这正是为什么：如果神经网络的每一层都是仿射的而不使用激活函数，那么无论堆叠多少层，网络的表达能力都等同于单层。我们可以把仿射层看作拉伸、旋转或平移的运算，或这些运算的组合。复合多个仿射变换等价于连续的拉伸 + 旋转 + 平移，最终仍归结为一次整体的拉伸 + 旋转 + 平移。因此，复合仿射变换并不会增加非线性的自由度，也无法表示曲线或复杂函数。",[11,42429,42430],{},"神经网络的每一层都是一个仿射映射：",[15,42432,42434],{"className":42433,"translate":19},[18],[15,42435,42437,42467],{"className":42436,"translate":19},[23],[15,42438,42440],{"className":42439},[27],[29,42441,42442],{"xmlns":31,"display":32},[34,42443,42444,42464],{},[37,42445,42446,42448,42450,42452,42454,42456,42458,42460,42462],{},[43,42447,36569],{},[50,42449,53],{"stretchy":52},[43,42451,56],{},[50,42453,59],{"stretchy":52},[50,42455,62],{},[43,42457,32000],{},[43,42459,56],{},[50,42461,2682],{},[43,42463,8699],{},[109,42465,42466],{"encoding":111},"f(x) = Wx + b",[15,42468,42470,42497,42518],{"className":42469,"ariaHidden":72},[116],[15,42471,42473,42476,42479,42482,42485,42488,42491,42494],{"className":42472},[120],[15,42474],{"className":42475,"style":125},[124],[15,42477,36569],{"className":42478,"style":36584},[129,133],[15,42480,53],{"className":42481},[188],[15,42483,56],{"className":42484},[129,133],[15,42486,59],{"className":42487},[195],[15,42489],{"className":42490,"style":199},[93],[15,42492,62],{"className":42493},[203],[15,42495],{"className":42496,"style":199},[93],[15,42498,42500,42503,42506,42509,42512,42515],{"className":42499},[120],[15,42501],{"className":42502,"style":39511},[124],[15,42504,32000],{"className":42505,"style":12449},[129,133],[15,42507,56],{"className":42508},[129,133],[15,42510],{"className":42511,"style":3277},[93],[15,42513,2682],{"className":42514},[2436],[15,42516],{"className":42517,"style":3277},[93],[15,42519,42521,42524],{"className":42520},[120],[15,42522],{"className":42523,"style":522},[124],[15,42525,8699],{"className":42526},[129,133],[21730,42528,42529,42692],{},[21733,42530,14831,42531,42559,42560,42662,42663,42691],{},[15,42532,42534,42547],{"className":42533,"translate":19},[23],[15,42535,42537],{"className":42536},[27],[29,42538,42539],{"xmlns":31},[34,42540,42541,42545],{},[37,42542,42543],{},[43,42544,32000],{},[109,42546,32000],{"encoding":111},[15,42548,42550],{"className":42549,"ariaHidden":72},[116],[15,42551,42553,42556],{"className":42552},[120],[15,42554],{"className":42555,"style":610},[124],[15,42557,32000],{"className":42558,"style":12449},[129,133]," 可逆（即 ",[15,42561,42563,42590],{"className":42562,"translate":19},[23],[15,42564,42566],{"className":42565},[27],[29,42567,42568],{"xmlns":31},[34,42569,42570,42587],{},[37,42571,42572,42575,42577,42579,42581,42583,42585],{},[43,42573,42574],{},"det",[50,42576,69],{},[50,42578,53],{"stretchy":52},[43,42580,32000],{},[50,42582,59],{"stretchy":52},[50,42584,1418],{"mathvariant":65},[308,42586,541],{},[109,42588,42589],{"encoding":111},"\\det(W) \\neq 0",[15,42591,42593,42653],{"className":42592,"ariaHidden":72},[116],[15,42594,42596,42599,42602,42605,42608,42611,42614,42650],{"className":42595},[120],[15,42597],{"className":42598,"style":125},[124],[15,42600,42574],{"className":42601},[217],[15,42603,53],{"className":42604},[188],[15,42606,32000],{"className":42607,"style":12449},[129,133],[15,42609,59],{"className":42610},[195],[15,42612],{"className":42613,"style":199},[93],[15,42615,42617,42644,42647],{"className":42616},[203],[15,42618,42620],{"className":42619},[203],[15,42621,42623],{"className":42622},[129,1670],[15,42624,42626],{"className":42625},[1674],[15,42627,42629,42632,42641],{"className":42628},[1678],[15,42630],{"className":42631,"style":1682},[124],[15,42633,42635],{"className":42634},[1686],[15,42636,42638],{"className":42637},[129],[15,42639,1693],{"className":42640},[203],[15,42642],{"className":42643},[1697],[15,42645],{"className":42646},[93,1701],[15,42648,62],{"className":42649},[203],[15,42651],{"className":42652,"style":199},[93],[15,42654,42656,42659],{"className":42655},[120],[15,42657],{"className":42658,"style":4886},[124],[15,42660,541],{"className":42661},[129],"），则 ",[15,42664,42666,42679],{"className":42665,"translate":19},[23],[15,42667,42669],{"className":42668},[27],[29,42670,42671],{"xmlns":31},[34,42672,42673,42677],{},[37,42674,42675],{},[43,42676,36569],{},[109,42678,36569],{"encoding":111},[15,42680,42682],{"className":42681,"ariaHidden":72},[116],[15,42683,42685,42688],{"className":42684},[120],[15,42686],{"className":42687,"style":1682},[124],[15,42689,36569],{"className":42690,"style":36584},[129,133]," 是双射且连续的。",[21733,42693,42694],{},"其逆映射为：",[15,42696,42698],{"className":42697,"translate":19},[18],[15,42699,42701,42751],{"className":42700,"translate":19},[23],[15,42702,42704],{"className":42703},[27],[29,42705,42706],{"xmlns":31,"display":32},[34,42707,42708,42748],{},[37,42709,42710,42720,42722,42724,42726,42728,42738,42740,42742,42744,42746],{},[305,42711,42712,42714],{},[43,42713,36569],{},[37,42715,42716,42718],{},[50,42717,315],{},[308,42719,318],{},[50,42721,53],{"stretchy":52},[43,42723,857],{},[50,42725,59],{"stretchy":52},[50,42727,62],{},[305,42729,42730,42732],{},[43,42731,13097],{},[37,42733,42734,42736],{},[50,42735,315],{},[308,42737,318],{},[50,42739,53],{"stretchy":52},[43,42741,857],{},[50,42743,315],{},[43,42745,8699],{},[50,42747,59],{"stretchy":52},[109,42749,42750],{"encoding":111},"f^{-1}(y) = A^{-1}(y - b)",[15,42752,42754,42813,42869],{"className":42753,"ariaHidden":72},[116],[15,42755,42757,42760,42795,42798,42801,42804,42807,42810],{"className":42756},[120],[15,42758],{"className":42759,"style":40920},[124],[15,42761,42763,42766],{"className":42762},[129],[15,42764,36569],{"className":42765,"style":36584},[129,133],[15,42767,42769],{"className":42768},[137],[15,42770,42772],{"className":42771},[141],[15,42773,42775],{"className":42774},[146],[15,42776,42778],{"className":42777,"style":40991},[150],[15,42779,42780,42783],{"style":1272},[15,42781],{"className":42782,"style":159},[158],[15,42784,42786],{"className":42785},[163,164,165,166],[15,42787,42789,42792],{"className":42788},[129,166],[15,42790,315],{"className":42791},[129,166],[15,42793,318],{"className":42794},[129,166],[15,42796,53],{"className":42797},[188],[15,42799,857],{"className":42800,"style":887},[129,133],[15,42802,59],{"className":42803},[195],[15,42805],{"className":42806,"style":199},[93],[15,42808,62],{"className":42809},[203],[15,42811],{"className":42812,"style":199},[93],[15,42814,42816,42819,42854,42857,42860,42863,42866],{"className":42815},[120],[15,42817],{"className":42818,"style":40920},[124],[15,42820,42822,42825],{"className":42821},[129],[15,42823,13097],{"className":42824},[129,133],[15,42826,42828],{"className":42827},[137],[15,42829,42831],{"className":42830},[141],[15,42832,42834],{"className":42833},[146],[15,42835,42837],{"className":42836,"style":40991},[150],[15,42838,42839,42842],{"style":1272},[15,42840],{"className":42841,"style":159},[158],[15,42843,42845],{"className":42844},[163,164,165,166],[15,42846,42848,42851],{"className":42847},[129,166],[15,42849,315],{"className":42850},[129,166],[15,42852,318],{"className":42853},[129,166],[15,42855,53],{"className":42856},[188],[15,42858,857],{"className":42859,"style":887},[129,133],[15,42861],{"className":42862,"style":3277},[93],[15,42864,315],{"className":42865},[2436],[15,42867],{"className":42868,"style":3277},[93],[15,42870,42872,42875,42878],{"className":42871},[120],[15,42873],{"className":42874,"style":125},[124],[15,42876,8699],{"className":42877},[129,133],[15,42879,59],{"className":42880},[195],[11,42882,42883,42884,42912],{},"它同样是仿射且连续的，因此是一个同胚（homeomorphism）。但这里有一个核心条件：",[15,42885,42887,42900],{"className":42886,"translate":19},[23],[15,42888,42890],{"className":42889},[27],[29,42891,42892],{"xmlns":31},[34,42893,42894,42898],{},[37,42895,42896],{},[43,42897,32000],{},[109,42899,32000],{"encoding":111},[15,42901,42903],{"className":42902,"ariaHidden":72},[116],[15,42904,42906,42909],{"className":42905},[120],[15,42907],{"className":42908,"style":610},[124],[15,42910,32000],{"className":42911,"style":12449},[129,133]," 必须可逆。",[11,42914,42915],{},"复合多个仿射变换等价于复合多个同胚。假设有多个连续的可逆仿射映射：",[15,42917,42919],{"className":42918,"translate":19},[18],[15,42920,42922,42967],{"className":42921,"translate":19},[23],[15,42923,42925],{"className":42924},[27],[29,42926,42927],{"xmlns":31,"display":32},[34,42928,42929,42964],{},[37,42930,42931,42937,42940,42952,42954,42956,42958],{},[40,42932,42933,42935],{},[43,42934,36569],{},[43,42936,39326],{},[50,42938,42939],{},"∘",[40,42941,42942,42944],{},[43,42943,36569],{},[37,42945,42946,42948,42950],{},[43,42947,39326],{},[50,42949,315],{},[308,42951,318],{},[50,42953,42939],{},[50,42955,31670],{},[50,42957,42939],{},[40,42959,42960,42962],{},[43,42961,36569],{},[308,42963,318],{},[109,42965,42966],{"encoding":111},"f_n \\circ f_{n-1} \\circ \\cdots \\circ f_1",[15,42968,42970,43025,43091,43109],{"className":42969,"ariaHidden":72},[116],[15,42971,42973,42976,43016,43019,43022],{"className":42972},[120],[15,42974],{"className":42975,"style":1682},[124],[15,42977,42979,42982],{"className":42978},[129],[15,42980,36569],{"className":42981,"style":36584},[129,133],[15,42983,42985],{"className":42984},[137],[15,42986,42988,43008],{"className":42987},[141,142],[15,42989,42991,43005],{"className":42990},[146],[15,42992,42994],{"className":42993,"style":1163},[150],[15,42995,42996,42999],{"style":39822},[15,42997],{"className":42998,"style":159},[158],[15,43000,43002],{"className":43001},[163,164,165,166],[15,43003,39326],{"className":43004},[129,133,166],[15,43006,175],{"className":43007},[174],[15,43009,43011],{"className":43010},[146],[15,43012,43014],{"className":43013,"style":182},[150],[15,43015],{},[15,43017],{"className":43018,"style":3277},[93],[15,43020,42939],{"className":43021},[2436],[15,43023],{"className":43024,"style":3277},[93],[15,43026,43028,43032,43082,43085,43088],{"className":43027},[120],[15,43029],{"className":43030,"style":43031},[124],"height:0.9028em;vertical-align:-0.2083em;",[15,43033,43035,43038],{"className":43034},[129],[15,43036,36569],{"className":43037,"style":36584},[129,133],[15,43039,43041],{"className":43040},[137],[15,43042,43044,43073],{"className":43043},[141,142],[15,43045,43047,43070],{"className":43046},[146],[15,43048,43050],{"className":43049,"style":1051},[150],[15,43051,43052,43055],{"style":39822},[15,43053],{"className":43054,"style":159},[158],[15,43056,43058],{"className":43057},[163,164,165,166],[15,43059,43061,43064,43067],{"className":43060},[129,166],[15,43062,39326],{"className":43063},[129,133,166],[15,43065,315],{"className":43066},[2436,166],[15,43068,318],{"className":43069},[129,166],[15,43071,175],{"className":43072},[174],[15,43074,43076],{"className":43075},[146],[15,43077,43080],{"className":43078,"style":43079},[150],"height:0.2083em;",[15,43081],{},[15,43083],{"className":43084,"style":3277},[93],[15,43086,42939],{"className":43087},[2436],[15,43089],{"className":43090,"style":3277},[93],[15,43092,43094,43097,43100,43103,43106],{"className":43093},[120],[15,43095],{"className":43096,"style":11098},[124],[15,43098,31670],{"className":43099},[449],[15,43101],{"className":43102,"style":3277},[93],[15,43104,42939],{"className":43105},[2436],[15,43107],{"className":43108,"style":3277},[93],[15,43110,43112,43115],{"className":43111},[120],[15,43113],{"className":43114,"style":1682},[124],[15,43116,43118,43121],{"className":43117},[129],[15,43119,36569],{"className":43120,"style":36584},[129,133],[15,43122,43124],{"className":43123},[137],[15,43125,43127,43147],{"className":43126},[141,142],[15,43128,43130,43144],{"className":43129},[146],[15,43131,43133],{"className":43132,"style":1051},[150],[15,43134,43135,43138],{"style":39822},[15,43136],{"className":43137,"style":159},[158],[15,43139,43141],{"className":43140},[163,164,165,166],[15,43142,318],{"className":43143},[129,166],[15,43145,175],{"className":43146},[174],[15,43148,43150],{"className":43149},[146],[15,43151,43153],{"className":43152,"style":182},[150],[15,43154],{},[11,43156,43157,43158,43228],{},"每个 ",[15,43159,43161,43179],{"className":43160,"translate":19},[23],[15,43162,43164],{"className":43163},[27],[29,43165,43166],{"xmlns":31},[34,43167,43168,43176],{},[37,43169,43170],{},[40,43171,43172,43174],{},[43,43173,36569],{},[43,43175,1459],{},[109,43177,43178],{"encoding":111},"f_i",[15,43180,43182],{"className":43181,"ariaHidden":72},[116],[15,43183,43185,43188],{"className":43184},[120],[15,43186],{"className":43187,"style":1682},[124],[15,43189,43191,43194],{"className":43190},[129],[15,43192,36569],{"className":43193,"style":36584},[129,133],[15,43195,43197],{"className":43196},[137],[15,43198,43200,43220],{"className":43199},[141,142],[15,43201,43203,43217],{"className":43202},[146],[15,43204,43206],{"className":43205,"style":1981},[150],[15,43207,43208,43211],{"style":39822},[15,43209],{"className":43210,"style":159},[158],[15,43212,43214],{"className":43213},[163,164,165,166],[15,43215,1459],{"className":43216},[129,133,166],[15,43218,175],{"className":43219},[174],[15,43221,43223],{"className":43222},[146],[15,43224,43226],{"className":43225,"style":182},[150],[15,43227],{}," 都是同胚，而同胚的复合仍然是同胚。因此，最终的映射也是同胚。",[11,43230,43231,43232,43261],{},"于是我们得出如下结论：设 ",[15,43233,43235,43249],{"className":43234,"translate":19},[23],[15,43236,43238],{"className":43237},[27],[29,43239,43240],{"xmlns":31},[34,43241,43242,43247],{},[37,43243,43244],{},[43,43245,43246],{},"M",[109,43248,43246],{"encoding":111},[15,43250,43252],{"className":43251,"ariaHidden":72},[116],[15,43253,43255,43258],{"className":43254},[120],[15,43256],{"className":43257,"style":610},[124],[15,43259,43246],{"className":43260,"style":12597},[129,133]," 为原始流形：",[21730,43263,43264,43267],{},[21733,43265,43266],{},"连续且双射的仿射映射，保持流形结构（同胚）。",[21733,43268,43269,43270,43298],{},"连续复合多个这样的仿射映射，得到的流形与 ",[15,43271,43273,43286],{"className":43272,"translate":19},[23],[15,43274,43276],{"className":43275},[27],[29,43277,43278],{"xmlns":31},[34,43279,43280,43284],{},[37,43281,43282],{},[43,43283,43246],{},[109,43285,43246],{"encoding":111},[15,43287,43289],{"className":43288,"ariaHidden":72},[116],[15,43290,43292,43295],{"className":43291},[120],[15,43293],{"className":43294,"style":610},[124],[15,43296,43246],{"className":43297,"style":12597},[129,133]," 同胚。",[11,43300,43301],{},"换句话说：「连续线性 + 偏置的变换不会改变流形的拓扑，只会对其进行拉伸、旋转、平移或剪切。」",[11,43303,43304],{},"这就是为什么仅使用线性层无法「弯曲」数据流形：在嵌套的线性层下，数据始终保持其「原始形状」（拓扑不变）。如果决策边界需要切开一个复杂流形，就必须引入非线性（激活函数）。",[11,43306,43307],{},"这也是深度学习的目标。如果原始流形的结构对于给定任务而言不可分，而学习目标又要求可分性，那么原始流形与目标流形必然不是同胚的。",[11,43309,43310],{},"在从原始流形到目标流形的变换中，要失去同胚性就必须摆脱线性、仿射的变换。因为仅靠仿射变换（更一般地说是同胚变换）通常不足以解开类别。深度网络往往需要通过非同胚的非线性操作来重构数据流形，使其最终线性可分。",[11,43312,43313],{},"我们需要考虑如何让仿射变换的复合变得非线性。这一洞见正是神经网络几何解释的核心：",[6559,43315,43316],{},[11,43317,43318],{},"线性层只能执行保持拓扑的仿射变换；非线性层则允许拓扑改变，从而使神经网络能够表示复杂函数。",[570,43320,43321],{"id":43321},"仿射层的激活",[11,43323,42430],{},[15,43325,43327],{"className":43326,"translate":19},[18],[15,43328,43330,43372],{"className":43329,"translate":19},[23],[15,43331,43333],{"className":43332},[27],[29,43334,43335],{"xmlns":31,"display":32},[34,43336,43337,43369],{},[37,43338,43339,43345,43347,43349,43351,43353,43359,43361,43363],{},[40,43340,43341,43343],{},[43,43342,34146],{},[43,43344,1459],{},[50,43346,53],{"stretchy":52},[43,43348,56],{},[50,43350,59],{"stretchy":52},[50,43352,62],{},[40,43354,43355,43357],{},[43,43356,32000],{},[43,43358,1459],{},[43,43360,56],{},[50,43362,2682],{},[40,43364,43365,43367],{},[43,43366,8699],{},[43,43368,1459],{},[109,43370,43371],{"encoding":111},"l_i(x) = W_i x + b_i",[15,43373,43375,43440,43498],{"className":43374,"ariaHidden":72},[116],[15,43376,43378,43381,43422,43425,43428,43431,43434,43437],{"className":43377},[120],[15,43379],{"className":43380,"style":125},[124],[15,43382,43384,43387],{"className":43383},[129],[15,43385,34146],{"className":43386,"style":34204},[129,133],[15,43388,43390],{"className":43389},[137],[15,43391,43393,43414],{"className":43392},[141,142],[15,43394,43396,43411],{"className":43395},[146],[15,43397,43399],{"className":43398,"style":1981},[150],[15,43400,43402,43405],{"style":43401},"top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;",[15,43403],{"className":43404,"style":159},[158],[15,43406,43408],{"className":43407},[163,164,165,166],[15,43409,1459],{"className":43410},[129,133,166],[15,43412,175],{"className":43413},[174],[15,43415,43417],{"className":43416},[146],[15,43418,43420],{"className":43419,"style":182},[150],[15,43421],{},[15,43423,53],{"className":43424},[188],[15,43426,56],{"className":43427},[129,133],[15,43429,59],{"className":43430},[195],[15,43432],{"className":43433,"style":199},[93],[15,43435,62],{"className":43436},[203],[15,43438],{"className":43439,"style":199},[93],[15,43441,43443,43446,43486,43489,43492,43495],{"className":43442},[120],[15,43444],{"className":43445,"style":20229},[124],[15,43447,43449,43452],{"className":43448},[129],[15,43450,32000],{"className":43451,"style":12449},[129,133],[15,43453,43455],{"className":43454},[137],[15,43456,43458,43478],{"className":43457},[141,142],[15,43459,43461,43475],{"className":43460},[146],[15,43462,43464],{"className":43463,"style":1981},[150],[15,43465,43466,43469],{"style":34239},[15,43467],{"className":43468,"style":159},[158],[15,43470,43472],{"className":43471},[163,164,165,166],[15,43473,1459],{"className":43474},[129,133,166],[15,43476,175],{"className":43477},[174],[15,43479,43481],{"className":43480},[146],[15,43482,43484],{"className":43483,"style":182},[150],[15,43485],{},[15,43487,56],{"className":43488},[129,133],[15,43490],{"className":43491,"style":3277},[93],[15,43493,2682],{"className":43494},[2436],[15,43496],{"className":43497,"style":3277},[93],[15,43499,43501,43504],{"className":43500},[120],[15,43502],{"className":43503,"style":2183},[124],[15,43505,43507,43510],{"className":43506},[129],[15,43508,8699],{"className":43509},[129,133],[15,43511,43513],{"className":43512},[137],[15,43514,43516,43536],{"className":43515},[141,142],[15,43517,43519,43533],{"className":43518},[146],[15,43520,43522],{"className":43521,"style":1981},[150],[15,43523,43524,43527],{"style":154},[15,43525],{"className":43526,"style":159},[158],[15,43528,43530],{"className":43529},[163,164,165,166],[15,43531,1459],{"className":43532},[129,133,166],[15,43534,175],{"className":43535},[174],[15,43537,43539],{"className":43538},[146],[15,43540,43542],{"className":43541,"style":182},[150],[15,43543],{},[11,43545,28697,43546,43574,43575,43603,43604,43632],{},[15,43547,43549,43562],{"className":43548,"translate":19},[23],[15,43550,43552],{"className":43551},[27],[29,43553,43554],{"xmlns":31},[34,43555,43556,43560],{},[37,43557,43558],{},[43,43559,1459],{},[109,43561,1459],{"encoding":111},[15,43563,43565],{"className":43564,"ariaHidden":72},[116],[15,43566,43568,43571],{"className":43567},[120],[15,43569],{"className":43570,"style":16111},[124],[15,43572,1459],{"className":43573},[129,133]," 表示第 ",[15,43576,43578,43591],{"className":43577,"translate":19},[23],[15,43579,43581],{"className":43580},[27],[29,43582,43583],{"xmlns":31},[34,43584,43585,43589],{},[37,43586,43587],{},[43,43588,1459],{},[109,43590,1459],{"encoding":111},[15,43592,43594],{"className":43593,"ariaHidden":72},[116],[15,43595,43597,43600],{"className":43596},[120],[15,43598],{"className":43599,"style":16111},[124],[15,43601,1459],{"className":43602},[129,133]," 层。",[15,43605,43607,43620],{"className":43606,"translate":19},[23],[15,43608,43610],{"className":43609},[27],[29,43611,43612],{"xmlns":31},[34,43613,43614,43618],{},[37,43615,43616],{},[43,43617,39326],{},[109,43619,39326],{"encoding":111},[15,43621,43623],{"className":43622,"ariaHidden":72},[116],[15,43624,43626,43629],{"className":43625},[120],[15,43627],{"className":43628,"style":4383},[124],[15,43630,39326],{"className":43631},[129,133]," 层仿射映射的嵌套复合为：",[15,43634,43636],{"className":43635,"translate":19},[18],[15,43637,43639,43697],{"className":43638,"translate":19},[23],[15,43640,43642],{"className":43641},[27],[29,43643,43644],{"xmlns":31,"display":32},[34,43645,43646,43694],{},[37,43647,43648,43654,43656,43668,43670,43682,43684,43686,43688],{},[40,43649,43650,43652],{},[43,43651,34146],{},[43,43653,39326],{},[50,43655,42939],{},[40,43657,43658,43660],{},[43,43659,34146],{},[37,43661,43662,43664,43666],{},[43,43663,39326],{},[50,43665,315],{},[308,43667,318],{},[50,43669,42939],{},[40,43671,43672,43674],{},[43,43673,34146],{},[37,43675,43676,43678,43680],{},[43,43677,39326],{},[50,43679,315],{},[308,43681,310],{},[50,43683,42939],{},[50,43685,31670],{},[50,43687,42939],{},[40,43689,43690,43692],{},[43,43691,34146],{},[308,43693,541],{},[109,43695,43696],{"encoding":111},"l_n \\circ l_{n-1} \\circ l_{n-2} \\circ \\dots \\circ l_0",[15,43698,43700,43755,43819,43883,43901],{"className":43699,"ariaHidden":72},[116],[15,43701,43703,43706,43746,43749,43752],{"className":43702},[120],[15,43704],{"className":43705,"style":2183},[124],[15,43707,43709,43712],{"className":43708},[129],[15,43710,34146],{"className":43711,"style":34204},[129,133],[15,43713,43715],{"className":43714},[137],[15,43716,43718,43738],{"className":43717},[141,142],[15,43719,43721,43735],{"className":43720},[146],[15,43722,43724],{"className":43723,"style":1163},[150],[15,43725,43726,43729],{"style":43401},[15,43727],{"className":43728,"style":159},[158],[15,43730,43732],{"className":43731},[163,164,165,166],[15,43733,39326],{"className":43734},[129,133,166],[15,43736,175],{"className":43737},[174],[15,43739,43741],{"className":43740},[146],[15,43742,43744],{"className":43743,"style":182},[150],[15,43745],{},[15,43747],{"className":43748,"style":3277},[93],[15,43750,42939],{"className":43751},[2436],[15,43753],{"className":43754,"style":3277},[93],[15,43756,43758,43761,43810,43813,43816],{"className":43757},[120],[15,43759],{"className":43760,"style":43031},[124],[15,43762,43764,43767],{"className":43763},[129],[15,43765,34146],{"className":43766,"style":34204},[129,133],[15,43768,43770],{"className":43769},[137],[15,43771,43773,43802],{"className":43772},[141,142],[15,43774,43776,43799],{"className":43775},[146],[15,43777,43779],{"className":43778,"style":1051},[150],[15,43780,43781,43784],{"style":43401},[15,43782],{"className":43783,"style":159},[158],[15,43785,43787],{"className":43786},[163,164,165,166],[15,43788,43790,43793,43796],{"className":43789},[129,166],[15,43791,39326],{"className":43792},[129,133,166],[15,43794,315],{"className":43795},[2436,166],[15,43797,318],{"className":43798},[129,166],[15,43800,175],{"className":43801},[174],[15,43803,43805],{"className":43804},[146],[15,43806,43808],{"className":43807,"style":43079},[150],[15,43809],{},[15,43811],{"className":43812,"style":3277},[93],[15,43814,42939],{"className":43815},[2436],[15,43817],{"className":43818,"style":3277},[93],[15,43820,43822,43825,43874,43877,43880],{"className":43821},[120],[15,43823],{"className":43824,"style":43031},[124],[15,43826,43828,43831],{"className":43827},[129],[15,43829,34146],{"className":43830,"style":34204},[129,133],[15,43832,43834],{"className":43833},[137],[15,43835,43837,43866],{"className":43836},[141,142],[15,43838,43840,43863],{"className":43839},[146],[15,43841,43843],{"className":43842,"style":1051},[150],[15,43844,43845,43848],{"style":43401},[15,43846],{"className":43847,"style":159},[158],[15,43849,43851],{"className":43850},[163,164,165,166],[15,43852,43854,43857,43860],{"className":43853},[129,166],[15,43855,39326],{"className":43856},[129,133,166],[15,43858,315],{"className":43859},[2436,166],[15,43861,310],{"className":43862},[129,166],[15,43864,175],{"className":43865},[174],[15,43867,43869],{"className":43868},[146],[15,43870,43872],{"className":43871,"style":43079},[150],[15,43873],{},[15,43875],{"className":43876,"style":3277},[93],[15,43878,42939],{"className":43879},[2436],[15,43881],{"className":43882,"style":3277},[93],[15,43884,43886,43889,43892,43895,43898],{"className":43885},[120],[15,43887],{"className":43888,"style":11098},[124],[15,43890,31670],{"className":43891},[449],[15,43893],{"className":43894,"style":3277},[93],[15,43896,42939],{"className":43897},[2436],[15,43899],{"className":43900,"style":3277},[93],[15,43902,43904,43907],{"className":43903},[120],[15,43905],{"className":43906,"style":2183},[124],[15,43908,43910,43913],{"className":43909},[129],[15,43911,34146],{"className":43912,"style":34204},[129,133],[15,43914,43916],{"className":43915},[137],[15,43917,43919,43939],{"className":43918},[141,142],[15,43920,43922,43936],{"className":43921},[146],[15,43923,43925],{"className":43924,"style":1051},[150],[15,43926,43927,43930],{"style":43401},[15,43928],{"className":43929,"style":159},[158],[15,43931,43933],{"className":43932},[163,164,165,166],[15,43934,541],{"className":43935},[129,166],[15,43937,175],{"className":43938},[174],[15,43940,43942],{"className":43941},[146],[15,43943,43945],{"className":43944,"style":182},[150],[15,43946],{},[11,43948,43949],{},"如果不引入激活函数，上述结果仍然是仿射函数。",[11,43951,43952],{},"证明很简单：",[15,43954,43956],{"className":43955,"translate":19},[18],[15,43957,43959,44075],{"className":43958,"translate":19},[23],[15,43960,43962],{"className":43961},[27],[29,43963,43964],{"xmlns":31,"display":32},[34,43965,43966,44072],{},[37,43967,43968,43974,43976,43982,43984,43986,43988,43990,43992,43998,44000,44006,44008,44010,44016,44018,44020,44026,44028,44030,44036,44042,44044,44046,44048,44050,44056,44062,44064,44070],{},[40,43969,43970,43972],{},[43,43971,34146],{},[308,43973,310],{},[50,43975,53],{"stretchy":52},[40,43977,43978,43980],{},[43,43979,34146],{},[308,43981,318],{},[50,43983,53],{"stretchy":52},[43,43985,56],{},[50,43987,59],{"stretchy":52},[50,43989,59],{"stretchy":52},[50,43991,62],{},[40,43993,43994,43996],{},[43,43995,32000],{},[308,43997,310],{},[50,43999,53],{"stretchy":52},[40,44001,44002,44004],{},[43,44003,32000],{},[308,44005,318],{},[43,44007,56],{},[50,44009,2682],{},[40,44011,44012,44014],{},[43,44013,8699],{},[308,44015,318],{},[50,44017,59],{"stretchy":52},[50,44019,2682],{},[40,44021,44022,44024],{},[43,44023,8699],{},[308,44025,310],{},[50,44027,62],{},[50,44029,53],{"stretchy":52},[40,44031,44032,44034],{},[43,44033,32000],{},[308,44035,310],{},[40,44037,44038,44040],{},[43,44039,32000],{},[308,44041,318],{},[50,44043,59],{"stretchy":52},[43,44045,56],{},[50,44047,2682],{},[50,44049,53],{"stretchy":52},[40,44051,44052,44054],{},[43,44053,32000],{},[308,44055,310],{},[40,44057,44058,44060],{},[43,44059,8699],{},[308,44061,318],{},[50,44063,2682],{},[40,44065,44066,44068],{},[43,44067,8699],{},[308,44069,310],{},[50,44071,59],{"stretchy":52},[109,44073,44074],{"encoding":111},"l_2(l_1(x)) = W_2(W_1x+b_1)+b_2 = (W_2W_1)x+(W_2b_1+b_2)",[15,44076,44078,44185,44286,44344,44399,44503,44601],{"className":44077,"ariaHidden":72},[116],[15,44079,44081,44084,44124,44127,44167,44170,44173,44176,44179,44182],{"className":44080},[120],[15,44082],{"className":44083,"style":125},[124],[15,44085,44087,44090],{"className":44086},[129],[15,44088,34146],{"className":44089,"style":34204},[129,133],[15,44091,44093],{"className":44092},[137],[15,44094,44096,44116],{"className":44095},[141,142],[15,44097,44099,44113],{"className":44098},[146],[15,44100,44102],{"className":44101,"style":1051},[150],[15,44103,44104,44107],{"style":43401},[15,44105],{"className":44106,"style":159},[158],[15,44108,44110],{"className":44109},[163,164,165,166],[15,44111,310],{"className":44112},[129,166],[15,44114,175],{"className":44115},[174],[15,44117,44119],{"className":44118},[146],[15,44120,44122],{"className":44121,"style":182},[150],[15,44123],{},[15,44125,53],{"className":44126},[188],[15,44128,44130,44133],{"className":44129},[129],[15,44131,34146],{"className":44132,"style":34204},[129,133],[15,44134,44136],{"className":44135},[137],[15,44137,44139,44159],{"className":44138},[141,142],[15,44140,44142,44156],{"className":44141},[146],[15,44143,44145],{"className":44144,"style":1051},[150],[15,44146,44147,44150],{"style":43401},[15,44148],{"className":44149,"style":159},[158],[15,44151,44153],{"className":44152},[163,164,165,166],[15,44154,318],{"className":44155},[129,166],[15,44157,175],{"className":44158},[174],[15,44160,44162],{"className":44161},[146],[15,44163,44165],{"className":44164,"style":182},[150],[15,44166],{},[15,44168,53],{"className":44169},[188],[15,44171,56],{"className":44172},[129,133],[15,44174,6149],{"className":44175},[195],[15,44177],{"className":44178,"style":199},[93],[15,44180,62],{"className":44181},[203],[15,44183],{"className":44184,"style":199},[93],[15,44186,44188,44191,44231,44234,44274,44277,44280,44283],{"className":44187},[120],[15,44189],{"className":44190,"style":125},[124],[15,44192,44194,44197],{"className":44193},[129],[15,44195,32000],{"className":44196,"style":12449},[129,133],[15,44198,44200],{"className":44199},[137],[15,44201,44203,44223],{"className":44202},[141,142],[15,44204,44206,44220],{"className":44205},[146],[15,44207,44209],{"className":44208,"style":1051},[150],[15,44210,44211,44214],{"style":34239},[15,44212],{"className":44213,"style":159},[158],[15,44215,44217],{"className":44216},[163,164,165,166],[15,44218,310],{"className":44219},[129,166],[15,44221,175],{"className":44222},[174],[15,44224,44226],{"className":44225},[146],[15,44227,44229],{"className":44228,"style":182},[150],[15,44230],{},[15,44232,53],{"className":44233},[188],[15,44235,44237,44240],{"className":44236},[129],[15,44238,32000],{"className":44239,"style":12449},[129,133],[15,44241,44243],{"className":44242},[137],[15,44244,44246,44266],{"className":44245},[141,142],[15,44247,44249,44263],{"className":44248},[146],[15,44250,44252],{"className":44251,"style":1051},[150],[15,44253,44254,44257],{"style":34239},[15,44255],{"className":44256,"style":159},[158],[15,44258,44260],{"className":44259},[163,164,165,166],[15,44261,318],{"className":44262},[129,166],[15,44264,175],{"className":44265},[174],[15,44267,44269],{"className":44268},[146],[15,44270,44272],{"className":44271,"style":182},[150],[15,44273],{},[15,44275,56],{"className":44276},[129,133],[15,44278],{"className":44279,"style":3277},[93],[15,44281,2682],{"className":44282},[2436],[15,44284],{"className":44285,"style":3277},[93],[15,44287,44289,44292,44332,44335,44338,44341],{"className":44288},[120],[15,44290],{"className":44291,"style":125},[124],[15,44293,44295,44298],{"className":44294},[129],[15,44296,8699],{"className":44297},[129,133],[15,44299,44301],{"className":44300},[137],[15,44302,44304,44324],{"className":44303},[141,142],[15,44305,44307,44321],{"className":44306},[146],[15,44308,44310],{"className":44309,"style":1051},[150],[15,44311,44312,44315],{"style":154},[15,44313],{"className":44314,"style":159},[158],[15,44316,44318],{"className":44317},[163,164,165,166],[15,44319,318],{"className":44320},[129,166],[15,44322,175],{"className":44323},[174],[15,44325,44327],{"className":44326},[146],[15,44328,44330],{"className":44329,"style":182},[150],[15,44331],{},[15,44333,59],{"className":44334},[195],[15,44336],{"className":44337,"style":3277},[93],[15,44339,2682],{"className":44340},[2436],[15,44342],{"className":44343,"style":3277},[93],[15,44345,44347,44350,44390,44393,44396],{"className":44346},[120],[15,44348],{"className":44349,"style":2183},[124],[15,44351,44353,44356],{"className":44352},[129],[15,44354,8699],{"className":44355},[129,133],[15,44357,44359],{"className":44358},[137],[15,44360,44362,44382],{"className":44361},[141,142],[15,44363,44365,44379],{"className":44364},[146],[15,44366,44368],{"className":44367,"style":1051},[150],[15,44369,44370,44373],{"style":154},[15,44371],{"className":44372,"style":159},[158],[15,44374,44376],{"className":44375},[163,164,165,166],[15,44377,310],{"className":44378},[129,166],[15,44380,175],{"className":44381},[174],[15,44383,44385],{"className":44384},[146],[15,44386,44388],{"className":44387,"style":182},[150],[15,44389],{},[15,44391],{"className":44392,"style":199},[93],[15,44394,62],{"className":44395},[203],[15,44397],{"className":44398,"style":199},[93],[15,44400,44402,44405,44408,44448,44488,44491,44494,44497,44500],{"className":44401},[120],[15,44403],{"className":44404,"style":125},[124],[15,44406,53],{"className":44407},[188],[15,44409,44411,44414],{"className":44410},[129],[15,44412,32000],{"className":44413,"style":12449},[129,133],[15,44415,44417],{"className":44416},[137],[15,44418,44420,44440],{"className":44419},[141,142],[15,44421,44423,44437],{"className":44422},[146],[15,44424,44426],{"className":44425,"style":1051},[150],[15,44427,44428,44431],{"style":34239},[15,44429],{"className":44430,"style":159},[158],[15,44432,44434],{"className":44433},[163,164,165,166],[15,44435,310],{"className":44436},[129,166],[15,44438,175],{"className":44439},[174],[15,44441,44443],{"className":44442},[146],[15,44444,44446],{"className":44445,"style":182},[150],[15,44447],{},[15,44449,44451,44454],{"className":44450},[129],[15,44452,32000],{"className":44453,"style":12449},[129,133],[15,44455,44457],{"className":44456},[137],[15,44458,44460,44480],{"className":44459},[141,142],[15,44461,44463,44477],{"className":44462},[146],[15,44464,44466],{"className":44465,"style":1051},[150],[15,44467,44468,44471],{"style":34239},[15,44469],{"className":44470,"style":159},[158],[15,44472,44474],{"className":44473},[163,164,165,166],[15,44475,318],{"className":44476},[129,166],[15,44478,175],{"className":44479},[174],[15,44481,44483],{"className":44482},[146],[15,44484,44486],{"className":44485,"style":182},[150],[15,44487],{},[15,44489,59],{"className":44490},[195],[15,44492,56],{"className":44493},[129,133],[15,44495],{"className":44496,"style":3277},[93],[15,44498,2682],{"className":44499},[2436],[15,44501],{"className":44502,"style":3277},[93],[15,44504,44506,44509,44512,44552,44592,44595,44598],{"className":44505},[120],[15,44507],{"className":44508,"style":125},[124],[15,44510,53],{"className":44511},[188],[15,44513,44515,44518],{"className":44514},[129],[15,44516,32000],{"className":44517,"style":12449},[129,133],[15,44519,44521],{"className":44520},[137],[15,44522,44524,44544],{"className":44523},[141,142],[15,44525,44527,44541],{"className":44526},[146],[15,44528,44530],{"className":44529,"style":1051},[150],[15,44531,44532,44535],{"style":34239},[15,44533],{"className":44534,"style":159},[158],[15,44536,44538],{"className":44537},[163,164,165,166],[15,44539,310],{"className":44540},[129,166],[15,44542,175],{"className":44543},[174],[15,44545,44547],{"className":44546},[146],[15,44548,44550],{"className":44549,"style":182},[150],[15,44551],{},[15,44553,44555,44558],{"className":44554},[129],[15,44556,8699],{"className":44557},[129,133],[15,44559,44561],{"className":44560},[137],[15,44562,44564,44584],{"className":44563},[141,142],[15,44565,44567,44581],{"className":44566},[146],[15,44568,44570],{"className":44569,"style":1051},[150],[15,44571,44572,44575],{"style":154},[15,44573],{"className":44574,"style":159},[158],[15,44576,44578],{"className":44577},[163,164,165,166],[15,44579,318],{"className":44580},[129,166],[15,44582,175],{"className":44583},[174],[15,44585,44587],{"className":44586},[146],[15,44588,44590],{"className":44589,"style":182},[150],[15,44591],{},[15,44593],{"className":44594,"style":3277},[93],[15,44596,2682],{"className":44597},[2436],[15,44599],{"className":44600,"style":3277},[93],[15,44602,44604,44607,44647],{"className":44603},[120],[15,44605],{"className":44606,"style":125},[124],[15,44608,44610,44613],{"className":44609},[129],[15,44611,8699],{"className":44612},[129,133],[15,44614,44616],{"className":44615},[137],[15,44617,44619,44639],{"className":44618},[141,142],[15,44620,44622,44636],{"className":44621},[146],[15,44623,44625],{"className":44624,"style":1051},[150],[15,44626,44627,44630],{"style":154},[15,44628],{"className":44629,"style":159},[158],[15,44631,44633],{"className":44632},[163,164,165,166],[15,44634,310],{"className":44635},[129,166],[15,44637,175],{"className":44638},[174],[15,44640,44642],{"className":44641},[146],[15,44643,44645],{"className":44644,"style":182},[150],[15,44646],{},[15,44648,59],{"className":44649},[195],[11,44651,44652,44653,44708],{},"它仍然保持 ",[15,44654,44656,44675],{"className":44655,"translate":19},[23],[15,44657,44659],{"className":44658},[27],[29,44660,44661],{"xmlns":31},[34,44662,44663,44673],{},[37,44664,44665,44667,44669,44671],{},[43,44666,13097],{},[43,44668,56],{},[50,44670,2682],{},[43,44672,8699],{},[109,44674,41428],{"encoding":111},[15,44676,44678,44699],{"className":44677,"ariaHidden":72},[116],[15,44679,44681,44684,44687,44690,44693,44696],{"className":44680},[120],[15,44682],{"className":44683,"style":39511},[124],[15,44685,13097],{"className":44686},[129,133],[15,44688,56],{"className":44689},[129,133],[15,44691],{"className":44692,"style":3277},[93],[15,44694,2682],{"className":44695},[2436],[15,44697],{"className":44698,"style":3277},[93],[15,44700,44702,44705],{"className":44701},[120],[15,44703],{"className":44704,"style":522},[124],[15,44706,8699],{"className":44707},[129,133]," 的形式。",[11,44710,44711],{},"由归纳法可得：",[15,44713,44715],{"className":44714,"translate":19},[18],[15,44716,44718,44758],{"className":44717,"translate":19},[23],[15,44719,44721],{"className":44720},[27],[29,44722,44723],{"xmlns":31,"display":32},[34,44724,44725,44755],{},[37,44726,44727,44733,44735,44737,44739,44745,44747,44749,44751,44753],{},[40,44728,44729,44731],{},[43,44730,34146],{},[43,44732,39326],{},[50,44734,42939],{},[50,44736,31670],{},[50,44738,42939],{},[40,44740,44741,44743],{},[43,44742,34146],{},[308,44744,541],{},[50,44746,62],{},[43,44748,32000],{},[43,44750,56],{},[50,44752,2682],{},[43,44754,8699],{},[109,44756,44757],{"encoding":111},"l_n \\circ \\cdots \\circ l_0 = Wx + b",[15,44759,44761,44816,44834,44889,44910],{"className":44760,"ariaHidden":72},[116],[15,44762,44764,44767,44807,44810,44813],{"className":44763},[120],[15,44765],{"className":44766,"style":2183},[124],[15,44768,44770,44773],{"className":44769},[129],[15,44771,34146],{"className":44772,"style":34204},[129,133],[15,44774,44776],{"className":44775},[137],[15,44777,44779,44799],{"className":44778},[141,142],[15,44780,44782,44796],{"className":44781},[146],[15,44783,44785],{"className":44784,"style":1163},[150],[15,44786,44787,44790],{"style":43401},[15,44788],{"className":44789,"style":159},[158],[15,44791,44793],{"className":44792},[163,164,165,166],[15,44794,39326],{"className":44795},[129,133,166],[15,44797,175],{"className":44798},[174],[15,44800,44802],{"className":44801},[146],[15,44803,44805],{"className":44804,"style":182},[150],[15,44806],{},[15,44808],{"className":44809,"style":3277},[93],[15,44811,42939],{"className":44812},[2436],[15,44814],{"className":44815,"style":3277},[93],[15,44817,44819,44822,44825,44828,44831],{"className":44818},[120],[15,44820],{"className":44821,"style":11098},[124],[15,44823,31670],{"className":44824},[449],[15,44826],{"className":44827,"style":3277},[93],[15,44829,42939],{"className":44830},[2436],[15,44832],{"className":44833,"style":3277},[93],[15,44835,44837,44840,44880,44883,44886],{"className":44836},[120],[15,44838],{"className":44839,"style":2183},[124],[15,44841,44843,44846],{"className":44842},[129],[15,44844,34146],{"className":44845,"style":34204},[129,133],[15,44847,44849],{"className":44848},[137],[15,44850,44852,44872],{"className":44851},[141,142],[15,44853,44855,44869],{"className":44854},[146],[15,44856,44858],{"className":44857,"style":1051},[150],[15,44859,44860,44863],{"style":43401},[15,44861],{"className":44862,"style":159},[158],[15,44864,44866],{"className":44865},[163,164,165,166],[15,44867,541],{"className":44868},[129,166],[15,44870,175],{"className":44871},[174],[15,44873,44875],{"className":44874},[146],[15,44876,44878],{"className":44877,"style":182},[150],[15,44879],{},[15,44881],{"className":44882,"style":199},[93],[15,44884,62],{"className":44885},[203],[15,44887],{"className":44888,"style":199},[93],[15,44890,44892,44895,44898,44901,44904,44907],{"className":44891},[120],[15,44893],{"className":44894,"style":39511},[124],[15,44896,32000],{"className":44897,"style":12449},[129,133],[15,44899,56],{"className":44900},[129,133],[15,44902],{"className":44903,"style":3277},[93],[15,44905,2682],{"className":44906},[2436],[15,44908],{"className":44909,"style":3277},[93],[15,44911,44913,44916],{"className":44912},[120],[15,44914],{"className":44915,"style":522},[124],[15,44917,8699],{"className":44918},[129,133],[11,44920,44921],{},"因此，网络的深度完全失去了意义。",[11,44923,44924],{},"为了引入非线性，我们可以在每一层上添加激活函数：",[15,44926,44928],{"className":44927,"translate":19},[18],[15,44929,44931,45033],{"className":44930,"translate":19},[23],[15,44932,44934],{"className":44933},[27],[29,44935,44936],{"xmlns":31,"display":32},[34,44937,44938,45030],{},[37,44939,44940,44942,44944,44946,44948,44950,44952,44958,44960,44966,44968,44970,44972,44984,44986,44998,45000,45002,45004,45006,45008,45014,45016,45022,45024,45026,45028],{},[43,44941,36569],{},[50,44943,53],{"stretchy":52},[43,44945,56],{},[50,44947,59],{"stretchy":52},[50,44949,62],{},[50,44951,53],{"stretchy":52},[40,44953,44954,44956],{},[43,44955,27294],{},[43,44957,39326],{},[50,44959,42939],{},[40,44961,44962,44964],{},[43,44963,34146],{},[43,44965,39326],{},[50,44967,59],{"stretchy":52},[50,44969,42939],{},[50,44971,53],{"stretchy":52},[40,44973,44974,44976],{},[43,44975,27294],{},[37,44977,44978,44980,44982],{},[43,44979,39326],{},[50,44981,315],{},[308,44983,318],{},[50,44985,42939],{},[40,44987,44988,44990],{},[43,44989,34146],{},[37,44991,44992,44994,44996],{},[43,44993,39326],{},[50,44995,315],{},[308,44997,318],{},[50,44999,59],{"stretchy":52},[50,45001,42939],{},[50,45003,31670],{},[50,45005,42939],{},[50,45007,53],{"stretchy":52},[40,45009,45010,45012],{},[43,45011,27294],{},[308,45013,541],{},[50,45015,42939],{},[40,45017,45018,45020],{},[43,45019,34146],{},[308,45021,541],{},[50,45023,59],{"stretchy":52},[50,45025,53],{"stretchy":52},[43,45027,56],{},[50,45029,59],{"stretchy":52},[109,45031,45032],{"encoding":111},"f(x) = (\\sigma_n \\circ l_n) \\circ (\\sigma_{n-1} \\circ l_{n-1}) \\circ \\cdots \\circ (\\sigma_0 \\circ l_0)(x)",[15,45034,45036,45063,45121,45179,45246,45313,45331,45389],{"className":45035,"ariaHidden":72},[116],[15,45037,45039,45042,45045,45048,45051,45054,45057,45060],{"className":45038},[120],[15,45040],{"className":45041,"style":125},[124],[15,45043,36569],{"className":45044,"style":36584},[129,133],[15,45046,53],{"className":45047},[188],[15,45049,56],{"className":45050},[129,133],[15,45052,59],{"className":45053},[195],[15,45055],{"className":45056,"style":199},[93],[15,45058,62],{"className":45059},[203],[15,45061],{"className":45062,"style":199},[93],[15,45064,45066,45069,45072,45112,45115,45118],{"className":45065},[120],[15,45067],{"className":45068,"style":125},[124],[15,45070,53],{"className":45071},[188],[15,45073,45075,45078],{"className":45074},[129],[15,45076,27294],{"className":45077,"style":887},[129,133],[15,45079,45081],{"className":45080},[137],[15,45082,45084,45104],{"className":45083},[141,142],[15,45085,45087,45101],{"className":45086},[146],[15,45088,45090],{"className":45089,"style":1163},[150],[15,45091,45092,45095],{"style":1100},[15,45093],{"className":45094,"style":159},[158],[15,45096,45098],{"className":45097},[163,164,165,166],[15,45099,39326],{"className":45100},[129,133,166],[15,45102,175],{"className":45103},[174],[15,45105,45107],{"className":45106},[146],[15,45108,45110],{"className":45109,"style":182},[150],[15,45111],{},[15,45113],{"className":45114,"style":3277},[93],[15,45116,42939],{"className":45117},[2436],[15,45119],{"className":45120,"style":3277},[93],[15,45122,45124,45127,45167,45170,45173,45176],{"className":45123},[120],[15,45125],{"className":45126,"style":125},[124],[15,45128,45130,45133],{"className":45129},[129],[15,45131,34146],{"className":45132,"style":34204},[129,133],[15,45134,45136],{"className":45135},[137],[15,45137,45139,45159],{"className":45138},[141,142],[15,45140,45142,45156],{"className":45141},[146],[15,45143,45145],{"className":45144,"style":1163},[150],[15,45146,45147,45150],{"style":43401},[15,45148],{"className":45149,"style":159},[158],[15,45151,45153],{"className":45152},[163,164,165,166],[15,45154,39326],{"className":45155},[129,133,166],[15,45157,175],{"className":45158},[174],[15,45160,45162],{"className":45161},[146],[15,45163,45165],{"className":45164,"style":182},[150],[15,45166],{},[15,45168,59],{"className":45169},[195],[15,45171],{"className":45172,"style":3277},[93],[15,45174,42939],{"className":45175},[2436],[15,45177],{"className":45178,"style":3277},[93],[15,45180,45182,45185,45188,45237,45240,45243],{"className":45181},[120],[15,45183],{"className":45184,"style":125},[124],[15,45186,53],{"className":45187},[188],[15,45189,45191,45194],{"className":45190},[129],[15,45192,27294],{"className":45193,"style":887},[129,133],[15,45195,45197],{"className":45196},[137],[15,45198,45200,45229],{"className":45199},[141,142],[15,45201,45203,45226],{"className":45202},[146],[15,45204,45206],{"className":45205,"style":1051},[150],[15,45207,45208,45211],{"style":1100},[15,45209],{"className":45210,"style":159},[158],[15,45212,45214],{"className":45213},[163,164,165,166],[15,45215,45217,45220,45223],{"className":45216},[129,166],[15,45218,39326],{"className":45219},[129,133,166],[15,45221,315],{"className":45222},[2436,166],[15,45224,318],{"className":45225},[129,166],[15,45227,175],{"className":45228},[174],[15,45230,45232],{"className":45231},[146],[15,45233,45235],{"className":45234,"style":43079},[150],[15,45236],{},[15,45238],{"className":45239,"style":3277},[93],[15,45241,42939],{"className":45242},[2436],[15,45244],{"className":45245,"style":3277},[93],[15,45247,45249,45252,45301,45304,45307,45310],{"className":45248},[120],[15,45250],{"className":45251,"style":125},[124],[15,45253,45255,45258],{"className":45254},[129],[15,45256,34146],{"className":45257,"style":34204},[129,133],[15,45259,45261],{"className":45260},[137],[15,45262,45264,45293],{"className":45263},[141,142],[15,45265,45267,45290],{"className":45266},[146],[15,45268,45270],{"className":45269,"style":1051},[150],[15,45271,45272,45275],{"style":43401},[15,45273],{"className":45274,"style":159},[158],[15,45276,45278],{"className":45277},[163,164,165,166],[15,45279,45281,45284,45287],{"className":45280},[129,166],[15,45282,39326],{"className":45283},[129,133,166],[15,45285,315],{"className":45286},[2436,166],[15,45288,318],{"className":45289},[129,166],[15,45291,175],{"className":45292},[174],[15,45294,45296],{"className":45295},[146],[15,45297,45299],{"className":45298,"style":43079},[150],[15,45300],{},[15,45302,59],{"className":45303},[195],[15,45305],{"className":45306,"style":3277},[93],[15,45308,42939],{"className":45309},[2436],[15,45311],{"className":45312,"style":3277},[93],[15,45314,45316,45319,45322,45325,45328],{"className":45315},[120],[15,45317],{"className":45318,"style":11098},[124],[15,45320,31670],{"className":45321},[449],[15,45323],{"className":45324,"style":3277},[93],[15,45326,42939],{"className":45327},[2436],[15,45329],{"className":45330,"style":3277},[93],[15,45332,45334,45337,45340,45380,45383,45386],{"className":45333},[120],[15,45335],{"className":45336,"style":125},[124],[15,45338,53],{"className":45339},[188],[15,45341,45343,45346],{"className":45342},[129],[15,45344,27294],{"className":45345,"style":887},[129,133],[15,45347,45349],{"className":45348},[137],[15,45350,45352,45372],{"className":45351},[141,142],[15,45353,45355,45369],{"className":45354},[146],[15,45356,45358],{"className":45357,"style":1051},[150],[15,45359,45360,45363],{"style":1100},[15,45361],{"className":45362,"style":159},[158],[15,45364,45366],{"className":45365},[163,164,165,166],[15,45367,541],{"className":45368},[129,166],[15,45370,175],{"className":45371},[174],[15,45373,45375],{"className":45374},[146],[15,45376,45378],{"className":45377,"style":182},[150],[15,45379],{},[15,45381],{"className":45382,"style":3277},[93],[15,45384,42939],{"className":45385},[2436],[15,45387],{"className":45388,"style":3277},[93],[15,45390,45392,45395,45435,45438,45441,45444],{"className":45391},[120],[15,45393],{"className":45394,"style":125},[124],[15,45396,45398,45401],{"className":45397},[129],[15,45399,34146],{"className":45400,"style":34204},[129,133],[15,45402,45404],{"className":45403},[137],[15,45405,45407,45427],{"className":45406},[141,142],[15,45408,45410,45424],{"className":45409},[146],[15,45411,45413],{"className":45412,"style":1051},[150],[15,45414,45415,45418],{"style":43401},[15,45416],{"className":45417,"style":159},[158],[15,45419,45421],{"className":45420},[163,164,165,166],[15,45422,541],{"className":45423},[129,166],[15,45425,175],{"className":45426},[174],[15,45428,45430],{"className":45429},[146],[15,45431,45433],{"className":45432,"style":182},[150],[15,45434],{},[15,45436,59],{"className":45437},[195],[15,45439,53],{"className":45440},[188],[15,45442,56],{"className":45443},[129,133],[15,45445,59],{"className":45446},[195],[11,45448,45449],{},"其中",[15,45451,45453],{"className":45452,"translate":19},[18],[15,45454,45456,45498],{"className":45455,"translate":19},[23],[15,45457,45459],{"className":45458},[27],[29,45460,45461],{"xmlns":31,"display":32},[34,45462,45463,45495],{},[37,45464,45465,45471,45473,45475,45477,45479,45485,45487,45489],{},[40,45466,45467,45469],{},[43,45468,34146],{},[43,45470,1459],{},[50,45472,53],{"stretchy":52},[43,45474,56],{},[50,45476,59],{"stretchy":52},[50,45478,62],{},[40,45480,45481,45483],{},[43,45482,32000],{},[43,45484,1459],{},[43,45486,56],{},[50,45488,2682],{},[40,45490,45491,45493],{},[43,45492,8699],{},[43,45494,1459],{},[109,45496,45497],{"encoding":111},"l_i(x)=W_i x+b_i",[15,45499,45501,45565,45623],{"className":45500,"ariaHidden":72},[116],[15,45502,45504,45507,45547,45550,45553,45556,45559,45562],{"className":45503},[120],[15,45505],{"className":45506,"style":125},[124],[15,45508,45510,45513],{"className":45509},[129],[15,45511,34146],{"className":45512,"style":34204},[129,133],[15,45514,45516],{"className":45515},[137],[15,45517,45519,45539],{"className":45518},[141,142],[15,45520,45522,45536],{"className":45521},[146],[15,45523,45525],{"className":45524,"style":1981},[150],[15,45526,45527,45530],{"style":43401},[15,45528],{"className":45529,"style":159},[158],[15,45531,45533],{"className":45532},[163,164,165,166],[15,45534,1459],{"className":45535},[129,133,166],[15,45537,175],{"className":45538},[174],[15,45540,45542],{"className":45541},[146],[15,45543,45545],{"className":45544,"style":182},[150],[15,45546],{},[15,45548,53],{"className":45549},[188],[15,45551,56],{"className":45552},[129,133],[15,45554,59],{"className":45555},[195],[15,45557],{"className":45558,"style":199},[93],[15,45560,62],{"className":45561},[203],[15,45563],{"className":45564,"style":199},[93],[15,45566,45568,45571,45611,45614,45617,45620],{"className":45567},[120],[15,45569],{"className":45570,"style":20229},[124],[15,45572,45574,45577],{"className":45573},[129],[15,45575,32000],{"className":45576,"style":12449},[129,133],[15,45578,45580],{"className":45579},[137],[15,45581,45583,45603],{"className":45582},[141,142],[15,45584,45586,45600],{"className":45585},[146],[15,45587,45589],{"className":45588,"style":1981},[150],[15,45590,45591,45594],{"style":34239},[15,45592],{"className":45593,"style":159},[158],[15,45595,45597],{"className":45596},[163,164,165,166],[15,45598,1459],{"className":45599},[129,133,166],[15,45601,175],{"className":45602},[174],[15,45604,45606],{"className":45605},[146],[15,45607,45609],{"className":45608,"style":182},[150],[15,45610],{},[15,45612,56],{"className":45613},[129,133],[15,45615],{"className":45616,"style":3277},[93],[15,45618,2682],{"className":45619},[2436],[15,45621],{"className":45622,"style":3277},[93],[15,45624,45626,45629],{"className":45625},[120],[15,45627],{"className":45628,"style":2183},[124],[15,45630,45632,45635],{"className":45631},[129],[15,45633,8699],{"className":45634},[129,133],[15,45636,45638],{"className":45637},[137],[15,45639,45641,45661],{"className":45640},[141,142],[15,45642,45644,45658],{"className":45643},[146],[15,45645,45647],{"className":45646,"style":1981},[150],[15,45648,45649,45652],{"style":154},[15,45650],{"className":45651,"style":159},[158],[15,45653,45655],{"className":45654},[163,164,165,166],[15,45656,1459],{"className":45657},[129,133,166],[15,45659,175],{"className":45660},[174],[15,45662,45664],{"className":45663},[146],[15,45665,45667],{"className":45666,"style":182},[150],[15,45668],{},[11,45670,45671],{},"展开后即为：",[15,45673,45675],{"className":45674,"translate":19},[18],[15,45676,45678,45797],{"className":45677,"translate":19},[23],[15,45679,45681],{"className":45680},[27],[29,45682,45683],{"xmlns":31,"display":32},[34,45684,45685,45794],{},[37,45686,45687,45689,45691,45693,45695,45697,45703,45706,45712,45724,45726,45738,45740,45746,45748,45754,45756,45758,45764,45766,45768,45770,45782,45784,45786,45792],{},[43,45688,36569],{},[50,45690,53],{"stretchy":52},[43,45692,56],{},[50,45694,59],{"stretchy":52},[50,45696,62],{},[40,45698,45699,45701],{},[43,45700,27294],{},[43,45702,39326],{},[50,45704,53],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},"1.8em",[40,45707,45708,45710],{},[43,45709,32000],{},[43,45711,39326],{},[40,45713,45714,45716],{},[43,45715,27294],{},[37,45717,45718,45720,45722],{},[43,45719,39326],{},[50,45721,315],{},[308,45723,318],{},[50,45725,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,45727,45728,45730],{},[43,45729,32000],{},[37,45731,45732,45734,45736],{},[43,45733,39326],{},[50,45735,315],{},[308,45737,318],{},[50,45739,31670],{},[40,45741,45742,45744],{},[43,45743,27294],{},[308,45745,541],{},[50,45747,53],{"stretchy":52},[40,45749,45750,45752],{},[43,45751,32000],{},[308,45753,541],{},[43,45755,56],{},[50,45757,2682],{},[40,45759,45760,45762],{},[43,45761,8699],{},[308,45763,541],{},[50,45765,59],{"stretchy":52},[50,45767,31670],{},[50,45769,2682],{},[40,45771,45772,45774],{},[43,45773,8699],{},[37,45775,45776,45778,45780],{},[43,45777,39326],{},[50,45779,315],{},[308,45781,318],{},[50,45783,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,45785,2682],{},[40,45787,45788,45790],{},[43,45789,8699],{},[43,45791,39326],{},[50,45793,59],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[109,45795,45796],{"encoding":111},"f(x) = \\sigma_n \\Bigl( W_n \\sigma_{n-1} \\bigl( W_{n-1} \\cdots \\sigma_0(W_0x+b_0) \\cdots +b_{n-1} \\bigr) + b_n \\Bigr)",[15,45798,45800,45827,46128,46192,46262],{"className":45799,"ariaHidden":72},[116],[15,45801,45803,45806,45809,45812,45815,45818,45821,45824],{"className":45802},[120],[15,45804],{"className":45805,"style":125},[124],[15,45807,36569],{"className":45808,"style":36584},[129,133],[15,45810,53],{"className":45811},[188],[15,45813,56],{"className":45814},[129,133],[15,45816,59],{"className":45817},[195],[15,45819],{"className":45820,"style":199},[93],[15,45822,62],{"className":45823},[203],[15,45825],{"className":45826,"style":199},[93],[15,45828,45830,45834,45874,45880,45920,45969,45975,46024,46027,46030,46033,46073,46076,46116,46119,46122,46125],{"className":45829},[120],[15,45831],{"className":45832,"style":45833},[124],"height:1.8em;vertical-align:-0.65em;",[15,45835,45837,45840],{"className":45836},[129],[15,45838,27294],{"className":45839,"style":887},[129,133],[15,45841,45843],{"className":45842},[137],[15,45844,45846,45866],{"className":45845},[141,142],[15,45847,45849,45863],{"className":45848},[146],[15,45850,45852],{"className":45851,"style":1163},[150],[15,45853,45854,45857],{"style":1100},[15,45855],{"className":45856,"style":159},[158],[15,45858,45860],{"className":45859},[163,164,165,166],[15,45861,39326],{"className":45862},[129,133,166],[15,45864,175],{"className":45865},[174],[15,45867,45869],{"className":45868},[146],[15,45870,45872],{"className":45871,"style":182},[150],[15,45873],{},[15,45875,45877],{"className":45876},[188],[15,45878,53],{"className":45879},[228,24461],[15,45881,45883,45886],{"className":45882},[129],[15,45884,32000],{"className":45885,"style":12449},[129,133],[15,45887,45889],{"className":45888},[137],[15,45890,45892,45912],{"className":45891},[141,142],[15,45893,45895,45909],{"className":45894},[146],[15,45896,45898],{"className":45897,"style":1163},[150],[15,45899,45900,45903],{"style":34239},[15,45901],{"className":45902,"style":159},[158],[15,45904,45906],{"className":45905},[163,164,165,166],[15,45907,39326],{"className":45908},[129,133,166],[15,45910,175],{"className":45911},[174],[15,45913,45915],{"className":45914},[146],[15,45916,45918],{"className":45917,"style":182},[150],[15,45919],{},[15,45921,45923,45926],{"className":45922},[129],[15,45924,27294],{"className":45925,"style":887},[129,133],[15,45927,45929],{"className":45928},[137],[15,45930,45932,45961],{"className":45931},[141,142],[15,45933,45935,45958],{"className":45934},[146],[15,45936,45938],{"className":45937,"style":1051},[150],[15,45939,45940,45943],{"style":1100},[15,45941],{"className":45942,"style":159},[158],[15,45944,45946],{"className":45945},[163,164,165,166],[15,45947,45949,45952,45955],{"className":45948},[129,166],[15,45950,39326],{"className":45951},[129,133,166],[15,45953,315],{"className":45954},[2436,166],[15,45956,318],{"className":45957},[129,166],[15,45959,175],{"className":45960},[174],[15,45962,45964],{"className":45963},[146],[15,45965,45967],{"className":45966,"style":43079},[150],[15,45968],{},[15,45970,45972],{"className":45971},[188],[15,45973,53],{"className":45974},[228,229],[15,45976,45978,45981],{"className":45977},[129],[15,45979,32000],{"className":45980,"style":12449},[129,133],[15,45982,45984],{"className":45983},[137],[15,45985,45987,46016],{"className":45986},[141,142],[15,45988,45990,46013],{"className":45989},[146],[15,45991,45993],{"className":45992,"style":1051},[150],[15,45994,45995,45998],{"style":34239},[15,45996],{"className":45997,"style":159},[158],[15,45999,46001],{"className":46000},[163,164,165,166],[15,46002,46004,46007,46010],{"className":46003},[129,166],[15,46005,39326],{"className":46006},[129,133,166],[15,46008,315],{"className":46009},[2436,166],[15,46011,318],{"className":46012},[129,166],[15,46014,175],{"className":46015},[174],[15,46017,46019],{"className":46018},[146],[15,46020,46022],{"className":46021,"style":43079},[150],[15,46023],{},[15,46025],{"className":46026,"style":262},[93],[15,46028,31670],{"className":46029},[449],[15,46031],{"className":46032,"style":262},[93],[15,46034,46036,46039],{"className":46035},[129],[15,46037,27294],{"className":46038,"style":887},[129,133],[15,46040,46042],{"className":46041},[137],[15,46043,46045,46065],{"className":46044},[141,142],[15,46046,46048,46062],{"className":46047},[146],[15,46049,46051],{"className":46050,"style":1051},[150],[15,46052,46053,46056],{"style":1100},[15,46054],{"className":46055,"style":159},[158],[15,46057,46059],{"className":46058},[163,164,165,166],[15,46060,541],{"className":46061},[129,166],[15,46063,175],{"className":46064},[174],[15,46066,46068],{"className":46067},[146],[15,46069,46071],{"className":46070,"style":182},[150],[15,46072],{},[15,46074,53],{"className":46075},[188],[15,46077,46079,46082],{"className":46078},[129],[15,46080,32000],{"className":46081,"style":12449},[129,133],[15,46083,46085],{"className":46084},[137],[15,46086,46088,46108],{"className":46087},[141,142],[15,46089,46091,46105],{"className":46090},[146],[15,46092,46094],{"className":46093,"style":1051},[150],[15,46095,46096,46099],{"style":34239},[15,46097],{"className":46098,"style":159},[158],[15,46100,46102],{"className":46101},[163,164,165,166],[15,46103,541],{"className":46104},[129,166],[15,46106,175],{"className":46107},[174],[15,46109,46111],{"className":46110},[146],[15,46112,46114],{"className":46113,"style":182},[150],[15,46115],{},[15,46117,56],{"className":46118},[129,133],[15,46120],{"className":46121,"style":3277},[93],[15,46123,2682],{"className":46124},[2436],[15,46126],{"className":46127,"style":3277},[93],[15,46129,46131,46134,46174,46177,46180,46183,46186,46189],{"className":46130},[120],[15,46132],{"className":46133,"style":125},[124],[15,46135,46137,46140],{"className":46136},[129],[15,46138,8699],{"className":46139},[129,133],[15,46141,46143],{"className":46142},[137],[15,46144,46146,46166],{"className":46145},[141,142],[15,46147,46149,46163],{"className":46148},[146],[15,46150,46152],{"className":46151,"style":1051},[150],[15,46153,46154,46157],{"style":154},[15,46155],{"className":46156,"style":159},[158],[15,46158,46160],{"className":46159},[163,164,165,166],[15,46161,541],{"className":46162},[129,166],[15,46164,175],{"className":46165},[174],[15,46167,46169],{"className":46168},[146],[15,46170,46172],{"className":46171,"style":182},[150],[15,46173],{},[15,46175,59],{"className":46176},[195],[15,46178],{"className":46179,"style":262},[93],[15,46181,31670],{"className":46182},[449],[15,46184],{"className":46185,"style":3277},[93],[15,46187,2682],{"className":46188},[2436],[15,46190],{"className":46191,"style":3277},[93],[15,46193,46195,46198,46247,46253,46256,46259],{"className":46194},[120],[15,46196],{"className":46197,"style":213},[124],[15,46199,46201,46204],{"className":46200},[129],[15,46202,8699],{"className":46203},[129,133],[15,46205,46207],{"className":46206},[137],[15,46208,46210,46239],{"className":46209},[141,142],[15,46211,46213,46236],{"className":46212},[146],[15,46214,46216],{"className":46215,"style":1051},[150],[15,46217,46218,46221],{"style":154},[15,46219],{"className":46220,"style":159},[158],[15,46222,46224],{"className":46223},[163,164,165,166],[15,46225,46227,46230,46233],{"className":46226},[129,166],[15,46228,39326],{"className":46229},[129,133,166],[15,46231,315],{"className":46232},[2436,166],[15,46234,318],{"className":46235},[129,166],[15,46237,175],{"className":46238},[174],[15,46240,46242],{"className":46241},[146],[15,46243,46245],{"className":46244,"style":43079},[150],[15,46246],{},[15,46248,46250],{"className":46249},[195],[15,46251,59],{"className":46252},[228,229],[15,46254],{"className":46255,"style":3277},[93],[15,46257,2682],{"className":46258},[2436],[15,46260],{"className":46261,"style":3277},[93],[15,46263,46265,46268,46308],{"className":46264},[120],[15,46266],{"className":46267,"style":45833},[124],[15,46269,46271,46274],{"className":46270},[129],[15,46272,8699],{"className":46273},[129,133],[15,46275,46277],{"className":46276},[137],[15,46278,46280,46300],{"className":46279},[141,142],[15,46281,46283,46297],{"className":46282},[146],[15,46284,46286],{"className":46285,"style":1163},[150],[15,46287,46288,46291],{"style":154},[15,46289],{"className":46290,"style":159},[158],[15,46292,46294],{"className":46293},[163,164,165,166],[15,46295,39326],{"className":46296},[129,133,166],[15,46298,175],{"className":46299},[174],[15,46301,46303],{"className":46302},[146],[15,46304,46306],{"className":46305,"style":182},[150],[15,46307],{},[15,46309,46311],{"className":46310},[195],[15,46312,59],{"className":46313},[228,24461],[11,46315,46316],{},"这就是神经网络的标准形式。",[11,46318,46319],{},"在加入激活函数后：",[15,46321,46323],{"className":46322,"translate":19},[18],[15,46324,46326,46368],{"className":46325,"translate":19},[23],[15,46327,46329],{"className":46328},[27],[29,46330,46331],{"xmlns":31,"display":32},[34,46332,46333,46365],{},[37,46334,46335,46337,46339,46341,46347,46349,46351,46353,46355,46357,46363],{},[50,46336,53],{"stretchy":52},[43,46338,27294],{},[50,46340,42939],{},[40,46342,46343,46345],{},[43,46344,34146],{},[308,46346,310],{},[50,46348,59],{"stretchy":52},[50,46350,42939],{},[50,46352,53],{"stretchy":52},[43,46354,27294],{},[50,46356,42939],{},[40,46358,46359,46361],{},[43,46360,34146],{},[308,46362,318],{},[50,46364,59],{"stretchy":52},[109,46366,46367],{"encoding":111},"(\\sigma \\circ l_2) \\circ (\\sigma \\circ l_1)",[15,46369,46371,46392,46450,46471],{"className":46370,"ariaHidden":72},[116],[15,46372,46374,46377,46380,46383,46386,46389],{"className":46373},[120],[15,46375],{"className":46376,"style":125},[124],[15,46378,53],{"className":46379},[188],[15,46381,27294],{"className":46382,"style":887},[129,133],[15,46384],{"className":46385,"style":3277},[93],[15,46387,42939],{"className":46388},[2436],[15,46390],{"className":46391,"style":3277},[93],[15,46393,46395,46398,46438,46441,46444,46447],{"className":46394},[120],[15,46396],{"className":46397,"style":125},[124],[15,46399,46401,46404],{"className":46400},[129],[15,46402,34146],{"className":46403,"style":34204},[129,133],[15,46405,46407],{"className":46406},[137],[15,46408,46410,46430],{"className":46409},[141,142],[15,46411,46413,46427],{"className":46412},[146],[15,46414,46416],{"className":46415,"style":1051},[150],[15,46417,46418,46421],{"style":43401},[15,46419],{"className":46420,"style":159},[158],[15,46422,46424],{"className":46423},[163,164,165,166],[15,46425,310],{"className":46426},[129,166],[15,46428,175],{"className":46429},[174],[15,46431,46433],{"className":46432},[146],[15,46434,46436],{"className":46435,"style":182},[150],[15,46437],{},[15,46439,59],{"className":46440},[195],[15,46442],{"className":46443,"style":3277},[93],[15,46445,42939],{"className":46446},[2436],[15,46448],{"className":46449,"style":3277},[93],[15,46451,46453,46456,46459,46462,46465,46468],{"className":46452},[120],[15,46454],{"className":46455,"style":125},[124],[15,46457,53],{"className":46458},[188],[15,46460,27294],{"className":46461,"style":887},[129,133],[15,46463],{"className":46464,"style":3277},[93],[15,46466,42939],{"className":46467},[2436],[15,46469],{"className":46470,"style":3277},[93],[15,46472,46474,46477,46517],{"className":46473},[120],[15,46475],{"className":46476,"style":125},[124],[15,46478,46480,46483],{"className":46479},[129],[15,46481,34146],{"className":46482,"style":34204},[129,133],[15,46484,46486],{"className":46485},[137],[15,46487,46489,46509],{"className":46488},[141,142],[15,46490,46492,46506],{"className":46491},[146],[15,46493,46495],{"className":46494,"style":1051},[150],[15,46496,46497,46500],{"style":43401},[15,46498],{"className":46499,"style":159},[158],[15,46501,46503],{"className":46502},[163,164,165,166],[15,46504,318],{"className":46505},[129,166],[15,46507,175],{"className":46508},[174],[15,46510,46512],{"className":46511},[146],[15,46513,46515],{"className":46514,"style":182},[150],[15,46516],{},[15,46518,59],{"className":46519},[195],[11,46521,46522],{},"无法化简为",[15,46524,46526],{"className":46525,"translate":19},[18],[15,46527,46529,46549],{"className":46528,"translate":19},[23],[15,46530,46532],{"className":46531},[27],[29,46533,46534],{"xmlns":31,"display":32},[34,46535,46536,46546],{},[37,46537,46538,46540,46542,46544],{},[43,46539,32000],{},[43,46541,56],{},[50,46543,2682],{},[43,46545,8699],{},[109,46547,46548],{"encoding":111},"Wx+b",[15,46550,46552,46573],{"className":46551,"ariaHidden":72},[116],[15,46553,46555,46558,46561,46564,46567,46570],{"className":46554},[120],[15,46556],{"className":46557,"style":39511},[124],[15,46559,32000],{"className":46560,"style":12449},[129,133],[15,46562,56],{"className":46563},[129,133],[15,46565],{"className":46566,"style":3277},[93],[15,46568,2682],{"className":46569},[2436],[15,46571],{"className":46572,"style":3277},[93],[15,46574,46576,46579],{"className":46575},[120],[15,46577],{"className":46578,"style":522},[124],[15,46580,8699],{"className":46581},[129,133],[11,46583,46584],{},"因为一般来说：",[15,46586,46588],{"className":46587,"translate":19},[18],[15,46589,46591,46641],{"className":46590,"translate":19},[23],[15,46592,46594],{"className":46593},[27],[29,46595,46596],{"xmlns":31,"display":32},[34,46597,46598,46638],{},[37,46599,46600,46602,46604,46606,46608,46610,46612,46614,46616,46622,46624,46626,46628,46630,46632],{},[43,46601,27294],{},[50,46603,53],{"stretchy":52},[43,46605,13097],{},[43,46607,56],{},[50,46609,2682],{},[43,46611,8699],{},[50,46613,59],{"stretchy":52},[50,46615,1418],{"mathvariant":65},[305,46617,46618,46620],{},[43,46619,13097],{},[50,46621,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[43,46623,27294],{},[50,46625,53],{"stretchy":52},[43,46627,56],{},[50,46629,59],{"stretchy":52},[50,46631,2682],{},[305,46633,46634,46636],{},[43,46635,8699],{},[50,46637,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[109,46639,46640],{"encoding":111},"\\sigma(Ax+b) \\neq A'\\sigma(x)+b'",[15,46642,46644,46671,46725,46785],{"className":46643,"ariaHidden":72},[116],[15,46645,46647,46650,46653,46656,46659,46662,46665,46668],{"className":46646},[120],[15,46648],{"className":46649,"style":125},[124],[15,46651,27294],{"className":46652,"style":887},[129,133],[15,46654,53],{"className":46655},[188],[15,46657,13097],{"className":46658},[129,133],[15,46660,56],{"className":46661},[129,133],[15,46663],{"className":46664,"style":3277},[93],[15,46666,2682],{"className":46667},[2436],[15,46669],{"className":46670,"style":3277},[93],[15,46672,46674,46677,46680,46683,46686,46722],{"className":46673},[120],[15,46675],{"className":46676,"style":125},[124],[15,46678,8699],{"className":46679},[129,133],[15,46681,59],{"className":46682},[195],[15,46684],{"className":46685,"style":199},[93],[15,46687,46689,46716,46719],{"className":46688},[203],[15,46690,46692],{"className":46691},[203],[15,46693,46695],{"className":46694},[129,1670],[15,46696,46698],{"className":46697},[1674],[15,46699,46701,46704,46713],{"className":46700},[1678],[15,46702],{"className":46703,"style":1682},[124],[15,46705,46707],{"className":46706},[1686],[15,46708,46710],{"className":46709},[129],[15,46711,1693],{"className":46712},[203],[15,46714],{"className":46715},[1697],[15,46717],{"className":46718},[93,1701],[15,46720,62],{"className":46721},[203],[15,46723],{"className":46724,"style":199},[93],[15,46726,46728,46732,46764,46767,46770,46773,46776,46779,46782],{"className":46727},[120],[15,46729],{"className":46730,"style":46731},[124],"height:1.0519em;vertical-align:-0.25em;",[15,46733,46735,46738],{"className":46734},[129],[15,46736,13097],{"className":46737},[129,133],[15,46739,46741],{"className":46740},[137],[15,46742,46744],{"className":46743},[141],[15,46745,46747],{"className":46746},[146],[15,46748,46750],{"className":46749,"style":28137},[150],[15,46751,46752,46755],{"style":1272},[15,46753],{"className":46754,"style":159},[158],[15,46756,46758],{"className":46757},[163,164,165,166],[15,46759,46761],{"className":46760},[129,166],[15,46762,21510],{"className":46763},[129,166],[15,46765,27294],{"className":46766,"style":887},[129,133],[15,46768,53],{"className":46769},[188],[15,46771,56],{"className":46772},[129,133],[15,46774,59],{"className":46775},[195],[15,46777],{"className":46778,"style":3277},[93],[15,46780,2682],{"className":46781},[2436],[15,46783],{"className":46784,"style":3277},[93],[15,46786,46788,46791],{"className":46787},[120],[15,46789],{"className":46790,"style":28137},[124],[15,46792,46794,46797],{"className":46793},[129],[15,46795,8699],{"className":46796},[129,133],[15,46798,46800],{"className":46799},[137],[15,46801,46803],{"className":46802},[141],[15,46804,46806],{"className":46805},[146],[15,46807,46809],{"className":46808,"style":28137},[150],[15,46810,46811,46814],{"style":1272},[15,46812],{"className":46813,"style":159},[158],[15,46815,46817],{"className":46816},[163,164,165,166],[15,46818,46820],{"className":46819},[129,166],[15,46821,21510],{"className":46822},[129,166],[11,46824,46825],{},"激活函数破坏了仿射映射在复合下的封闭性。换言之，它打破了可加性与齐次性这两个关键特性。这本质上就是神经网络非线性表达能力的来源。",[11,46827,46828],{},"从前面讨论的流形视角来看，这可以进一步写成：",[15,46830,46832],{"className":46831,"translate":19},[18],[15,46833,46835,46934],{"className":46834,"translate":19},[23],[15,46836,46838],{"className":46837},[27],[29,46839,46840],{"xmlns":31,"display":32},[34,46841,46842,46931],{},[37,46843,46844,46846,46863,46869,46881,46889,46901,46907,46919,46927,46929],{},[43,46845,43246],{},[46847,46848,46849,46852],"mover",{},[50,46850,723],{"stretchy":72,"minsize":46851},"3.0em",[46853,46854,46857],"mpadded",{"width":46855,"lspace":46856},"+0.6em","0.3em",[40,46858,46859,46861],{},[43,46860,34146],{},[308,46862,541],{},[40,46864,46865,46867],{},[43,46866,43246],{},[308,46868,318],{},[46847,46870,46871,46873],{},[50,46872,723],{"stretchy":72,"minsize":46851},[46853,46874,46875],{"width":46855,"lspace":46856},[40,46876,46877,46879],{},[43,46878,27294],{},[308,46880,541],{},[23160,46882,46883,46885,46887],{},[43,46884,43246],{},[308,46886,318],{},[50,46888,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[46847,46890,46891,46893],{},[50,46892,723],{"stretchy":72,"minsize":46851},[46853,46894,46895],{"width":46855,"lspace":46856},[40,46896,46897,46899],{},[43,46898,34146],{},[308,46900,318],{},[40,46902,46903,46905],{},[43,46904,43246],{},[308,46906,310],{},[46847,46908,46909,46911],{},[50,46910,723],{"stretchy":72,"minsize":46851},[46853,46912,46913],{"width":46855,"lspace":46856},[40,46914,46915,46917],{},[43,46916,27294],{},[308,46918,318],{},[23160,46920,46921,46923,46925],{},[43,46922,43246],{},[308,46924,310],{},[50,46926,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[50,46928,723],{},[50,46930,31670],{},[109,46932,46933],{"encoding":111},"M \\xrightarrow{l_0} M_1 \\xrightarrow{\\sigma_0} M_1' \\xrightarrow{l_1} M_2 \\xrightarrow{\\sigma_1} M_2' \\rightarrow \\cdots",[15,46935,46937,47053,47195,47350,47489,47559],{"className":46936,"ariaHidden":72},[116],[15,46938,46940,46944,46947,46950,47050],{"className":46939},[120],[15,46941],{"className":46942,"style":46943},[124],"height:1.1191em;vertical-align:-0.011em;",[15,46945,43246],{"className":46946,"style":12597},[129,133],[15,46948],{"className":46949,"style":199},[93],[15,46951,46954],{"className":46952},[203,46953],"x-arrow",[15,46955,46957,47041],{"className":46956},[141,142],[15,46958,46960,47038],{"className":46959},[146],[15,46961,46964,47019],{"className":46962,"style":46963},[150],"height:1.1081em;",[15,46965,46967,46970],{"style":46966},"top:-3.322em;",[15,46968],{"className":46969,"style":159},[158],[15,46971,46974],{"className":46972},[163,164,165,166,46973],"x-arrow-pad",[15,46975,46977],{"className":46976},[129,166],[15,46978,46980,46983],{"className":46979},[129,166],[15,46981,34146],{"className":46982,"style":34204},[129,133,166],[15,46984,46986],{"className":46985},[137],[15,46987,46989,47011],{"className":46988},[141,142],[15,46990,46992,47008],{"className":46991},[146],[15,46993,46996],{"className":46994,"style":46995},[150],"height:0.3173em;",[15,46997,46999,47002],{"style":46998},"top:-2.357em;margin-left:-0.0197em;margin-right:0.0714em;",[15,47000],{"className":47001,"style":15987},[158],[15,47003,47005],{"className":47004},[163,15991,229,166],[15,47006,541],{"className":47007},[129,166],[15,47009,175],{"className":47010},[174],[15,47012,47014],{"className":47013},[146],[15,47015,47017],{"className":47016,"style":16004},[150],[15,47018],{},[15,47020,47023,47026],{"className":47021,"style":47022},[29137],"top:-2.689em;",[15,47024],{"className":47025,"style":159},[158],[15,47027,47030],{"className":47028,"style":47029},[29282],"height:0.522em;min-width:1.469em;",[4455,47031,47035],{"xmlns":4457,"width":29286,"height":47032,"viewBox":47033,"preserveAspectRatio":47034},"0.522em","0 0 400000 522","xMaxYMin slice",[4461,47036],{"d":47037},"M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",[15,47039,175],{"className":47040},[174],[15,47042,47044],{"className":47043},[146],[15,47045,47048],{"className":47046,"style":47047},[150],"height:0.011em;",[15,47049],{},[15,47051],{"className":47052,"style":199},[93],[15,47054,47056,47060,47101,47104,47192],{"className":47055},[120],[15,47057],{"className":47058,"style":47059},[124],"height:1.0734em;vertical-align:-0.15em;",[15,47061,47063,47066],{"className":47062},[129],[15,47064,43246],{"className":47065,"style":12597},[129,133],[15,47067,47069],{"className":47068},[137],[15,47070,47072,47093],{"className":47071},[141,142],[15,47073,47075,47090],{"className":47074},[146],[15,47076,47078],{"className":47077,"style":1051},[150],[15,47079,47081,47084],{"style":47080},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[15,47082],{"className":47083,"style":159},[158],[15,47085,47087],{"className":47086},[163,164,165,166],[15,47088,318],{"className":47089},[129,166],[15,47091,175],{"className":47092},[174],[15,47094,47096],{"className":47095},[146],[15,47097,47099],{"className":47098,"style":182},[150],[15,47100],{},[15,47102],{"className":47103,"style":199},[93],[15,47105,47107],{"className":47106},[203,46953],[15,47108,47110,47184],{"className":47109},[141,142],[15,47111,47113,47181],{"className":47112},[146],[15,47114,47117,47168],{"className":47115,"style":47116},[150],"height:0.9234em;",[15,47118,47119,47122],{"style":46966},[15,47120],{"className":47121,"style":159},[158],[15,47123,47125],{"className":47124},[163,164,165,166,46973],[15,47126,47128],{"className":47127},[129,166],[15,47129,47131,47134],{"className":47130},[129,166],[15,47132,27294],{"className":47133,"style":887},[129,133,166],[15,47135,47137],{"className":47136},[137],[15,47138,47140,47160],{"className":47139},[141,142],[15,47141,47143,47157],{"className":47142},[146],[15,47144,47146],{"className":47145,"style":46995},[150],[15,47147,47148,47151],{"style":15983},[15,47149],{"className":47150,"style":15987},[158],[15,47152,47154],{"className":47153},[163,15991,229,166],[15,47155,541],{"className":47156},[129,166],[15,47158,175],{"className":47159},[174],[15,47161,47163],{"className":47162},[146],[15,47164,47166],{"className":47165,"style":16004},[150],[15,47167],{},[15,47169,47171,47174],{"className":47170,"style":47022},[29137],[15,47172],{"className":47173,"style":159},[158],[15,47175,47177],{"className":47176,"style":47029},[29282],[4455,47178,47179],{"xmlns":4457,"width":29286,"height":47032,"viewBox":47033,"preserveAspectRatio":47034},[4461,47180],{"d":47037},[15,47182,175],{"className":47183},[174],[15,47185,47187],{"className":47186},[146],[15,47188,47190],{"className":47189,"style":47047},[150],[15,47191],{},[15,47193],{"className":47194,"style":199},[93],[15,47196,47198,47202,47257,47260,47347],{"className":47197},[120],[15,47199],{"className":47200,"style":47201},[124],"height:1.3551em;vertical-align:-0.247em;",[15,47203,47205,47208],{"className":47204},[129],[15,47206,43246],{"className":47207,"style":12597},[129,133],[15,47209,47211],{"className":47210},[137],[15,47212,47214,47249],{"className":47213},[141,142],[15,47215,47217,47246],{"className":47216},[146],[15,47218,47220,47232],{"className":47219,"style":28137},[150],[15,47221,47223,47226],{"style":47222},"top:-2.453em;margin-left:-0.109em;margin-right:0.05em;",[15,47224],{"className":47225,"style":159},[158],[15,47227,47229],{"className":47228},[163,164,165,166],[15,47230,318],{"className":47231},[129,166],[15,47233,47234,47237],{"style":1272},[15,47235],{"className":47236,"style":159},[158],[15,47238,47240],{"className":47239},[163,164,165,166],[15,47241,47243],{"className":47242},[129,166],[15,47244,21510],{"className":47245},[129,166],[15,47247,175],{"className":47248},[174],[15,47250,47252],{"className":47251},[146],[15,47253,47255],{"className":47254,"style":28173},[150],[15,47256],{},[15,47258],{"className":47259,"style":199},[93],[15,47261,47263],{"className":47262},[203,46953],[15,47264,47266,47339],{"className":47265},[141,142],[15,47267,47269,47336],{"className":47268},[146],[15,47270,47272,47323],{"className":47271,"style":46963},[150],[15,47273,47274,47277],{"style":46966},[15,47275],{"className":47276,"style":159},[158],[15,47278,47280],{"className":47279},[163,164,165,166,46973],[15,47281,47283],{"className":47282},[129,166],[15,47284,47286,47289],{"className":47285},[129,166],[15,47287,34146],{"className":47288,"style":34204},[129,133,166],[15,47290,47292],{"className":47291},[137],[15,47293,47295,47315],{"className":47294},[141,142],[15,47296,47298,47312],{"className":47297},[146],[15,47299,47301],{"className":47300,"style":46995},[150],[15,47302,47303,47306],{"style":46998},[15,47304],{"className":47305,"style":15987},[158],[15,47307,47309],{"className":47308},[163,15991,229,166],[15,47310,318],{"className":47311},[129,166],[15,47313,175],{"className":47314},[174],[15,47316,47318],{"className":47317},[146],[15,47319,47321],{"className":47320,"style":16004},[150],[15,47322],{},[15,47324,47326,47329],{"className":47325,"style":47022},[29137],[15,47327],{"className":47328,"style":159},[158],[15,47330,47332],{"className":47331,"style":47029},[29282],[4455,47333,47334],{"xmlns":4457,"width":29286,"height":47032,"viewBox":47033,"preserveAspectRatio":47034},[4461,47335],{"d":47037},[15,47337,175],{"className":47338},[174],[15,47340,47342],{"className":47341},[146],[15,47343,47345],{"className":47344,"style":47047},[150],[15,47346],{},[15,47348],{"className":47349,"style":199},[93],[15,47351,47353,47356,47396,47399,47486],{"className":47352},[120],[15,47354],{"className":47355,"style":47059},[124],[15,47357,47359,47362],{"className":47358},[129],[15,47360,43246],{"className":47361,"style":12597},[129,133],[15,47363,47365],{"className":47364},[137],[15,47366,47368,47388],{"className":47367},[141,142],[15,47369,47371,47385],{"className":47370},[146],[15,47372,47374],{"className":47373,"style":1051},[150],[15,47375,47376,47379],{"style":47080},[15,47377],{"className":47378,"style":159},[158],[15,47380,47382],{"className":47381},[163,164,165,166],[15,47383,310],{"className":47384},[129,166],[15,47386,175],{"className":47387},[174],[15,47389,47391],{"className":47390},[146],[15,47392,47394],{"className":47393,"style":182},[150],[15,47395],{},[15,47397],{"className":47398,"style":199},[93],[15,47400,47402],{"className":47401},[203,46953],[15,47403,47405,47478],{"className":47404},[141,142],[15,47406,47408,47475],{"className":47407},[146],[15,47409,47411,47462],{"className":47410,"style":47116},[150],[15,47412,47413,47416],{"style":46966},[15,47414],{"className":47415,"style":159},[158],[15,47417,47419],{"className":47418},[163,164,165,166,46973],[15,47420,47422],{"className":47421},[129,166],[15,47423,47425,47428],{"className":47424},[129,166],[15,47426,27294],{"className":47427,"style":887},[129,133,166],[15,47429,47431],{"className":47430},[137],[15,47432,47434,47454],{"className":47433},[141,142],[15,47435,47437,47451],{"className":47436},[146],[15,47438,47440],{"className":47439,"style":46995},[150],[15,47441,47442,47445],{"style":15983},[15,47443],{"className":47444,"style":15987},[158],[15,47446,47448],{"className":47447},[163,15991,229,166],[15,47449,318],{"className":47450},[129,166],[15,47452,175],{"className":47453},[174],[15,47455,47457],{"className":47456},[146],[15,47458,47460],{"className":47459,"style":16004},[150],[15,47461],{},[15,47463,47465,47468],{"className":47464,"style":47022},[29137],[15,47466],{"className":47467,"style":159},[158],[15,47469,47471],{"className":47470,"style":47029},[29282],[4455,47472,47473],{"xmlns":4457,"width":29286,"height":47032,"viewBox":47033,"preserveAspectRatio":47034},[4461,47474],{"d":47037},[15,47476,175],{"className":47477},[174],[15,47479,47481],{"className":47480},[146],[15,47482,47484],{"className":47483,"style":47047},[150],[15,47485],{},[15,47487],{"className":47488,"style":199},[93],[15,47490,47492,47496,47550,47553,47556],{"className":47491},[120],[15,47493],{"className":47494,"style":47495},[124],"height:1.0489em;vertical-align:-0.247em;",[15,47497,47499,47502],{"className":47498},[129],[15,47500,43246],{"className":47501,"style":12597},[129,133],[15,47503,47505],{"className":47504},[137],[15,47506,47508,47542],{"className":47507},[141,142],[15,47509,47511,47539],{"className":47510},[146],[15,47512,47514,47525],{"className":47513,"style":28137},[150],[15,47515,47516,47519],{"style":47222},[15,47517],{"className":47518,"style":159},[158],[15,47520,47522],{"className":47521},[163,164,165,166],[15,47523,310],{"className":47524},[129,166],[15,47526,47527,47530],{"style":1272},[15,47528],{"className":47529,"style":159},[158],[15,47531,47533],{"className":47532},[163,164,165,166],[15,47534,47536],{"className":47535},[129,166],[15,47537,21510],{"className":47538},[129,166],[15,47540,175],{"className":47541},[174],[15,47543,47545],{"className":47544},[146],[15,47546,47548],{"className":47547,"style":28173},[150],[15,47549],{},[15,47551],{"className":47552,"style":199},[93],[15,47554,723],{"className":47555},[203],[15,47557],{"className":47558,"style":199},[93],[15,47560,47562,47565],{"className":47561},[120],[15,47563],{"className":47564,"style":31796},[124],[15,47566,31670],{"className":47567},[449],[11,47569,47570],{},"其中：仿射层负责拉伸、旋转、平移、剪切；激活层负责对流形进行折叠、裁剪、压缩与重构。",[11,47572,47573,47574,47644],{},"如果所有 ",[15,47575,47577,47595],{"className":47576,"translate":19},[23],[15,47578,47580],{"className":47579},[27],[29,47581,47582],{"xmlns":31},[34,47583,47584,47592],{},[37,47585,47586],{},[40,47587,47588,47590],{},[43,47589,32000],{},[43,47591,1459],{},[109,47593,47594],{"encoding":111},"W_i",[15,47596,47598],{"className":47597,"ariaHidden":72},[116],[15,47599,47601,47604],{"className":47600},[120],[15,47602],{"className":47603,"style":20229},[124],[15,47605,47607,47610],{"className":47606},[129],[15,47608,32000],{"className":47609,"style":12449},[129,133],[15,47611,47613],{"className":47612},[137],[15,47614,47616,47636],{"className":47615},[141,142],[15,47617,47619,47633],{"className":47618},[146],[15,47620,47622],{"className":47621,"style":1981},[150],[15,47623,47624,47627],{"style":34239},[15,47625],{"className":47626,"style":159},[158],[15,47628,47630],{"className":47629},[163,164,165,166],[15,47631,1459],{"className":47632},[129,133,166],[15,47634,175],{"className":47635},[174],[15,47637,47639],{"className":47638},[146],[15,47640,47642],{"className":47641,"style":182},[150],[15,47643],{}," 都可逆，那么仿射层本身只是同胚变换，不会改变流形的拓扑。真正让流形逐步变得可分的，往往正是激活函数引入的非线性形变。",[11,47646,47647],{},"因此，从几何角度看，神经网络可以理解为作用在数据流形上的仿射变换与非线性流形操作的交替组合。这一视角与许多现代流形学习和几何深度学习的解释相一致。",[570,47649,47650],{"id":47650},"激活函数",[11,47652,47653],{},"激活函数并不存在「唯一最佳」的选择。不过，从神经网络的可训练性与表达能力来看，一个好的激活函数通常应满足几个核心特征：",[575,47655,47656],{"id":47656},"非线性",[11,47658,47659],{},"如前所述，需要非线性来打破堆叠仿射层的线性封闭性，使多层网络能够逼近任意函数。",[11,47661,47662],{},"数学要求：不能满足线性组合的封闭性：",[15,47664,47666],{"className":47665,"translate":19},[18],[15,47667,47669,47721],{"className":47668,"translate":19},[23],[15,47670,47672],{"className":47671},[27],[29,47673,47674],{"xmlns":31,"display":32},[34,47675,47676,47718],{},[37,47677,47678,47680,47682,47684,47686,47688,47690,47692,47694,47696,47698,47700,47702,47704,47706,47708,47710,47712,47714,47716],{},[43,47679,27294],{},[50,47681,53],{"stretchy":52},[43,47683,8690],{},[43,47685,56],{},[50,47687,2682],{},[43,47689,8699],{},[43,47691,857],{},[50,47693,59],{"stretchy":52},[50,47695,1418],{"mathvariant":65},[43,47697,8690],{},[43,47699,27294],{},[50,47701,53],{"stretchy":52},[43,47703,56],{},[50,47705,59],{"stretchy":52},[50,47707,2682],{},[43,47709,8699],{},[43,47711,27294],{},[50,47713,53],{"stretchy":52},[43,47715,857],{},[50,47717,59],{"stretchy":52},[109,47719,47720],{"encoding":111},"\\sigma(a x + b y) \\neq a \\sigma(x) + b \\sigma(y)",[15,47722,47724,47751,47808,47836],{"className":47723,"ariaHidden":72},[116],[15,47725,47727,47730,47733,47736,47739,47742,47745,47748],{"className":47726},[120],[15,47728],{"className":47729,"style":125},[124],[15,47731,27294],{"className":47732,"style":887},[129,133],[15,47734,53],{"className":47735},[188],[15,47737,8690],{"className":47738},[129,133],[15,47740,56],{"className":47741},[129,133],[15,47743],{"className":47744,"style":3277},[93],[15,47746,2682],{"className":47747},[2436],[15,47749],{"className":47750,"style":3277},[93],[15,47752,47754,47757,47760,47763,47766,47769,47805],{"className":47753},[120],[15,47755],{"className":47756,"style":125},[124],[15,47758,8699],{"className":47759},[129,133],[15,47761,857],{"className":47762,"style":887},[129,133],[15,47764,59],{"className":47765},[195],[15,47767],{"className":47768,"style":199},[93],[15,47770,47772,47799,47802],{"className":47771},[203],[15,47773,47775],{"className":47774},[203],[15,47776,47778],{"className":47777},[129,1670],[15,47779,47781],{"className":47780},[1674],[15,47782,47784,47787,47796],{"className":47783},[1678],[15,47785],{"className":47786,"style":1682},[124],[15,47788,47790],{"className":47789},[1686],[15,47791,47793],{"className":47792},[129],[15,47794,1693],{"className":47795},[203],[15,47797],{"className":47798},[1697],[15,47800],{"className":47801},[93,1701],[15,47803,62],{"className":47804},[203],[15,47806],{"className":47807,"style":199},[93],[15,47809,47811,47814,47818,47821,47824,47827,47830,47833],{"className":47810},[120],[15,47812],{"className":47813,"style":125},[124],[15,47815,47817],{"className":47816,"style":887},[129,133],"aσ",[15,47819,53],{"className":47820},[188],[15,47822,56],{"className":47823},[129,133],[15,47825,59],{"className":47826},[195],[15,47828],{"className":47829,"style":3277},[93],[15,47831,2682],{"className":47832},[2436],[15,47834],{"className":47835,"style":3277},[93],[15,47837,47839,47842,47846,47849,47852],{"className":47838},[120],[15,47840],{"className":47841,"style":125},[124],[15,47843,47845],{"className":47844,"style":887},[129,133],"bσ",[15,47847,53],{"className":47848},[188],[15,47850,857],{"className":47851,"style":887},[129,133],[15,47853,59],{"className":47854},[195],[11,47856,47857],{},"例如：ReLU、Tanh、Sigmoid、GELU。",[11,47859,47860],{},"线性函数并不适用，因为多个线性层的复合仍然是线性的，这会退化回仿射复合，从而限制表达能力。",[575,47862,47863],{"id":47863},"可微性",[11,47865,47866],{},"由于神经网络的核心依赖反向传播来确定权重，激活函数必须在其定义域的大部分区域可微，以便在反向传播过程中计算梯度。对于不可微的点（例如 ReLU 在 0 处），通常使用次梯度（subgradient）。此外，导数的性质会影响梯度流动：梯度过小会导致梯度消失（如 Sigmoid），梯度过大则会导致梯度爆炸。",[575,47868,47869],{"id":47869},"连续性",[11,47871,47872],{},"确保网络输出随输入连续变化，否则训练会变得不稳定。存在少数可控的不连续点通常是可以接受的（例如 ReLU 只有一个不连续点，实际影响很小）。",[11,47874,47875],{},"在上述性质中，满足条件只是「入场券」，而不是「成功的保证」。",{"title":36432,"searchDepth":36433,"depth":36433,"links":47877},[47878,47882,47883,47884,47885],{"id":36487,"depth":36433,"text":36487,"children":47879},[47880,47881],{"id":36587,"depth":36438,"text":36587},{"id":37532,"depth":36438,"text":37532},{"id":39300,"depth":36433,"text":39300},{"id":41465,"depth":36433,"text":41465},{"id":43321,"depth":36433,"text":43321},{"id":47650,"depth":36433,"text":47650,"children":47886},[47887,47888,47889],{"id":47656,"depth":36438,"text":47656},{"id":47863,"depth":36438,"text":47863},{"id":47869,"depth":36438,"text":47869},{},"\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations",{"title":36482,"description":36432},"blog\u002F2026\u002F2026-04-06-nonlinearity-with-activations","线性要求满足可加性与齐次性，但神经网络需要非线性才能摆脱简单的仿射变换。仿射层只能对数据进行拉伸和旋转，而不改变其拓扑结构；激活函数则能实现复杂的形变，使网络能够解开数据流形，有效地学习复杂模式。",[36476,36477],"2026-04-06T00:00:00+08:00","QD4VuoWBrJvdynBTTdzfxaPKlkRa5oc7977AIYkpnDg",{"id":47899,"title":47900,"body":47901,"description":47905,"draft":36466,"enableComment":36467,"extension":36468,"image":36432,"important":36466,"location":36469,"meta":93102,"navigation":36467,"ogImage":36469,"path":93103,"seo":93104,"stem":93105,"summary":93106,"tags":93107,"time":93108,"__hash__":93109},"blog\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa.md","Fano 不等式与 Le Cam 方法",{"type":8,"value":47902,"toc":93070},[47903,47906,48455,48458,48462,48684,48687,49072,49079,49085,49088,49092,49096,49310,49611,49965,50692,50787,51173,51238,51620,51624,51685,52023,52026,52210,52217,52221,52459,52697,52858,52861,52865,52869,52872,54062,54065,54269,54487,54951,54954,55614,55617,56535,56620,56737,57579,57785,58036,58771,58774,59404,60181,60185,60188,60526,60529,60966,60969,61713,62017,62055,62059,62284,62956,63287,63869,63947,63951,63983,64329,64996,65086,65090,65094,65906,65909,66323,66445,66781,66784,67157,67161,67164,69116,69119,69383,69621,70209,70325,70581,70816,71306,71338,71879,72431,72806,72809,73470,73473,74163,74369,74373,74380,74880,74974,75366,75669,75673,75677,75709,77494,77703,78187,78535,78754,78757,79265,79453,79908,79911,79915,80063,80399,80843,80846,81637,81673,81677,81754,81757,82102,82252,82320,82604,82983,83367,83370,83746,83886,84521,84627,84659,84771,84773,85294,85609,85614,85788,85794,86105,86365,86559,86858,87248,87576,88282,88611,88993,89134,89137,90090,90202,90533,90659,90955,90958,91238,91242,92280,92283,92287,92472,92475,92612,92615,92808,92810,92878,92968,92971],[11,47904,47905],{},"统计学的下界证明有一个几乎统一的套路：先把估计问题换成一个检验问题，再证明这个检验问题很难。",[11,47907,21938,47908,48100,48101,48129,48130,48166,48167,48202,48203,48238,48239,10179,48332,48425,48426,48454],{},[15,47909,47911,47955],{"className":47910,"translate":19},[23],[15,47912,47914],{"className":47913},[27],[29,47915,47916],{"xmlns":31},[34,47917,47918,47952],{},[37,47919,47920,47926,47928,47930,47932,47938,47940,47942,47944,47946,47948,47950],{},[40,47921,47922,47924],{},[43,47923,597],{},[308,47925,318],{},[50,47927,91],{"separator":72},[50,47929,335],{},[50,47931,91],{"separator":72},[40,47933,47934,47936],{},[43,47935,597],{},[43,47937,39326],{},[50,47939,991],{},[43,47941,12561],{},[50,47943,53],{"stretchy":52},[43,47945,48],{},[50,47947,91],{"separator":72},[308,47949,318],{},[50,47951,59],{"stretchy":52},[109,47953,47954],{"encoding":111},"X_1,\\dots,X_n\\sim N(\\theta,1)",[15,47956,47958,48073],{"className":47957,"ariaHidden":72},[116],[15,47959,47961,47964,48006,48009,48012,48015,48018,48021,48024,48064,48067,48070],{"className":47960},[120],[15,47962],{"className":47963,"style":34366},[124],[15,47965,47967,47971],{"className":47966},[129],[15,47968,597],{"className":47969,"style":47970},[129,133],"margin-right:0.0785em;",[15,47972,47974],{"className":47973},[137],[15,47975,47977,47998],{"className":47976},[141,142],[15,47978,47980,47995],{"className":47979},[146],[15,47981,47983],{"className":47982,"style":1051},[150],[15,47984,47986,47989],{"style":47985},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[15,47987],{"className":47988,"style":159},[158],[15,47990,47992],{"className":47991},[163,164,165,166],[15,47993,318],{"className":47994},[129,166],[15,47996,175],{"className":47997},[174],[15,47999,48001],{"className":48000},[146],[15,48002,48004],{"className":48003,"style":182},[150],[15,48005],{},[15,48007,91],{"className":48008},[254],[15,48010],{"className":48011,"style":262},[93],[15,48013,335],{"className":48014},[449],[15,48016],{"className":48017,"style":262},[93],[15,48019,91],{"className":48020},[254],[15,48022],{"className":48023,"style":262},[93],[15,48025,48027,48030],{"className":48026},[129],[15,48028,597],{"className":48029,"style":47970},[129,133],[15,48031,48033],{"className":48032},[137],[15,48034,48036,48056],{"className":48035},[141,142],[15,48037,48039,48053],{"className":48038},[146],[15,48040,48042],{"className":48041,"style":1163},[150],[15,48043,48044,48047],{"style":47985},[15,48045],{"className":48046,"style":159},[158],[15,48048,48050],{"className":48049},[163,164,165,166],[15,48051,39326],{"className":48052},[129,133,166],[15,48054,175],{"className":48055},[174],[15,48057,48059],{"className":48058},[146],[15,48060,48062],{"className":48061,"style":182},[150],[15,48063],{},[15,48065],{"className":48066,"style":199},[93],[15,48068,991],{"className":48069},[203],[15,48071],{"className":48072,"style":199},[93],[15,48074,48076,48079,48082,48085,48088,48091,48094,48097],{"className":48075},[120],[15,48077],{"className":48078,"style":125},[124],[15,48080,12561],{"className":48081,"style":12597},[129,133],[15,48083,53],{"className":48084},[188],[15,48086,48],{"className":48087,"style":170},[129,133],[15,48089,91],{"className":48090},[254],[15,48092],{"className":48093,"style":262},[93],[15,48095,318],{"className":48096},[129],[15,48098,59],{"className":48099},[195],"，要估计 ",[15,48102,48104,48117],{"className":48103,"translate":19},[23],[15,48105,48107],{"className":48106},[27],[29,48108,48109],{"xmlns":31},[34,48110,48111,48115],{},[37,48112,48113],{},[43,48114,48],{},[109,48116,512],{"encoding":111},[15,48118,48120],{"className":48119,"ariaHidden":72},[116],[15,48121,48123,48126],{"className":48122},[120],[15,48124],{"className":48125,"style":522},[124],[15,48127,48],{"className":48128,"style":170},[129,133],"。样本均值给出风险 ",[15,48131,48133,48151],{"className":48132,"translate":19},[23],[15,48134,48136],{"className":48135},[27],[29,48137,48138],{"xmlns":31},[34,48139,48140,48148],{},[37,48141,48142,48144,48146],{},[308,48143,318],{},[43,48145,544],{"mathvariant":65},[43,48147,39326],{},[109,48149,48150],{"encoding":111},"1\u002Fn",[15,48152,48154],{"className":48153,"ariaHidden":72},[116],[15,48155,48157,48160,48163],{"className":48156},[120],[15,48158],{"className":48159,"style":125},[124],[15,48161,29852],{"className":48162},[129],[15,48164,39326],{"className":48165},[129,133],"。这个 ",[15,48168,48170,48187],{"className":48169,"translate":19},[23],[15,48171,48173],{"className":48172},[27],[29,48174,48175],{"xmlns":31},[34,48176,48177,48185],{},[37,48178,48179,48181,48183],{},[308,48180,318],{},[43,48182,544],{"mathvariant":65},[43,48184,39326],{},[109,48186,48150],{"encoding":111},[15,48188,48190],{"className":48189,"ariaHidden":72},[116],[15,48191,48193,48196,48199],{"className":48192},[120],[15,48194],{"className":48195,"style":125},[124],[15,48197,29852],{"className":48198},[129],[15,48200,39326],{"className":48201},[129,133]," 是不是最优的？也就是说，是否存在某个估计量，能让最坏情况下的风险比 ",[15,48204,48206,48223],{"className":48205,"translate":19},[23],[15,48207,48209],{"className":48208},[27],[29,48210,48211],{"xmlns":31},[34,48212,48213,48221],{},[37,48214,48215,48217,48219],{},[308,48216,318],{},[43,48218,544],{"mathvariant":65},[43,48220,39326],{},[109,48222,48150],{"encoding":111},[15,48224,48226],{"className":48225,"ariaHidden":72},[116],[15,48227,48229,48232,48235],{"className":48228},[120],[15,48230],{"className":48231,"style":125},[124],[15,48233,29852],{"className":48234},[129],[15,48236,39326],{"className":48237},[129,133]," 更小？Le Cam 的两点方法给出的回答是：不存在。它的论证只用了两个候选参数 ",[15,48240,48242,48264],{"className":48241,"translate":19},[23],[15,48243,48245],{"className":48244},[27],[29,48246,48247],{"xmlns":31},[34,48248,48249,48261],{},[37,48250,48251,48257,48259],{},[40,48252,48253,48255],{},[43,48254,48],{},[308,48256,541],{},[50,48258,62],{},[308,48260,541],{},[109,48262,48263],{"encoding":111},"\\theta_0=0",[15,48265,48267,48323],{"className":48266,"ariaHidden":72},[116],[15,48268,48270,48273,48314,48317,48320],{"className":48269},[120],[15,48271],{"className":48272,"style":2183},[124],[15,48274,48276,48279],{"className":48275},[129],[15,48277,48],{"className":48278,"style":170},[129,133],[15,48280,48282],{"className":48281},[137],[15,48283,48285,48306],{"className":48284},[141,142],[15,48286,48288,48303],{"className":48287},[146],[15,48289,48291],{"className":48290,"style":1051},[150],[15,48292,48294,48297],{"style":48293},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[15,48295],{"className":48296,"style":159},[158],[15,48298,48300],{"className":48299},[163,164,165,166],[15,48301,541],{"className":48302},[129,166],[15,48304,175],{"className":48305},[174],[15,48307,48309],{"className":48308},[146],[15,48310,48312],{"className":48311,"style":182},[150],[15,48313],{},[15,48315],{"className":48316,"style":199},[93],[15,48318,62],{"className":48319},[203],[15,48321],{"className":48322,"style":199},[93],[15,48324,48326,48329],{"className":48325},[120],[15,48327],{"className":48328,"style":4886},[124],[15,48330,541],{"className":48331},[129],[15,48333,48335,48358],{"className":48334,"translate":19},[23],[15,48336,48338],{"className":48337},[27],[29,48339,48340],{"xmlns":31},[34,48341,48342,48355],{},[37,48343,48344,48350,48352],{},[40,48345,48346,48348],{},[43,48347,48],{},[308,48349,318],{},[50,48351,62],{},[43,48353,48354],{"mathvariant":65},"Δ",[109,48356,48357],{"encoding":111},"\\theta_1=\\Delta",[15,48359,48361,48416],{"className":48360,"ariaHidden":72},[116],[15,48362,48364,48367,48407,48410,48413],{"className":48363},[120],[15,48365],{"className":48366,"style":2183},[124],[15,48368,48370,48373],{"className":48369},[129],[15,48371,48],{"className":48372,"style":170},[129,133],[15,48374,48376],{"className":48375},[137],[15,48377,48379,48399],{"className":48378},[141,142],[15,48380,48382,48396],{"className":48381},[146],[15,48383,48385],{"className":48384,"style":1051},[150],[15,48386,48387,48390],{"style":48293},[15,48388],{"className":48389,"style":159},[158],[15,48391,48393],{"className":48392},[163,164,165,166],[15,48394,318],{"className":48395},[129,166],[15,48397,175],{"className":48398},[174],[15,48400,48402],{"className":48401},[146],[15,48403,48405],{"className":48404,"style":182},[150],[15,48406],{},[15,48408],{"className":48409,"style":199},[93],[15,48411,62],{"className":48412},[203],[15,48414],{"className":48415,"style":199},[93],[15,48417,48419,48422],{"className":48418},[120],[15,48420],{"className":48421,"style":610},[124],[15,48423,48354],{"className":48424},[129],"——如果连这两个点都分不清，那就更不可能把 ",[15,48427,48429,48442],{"className":48428,"translate":19},[23],[15,48430,48432],{"className":48431},[27],[29,48433,48434],{"xmlns":31},[34,48435,48436,48440],{},[37,48437,48438],{},[43,48439,48],{},[109,48441,512],{"encoding":111},[15,48443,48445],{"className":48444,"ariaHidden":72},[116],[15,48446,48448,48451],{"className":48447},[120],[15,48449],{"className":48450,"style":522},[124],[15,48452,48],{"className":48453,"style":170},[129,133]," 估准。",[11,48456,48457],{},"本文介绍这套语言里最常用的三个工具：Le Cam 的两点方法、Fano 不等式，以及 Assouad 的超立方体引理。三者的共同点是：先选一个有限的「困难子问题」，把估计下界转化为检验下界，再用总变差、KL 散度或互信息来度量「有多难分」。",[570,48459,48461],{"id":48460},"一极小极大下界在说什么","一、极小极大：下界在说什么",[11,48463,48464,48465,48532,48533,48536,48537,14475,48565,48625,48626,48654,48655,48683],{},"一个估计量 ",[15,48466,48468,48487],{"className":48467,"translate":19},[23],[15,48469,48471],{"className":48470},[27],[29,48472,48473],{"xmlns":31},[34,48474,48475,48484],{},[37,48476,48477],{},[46847,48478,48479,48481],{"accent":72},[43,48480,48],{},[50,48482,48483],{},"^",[109,48485,48486],{"encoding":111},"\\hat\\theta",[15,48488,48490],{"className":48489,"ariaHidden":72},[116],[15,48491,48493,48497],{"className":48492},[120],[15,48494],{"className":48495,"style":48496},[124],"height:0.9579em;",[15,48498,48501],{"className":48499},[129,48500],"accent",[15,48502,48504],{"className":48503},[141],[15,48505,48507],{"className":48506},[146],[15,48508,48510,48518],{"className":48509,"style":48496},[150],[15,48511,48512,48515],{"style":2287},[15,48513],{"className":48514,"style":1823},[158],[15,48516,48],{"className":48517,"style":170},[129,133],[15,48519,48521,48524],{"style":48520},"top:-3.2634em;",[15,48522],{"className":48523,"style":1823},[158],[15,48525,48529],{"className":48526,"style":48528},[48527],"accent-body","left:-0.1667em;",[15,48530,48483],{"className":48531},[129]," 的优劣，通常用它的",[698,48534,48535],{},"风险","衡量：在真实参数为 ",[15,48538,48540,48553],{"className":48539,"translate":19},[23],[15,48541,48543],{"className":48542},[27],[29,48544,48545],{"xmlns":31},[34,48546,48547,48551],{},[37,48548,48549],{},[43,48550,48],{},[109,48552,512],{"encoding":111},[15,48554,48556],{"className":48555,"ariaHidden":72},[116],[15,48557,48559,48562],{"className":48558},[120],[15,48560],{"className":48561,"style":522},[124],[15,48563,48],{"className":48564,"style":170},[129,133],[15,48566,48568,48585],{"className":48567,"translate":19},[23],[15,48569,48571],{"className":48570},[27],[29,48572,48573],{"xmlns":31},[34,48574,48575,48583],{},[37,48576,48577],{},[46847,48578,48579,48581],{"accent":72},[43,48580,48],{},[50,48582,48483],{},[109,48584,48486],{"encoding":111},[15,48586,48588],{"className":48587,"ariaHidden":72},[116],[15,48589,48591,48594],{"className":48590},[120],[15,48592],{"className":48593,"style":48496},[124],[15,48595,48597],{"className":48596},[129,48500],[15,48598,48600],{"className":48599},[141],[15,48601,48603],{"className":48602},[146],[15,48604,48606,48614],{"className":48605,"style":48496},[150],[15,48607,48608,48611],{"style":2287},[15,48609],{"className":48610,"style":1823},[158],[15,48612,48],{"className":48613,"style":170},[129,133],[15,48615,48616,48619],{"style":48520},[15,48617],{"className":48618,"style":1823},[158],[15,48620,48622],{"className":48621,"style":48528},[48527],[15,48623,48483],{"className":48624},[129]," 的平均损失。但真实参数是未知的，所以我们不能只关心某一个 ",[15,48627,48629,48642],{"className":48628,"translate":19},[23],[15,48630,48632],{"className":48631},[27],[29,48633,48634],{"xmlns":31},[34,48635,48636,48640],{},[37,48637,48638],{},[43,48639,48],{},[109,48641,512],{"encoding":111},[15,48643,48645],{"className":48644,"ariaHidden":72},[116],[15,48646,48648,48651],{"className":48647},[120],[15,48649],{"className":48650,"style":522},[124],[15,48652,48],{"className":48653,"style":170},[129,133]," 下的风险。极小极大（minimax）的立场是：先让大自然挑一个最不利的 ",[15,48656,48658,48671],{"className":48657,"translate":19},[23],[15,48659,48661],{"className":48660},[27],[29,48662,48663],{"xmlns":31},[34,48664,48665,48669],{},[37,48666,48667],{},[43,48668,48],{},[109,48670,512],{"encoding":111},[15,48672,48674],{"className":48673,"ariaHidden":72},[116],[15,48675,48677,48680],{"className":48676},[120],[15,48678],{"className":48679,"style":522},[124],[15,48681,48],{"className":48682,"style":170},[129,133],"，再让统计学家挑估计量，看统计学家能保证什么。",[11,48685,48686],{},"用博弈的话说，极小极大风险是",[15,48688,48690],{"className":48689,"translate":19},[18],[15,48691,48693,48770],{"className":48692,"translate":19},[23],[15,48694,48696],{"className":48695},[27],[29,48697,48698],{"xmlns":31,"display":32},[34,48699,48700,48767],{},[37,48701,48702,48717,48733,48735,48745,48747,48749,48751,48753,48755,48761,48763,48765],{},[2140,48703,48704,48711],{},[37,48705,48706,48709],{},[43,48707,48708],{},"inf",[50,48710,69],{},[46847,48712,48713,48715],{"accent":72},[43,48714,48],{},[50,48716,48483],{},[2140,48718,48719,48725],{},[37,48720,48721,48723],{},[43,48722,2932],{},[50,48724,69],{},[37,48726,48727,48729,48731],{},[43,48728,48],{},[50,48730,100],{},[43,48732,30875],{"mathvariant":65},[6856,48734,6858],{},[40,48736,48737,48739],{},[43,48738,26574],{"mathvariant":103},[40,48740,48741,48743],{},[43,48742,1388],{},[43,48744,48],{},[50,48746,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,48748,27529],{"mathvariant":65},[50,48750,53],{"stretchy":52},[43,48752,48],{},[50,48754,91],{"separator":72},[46847,48756,48757,48759],{"accent":72},[43,48758,48],{},[50,48760,48483],{},[50,48762,59],{"stretchy":52},[50,48764,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,48766,107],{"mathvariant":65},[109,48768,48769],{"encoding":111},"\\inf_{\\hat\\theta}\\sup_{\\theta\\in\\Theta}\\ \\mathbb E_{P_\\theta}\\bigl[\\ell(\\theta,\\hat\\theta)\\bigr].",[15,48771,48773],{"className":48772,"ariaHidden":72},[116],[15,48774,48776,48780,48863,48866,48919,48922,48925,49008,49014,49017,49020,49023,49026,49029,49060,49063,49069],{"className":48775},[120],[15,48777],{"className":48778,"style":48779},[124],"height:1.9318em;vertical-align:-0.9739em;",[15,48781,48783],{"className":48782},[217,1874],[15,48784,48786,48854],{"className":48785},[141,142],[15,48787,48789,48851],{"className":48788},[146],[15,48790,48792,48837],{"className":48791,"style":522},[150],[15,48793,48795,48798],{"style":48794},"top:-2.1635em;margin-left:0em;",[15,48796],{"className":48797,"style":1823},[158],[15,48799,48801],{"className":48800},[163,164,165,166],[15,48802,48804],{"className":48803},[129,166],[15,48805,48807],{"className":48806},[129,48500,166],[15,48808,48810],{"className":48809},[141],[15,48811,48813],{"className":48812},[146],[15,48814,48816,48825],{"className":48815,"style":48496},[150],[15,48817,48819,48822],{"style":48818},"top:-2.7em;",[15,48820],{"className":48821,"style":159},[158],[15,48823,48],{"className":48824,"style":170},[129,133,166],[15,48826,48828,48831],{"style":48827},"top:-2.9634em;",[15,48829],{"className":48830,"style":159},[158],[15,48832,48834],{"className":48833,"style":48528},[48527],[15,48835,48483],{"className":48836},[129,166],[15,48838,48839,48842],{"style":2287},[15,48840],{"className":48841,"style":1823},[158],[15,48843,48844],{},[15,48845,48847,48848],{"className":48846},[217],"in",[15,48849,36569],{"style":48850},"margin-right:0.0778em;",[15,48852,175],{"className":48853},[174],[15,48855,48857],{"className":48856},[146],[15,48858,48861],{"className":48859,"style":48860},[150],"height:0.9365em;",[15,48862],{},[15,48864],{"className":48865,"style":262},[93],[15,48867,48869],{"className":48868},[217,1874],[15,48870,48872,48911],{"className":48871},[141,142],[15,48873,48875,48908],{"className":48874},[146],[15,48876,48878,48898],{"className":48877,"style":4383},[150],[15,48879,48880,48883],{"style":4386},[15,48881],{"className":48882,"style":1823},[158],[15,48884,48886],{"className":48885},[163,164,165,166],[15,48887,48889,48892,48895],{"className":48888},[129,166],[15,48890,48],{"className":48891,"style":170},[129,133,166],[15,48893,100],{"className":48894},[203,166],[15,48896,30875],{"className":48897},[129,166],[15,48899,48900,48903],{"style":2287},[15,48901],{"className":48902,"style":1823},[158],[15,48904,48905],{},[15,48906,2932],{"className":48907},[217],[15,48909,175],{"className":48910},[174],[15,48912,48914],{"className":48913},[146],[15,48915,48917],{"className":48916,"style":4424},[150],[15,48918],{},[15,48920,6858],{"className":48921},[93],[15,48923],{"className":48924,"style":262},[93],[15,48926,48928,48931],{"className":48927},[129],[15,48929,26574],{"className":48930},[129,285],[15,48932,48934],{"className":48933},[137],[15,48935,48937,48999],{"className":48936},[141,142],[15,48938,48940,48996],{"className":48939},[146],[15,48941,48943],{"className":48942,"style":1525},[150],[15,48944,48945,48948],{"style":154},[15,48946],{"className":48947,"style":159},[158],[15,48949,48951],{"className":48950},[163,164,165,166],[15,48952,48954],{"className":48953},[129,166],[15,48955,48957,48960],{"className":48956},[129,166],[15,48958,1388],{"className":48959,"style":12449},[129,133,166],[15,48961,48963],{"className":48962},[137],[15,48964,48966,48987],{"className":48965},[141,142],[15,48967,48969,48984],{"className":48968},[146],[15,48970,48972],{"className":48971,"style":1594},[150],[15,48973,48975,48978],{"style":48974},"top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;",[15,48976],{"className":48977,"style":15987},[158],[15,48979,48981],{"className":48980},[163,15991,229,166],[15,48982,48],{"className":48983,"style":170},[129,133,166],[15,48985,175],{"className":48986},[174],[15,48988,48990],{"className":48989},[146],[15,48991,48994],{"className":48992,"style":48993},[150],"height:0.1512em;",[15,48995],{},[15,48997,175],{"className":48998},[174],[15,49000,49002],{"className":49001},[146],[15,49003,49006],{"className":49004,"style":49005},[150],"height:0.2559em;",[15,49007],{},[15,49009,49011],{"className":49010},[188],[15,49012,1407],{"className":49013},[228,229],[15,49015,27529],{"className":49016},[129],[15,49018,53],{"className":49019},[188],[15,49021,48],{"className":49022,"style":170},[129,133],[15,49024,91],{"className":49025},[254],[15,49027],{"className":49028,"style":262},[93],[15,49030,49032],{"className":49031},[129,48500],[15,49033,49035],{"className":49034},[141],[15,49036,49038],{"className":49037},[146],[15,49039,49041,49049],{"className":49040,"style":48496},[150],[15,49042,49043,49046],{"style":2287},[15,49044],{"className":49045,"style":1823},[158],[15,49047,48],{"className":49048,"style":170},[129,133],[15,49050,49051,49054],{"style":48520},[15,49052],{"className":49053,"style":1823},[158],[15,49055,49057],{"className":49056,"style":48528},[48527],[15,49058,48483],{"className":49059},[129],[15,49061,59],{"className":49062},[195],[15,49064,49066],{"className":49065},[195],[15,49067,1423],{"className":49068},[228,229],[15,49070,107],{"className":49071},[129],[11,49073,49074,49075,49078],{},"这个量回答的是：",[698,49076,49077],{},"在所有可能的参数中，最坏情况下的风险最小能压到多少。"," 一个上界对应一个具体的估计量；一个下界则对应一句否定：无论你多聪明，都不可能比这个数更好。",[11,49080,49081,49082,49084],{},"下界为什么难？因为你要同时对",[698,49083,7250],{},"估计量说话。逐一检查是不可能的。所以下界证明总是走同一条路：把参数空间缩小到一个有限的、精心挑选的子集，使得在这个子集上「区分参数」本身就已经很难；如果区分都做不到，估计就更做不到。",[11,49086,49087],{},"这就是「归约到检验」。接下来的三种方法，区别只在于子集有多大、用什么距离度量「难分」。",[570,49089,49091],{"id":49090},"二决策论框架","二、决策论框架",[575,49093,49095],{"id":49094},"_21-模型损失与风险","2.1 模型、损失与风险",[11,49097,21938,49098,49222,49223,49251,49252,49280,49281,49309],{},[15,49099,49101,49131],{"className":49100,"translate":19},[23],[15,49102,49104],{"className":49103},[27],[29,49105,49106],{"xmlns":31},[34,49107,49108,49128],{},[37,49109,49110,49112,49118,49120,49122,49124,49126],{},[50,49111,636],{"stretchy":52},[40,49113,49114,49116],{},[43,49115,1388],{},[43,49117,48],{},[50,49119,718],{},[43,49121,48],{},[50,49123,100],{},[43,49125,30875],{"mathvariant":65},[50,49127,645],{"stretchy":52},[109,49129,49130],{"encoding":111},"\\{P_\\theta:\\theta\\in\\Theta\\}",[15,49132,49134,49192,49210],{"className":49133,"ariaHidden":72},[116],[15,49135,49137,49140,49143,49183,49186,49189],{"className":49136},[120],[15,49138],{"className":49139,"style":125},[124],[15,49141,636],{"className":49142},[188],[15,49144,49146,49149],{"className":49145},[129],[15,49147,1388],{"className":49148,"style":12449},[129,133],[15,49150,49152],{"className":49151},[137],[15,49153,49155,49175],{"className":49154},[141,142],[15,49156,49158,49172],{"className":49157},[146],[15,49159,49161],{"className":49160,"style":151},[150],[15,49162,49163,49166],{"style":34239},[15,49164],{"className":49165,"style":159},[158],[15,49167,49169],{"className":49168},[163,164,165,166],[15,49170,48],{"className":49171,"style":170},[129,133,166],[15,49173,175],{"className":49174},[174],[15,49176,49178],{"className":49177},[146],[15,49179,49181],{"className":49180,"style":182},[150],[15,49182],{},[15,49184],{"className":49185,"style":199},[93],[15,49187,718],{"className":49188},[203],[15,49190],{"className":49191,"style":199},[93],[15,49193,49195,49198,49201,49204,49207],{"className":49194},[120],[15,49196],{"className":49197,"style":2405},[124],[15,49199,48],{"className":49200,"style":170},[129,133],[15,49202],{"className":49203,"style":199},[93],[15,49205,100],{"className":49206},[203],[15,49208],{"className":49209,"style":199},[93],[15,49211,49213,49216,49219],{"className":49212},[120],[15,49214],{"className":49215,"style":125},[124],[15,49217,30875],{"className":49218},[129],[15,49220,645],{"className":49221},[195]," 是样本空间 ",[15,49224,49226,49239],{"className":49225,"translate":19},[23],[15,49227,49229],{"className":49228},[27],[29,49230,49231],{"xmlns":31},[34,49232,49233,49237],{},[37,49234,49235],{},[43,49236,597],{"mathvariant":596},[109,49238,600],{"encoding":111},[15,49240,49242],{"className":49241,"ariaHidden":72},[116],[15,49243,49245,49248],{"className":49244},[120],[15,49246],{"className":49247,"style":610},[124],[15,49249,597],{"className":49250,"style":615},[129,614]," 上的一族分布，参数空间 ",[15,49253,49255,49268],{"className":49254,"translate":19},[23],[15,49256,49258],{"className":49257},[27],[29,49259,49260],{"xmlns":31},[34,49261,49262,49266],{},[37,49263,49264],{},[43,49265,30875],{"mathvariant":65},[109,49267,31304],{"encoding":111},[15,49269,49271],{"className":49270,"ariaHidden":72},[116],[15,49272,49274,49277],{"className":49273},[120],[15,49275],{"className":49276,"style":610},[124],[15,49278,30875],{"className":49279},[129]," 可以是任意集合。我们观测 ",[15,49282,49284,49297],{"className":49283,"translate":19},[23],[15,49285,49287],{"className":49286},[27],[29,49288,49289],{"xmlns":31},[34,49290,49291,49295],{},[37,49292,49293],{},[43,49294,39326],{},[109,49296,39326],{"encoding":111},[15,49298,49300],{"className":49299,"ariaHidden":72},[116],[15,49301,49303,49306],{"className":49302},[120],[15,49304],{"className":49305,"style":4383},[124],[15,49307,39326],{"className":49308},[129,133]," 个独立同分布的样本",[15,49311,49313],{"className":49312,"translate":19},[18],[15,49314,49316,49375],{"className":49315,"translate":19},[23],[15,49317,49319],{"className":49318},[27],[29,49320,49321],{"xmlns":31,"display":32},[34,49322,49323,49372],{},[37,49324,49325,49331,49333,49335,49341,49343,49345,49347,49353,49355,49357,49370],{},[305,49326,49327,49329],{},[43,49328,597],{},[43,49330,39326],{},[50,49332,62],{},[50,49334,53],{"stretchy":52},[40,49336,49337,49339],{},[43,49338,597],{},[308,49340,318],{},[50,49342,91],{"separator":72},[50,49344,335],{},[50,49346,91],{"separator":72},[40,49348,49349,49351],{},[43,49350,597],{},[43,49352,39326],{},[50,49354,59],{"stretchy":52},[50,49356,991],{},[23160,49358,49359,49361,49363],{},[43,49360,1388],{},[43,49362,48],{},[37,49364,49365,49368],{},[50,49366,49367],{},"⊗",[43,49369,39326],{},[50,49371,91],{"separator":72},[109,49373,49374],{"encoding":111},"X^n=(X_1,\\dots,X_n)\\sim P_\\theta^{\\otimes n},",[15,49376,49378,49422,49541],{"className":49377,"ariaHidden":72},[116],[15,49379,49381,49384,49413,49416,49419],{"className":49380},[120],[15,49382],{"className":49383,"style":1251},[124],[15,49385,49387,49390],{"className":49386},[129],[15,49388,597],{"className":49389,"style":47970},[129,133],[15,49391,49393],{"className":49392},[137],[15,49394,49396],{"className":49395},[141],[15,49397,49399],{"className":49398},[146],[15,49400,49402],{"className":49401,"style":1251},[150],[15,49403,49404,49407],{"style":1272},[15,49405],{"className":49406,"style":159},[158],[15,49408,49410],{"className":49409},[163,164,165,166],[15,49411,39326],{"className":49412},[129,133,166],[15,49414],{"className":49415,"style":199},[93],[15,49417,62],{"className":49418},[203],[15,49420],{"className":49421,"style":199},[93],[15,49423,49425,49428,49431,49471,49474,49477,49480,49483,49486,49489,49529,49532,49535,49538],{"className":49424},[120],[15,49426],{"className":49427,"style":125},[124],[15,49429,53],{"className":49430},[188],[15,49432,49434,49437],{"className":49433},[129],[15,49435,597],{"className":49436,"style":47970},[129,133],[15,49438,49440],{"className":49439},[137],[15,49441,49443,49463],{"className":49442},[141,142],[15,49444,49446,49460],{"className":49445},[146],[15,49447,49449],{"className":49448,"style":1051},[150],[15,49450,49451,49454],{"style":47985},[15,49452],{"className":49453,"style":159},[158],[15,49455,49457],{"className":49456},[163,164,165,166],[15,49458,318],{"className":49459},[129,166],[15,49461,175],{"className":49462},[174],[15,49464,49466],{"className":49465},[146],[15,49467,49469],{"className":49468,"style":182},[150],[15,49470],{},[15,49472,91],{"className":49473},[254],[15,49475],{"className":49476,"style":262},[93],[15,49478,335],{"className":49479},[449],[15,49481],{"className":49482,"style":262},[93],[15,49484,91],{"className":49485},[254],[15,49487],{"className":49488,"style":262},[93],[15,49490,49492,49495],{"className":49491},[129],[15,49493,597],{"className":49494,"style":47970},[129,133],[15,49496,49498],{"className":49497},[137],[15,49499,49501,49521],{"className":49500},[141,142],[15,49502,49504,49518],{"className":49503},[146],[15,49505,49507],{"className":49506,"style":1163},[150],[15,49508,49509,49512],{"style":47985},[15,49510],{"className":49511,"style":159},[158],[15,49513,49515],{"className":49514},[163,164,165,166],[15,49516,39326],{"className":49517},[129,133,166],[15,49519,175],{"className":49520},[174],[15,49522,49524],{"className":49523},[146],[15,49525,49527],{"className":49526,"style":182},[150],[15,49528],{},[15,49530,59],{"className":49531},[195],[15,49533],{"className":49534,"style":199},[93],[15,49536,991],{"className":49537},[203],[15,49539],{"className":49540,"style":199},[93],[15,49542,49544,49548,49608],{"className":49543},[120],[15,49545],{"className":49546,"style":49547},[124],"height:1.1128em;vertical-align:-0.2914em;",[15,49549,49551,49554],{"className":49550},[129],[15,49552,1388],{"className":49553,"style":12449},[129,133],[15,49555,49557],{"className":49556},[137],[15,49558,49560,49599],{"className":49559},[141,142],[15,49561,49563,49596],{"className":49562},[146],[15,49564,49567,49579],{"className":49565,"style":49566},[150],"height:0.8213em;",[15,49568,49570,49573],{"style":49569},"top:-2.4086em;margin-left:-0.1389em;margin-right:0.05em;",[15,49571],{"className":49572,"style":159},[158],[15,49574,49576],{"className":49575},[163,164,165,166],[15,49577,48],{"className":49578,"style":170},[129,133,166],[15,49580,49581,49584],{"style":1272},[15,49582],{"className":49583,"style":159},[158],[15,49585,49587],{"className":49586},[163,164,165,166],[15,49588,49590,49593],{"className":49589},[129,166],[15,49591,49367],{"className":49592},[129,166],[15,49594,39326],{"className":49595},[129,133,166],[15,49597,175],{"className":49598},[174],[15,49600,49602],{"className":49601},[146],[15,49603,49606],{"className":49604,"style":49605},[150],"height:0.2914em;",[15,49607],{},[15,49609,91],{"className":49610},[254],[11,49612,49613,49614,49617,49618,1313,49753,804,49756,49874,49875,49935,49936,49964],{},"一个",[698,49615,49616],{},"估计量","是任意可测映射 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度量估计值 ",[15,49876,49878,49895],{"className":49877,"translate":19},[23],[15,49879,49881],{"className":49880},[27],[29,49882,49883],{"xmlns":31},[34,49884,49885,49893],{},[37,49886,49887],{},[46847,49888,49889,49891],{"accent":72},[43,49890,48],{},[50,49892,48483],{},[109,49894,48486],{"encoding":111},[15,49896,49898],{"className":49897,"ariaHidden":72},[116],[15,49899,49901,49904],{"className":49900},[120],[15,49902],{"className":49903,"style":48496},[124],[15,49905,49907],{"className":49906},[129,48500],[15,49908,49910],{"className":49909},[141],[15,49911,49913],{"className":49912},[146],[15,49914,49916,49924],{"className":49915,"style":48496},[150],[15,49917,49918,49921],{"style":2287},[15,49919],{"className":49920,"style":1823},[158],[15,49922,48],{"className":49923,"style":170},[129,133],[15,49925,49926,49929],{"style":48520},[15,49927],{"className":49928,"style":1823},[158],[15,49930,49932],{"className":49931,"style":48528},[48527],[15,49933,48483],{"className":49934},[129]," 与真值 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损失 ",[15,50480,50482,50528],{"className":50481,"translate":19},[23],[15,50483,50485],{"className":50484},[27],[29,50486,50487],{"xmlns":31},[34,50488,50489,50525],{},[37,50490,50491,50493,50495,50497,50499,50505,50507,50509,50511,50513,50515,50517,50523],{},[43,50492,27529],{"mathvariant":65},[50,50494,53],{"stretchy":52},[43,50496,48],{},[50,50498,91],{"separator":72},[46847,50500,50501,50503],{"accent":72},[43,50502,48],{},[50,50504,48483],{},[50,50506,59],{"stretchy":52},[50,50508,62],{},[308,50510,318],{"mathvariant":1468},[50,50512,1407],{"stretchy":52},[43,50514,48],{},[50,50516,1418],{"mathvariant":65},[46847,50518,50519,50521],{"accent":72},[43,50520,48],{},[50,50522,48483],{},[50,50524,1423],{"stretchy":52},[109,50526,50527],{"encoding":111},"\\ell(\\theta,\\hat\\theta)=\\mathbf 1[\\theta\\neq\\hat\\theta]",[15,50529,50531,50595,50652],{"className":50530,"ariaHidden":72},[116],[15,50532,50534,50537,50540,50543,50546,50549,50552,50583,50586,50589,50592],{"className":50533},[120],[15,50535],{"className":50536,"style":50032},[124],[15,50538,27529],{"className":50539},[129],[15,50541,53],{"className":50542},[188],[15,50544,48],{"className":50545,"style":170},[129,133],[15,50547,91],{"className":50548},[254],[15,50550],{"className":50551,"style":262},[93],[15,50553,50555],{"className":50554},[129,48500],[15,50556,50558],{"className":50557},[141],[15,50559,50561],{"className":50560},[146],[15,50562,50564,50572],{"className":50563,"style":48496},[150],[15,50565,50566,50569],{"style":2287},[15,50567],{"className":50568,"style":1823},[158],[15,50570,48],{"className":50571,"style":170},[129,133],[15,50573,50574,50577],{"style":48520},[15,50575],{"className":50576,"style":1823},[158],[15,50578,50580],{"className":50579,"style":48528},[48527],[15,50581,48483],{"className":50582},[129],[15,50584,59],{"className":50585},[195],[15,50587],{"className":50588,"style":199},[93],[15,50590,62],{"className":50591},[203],[15,50593],{"className":50594,"style":199},[93],[15,50596,50598,50601,50604,50607,50610,50613,50649],{"className":50597},[120],[15,50599],{"className":50600,"style":125},[124],[15,50602,318],{"className":50603},[129,1950],[15,50605,1407],{"className":50606},[188],[15,50608,48],{"className":50609,"style":170},[129,133],[15,50611],{"className":50612,"style":199},[93],[15,50614,50616,50643,50646],{"className":50615},[203],[15,50617,50619],{"className":50618},[203],[15,50620,50622],{"className":50621},[129,1670],[15,50623,50625],{"className":50624},[1674],[15,50626,50628,50631,50640],{"className":50627},[1678],[15,50629],{"className":50630,"style":1682},[124],[15,50632,50634],{"className":50633},[1686],[15,50635,50637],{"className":50636},[129],[15,50638,1693],{"className":50639},[203],[15,50641],{"className":50642},[1697],[15,50644],{"className":50645},[93,1701],[15,50647,62],{"className":50648},[203],[15,50650],{"className":50651,"style":199},[93],[15,50653,50655,50658,50689],{"className":50654},[120],[15,50656],{"className":50657,"style":50032},[124],[15,50659,50661],{"className":50660},[129,48500],[15,50662,50664],{"className":50663},[141],[15,50665,50667],{"className":50666},[146],[15,50668,50670,50678],{"className":50669,"style":48496},[150],[15,50671,50672,50675],{"style":2287},[15,50673],{"className":50674,"style":1823},[158],[15,50676,48],{"className":50677,"style":170},[129,133],[15,50679,50680,50683],{"style":48520},[15,50681],{"className":50682,"style":1823},[158],[15,50684,50686],{"className":50685,"style":48528},[48527],[15,50687,48483],{"className":50688},[129],[15,50690,1423],{"className":50691},[195],[11,50693,50694,50695,5831,50755,50783,50784,50786],{},"估计量 ",[15,50696,50698,50715],{"className":50697,"translate":19},[23],[15,50699,50701],{"className":50700},[27],[29,50702,50703],{"xmlns":31},[34,50704,50705,50713],{},[37,50706,50707],{},[46847,50708,50709,50711],{"accent":72},[43,50710,48],{},[50,50712,48483],{},[109,50714,48486],{"encoding":111},[15,50716,50718],{"className":50717,"ariaHidden":72},[116],[15,50719,50721,50724],{"className":50720},[120],[15,50722],{"className":50723,"style":48496},[124],[15,50725,50727],{"className":50726},[129,48500],[15,50728,50730],{"className":50729},[141],[15,50731,50733],{"className":50732},[146],[15,50734,50736,50744],{"className":50735,"style":48496},[150],[15,50737,50738,50741],{"style":2287},[15,50739],{"className":50740,"style":1823},[158],[15,50742,48],{"className":50743,"style":170},[129,133],[15,50745,50746,50749],{"style":48520},[15,50747],{"className":50748,"style":1823},[158],[15,50750,50752],{"className":50751,"style":48528},[48527],[15,50753,48483],{"className":50754},[129],[15,50756,50758,50771],{"className":50757,"translate":19},[23],[15,50759,50761],{"className":50760},[27],[29,50762,50763],{"xmlns":31},[34,50764,50765,50769],{},[37,50766,50767],{},[43,50768,48],{},[109,50770,512],{"encoding":111},[15,50772,50774],{"className":50773,"ariaHidden":72},[116],[15,50775,50777,50780],{"className":50776},[120],[15,50778],{"className":50779,"style":522},[124],[15,50781,48],{"className":50782,"style":170},[129,133]," 处的",[698,50785,48535],{},"是",[15,50788,50790],{"className":50789,"translate":19},[18],[15,50791,50793,50871],{"className":50792,"translate":19},[23],[15,50794,50796],{"className":50795},[27],[29,50797,50798],{"xmlns":31,"display":32},[34,50799,50800,50868],{},[37,50801,50802,50808,50810,50816,50818,50820,50836,50838,50840,50842,50844,50846,50852,50854,50860,50862,50864,50866],{},[40,50803,50804,50806],{},[43,50805,104],{},[43,50807,48],{},[50,50809,53],{"stretchy":52},[46847,50811,50812,50814],{"accent":72},[43,50813,48],{},[50,50815,48483],{},[50,50817,59],{"stretchy":52},[50,50819,62],{},[40,50821,50822,50824],{},[43,50823,26574],{"mathvariant":103},[23160,50825,50826,50828,50830],{},[43,50827,1388],{},[43,50829,48],{},[37,50831,50832,50834],{},[50,50833,49367],{},[43,50835,39326],{},[50,50837,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,50839,27529],{"mathvariant":65},[50,50841,53],{"stretchy":52},[43,50843,48],{},[50,50845,91],{"separator":72},[46847,50847,50848,50850],{"accent":72},[43,50849,48],{},[50,50851,48483],{},[50,50853,53],{"stretchy":52},[305,50855,50856,50858],{},[43,50857,597],{},[43,50859,39326],{},[50,50861,59],{"stretchy":52},[50,50863,59],{"stretchy":52},[50,50865,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,50867,107],{"mathvariant":65},[109,50869,50870],{"encoding":111},"R_\\theta(\\hat\\theta)=\\mathbb E_{P_\\theta^{\\otimes n}}\\bigl[\\ell(\\theta,\\hat\\theta(X^n))\\bigr].",[15,50872,50874,50967],{"className":50873,"ariaHidden":72},[116],[15,50875,50877,50880,50921,50924,50955,50958,50961,50964],{"className":50876},[120],[15,50878],{"className":50879,"style":50032},[124],[15,50881,50883,50886],{"className":50882},[129],[15,50884,104],{"className":50885,"style":33871},[129,133],[15,50887,50889],{"className":50888},[137],[15,50890,50892,50913],{"className":50891},[141,142],[15,50893,50895,50910],{"className":50894},[146],[15,50896,50898],{"className":50897,"style":151},[150],[15,50899,50901,50904],{"style":50900},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[15,50902],{"className":50903,"style":159},[158],[15,50905,50907],{"className":50906},[163,164,165,166],[15,50908,48],{"className":50909,"style":170},[129,133,166],[15,50911,175],{"className":50912},[174],[15,50914,50916],{"className":50915},[146],[15,50917,50919],{"className":50918,"style":182},[150],[15,50920],{},[15,50922,53],{"className":50923},[188],[15,50925,50927],{"className":50926},[129,48500],[15,50928,50930],{"className":50929},[141],[15,50931,50933],{"className":50932},[146],[15,50934,50936,50944],{"className":50935,"style":48496},[150],[15,50937,50938,50941],{"style":2287},[15,50939],{"className":50940,"style":1823},[158],[15,50942,48],{"className":50943,"style":170},[129,133],[15,50945,50946,50949],{"style":48520},[15,50947],{"className":50948,"style":1823},[158],[15,50950,50952],{"className":50951,"style":48528},[48527],[15,50953,48483],{"className":50954},[129],[15,50956,59],{"className":50957},[195],[15,50959],{"className":50960,"style":199},[93],[15,50962,62],{"className":50963},[203],[15,50965],{"className":50966,"style":199},[93],[15,50968,50970,50974,51077,51083,51086,51089,51092,51095,51098,51129,51132,51161,51164,51170],{"className":50969},[120],[15,50971],{"className":50972,"style":50973},[124],"height:1.4308em;vertical-align:-0.4729em;",[15,50975,50977,50980],{"className":50976},[129],[15,50978,26574],{"className":50979},[129,285],[15,50981,50983],{"className":50982},[137],[15,50984,50986,51068],{"className":50985},[141,142],[15,50987,50989,51065],{"className":50988},[146],[15,50990,50992],{"className":50991,"style":1594},[150],[15,50993,50995,50998],{"style":50994},"top:-2.4701em;margin-left:0em;margin-right:0.05em;",[15,50996],{"className":50997,"style":159},[158],[15,50999,51001],{"className":51000},[163,164,165,166],[15,51002,51004],{"className":51003},[129,166],[15,51005,51007,51010],{"className":51006},[129,166],[15,51008,1388],{"className":51009,"style":12449},[129,133,166],[15,51011,51013],{"className":51012},[137],[15,51014,51016,51056],{"className":51015},[141,142],[15,51017,51019,51053],{"className":51018},[146],[15,51020,51023,51035],{"className":51021,"style":51022},[150],"height:0.821em;",[15,51024,51026,51029],{"style":51025},"top:-2.1528em;margin-left:-0.1389em;margin-right:0.0714em;",[15,51027],{"className":51028,"style":15987},[158],[15,51030,51032],{"className":51031},[163,15991,229,166],[15,51033,48],{"className":51034,"style":170},[129,133,166],[15,51036,51038,51041],{"style":51037},"top:-2.9043em;margin-right:0.0714em;",[15,51039],{"className":51040,"style":15987},[158],[15,51042,51044],{"className":51043},[163,15991,229,166],[15,51045,51047,51050],{"className":51046},[129,166],[15,51048,49367],{"className":51049},[129,166],[15,51051,39326],{"className":51052},[129,133,166],[15,51054,175],{"className":51055},[174],[15,51057,51059],{"className":51058},[146],[15,51060,51063],{"className":51061,"style":51062},[150],"height:0.3472em;",[15,51064],{},[15,51066,175],{"className":51067},[174],[15,51069,51071],{"className":51070},[146],[15,51072,51075],{"className":51073,"style":51074},[150],"height:0.4729em;",[15,51076],{},[15,51078,51080],{"className":51079},[188],[15,51081,1407],{"className":51082},[228,229],[15,51084,27529],{"className":51085},[129],[15,51087,53],{"className":51088},[188],[15,51090,48],{"className":51091,"style":170},[129,133],[15,51093,91],{"className":51094},[254],[15,51096],{"className":51097,"style":262},[93],[15,51099,51101],{"className":51100},[129,48500],[15,51102,51104],{"className":51103},[141],[15,51105,51107],{"className":51106},[146],[15,51108,51110,51118],{"className":51109,"style":48496},[150],[15,51111,51112,51115],{"style":2287},[15,51113],{"className":51114,"style":1823},[158],[15,51116,48],{"className":51117,"style":170},[129,133],[15,51119,51120,51123],{"style":48520},[15,51121],{"className":51122,"style":1823},[158],[15,51124,51126],{"className":51125,"style":48528},[48527],[15,51127,48483],{"className":51128},[129],[15,51130,53],{"className":51131},[188],[15,51133,51135,51138],{"className":51134},[129],[15,51136,597],{"className":51137,"style":47970},[129,133],[15,51139,51141],{"className":51140},[137],[15,51142,51144],{"className":51143},[141],[15,51145,51147],{"className":51146},[146],[15,51148,51150],{"className":51149,"style":1251},[150],[15,51151,51152,51155],{"style":1272},[15,51153],{"className":51154,"style":159},[158],[15,51156,51158],{"className":51157},[163,164,165,166],[15,51159,39326],{"className":51160},[129,133,166],[15,51162,6149],{"className":51163},[195],[15,51165,51167],{"className":51166},[195],[15,51168,1423],{"className":51169},[228,229],[15,51171,107],{"className":51172},[129],[11,51174,51175,51176,51179,51180,51208,51209,51237],{},"注意这里的期望是对",[698,51177,51178],{},"数据","取的，",[15,51181,51183,51196],{"className":51182,"translate":19},[23],[15,51184,51186],{"className":51185},[27],[29,51187,51188],{"xmlns":31},[34,51189,51190,51194],{},[37,51191,51192],{},[43,51193,48],{},[109,51195,512],{"encoding":111},[15,51197,51199],{"className":51198,"ariaHidden":72},[116],[15,51200,51202,51205],{"className":51201},[120],[15,51203],{"className":51204,"style":522},[124],[15,51206,48],{"className":51207,"style":170},[129,133]," 固定。极小极大风险则再对 ",[15,51210,51212,51225],{"className":51211,"translate":19},[23],[15,51213,51215],{"className":51214},[27],[29,51216,51217],{"xmlns":31},[34,51218,51219,51223],{},[37,51220,51221],{},[43,51222,48],{},[109,51224,512],{"encoding":111},[15,51226,51228],{"className":51227,"ariaHidden":72},[116],[15,51229,51231,51234],{"className":51230},[120],[15,51232],{"className":51233,"style":522},[124],[15,51235,48],{"className":51236,"style":170},[129,133]," 取最坏情况：",[15,51239,51241],{"className":51240,"translate":19},[18],[15,51242,51244,51322],{"className":51243,"translate":19},[23],[15,51245,51247],{"className":51246},[27],[29,51248,51249],{"xmlns":31,"display":32},[34,51250,51251,51319],{},[37,51252,51253,51259,51261,51263,51265,51267,51269,51271,51285,51301,51307,51309,51315,51317],{},[40,51254,51255,51257],{},[43,51256,43246],{"mathvariant":28910},[43,51258,39326],{},[50,51260,53],{"stretchy":52},[43,51262,30875],{"mathvariant":65},[50,51264,91],{"separator":72},[43,51266,27529],{"mathvariant":65},[50,51268,59],{"stretchy":52},[50,51270,62],{},[2140,51272,51273,51279],{},[37,51274,51275,51277],{},[43,51276,48708],{},[50,51278,69],{},[46847,51280,51281,51283],{"accent":72},[43,51282,48],{},[50,51284,48483],{},[2140,51286,51287,51293],{},[37,51288,51289,51291],{},[43,51290,2932],{},[50,51292,69],{},[37,51294,51295,51297,51299],{},[43,51296,48],{},[50,51298,100],{},[43,51300,30875],{"mathvariant":65},[40,51302,51303,51305],{},[43,51304,104],{},[43,51306,48],{},[50,51308,53],{"stretchy":52},[46847,51310,51311,51313],{"accent":72},[43,51312,48],{},[50,51314,48483],{},[50,51316,59],{"stretchy":52},[43,51318,107],{"mathvariant":65},[109,51320,51321],{"encoding":111},"\\mathfrak M_n(\\Theta,\\ell)=\\inf_{\\hat\\theta}\\sup_{\\theta\\in\\Theta}R_\\theta(\\hat\\theta).",[15,51323,51325,51398],{"className":51324,"ariaHidden":72},[116],[15,51326,51328,51331,51371,51374,51377,51380,51383,51386,51389,51392,51395],{"className":51327},[120],[15,51329],{"className":51330,"style":125},[124],[15,51332,51334,51337],{"className":51333},[129],[15,51335,43246],{"className":51336},[129,28937],[15,51338,51340],{"className":51339},[137],[15,51341,51343,51363],{"className":51342},[141,142],[15,51344,51346,51360],{"className":51345},[146],[15,51347,51349],{"className":51348,"style":1163},[150],[15,51350,51351,51354],{"style":154},[15,51352],{"className":51353,"style":159},[158],[15,51355,51357],{"className":51356},[163,164,165,166],[15,51358,39326],{"className":51359},[129,133,166],[15,51361,175],{"className":51362},[174],[15,51364,51366],{"className":51365},[146],[15,51367,51369],{"className":51368,"style":182},[150],[15,51370],{},[15,51372,53],{"className":51373},[188],[15,51375,30875],{"className":51376},[129],[15,51378,91],{"className":51379},[254],[15,51381],{"className":51382,"style":262},[93],[15,51384,27529],{"className":51385},[129],[15,51387,59],{"className":51388},[195],[15,51390],{"className":51391,"style":199},[93],[15,51393,62],{"className":51394},[203],[15,51396],{"className":51397,"style":199},[93],[15,51399,51401,51404,51481,51484,51537,51540,51580,51583,51614,51617],{"className":51400},[120],[15,51402],{"className":51403,"style":48779},[124],[15,51405,51407],{"className":51406},[217,1874],[15,51408,51410,51473],{"className":51409},[141,142],[15,51411,51413,51470],{"className":51412},[146],[15,51414,51416,51458],{"className":51415,"style":522},[150],[15,51417,51418,51421],{"style":48794},[15,51419],{"className":51420,"style":1823},[158],[15,51422,51424],{"className":51423},[163,164,165,166],[15,51425,51427],{"className":51426},[129,166],[15,51428,51430],{"className":51429},[129,48500,166],[15,51431,51433],{"className":51432},[141],[15,51434,51436],{"className":51435},[146],[15,51437,51439,51447],{"className":51438,"style":48496},[150],[15,51440,51441,51444],{"style":48818},[15,51442],{"className":51443,"style":159},[158],[15,51445,48],{"className":51446,"style":170},[129,133,166],[15,51448,51449,51452],{"style":48827},[15,51450],{"className":51451,"style":159},[158],[15,51453,51455],{"className":51454,"style":48528},[48527],[15,51456,48483],{"className":51457},[129,166],[15,51459,51460,51463],{"style":2287},[15,51461],{"className":51462,"style":1823},[158],[15,51464,51465],{},[15,51466,48847,51468],{"className":51467},[217],[15,51469,36569],{"style":48850},[15,51471,175],{"className":51472},[174],[15,51474,51476],{"className":51475},[146],[15,51477,51479],{"className":51478,"style":48860},[150],[15,51480],{},[15,51482],{"className":51483,"style":262},[93],[15,51485,51487],{"className":51486},[217,1874],[15,51488,51490,51529],{"className":51489},[141,142],[15,51491,51493,51526],{"className":51492},[146],[15,51494,51496,51516],{"className":51495,"style":4383},[150],[15,51497,51498,51501],{"style":4386},[15,51499],{"className":51500,"style":1823},[158],[15,51502,51504],{"className":51503},[163,164,165,166],[15,51505,51507,51510,51513],{"className":51506},[129,166],[15,51508,48],{"className":51509,"style":170},[129,133,166],[15,51511,100],{"className":51512},[203,166],[15,51514,30875],{"className":51515},[129,166],[15,51517,51518,51521],{"style":2287},[15,51519],{"className":51520,"style":1823},[158],[15,51522,51523],{},[15,51524,2932],{"className":51525},[217],[15,51527,175],{"className":51528},[174],[15,51530,51532],{"className":51531},[146],[15,51533,51535],{"className":51534,"style":4424},[150],[15,51536],{},[15,51538],{"className":51539,"style":262},[93],[15,51541,51543,51546],{"className":51542},[129],[15,51544,104],{"className":51545,"style":33871},[129,133],[15,51547,51549],{"className":51548},[137],[15,51550,51552,51572],{"className":51551},[141,142],[15,51553,51555,51569],{"className":51554},[146],[15,51556,51558],{"className":51557,"style":151},[150],[15,51559,51560,51563],{"style":50900},[15,51561],{"className":51562,"style":159},[158],[15,51564,51566],{"className":51565},[163,164,165,166],[15,51567,48],{"className":51568,"style":170},[129,133,166],[15,51570,175],{"className":51571},[174],[15,51573,51575],{"className":51574},[146],[15,51576,51578],{"className":51577,"style":182},[150],[15,51579],{},[15,51581,53],{"className":51582},[188],[15,51584,51586],{"className":51585},[129,48500],[15,51587,51589],{"className":51588},[141],[15,51590,51592],{"className":51591},[146],[15,51593,51595,51603],{"className":51594,"style":48496},[150],[15,51596,51597,51600],{"style":2287},[15,51598],{"className":51599,"style":1823},[158],[15,51601,48],{"className":51602,"style":170},[129,133],[15,51604,51605,51608],{"style":48520},[15,51606],{"className":51607,"style":1823},[158],[15,51609,51611],{"className":51610,"style":48528},[48527],[15,51612,48483],{"className":51613},[129],[15,51615,59],{"className":51616},[195],[15,51618,107],{"className":51619},[129],[575,51621,51623],{"id":51622},"_22-一个先验视角","2.2 一个先验视角",[11,51625,51626,51627,51655,51656,51684],{},"极小极大风险有一个贝叶斯下界。对任意 ",[15,51628,51630,51643],{"className":51629,"translate":19},[23],[15,51631,51633],{"className":51632},[27],[29,51634,51635],{"xmlns":31},[34,51636,51637,51641],{},[37,51638,51639],{},[43,51640,30875],{"mathvariant":65},[109,51642,31304],{"encoding":111},[15,51644,51646],{"className":51645,"ariaHidden":72},[116],[15,51647,51649,51652],{"className":51648},[120],[15,51650],{"className":51651,"style":610},[124],[15,51653,30875],{"className":51654},[129]," 上的先验 ",[15,51657,51659,51672],{"className":51658,"translate":19},[23],[15,51660,51662],{"className":51661},[27],[29,51663,51664],{"xmlns":31},[34,51665,51666,51670],{},[37,51667,51668],{},[43,51669,15763],{},[109,51671,16379],{"encoding":111},[15,51673,51675],{"className":51674,"ariaHidden":72},[116],[15,51676,51678,51681],{"className":51677},[120],[15,51679],{"className":51680,"style":4383},[124],[15,51682,15763],{"className":51683,"style":887},[129,133],"，记贝叶斯风险为",[15,51686,51688],{"className":51687,"translate":19},[18],[15,51689,51691,51764],{"className":51690,"translate":19},[23],[15,51692,51694],{"className":51693},[27],[29,51695,51696],{"xmlns":31,"display":32},[34,51697,51698,51761],{},[37,51699,51700,51706,51708,51710,51712,51714,51728,51731,51737,51739,51745,51747,51749,51751,51753,51755,51757,51759],{},[40,51701,51702,51704],{},[43,51703,33895],{"mathvariant":28910},[43,51705,39326],{},[50,51707,53],{"stretchy":52},[43,51709,15763],{},[50,51711,59],{"stretchy":52},[50,51713,62],{},[2140,51715,51716,51722],{},[37,51717,51718,51720],{},[43,51719,48708],{},[50,51721,69],{},[46847,51723,51724,51726],{"accent":72},[43,51725,48],{},[50,51727,48483],{},[50,51729,51730],{},"∫",[40,51732,51733,51735],{},[43,51734,104],{},[43,51736,48],{},[50,51738,53],{"stretchy":52},[46847,51740,51741,51743],{"accent":72},[43,51742,48],{},[50,51744,48483],{},[50,51746,59],{"stretchy":52},[6856,51748,15788],{},[43,51750,15763],{},[50,51752,53],{"stretchy":52},[43,51754,10376],{},[43,51756,48],{},[50,51758,59],{"stretchy":52},[43,51760,107],{"mathvariant":65},[109,51762,51763],{"encoding":111},"\\mathfrak B_n(\\pi)=\\inf_{\\hat\\theta}\\int R_\\theta(\\hat\\theta)\\,\\pi(d\\theta).",[15,51765,51767,51831],{"className":51766,"ariaHidden":72},[116],[15,51768,51770,51773,51813,51816,51819,51822,51825,51828],{"className":51769},[120],[15,51771],{"className":51772,"style":125},[124],[15,51774,51776,51779],{"className":51775},[129],[15,51777,33895],{"className":51778},[129,28937],[15,51780,51782],{"className":51781},[137],[15,51783,51785,51805],{"className":51784},[141,142],[15,51786,51788,51802],{"className":51787},[146],[15,51789,51791],{"className":51790,"style":1163},[150],[15,51792,51793,51796],{"style":154},[15,51794],{"className":51795,"style":159},[158],[15,51797,51799],{"className":51798},[163,164,165,166],[15,51800,39326],{"className":51801},[129,133,166],[15,51803,175],{"className":51804},[174],[15,51806,51808],{"className":51807},[146],[15,51809,51811],{"className":51810,"style":182},[150],[15,51812],{},[15,51814,53],{"className":51815},[188],[15,51817,15763],{"className":51818,"style":887},[129,133],[15,51820,59],{"className":51821},[195],[15,51823],{"className":51824,"style":199},[93],[15,51826,62],{"className":51827},[203],[15,51829],{"className":51830,"style":199},[93],[15,51832,51834,51838,51915,51918,51922,51925,51965,51968,51999,52002,52005,52008,52011,52014,52017,52020],{"className":51833},[120],[15,51835],{"className":51836,"style":51837},[124],"height:2.2965em;vertical-align:-0.9365em;",[15,51839,51841],{"className":51840},[217,1874],[15,51842,51844,51907],{"className":51843},[141,142],[15,51845,51847,51904],{"className":51846},[146],[15,51848,51850,51892],{"className":51849,"style":522},[150],[15,51851,51852,51855],{"style":48794},[15,51853],{"className":51854,"style":1823},[158],[15,51856,51858],{"className":51857},[163,164,165,166],[15,51859,51861],{"className":51860},[129,166],[15,51862,51864],{"className":51863},[129,48500,166],[15,51865,51867],{"className":51866},[141],[15,51868,51870],{"className":51869},[146],[15,51871,51873,51881],{"className":51872,"style":48496},[150],[15,51874,51875,51878],{"style":48818},[15,51876],{"className":51877,"style":159},[158],[15,51879,48],{"className":51880,"style":170},[129,133,166],[15,51882,51883,51886],{"style":48827},[15,51884],{"className":51885,"style":159},[158],[15,51887,51889],{"className":51888,"style":48528},[48527],[15,51890,48483],{"className":51891},[129,166],[15,51893,51894,51897],{"style":2287},[15,51895],{"className":51896,"style":1823},[158],[15,51898,51899],{},[15,51900,48847,51902],{"className":51901},[217],[15,51903,36569],{"style":48850},[15,51905,175],{"className":51906},[174],[15,51908,51910],{"className":51909},[146],[15,51911,51913],{"className":51912,"style":48860},[150],[15,51914],{},[15,51916],{"className":51917,"style":262},[93],[15,51919,51730],{"className":51920,"style":51921},[217,1918,1919],"margin-right:0.4445em;position:relative;top:-0.0011em;",[15,51923],{"className":51924,"style":262},[93],[15,51926,51928,51931],{"className":51927},[129],[15,51929,104],{"className":51930,"style":33871},[129,133],[15,51932,51934],{"className":51933},[137],[15,51935,51937,51957],{"className":51936},[141,142],[15,51938,51940,51954],{"className":51939},[146],[15,51941,51943],{"className":51942,"style":151},[150],[15,51944,51945,51948],{"style":50900},[15,51946],{"className":51947,"style":159},[158],[15,51949,51951],{"className":51950},[163,164,165,166],[15,51952,48],{"className":51953,"style":170},[129,133,166],[15,51955,175],{"className":51956},[174],[15,51958,51960],{"className":51959},[146],[15,51961,51963],{"className":51962,"style":182},[150],[15,51964],{},[15,51966,53],{"className":51967},[188],[15,51969,51971],{"className":51970},[129,48500],[15,51972,51974],{"className":51973},[141],[15,51975,51977],{"className":51976},[146],[15,51978,51980,51988],{"className":51979,"style":48496},[150],[15,51981,51982,51985],{"style":2287},[15,51983],{"className":51984,"style":1823},[158],[15,51986,48],{"className":51987,"style":170},[129,133],[15,51989,51990,51993],{"style":48520},[15,51991],{"className":51992,"style":1823},[158],[15,51994,51996],{"className":51995,"style":48528},[48527],[15,51997,48483],{"className":51998},[129],[15,52000,59],{"className":52001},[195],[15,52003],{"className":52004,"style":262},[93],[15,52006,15763],{"className":52007,"style":887},[129,133],[15,52009,53],{"className":52010},[188],[15,52012,10376],{"className":52013},[129,133],[15,52015,48],{"className":52016,"style":170},[129,133],[15,52018,59],{"className":52019},[195],[15,52021,107],{"className":52022},[129],[11,52024,52025],{},"由于「积分 ≤ 最大值」，",[15,52027,52029],{"className":52028,"translate":19},[18],[15,52030,52032,52076],{"className":52031,"translate":19},[23],[15,52033,52035],{"className":52034},[27],[29,52036,52037],{"xmlns":31,"display":32},[34,52038,52039,52073],{},[37,52040,52041,52047,52049,52051,52053,52055,52061,52063,52065,52067,52069,52071],{},[40,52042,52043,52045],{},[43,52044,33895],{"mathvariant":28910},[43,52046,39326],{},[50,52048,53],{"stretchy":52},[43,52050,15763],{},[50,52052,59],{"stretchy":52},[50,52054,2717],{},[40,52056,52057,52059],{},[43,52058,43246],{"mathvariant":28910},[43,52060,39326],{},[50,52062,53],{"stretchy":52},[43,52064,30875],{"mathvariant":65},[50,52066,91],{"separator":72},[43,52068,27529],{"mathvariant":65},[50,52070,59],{"stretchy":52},[43,52072,107],{"mathvariant":65},[109,52074,52075],{"encoding":111},"\\mathfrak B_n(\\pi)\\le \\mathfrak M_n(\\Theta,\\ell).",[15,52077,52079,52143],{"className":52078,"ariaHidden":72},[116],[15,52080,52082,52085,52125,52128,52131,52134,52137,52140],{"className":52081},[120],[15,52083],{"className":52084,"style":125},[124],[15,52086,52088,52091],{"className":52087},[129],[15,52089,33895],{"className":52090},[129,28937],[15,52092,52094],{"className":52093},[137],[15,52095,52097,52117],{"className":52096},[141,142],[15,52098,52100,52114],{"className":52099},[146],[15,52101,52103],{"className":52102,"style":1163},[150],[15,52104,52105,52108],{"style":154},[15,52106],{"className":52107,"style":159},[158],[15,52109,52111],{"className":52110},[163,164,165,166],[15,52112,39326],{"className":52113},[129,133,166],[15,52115,175],{"className":52116},[174],[15,52118,52120],{"className":52119},[146],[15,52121,52123],{"className":52122,"style":182},[150],[15,52124],{},[15,52126,53],{"className":52127},[188],[15,52129,15763],{"className":52130,"style":887},[129,133],[15,52132,59],{"className":52133},[195],[15,52135],{"className":52136,"style":199},[93],[15,52138,2717],{"className":52139},[203],[15,52141],{"className":52142,"style":199},[93],[15,52144,52146,52149,52189,52192,52195,52198,52201,52204,52207],{"className":52145},[120],[15,52147],{"className":52148,"style":125},[124],[15,52150,52152,52155],{"className":52151},[129],[15,52153,43246],{"className":52154},[129,28937],[15,52156,52158],{"className":52157},[137],[15,52159,52161,52181],{"className":52160},[141,142],[15,52162,52164,52178],{"className":52163},[146],[15,52165,52167],{"className":52166,"style":1163},[150],[15,52168,52169,52172],{"style":154},[15,52170],{"className":52171,"style":159},[158],[15,52173,52175],{"className":52174},[163,164,165,166],[15,52176,39326],{"className":52177},[129,133,166],[15,52179,175],{"className":52180},[174],[15,52182,52184],{"className":52183},[146],[15,52185,52187],{"className":52186,"style":182},[150],[15,52188],{},[15,52190,53],{"className":52191},[188],[15,52193,30875],{"className":52194},[129],[15,52196,91],{"className":52197},[254],[15,52199],{"className":52200,"style":262},[93],[15,52202,27529],{"className":52203},[129],[15,52205,59],{"className":52206},[195],[15,52208,107],{"className":52209},[129],[11,52211,52212,52213,52216],{},"所以，",[698,52214,52215],{},"任何一个先验下的贝叶斯风险，都是极小极大风险的下界","。这个观察非常有用：下界证明往往先选一个先验（通常是有限集上的均匀分布），再算贝叶斯风险。Le Cam 与 Fano 本质上都是在做这件事，只是把贝叶斯风险进一步转化成了检验误差。",[575,52218,52220],{"id":52219},"_23-归约到有限子问题","2.3 归约到有限子问题",[11,52222,52223,52224,7610],{},"对任意有限子集 ",[15,52225,52227,52273],{"className":52226,"translate":19},[23],[15,52228,52230],{"className":52229},[27],[29,52231,52232],{"xmlns":31},[34,52233,52234,52270],{},[37,52235,52236,52242,52244,52246,52252,52254,52256,52258,52264,52266,52268],{},[40,52237,52238,52240],{},[43,52239,30875],{"mathvariant":65},[308,52241,541],{},[50,52243,62],{},[50,52245,636],{"stretchy":52},[40,52247,52248,52250],{},[43,52249,48],{},[308,52251,318],{},[50,52253,91],{"separator":72},[50,52255,335],{},[50,52257,91],{"separator":72},[40,52259,52260,52262],{},[43,52261,48],{},[43,52263,43246],{},[50,52265,645],{"stretchy":52},[50,52267,5647],{},[43,52269,30875],{"mathvariant":65},[109,52271,52272],{"encoding":111},"\\Theta_0=\\{\\theta_1,\\dots,\\theta_M\\}\\subseteq\\Theta",[15,52274,52276,52331,52450],{"className":52275,"ariaHidden":72},[116],[15,52277,52279,52282,52322,52325,52328],{"className":52278},[120],[15,52280],{"className":52281,"style":20229},[124],[15,52283,52285,52288],{"className":52284},[129],[15,52286,30875],{"className":52287},[129],[15,52289,52291],{"className":52290},[137],[15,52292,52294,52314],{"className":52293},[141,142],[15,52295,52297,52311],{"className":52296},[146],[15,52298,52300],{"className":52299,"style":1051},[150],[15,52301,52302,52305],{"style":154},[15,52303],{"className":52304,"style":159},[158],[15,52306,52308],{"className":52307},[163,164,165,166],[15,52309,541],{"className":52310},[129,166],[15,52312,175],{"className":52313},[174],[15,52315,52317],{"className":52316},[146],[15,52318,52320],{"className":52319,"style":182},[150],[15,52321],{},[15,52323],{"className":52324,"style":199},[93],[15,52326,62],{"className":52327},[203],[15,52329],{"className":52330,"style":199},[93],[15,52332,52334,52337,52340,52380,52383,52386,52389,52392,52395,52398,52438,52441,52444,52447],{"className":52333},[120],[15,52335],{"className":52336,"style":125},[124],[15,52338,636],{"className":52339},[188],[15,52341,52343,52346],{"className":52342},[129],[15,52344,48],{"className":52345,"style":170},[129,133],[15,52347,52349],{"className":52348},[137],[15,52350,52352,52372],{"className":52351},[141,142],[15,52353,52355,52369],{"className":52354},[146],[15,52356,52358],{"className":52357,"style":1051},[150],[15,52359,52360,52363],{"style":48293},[15,52361],{"className":52362,"style":159},[158],[15,52364,52366],{"className":52365},[163,164,165,166],[15,52367,318],{"className":52368},[129,166],[15,52370,175],{"className":52371},[174],[15,52373,52375],{"className":52374},[146],[15,52376,52378],{"className":52377,"style":182},[150],[15,52379],{},[15,52381,91],{"className":52382},[254],[15,52384],{"className":52385,"style":262},[93],[15,52387,335],{"className":52388},[449],[15,52390],{"className":52391,"style":262},[93],[15,52393,91],{"className":52394},[254],[15,52396],{"className":52397,"style":262},[93],[15,52399,52401,52404],{"className":52400},[129],[15,52402,48],{"className":52403,"style":170},[129,133],[15,52405,52407],{"className":52406},[137],[15,52408,52410,52430],{"className":52409},[141,142],[15,52411,52413,52427],{"className":52412},[146],[15,52414,52416],{"className":52415,"style":1525},[150],[15,52417,52418,52421],{"style":48293},[15,52419],{"className":52420,"style":159},[158],[15,52422,52424],{"className":52423},[163,164,165,166],[15,52425,43246],{"className":52426,"style":12597},[129,133,166],[15,52428,175],{"className":52429},[174],[15,52431,52433],{"className":52432},[146],[15,52434,52436],{"className":52435,"style":182},[150],[15,52437],{},[15,52439,645],{"className":52440},[195],[15,52442],{"className":52443,"style":199},[93],[15,52445,5647],{"className":52446},[203],[15,52448],{"className":52449,"style":199},[93],[15,52451,52453,52456],{"className":52452},[120],[15,52454],{"className":52455,"style":610},[124],[15,52457,30875],{"className":52458},[129],[15,52460,52462],{"className":52461,"translate":19},[18],[15,52463,52465,52517],{"className":52464,"translate":19},[23],[15,52466,52468],{"className":52467},[27],[29,52469,52470],{"xmlns":31,"display":32},[34,52471,52472,52514],{},[37,52473,52474,52480,52482,52484,52486,52488,52490,52492,52498,52500,52506,52508,52510,52512],{},[40,52475,52476,52478],{},[43,52477,43246],{"mathvariant":28910},[43,52479,39326],{},[50,52481,53],{"stretchy":52},[43,52483,30875],{"mathvariant":65},[50,52485,91],{"separator":72},[43,52487,27529],{"mathvariant":65},[50,52489,59],{"stretchy":52},[50,52491,7473],{},[40,52493,52494,52496],{},[43,52495,43246],{"mathvariant":28910},[43,52497,39326],{},[50,52499,53],{"stretchy":52},[40,52501,52502,52504],{},[43,52503,30875],{"mathvariant":65},[308,52505,541],{},[50,52507,91],{"separator":72},[43,52509,27529],{"mathvariant":65},[50,52511,59],{"stretchy":52},[43,52513,107],{"mathvariant":65},[109,52515,52516],{"encoding":111},"\\mathfrak M_n(\\Theta,\\ell)\\ge \\mathfrak M_n(\\Theta_0,\\ell).",[15,52518,52520,52593],{"className":52519,"ariaHidden":72},[116],[15,52521,52523,52526,52566,52569,52572,52575,52578,52581,52584,52587,52590],{"className":52522},[120],[15,52524],{"className":52525,"style":125},[124],[15,52527,52529,52532],{"className":52528},[129],[15,52530,43246],{"className":52531},[129,28937],[15,52533,52535],{"className":52534},[137],[15,52536,52538,52558],{"className":52537},[141,142],[15,52539,52541,52555],{"className":52540},[146],[15,52542,52544],{"className":52543,"style":1163},[150],[15,52545,52546,52549],{"style":154},[15,52547],{"className":52548,"style":159},[158],[15,52550,52552],{"className":52551},[163,164,165,166],[15,52553,39326],{"className":52554},[129,133,166],[15,52556,175],{"className":52557},[174],[15,52559,52561],{"className":52560},[146],[15,52562,52564],{"className":52563,"style":182},[150],[15,52565],{},[15,52567,53],{"className":52568},[188],[15,52570,30875],{"className":52571},[129],[15,52573,91],{"className":52574},[254],[15,52576],{"className":52577,"style":262},[93],[15,52579,27529],{"className":52580},[129],[15,52582,59],{"className":52583},[195],[15,52585],{"className":52586,"style":199},[93],[15,52588,7473],{"className":52589},[203],[15,52591],{"className":52592,"style":199},[93],[15,52594,52596,52599,52639,52642,52682,52685,52688,52691,52694],{"className":52595},[120],[15,52597],{"className":52598,"style":125},[124],[15,52600,52602,52605],{"className":52601},[129],[15,52603,43246],{"className":52604},[129,28937],[15,52606,52608],{"className":52607},[137],[15,52609,52611,52631],{"className":52610},[141,142],[15,52612,52614,52628],{"className":52613},[146],[15,52615,52617],{"className":52616,"style":1163},[150],[15,52618,52619,52622],{"style":154},[15,52620],{"className":52621,"style":159},[158],[15,52623,52625],{"className":52624},[163,164,165,166],[15,52626,39326],{"className":52627},[129,133,166],[15,52629,175],{"className":52630},[174],[15,52632,52634],{"className":52633},[146],[15,52635,52637],{"className":52636,"style":182},[150],[15,52638],{},[15,52640,53],{"className":52641},[188],[15,52643,52645,52648],{"className":52644},[129],[15,52646,30875],{"className":52647},[129],[15,52649,52651],{"className":52650},[137],[15,52652,52654,52674],{"className":52653},[141,142],[15,52655,52657,52671],{"className":52656},[146],[15,52658,52660],{"className":52659,"style":1051},[150],[15,52661,52662,52665],{"style":154},[15,52663],{"className":52664,"style":159},[158],[15,52666,52668],{"className":52667},[163,164,165,166],[15,52669,541],{"className":52670},[129,166],[15,52672,175],{"className":52673},[174],[15,52675,52677],{"className":52676},[146],[15,52678,52680],{"className":52679,"style":182},[150],[15,52681],{},[15,52683,91],{"className":52684},[254],[15,52686],{"className":52687,"style":262},[93],[15,52689,27529],{"className":52690},[129],[15,52692,59],{"className":52693},[195],[15,52695,107],{"className":52696},[129],[11,52698,52699,52700,52728,52729,52799,52800,52828,52829,52857],{},"所以我们只需要在一个有限问题上证明下界。有限问题可以看成一场 ",[15,52701,52703,52716],{"className":52702,"translate":19},[23],[15,52704,52706],{"className":52705},[27],[29,52707,52708],{"xmlns":31},[34,52709,52710,52714],{},[37,52711,52712],{},[43,52713,43246],{},[109,52715,43246],{"encoding":111},[15,52717,52719],{"className":52718,"ariaHidden":72},[116],[15,52720,52722,52725],{"className":52721},[120],[15,52723],{"className":52724,"style":610},[124],[15,52726,43246],{"className":52727,"style":12597},[129,133]," 元检验：自然以某个先验选出 ",[15,52730,52732,52750],{"className":52731,"translate":19},[23],[15,52733,52735],{"className":52734},[27],[29,52736,52737],{"xmlns":31},[34,52738,52739,52747],{},[37,52740,52741],{},[40,52742,52743,52745],{},[43,52744,48],{},[43,52746,34524],{},[109,52748,52749],{"encoding":111},"\\theta_K",[15,52751,52753],{"className":52752,"ariaHidden":72},[116],[15,52754,52756,52759],{"className":52755},[120],[15,52757],{"className":52758,"style":2183},[124],[15,52760,52762,52765],{"className":52761},[129],[15,52763,48],{"className":52764,"style":170},[129,133],[15,52766,52768],{"className":52767},[137],[15,52769,52771,52791],{"className":52770},[141,142],[15,52772,52774,52788],{"className":52773},[146],[15,52775,52777],{"className":52776,"style":1525},[150],[15,52778,52779,52782],{"style":48293},[15,52780],{"className":52781,"style":159},[158],[15,52783,52785],{"className":52784},[163,164,165,166],[15,52786,34524],{"className":52787,"style":5665},[129,133,166],[15,52789,175],{"className":52790},[174],[15,52792,52794],{"className":52793},[146],[15,52795,52797],{"className":52796,"style":182},[150],[15,52798],{},"，统计学家看到数据后要猜出 ",[15,52801,52803,52816],{"className":52802,"translate":19},[23],[15,52804,52806],{"className":52805},[27],[29,52807,52808],{"xmlns":31},[34,52809,52810,52814],{},[37,52811,52812],{},[43,52813,34524],{},[109,52815,34524],{"encoding":111},[15,52817,52819],{"className":52818,"ariaHidden":72},[116],[15,52820,52822,52825],{"className":52821},[120],[15,52823],{"className":52824,"style":610},[124],[15,52826,34524],{"className":52827,"style":5665},[129,133],"。如果连「猜出是哪一个」都做不到，那么「把 ",[15,52830,52832,52845],{"className":52831,"translate":19},[23],[15,52833,52835],{"className":52834},[27],[29,52836,52837],{"xmlns":31},[34,52838,52839,52843],{},[37,52840,52841],{},[43,52842,48],{},[109,52844,512],{"encoding":111},[15,52846,52848],{"className":52847,"ariaHidden":72},[116],[15,52849,52851,52854],{"className":52850},[120],[15,52852],{"className":52853,"style":522},[124],[15,52855,48],{"className":52856,"style":170},[129,133]," 估到很小的损失」当然也做不到。",[11,52859,52860],{},"把这句话写成不等式，就是 Le Cam 与 Fano 的工具。",[570,52862,52864],{"id":52863},"三le-cam-两点方法","三、Le Cam 两点方法",[575,52866,52868],{"id":52867},"_31-定理与证明","3.1 定理与证明",[11,52870,52871],{},"先看最简单的情形：只有两个候选参数。",[6559,52873,52874,53282],{},[11,52875,52876,52879,52880,7610,53026,20198,53192,53221,53222,7610],{},[698,52877,52878],{},"定理 3.1（Le Cam 两点方法）"," 设 ",[15,52881,52883,52913],{"className":52882,"translate":19},[23],[15,52884,52886],{"className":52885},[27],[29,52887,52888],{"xmlns":31},[34,52889,52890,52910],{},[37,52891,52892,52898,52900,52906,52908],{},[40,52893,52894,52896],{},[43,52895,48],{},[308,52897,541],{},[50,52899,91],{"separator":72},[40,52901,52902,52904],{},[43,52903,48],{},[308,52905,318],{},[50,52907,100],{},[43,52909,30875],{"mathvariant":65},[109,52911,52912],{"encoding":111},"\\theta_0,\\theta_1\\in\\Theta",[15,52914,52916,53017],{"className":52915,"ariaHidden":72},[116],[15,52917,52919,52922,52962,52965,52968,53008,53011,53014],{"className":52918},[120],[15,52920],{"className":52921,"style":1682},[124],[15,52923,52925,52928],{"className":52924},[129],[15,52926,48],{"className":52927,"style":170},[129,133],[15,52929,52931],{"className":52930},[137],[15,52932,52934,52954],{"className":52933},[141,142],[15,52935,52937,52951],{"className":52936},[146],[15,52938,52940],{"className":52939,"style":1051},[150],[15,52941,52942,52945],{"style":48293},[15,52943],{"className":52944,"style":159},[158],[15,52946,52948],{"className":52947},[163,164,165,166],[15,52949,541],{"className":52950},[129,166],[15,52952,175],{"className":52953},[174],[15,52955,52957],{"className":52956},[146],[15,52958,52960],{"className":52959,"style":182},[150],[15,52961],{},[15,52963,91],{"className":52964},[254],[15,52966],{"className":52967,"style":262},[93],[15,52969,52971,52974],{"className":52970},[129],[15,52972,48],{"className":52973,"style":170},[129,133],[15,52975,52977],{"className":52976},[137],[15,52978,52980,53000],{"className":52979},[141,142],[15,52981,52983,52997],{"className":52982},[146],[15,52984,52986],{"className":52985,"style":1051},[150],[15,52987,52988,52991],{"style":48293},[15,52989],{"className":52990,"style":159},[158],[15,52992,52994],{"className":52993},[163,164,165,166],[15,52995,318],{"className":52996},[129,166],[15,52998,175],{"className":52999},[174],[15,53001,53003],{"className":53002},[146],[15,53004,53006],{"className":53005,"style":182},[150],[15,53007],{},[15,53009],{"className":53010,"style":199},[93],[15,53012,100],{"className":53013},[203],[15,53015],{"className":53016,"style":199},[93],[15,53018,53020,53023],{"className":53019},[120],[15,53021],{"className":53022,"style":610},[124],[15,53024,30875],{"className":53025},[129],[15,53027,53029,53067],{"className":53028,"translate":19},[23],[15,53030,53032],{"className":53031},[27],[29,53033,53034],{"xmlns":31},[34,53035,53036,53064],{},[37,53037,53038,53040,53042,53048,53050,53056,53058,53060,53062],{},[43,53039,27529],{"mathvariant":65},[50,53041,53],{"stretchy":52},[40,53043,53044,53046],{},[43,53045,48],{},[308,53047,541],{},[50,53049,91],{"separator":72},[40,53051,53052,53054],{},[43,53053,48],{},[308,53055,318],{},[50,53057,59],{"stretchy":52},[50,53059,7473],{},[308,53061,310],{},[43,53063,29575],{},[109,53065,53066],{"encoding":111},"\\ell(\\theta_0,\\theta_1)\\ge 2\\delta",[15,53068,53070,53180],{"className":53069,"ariaHidden":72},[116],[15,53071,53073,53076,53079,53082,53122,53125,53128,53168,53171,53174,53177],{"className":53072},[120],[15,53074],{"className":53075,"style":125},[124],[15,53077,27529],{"className":53078},[129],[15,53080,53],{"className":53081},[188],[15,53083,53085,53088],{"className":53084},[129],[15,53086,48],{"className":53087,"style":170},[129,133],[15,53089,53091],{"className":53090},[137],[15,53092,53094,53114],{"className":53093},[141,142],[15,53095,53097,53111],{"className":53096},[146],[15,53098,53100],{"className":53099,"style":1051},[150],[15,53101,53102,53105],{"style":48293},[15,53103],{"className":53104,"style":159},[158],[15,53106,53108],{"className":53107},[163,164,165,166],[15,53109,541],{"className":53110},[129,166],[15,53112,175],{"className":53113},[174],[15,53115,53117],{"className":53116},[146],[15,53118,53120],{"className":53119,"style":182},[150],[15,53121],{},[15,53123,91],{"className":53124},[254],[15,53126],{"className":53127,"style":262},[93],[15,53129,53131,53134],{"className":53130},[129],[15,53132,48],{"className":53133,"style":170},[129,133],[15,53135,53137],{"className":53136},[137],[15,53138,53140,53160],{"className":53139},[141,142],[15,53141,53143,53157],{"className":53142},[146],[15,53144,53146],{"className":53145,"style":1051},[150],[15,53147,53148,53151],{"style":48293},[15,53149],{"className":53150,"style":159},[158],[15,53152,53154],{"className":53153},[163,164,165,166],[15,53155,318],{"className":53156},[129,166],[15,53158,175],{"className":53159},[174],[15,53161,53163],{"className":53162},[146],[15,53164,53166],{"className":53165,"style":182},[150],[15,53167],{},[15,53169,59],{"className":53170},[195],[15,53172],{"className":53173,"style":199},[93],[15,53175,7473],{"className":53176},[203],[15,53178],{"className":53179,"style":199},[93],[15,53181,53183,53186,53189],{"className":53182},[120],[15,53184],{"className":53185,"style":522},[124],[15,53187,310],{"className":53188},[129],[15,53190,29575],{"className":53191,"style":29856},[129,133],[15,53193,53195,53209],{"className":53194,"translate":19},[23],[15,53196,53198],{"className":53197},[27],[29,53199,53200],{"xmlns":31},[34,53201,53202,53206],{},[37,53203,53204],{},[43,53205,27529],{"mathvariant":65},[109,53207,53208],{"encoding":111},"\\ell",[15,53210,53212],{"className":53211,"ariaHidden":72},[116],[15,53213,53215,53218],{"className":53214},[120],[15,53216],{"className":53217,"style":522},[124],[15,53219,27529],{"className":53220},[129]," 满足三角不等式。则对任意估计量 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E_{P_0}\\ell(\\theta_0,\\hat\\theta)+\\mathbb E_{P_1}\\ell(\\theta_1,\\hat\\theta)\n\\ge \\\\ \\delta\\bigl[1+P_0(\\psi=1)-P_1(\\psi=1)\\bigr]\n\\ge \\\\ \\delta\\bigl(1-\\lVert P_0-P_1\\rVert_{\\mathrm{TV}}\\bigr).",[15,55788,55790,55971,56149,56152,56179,56240,56261,56322,56346,56349,56376,56434],{"className":55789,"ariaHidden":72},[116],[15,55791,55793,55796,55876,55879,55882,55922,55925,55928,55959,55962,55965,55968],{"className":55792},[120],[15,55794],{"className":55795,"style":55091},[124],[15,55797,55799,55802],{"className":55798},[129],[15,55800,26574],{"className":55801},[129,285],[15,55803,55805],{"className":55804},[137],[15,55806,55808,55868],{"className":55807},[141,142],[15,55809,55811,55865],{"className":55810},[146],[15,55812,55814],{"className":55813,"style":1525},[150],[15,55815,55816,55819],{"style":154},[15,55817],{"className":55818,"style":159},[158],[15,55820,55822],{"className":55821},[163,164,165,166],[15,55823,55825],{"className":55824},[129,166],[15,55826,55828,55831],{"className":55827},[129,166],[15,55829,1388],{"className":55830,"style":12449},[129,133,166],[15,55832,55834],{"className":55833},[137],[15,55835,55837,55857],{"className":55836},[141,142],[15,55838,55840,55854],{"className":55839},[146],[15,55841,55843],{"className":55842,"style":46995},[150],[15,55844,55845,55848],{"style":55141},[15,55846],{"className":55847,"style":15987},[158],[15,55849,55851],{"className":55850},[163,15991,229,166],[15,55852,541],{"className":55853},[129,166],[15,55855,175],{"className":55856},[174],[15,55858,55860],{"className":55859},[146],[15,55861,55863],{"className":55862,"style":16004},[150],[15,55864],{},[15,55866,175],{"className":55867},[174],[15,55869,55871],{"className":55870},[146],[15,55872,55874],{"className":55873,"style":55171},[150],[15,55875],{},[15,55877,27529],{"className":55878},[129],[15,55880,53],{"className":55881},[188],[15,55883,55885,55888],{"className":55884},[129],[15,55886,48],{"className":55887,"style":170},[129,133],[15,55889,55891],{"className":55890},[137],[15,55892,55894,55914],{"className":55893},[141,142],[15,55895,55897,55911],{"className":55896},[146],[15,55898,55900],{"className":55899,"style":1051},[150],[15,55901,55902,55905],{"style":48293},[15,55903],{"className":55904,"style":159},[158],[15,55906,55908],{"className":55907},[163,164,165,166],[15,55909,541],{"className":55910},[129,166],[15,55912,175],{"className":55913},[174],[15,55915,55917],{"className":55916},[146],[15,55918,55920],{"className":55919,"style":182},[150],[15,55921],{},[15,55923,91],{"className":55924},[254],[15,55926],{"className":55927,"style":262},[93],[15,55929,55931],{"className":55930},[129,48500],[15,55932,55934],{"className":55933},[141],[15,55935,55937],{"className":55936},[146],[15,55938,55940,55948],{"className":55939,"style":48496},[150],[15,55941,55942,55945],{"style":2287},[15,55943],{"className":55944,"style":1823},[158],[15,55946,48],{"className":55947,"style":170},[129,133],[15,55949,55950,55953],{"style":48520},[15,55951],{"className":55952,"style":1823},[158],[15,55954,55956],{"className":55955,"style":48528},[48527],[15,55957,48483],{"className":55958},[129],[15,55960,59],{"className":55961},[195],[15,55963],{"className":55964,"style":3277},[93],[15,55966,2682],{"className":55967},[2436],[15,55969],{"className":55970,"style":3277},[93],[15,55972,55974,55977,56057,56060,56063,56103,56106,56109,56140,56143,56146],{"className":55973},[120],[15,55975],{"className":55976,"style":55091},[124],[15,55978,55980,55983],{"className":55979},[129],[15,55981,26574],{"className":55982},[129,285],[15,55984,55986],{"className":55985},[137],[15,55987,55989,56049],{"className":55988},[141,142],[15,55990,55992,56046],{"className":55991},[146],[15,55993,55995],{"className":55994,"style":1525},[150],[15,55996,55997,56000],{"style":154},[15,55998],{"className":55999,"style":159},[158],[15,56001,56003],{"className":56002},[163,164,165,166],[15,56004,56006],{"className":56005},[129,166],[15,56007,56009,56012],{"className":56008},[129,166],[15,56010,1388],{"className":56011,"style":12449},[129,133,166],[15,56013,56015],{"className":56014},[137],[15,56016,56018,56038],{"className":56017},[141,142],[15,56019,56021,56035],{"className":56020},[146],[15,56022,56024],{"className":56023,"style":46995},[150],[15,56025,56026,56029],{"style":55141},[15,56027],{"className":56028,"style":15987},[158],[15,56030,56032],{"className":56031},[163,15991,229,166],[15,56033,318],{"className":56034},[129,166],[15,56036,175],{"className":56037},[174],[15,56039,56041],{"className":56040},[146],[15,56042,56044],{"className":56043,"style":16004},[150],[15,56045],{},[15,56047,175],{"className":56048},[174],[15,56050,56052],{"className":56051},[146],[15,56053,56055],{"className":56054,"style":55171},[150],[15,56056],{},[15,56058,27529],{"className":56059},[129],[15,56061,53],{"className":56062},[188],[15,56064,56066,56069],{"className":56065},[129],[15,56067,48],{"className":56068,"style":170},[129,133],[15,56070,56072],{"className":56071},[137],[15,56073,56075,56095],{"className":56074},[141,142],[15,56076,56078,56092],{"className":56077},[146],[15,56079,56081],{"className":56080,"style":1051},[150],[15,56082,56083,56086],{"style":48293},[15,56084],{"className":56085,"style":159},[158],[15,56087,56089],{"className":56088},[163,164,165,166],[15,56090,318],{"className":56091},[129,166],[15,56093,175],{"className":56094},[174],[15,56096,56098],{"className":56097},[146],[15,56099,56101],{"className":56100,"style":182},[150],[15,56102],{},[15,56104,91],{"className":56105},[254],[15,56107],{"className":56108,"style":262},[93],[15,56110,56112],{"className":56111},[129,48500],[15,56113,56115],{"className":56114},[141],[15,56116,56118],{"className":56117},[146],[15,56119,56121,56129],{"className":56120,"style":48496},[150],[15,56122,56123,56126],{"style":2287},[15,56124],{"className":56125,"style":1823},[158],[15,56127,48],{"className":56128,"style":170},[129,133],[15,56130,56131,56134],{"style":48520},[15,56132],{"className":56133,"style":1823},[158],[15,56135,56137],{"className":56136,"style":48528},[48527],[15,56138,48483],{"className":56139},[129],[15,56141,59],{"className":56142},[195],[15,56144],{"className":56145,"style":199},[93],[15,56147,7473],{"className":56148},[203],[15,56150],{"className":56151},[93,24721],[15,56153,56155,56158,56161,56167,56170,56173,56176],{"className":56154},[120],[15,56156],{"className":56157,"style":213},[124],[15,56159,29575],{"className":56160,"style":29856},[129,133],[15,56162,56164],{"className":56163},[188],[15,56165,1407],{"className":56166},[228,229],[15,56168,318],{"className":56169},[129],[15,56171],{"className":56172,"style":3277},[93],[15,56174,2682],{"className":56175},[2436],[15,56177],{"className":56178,"style":3277},[93],[15,56180,56182,56185,56225,56228,56231,56234,56237],{"className":56181},[120],[15,56183],{"className":56184,"style":125},[124],[15,56186,56188,56191],{"className":56187},[129],[15,56189,1388],{"className":56190,"style":12449},[129,133],[15,56192,56194],{"className":56193},[137],[15,56195,56197,56217],{"className":56196},[141,142],[15,56198,56200,56214],{"className":56199},[146],[15,56201,56203],{"className":56202,"style":1051},[150],[15,56204,56205,56208],{"style":34239},[15,56206],{"className":56207,"style":159},[158],[15,56209,56211],{"className":56210},[163,164,165,166],[15,56212,541],{"className":56213},[129,166],[15,56215,175],{"className":56216},[174],[15,56218,56220],{"className":56219},[146],[15,56221,56223],{"className":56222,"style":182},[150],[15,56224],{},[15,56226,53],{"className":56227},[188],[15,56229,54082],{"className":56230,"style":887},[129,133],[15,56232],{"className":56233,"style":199},[93],[15,56235,62],{"className":56236},[203],[15,56238],{"className":56239,"style":199},[93],[15,56241,56243,56246,56249,56252,56255,56258],{"className":56242},[120],[15,56244],{"className":56245,"style":125},[124],[15,56247,318],{"className":56248},[129],[15,56250,59],{"className":56251},[195],[15,56253],{"className":56254,"style":3277},[93],[15,56256,315],{"className":56257},[2436],[15,56259],{"className":56260,"style":3277},[93],[15,56262,56264,56267,56307,56310,56313,56316,56319],{"className":56263},[120],[15,56265],{"className":56266,"style":125},[124],[15,56268,56270,56273],{"className":56269},[129],[15,56271,1388],{"className":56272,"style":12449},[129,133],[15,56274,56276],{"className":56275},[137],[15,56277,56279,56299],{"className":56278},[141,142],[15,56280,56282,56296],{"className":56281},[146],[15,56283,56285],{"className":56284,"style":1051},[150],[15,56286,56287,56290],{"style":34239},[15,56288],{"className":56289,"style":159},[158],[15,56291,56293],{"className":56292},[163,164,165,166],[15,56294,318],{"className":56295},[129,166],[15,56297,175],{"className":56298},[174],[15,56300,56302],{"className":56301},[146],[15,56303,56305],{"className":56304,"style":182},[150],[15,56306],{},[15,56308,53],{"className":56309},[188],[15,56311,54082],{"className":56312,"style":887},[129,133],[15,56314],{"className":56315,"style":199},[93],[15,56317,62],{"className":56318},[203],[15,56320],{"className":56321,"style":199},[93],[15,56323,56325,56328,56331,56334,56340,56343],{"className":56324},[120],[15,56326],{"className":56327,"style":213},[124],[15,56329,318],{"className":56330},[129],[15,56332,59],{"className":56333},[195],[15,56335,56337],{"className":56336},[195],[15,56338,1423],{"className":56339},[228,229],[15,56341],{"className":56342,"style":199},[93],[15,56344,7473],{"className":56345},[203],[15,56347],{"className":56348},[93,24721],[15,56350,56352,56355,56358,56364,56367,56370,56373],{"className":56351},[120],[15,56353],{"className":56354,"style":213},[124],[15,56356,29575],{"className":56357,"style":29856},[129,133],[15,56359,56361],{"className":56360},[188],[15,56362,53],{"className":56363},[228,229],[15,56365,318],{"className":56366},[129],[15,56368],{"className":56369,"style":3277},[93],[15,56371,315],{"className":56372},[2436],[15,56374],{"className":56375,"style":3277},[93],[15,56377,56379,56382,56385,56425,56428,56431],{"className":56378},[120],[15,56380],{"className":56381,"style":125},[124],[15,56383,33823],{"className":56384},[188],[15,56386,56388,56391],{"className":56387},[129],[15,56389,1388],{"className":56390,"style":12449},[129,133],[15,56392,56394],{"className":56393},[137],[15,56395,56397,56417],{"className":56396},[141,142],[15,56398,56400,56414],{"className":56399},[146],[15,56401,56403],{"className":56402,"style":1051},[150],[15,56404,56405,56408],{"style":34239},[15,56406],{"className":56407,"style":159},[158],[15,56409,56411],{"className":56410},[163,164,165,166],[15,56412,541],{"className":56413},[129,166],[15,56415,175],{"className":56416},[174],[15,56418,56420],{"className":56419},[146],[15,56421,56423],{"className":56422,"style":182},[150],[15,56424],{},[15,56426],{"className":56427,"style":3277},[93],[15,56429,315],{"className":56430},[2436],[15,56432],{"className":56433,"style":3277},[93],[15,56435,56437,56440,56480,56526,56532],{"className":56436},[120],[15,56438],{"className":56439,"style":213},[124],[15,56441,56443,56446],{"className":56442},[129],[15,56444,1388],{"className":56445,"style":12449},[129,133],[15,56447,56449],{"className":56448},[137],[15,56450,56452,56472],{"className":56451},[141,142],[15,56453,56455,56469],{"className":56454},[146],[15,56456,56458],{"className":56457,"style":1051},[150],[15,56459,56460,56463],{"style":34239},[15,56461],{"className":56462,"style":159},[158],[15,56464,56466],{"className":56465},[163,164,165,166],[15,56467,318],{"className":56468},[129,166],[15,56470,175],{"className":56471},[174],[15,56473,56475],{"className":56474},[146],[15,56476,56478],{"className":56477,"style":182},[150],[15,56479],{},[15,56481,56483,56486],{"className":56482},[195],[15,56484,33823],{"className":56485},[195],[15,56487,56489],{"className":56488},[137],[15,56490,56492,56518],{"className":56491},[141,142],[15,56493,56495,56515],{"className":56494},[146],[15,56496,56498],{"className":56497,"style":1525},[150],[15,56499,56500,56503],{"style":154},[15,56501],{"className":56502,"style":159},[158],[15,56504,56506],{"className":56505},[163,164,165,166],[15,56507,56509],{"className":56508},[129,166],[15,56510,56512],{"className":56511},[129,166],[15,56513,54040],{"className":56514,"style":2246},[129,221,166],[15,56516,175],{"className":56517},[174],[15,56519,56521],{"className":56520},[146],[15,56522,56524],{"className":56523,"style":182},[150],[15,56525],{},[15,56527,56529],{"className":56528},[195],[15,56530,59],{"className":56531},[228,229],[15,56533,107],{"className":56534},[129],[11,56536,56537,56538,56619],{},"最后，最大值不小于平均值，得到 ",[15,56539,56541,56573],{"className":56540,"translate":19},[23],[15,56542,56544],{"className":56543},[27],[29,56545,56546],{"xmlns":31},[34,56547,56548,56570],{},[37,56549,56550,56552,56554,56556,56558,56564,56566,56568],{},[43,56551,29575],{},[50,56553,53],{"stretchy":52},[308,56555,318],{},[50,56557,315],{},[37,56559,56560,56562],{},[43,56561,53409],{"mathvariant":65},[43,56563,53412],{"mathvariant":65},[50,56565,59],{"stretchy":52},[43,56567,544],{"mathvariant":65},[308,56569,310],{},[109,56571,56572],{"encoding":111},"\\delta(1-\\mathrm{TV})\u002F2",[15,56574,56576,56600],{"className":56575,"ariaHidden":72},[116],[15,56577,56579,56582,56585,56588,56591,56594,56597],{"className":56578},[120],[15,56580],{"className":56581,"style":125},[124],[15,56583,29575],{"className":56584,"style":29856},[129,133],[15,56586,53],{"className":56587},[188],[15,56589,318],{"className":56590},[129],[15,56592],{"className":56593,"style":3277},[93],[15,56595,315],{"className":56596},[2436],[15,56598],{"className":56599,"style":3277},[93],[15,56601,56603,56606,56612,56615],{"className":56602},[120],[15,56604],{"className":56605,"style":125},[124],[15,56607,56609],{"className":56608},[129],[15,56610,54040],{"className":56611,"style":2246},[129,221],[15,56613,59],{"className":56614},[195],[15,56616,56618],{"className":56617},[129],"\u002F2","。证毕。",[11,56621,56622,56623,30641,56651,56679,56680,56736],{},"如果取 ",[15,56624,56626,56639],{"className":56625,"translate":19},[23],[15,56627,56629],{"className":56628},[27],[29,56630,56631],{"xmlns":31},[34,56632,56633,56637],{},[37,56634,56635],{},[308,56636,541],{},[109,56638,541],{"encoding":111},[15,56640,56642],{"className":56641,"ariaHidden":72},[116],[15,56643,56645,56648],{"className":56644},[120],[15,56646],{"className":56647,"style":4886},[124],[15,56649,541],{"className":56650},[129],[15,56652,56654,56667],{"className":56653,"translate":19},[23],[15,56655,56657],{"className":56656},[27],[29,56658,56659],{"xmlns":31},[34,56660,56661,56665],{},[37,56662,56663],{},[308,56664,318],{},[109,56666,318],{"encoding":111},[15,56668,56670],{"className":56669,"ariaHidden":72},[116],[15,56671,56673,56676],{"className":56672},[120],[15,56674],{"className":56675,"style":4886},[124],[15,56677,318],{"className":56678},[129]," 损失、",[15,56681,56683,56705],{"className":56682,"translate":19},[23],[15,56684,56686],{"className":56685},[27],[29,56687,56688],{"xmlns":31},[34,56689,56690,56702],{},[37,56691,56692,56694,56696,56698,56700],{},[43,56693,29575],{},[50,56695,62],{},[308,56697,318],{},[43,56699,544],{"mathvariant":65},[308,56701,310],{},[109,56703,56704],{"encoding":111},"\\delta=1\u002F2",[15,56706,56708,56726],{"className":56707,"ariaHidden":72},[116],[15,56709,56711,56714,56717,56720,56723],{"className":56710},[120],[15,56712],{"className":56713,"style":522},[124],[15,56715,29575],{"className":56716,"style":29856},[129,133],[15,56718],{"className":56719,"style":199},[93],[15,56721,62],{"className":56722},[203],[15,56724],{"className":56725,"style":199},[93],[15,56727,56729,56732],{"className":56728},[120],[15,56730],{"className":56731,"style":125},[124],[15,56733,56735],{"className":56734},[129],"1\u002F2","，就得到最常见的检验形式：",[15,56738,56740],{"className":56739,"translate":19},[18],[15,56741,56743,56877],{"className":56742,"translate":19},[23],[15,56744,56746],{"className":56745},[27],[29,56747,56748],{"xmlns":31,"display":32},[34,56749,56750,56874],{},[37,56751,56752,56766,56786,56788,56800,56802,56808,56810,56816,56818,56820,56826,56828,56830,56832,56834,56846,56848,56860,56870,56872],{},[2140,56753,56754,56760],{},[37,56755,56756,56758],{},[43,56757,48708],{},[50,56759,69],{},[46847,56761,56762,56764],{"accent":72},[43,56763,48],{},[50,56765,48483],{},[2140,56767,56768,56774],{},[37,56769,56770,56772],{},[43,56771,19705],{},[50,56773,69],{},[37,56775,56776,56778,56780,56782,56784],{},[43,56777,1459],{},[50,56779,62],{},[308,56781,541],{},[50,56783,91],{"separator":72},[308,56785,318],{},[6856,56787,6858],{},[23160,56789,56790,56792,56794],{},[43,56791,1388],{},[43,56793,1459],{},[37,56795,56796,56798],{},[50,56797,49367],{},[43,56799,39326],{},[50,56801,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[46847,56803,56804,56806],{"accent":72},[43,56805,48],{},[50,56807,48483],{},[50,56809,1418],{"mathvariant":65},[40,56811,56812,56814],{},[43,56813,48],{},[43,56815,1459],{},[50,56817,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,56819,7473],{},[1443,56821,56822,56824],{},[308,56823,318],{},[308,56825,310],{},[50,56827,53],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[308,56829,318],{},[50,56831,315],{},[50,56833,33823],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[23160,56835,56836,56838,56840],{},[43,56837,1388],{},[308,56839,541],{},[37,56841,56842,56844],{},[50,56843,49367],{},[43,56845,39326],{},[50,56847,315],{},[23160,56849,56850,56852,56854],{},[43,56851,1388],{},[308,56853,318],{},[37,56855,56856,56858],{},[50,56857,49367],{},[43,56859,39326],{},[40,56861,56862,56864],{},[50,56863,33823],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[37,56865,56866,56868],{},[43,56867,53409],{"mathvariant":65},[43,56869,53412],{"mathvariant":65},[50,56871,59],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[43,56873,107],{"mathvariant":65},[109,56875,56876],{"encoding":111},"\\inf_{\\hat\\theta}\\max_{i=0,1}\\ P_i^{\\otimes n}\\bigl(\\hat\\theta\\neq\\theta_i\\bigr)\n\\ge\n\\frac12\\Bigl(1-\\bigl\\lVert P_0^{\\otimes n}-P_1^{\\otimes n}\\bigr\\rVert_{\\mathrm{TV}}\\Bigr).",[15,56878,56880,57170,57231,57318,57427],{"className":56879,"ariaHidden":72},[116],[15,56881,56883,56887,56964,56967,57026,57029,57032,57091,57097,57128,57131,57167],{"className":56882},[120],[15,56884],{"className":56885,"style":56886},[124],"height:1.8944em;vertical-align:-0.9365em;",[15,56888,56890],{"className":56889},[217,1874],[15,56891,56893,56956],{"className":56892},[141,142],[15,56894,56896,56953],{"className":56895},[146],[15,56897,56899,56941],{"className":56898,"style":522},[150],[15,56900,56901,56904],{"style":48794},[15,56902],{"className":56903,"style":1823},[158],[15,56905,56907],{"className":56906},[163,164,165,166],[15,56908,56910],{"className":56909},[129,166],[15,56911,56913],{"className":56912},[129,48500,166],[15,56914,56916],{"className":56915},[141],[15,56917,56919],{"className":56918},[146],[15,56920,56922,56930],{"className":56921,"style":48496},[150],[15,56923,56924,56927],{"style":48818},[15,56925],{"className":56926,"style":159},[158],[15,56928,48],{"className":56929,"style":170},[129,133,166],[15,56931,56932,56935],{"style":48827},[15,56933],{"className":56934,"style":159},[158],[15,56936,56938],{"className":56937,"style":48528},[48527],[15,56939,48483],{"className":56940},[129,166],[15,56942,56943,56946],{"style":2287},[15,56944],{"className":56945,"style":1823},[158],[15,56947,56948],{},[15,56949,48847,56951],{"className":56950},[217],[15,56952,36569],{"style":48850},[15,56954,175],{"className":56955},[174],[15,56957,56959],{"className":56958},[146],[15,56960,56962],{"className":56961,"style":48860},[150],[15,56963],{},[15,56965],{"className":56966,"style":262},[93],[15,56968,56970],{"className":56969},[217,1874],[15,56971,56973,57018],{"className":56972},[141,142],[15,56974,56976,57015],{"className":56975},[146],[15,56977,56979,57005],{"className":56978,"style":4383},[150],[15,56980,56981,56984],{"style":53444},[15,56982],{"className":56983,"style":1823},[158],[15,56985,56987],{"className":56986},[163,164,165,166],[15,56988,56990,56993,56996,56999,57002],{"className":56989},[129,166],[15,56991,1459],{"className":56992},[129,133,166],[15,56994,62],{"className":56995},[203,166],[15,56997,541],{"className":56998},[129,166],[15,57000,91],{"className":57001},[254,166],[15,57003,318],{"className":57004},[129,166],[15,57006,57007,57010],{"style":2287},[15,57008],{"className":57009,"style":1823},[158],[15,57011,57012],{},[15,57013,19705],{"className":57014},[217],[15,57016,175],{"className":57017},[174],[15,57019,57021],{"className":57020},[146],[15,57022,57024],{"className":57023,"style":53488},[150],[15,57025],{},[15,57027,6858],{"className":57028},[93],[15,57030],{"className":57031,"style":262},[93],[15,57033,57035,57038],{"className":57034},[129],[15,57036,1388],{"className":57037,"style":12449},[129,133],[15,57039,57041],{"className":57040},[137],[15,57042,57044,57082],{"className":57043},[141,142],[15,57045,57047,57079],{"className":57046},[146],[15,57048,57050,57062],{"className":57049,"style":49566},[150],[15,57051,57053,57056],{"style":57052},"top:-2.433em;margin-left:-0.1389em;margin-right:0.05em;",[15,57054],{"className":57055,"style":159},[158],[15,57057,57059],{"className":57058},[163,164,165,166],[15,57060,1459],{"className":57061},[129,133,166],[15,57063,57064,57067],{"style":1272},[15,57065],{"className":57066,"style":159},[158],[15,57068,57070],{"className":57069},[163,164,165,166],[15,57071,57073,57076],{"className":57072},[129,166],[15,57074,49367],{"className":57075},[129,166],[15,57077,39326],{"className":57078},[129,133,166],[15,57080,175],{"className":57081},[174],[15,57083,57085],{"className":57084},[146],[15,57086,57089],{"className":57087,"style":57088},[150],"height:0.267em;",[15,57090],{},[15,57092,57094],{"className":57093},[188],[15,57095,53],{"className":57096},[228,229],[15,57098,57100],{"className":57099},[129,48500],[15,57101,57103],{"className":57102},[141],[15,57104,57106],{"className":57105},[146],[15,57107,57109,57117],{"className":57108,"style":48496},[150],[15,57110,57111,57114],{"style":2287},[15,57112],{"className":57113,"style":1823},[158],[15,57115,48],{"className":57116,"style":170},[129,133],[15,57118,57119,57122],{"style":48520},[15,57120],{"className":57121,"style":1823},[158],[15,57123,57125],{"className":57124,"style":48528},[48527],[15,57126,48483],{"className":57127},[129],[15,57129],{"className":57130,"style":199},[93],[15,57132,57134,57161,57164],{"className":57133},[203],[15,57135,57137],{"className":57136},[203],[15,57138,57140],{"className":57139},[129,1670],[15,57141,57143],{"className":57142},[1674],[15,57144,57146,57149,57158],{"className":57145},[1678],[15,57147],{"className":57148,"style":1682},[124],[15,57150,57152],{"className":57151},[1686],[15,57153,57155],{"className":57154},[129],[15,57156,1693],{"className":57157},[203],[15,57159],{"className":57160},[1697],[15,57162],{"className":57163},[93,1701],[15,57165,62],{"className":57166},[203],[15,57168],{"className":57169,"style":199},[93],[15,57171,57173,57176,57216,57222,57225,57228],{"className":57172},[120],[15,57174],{"className":57175,"style":213},[124],[15,57177,57179,57182],{"className":57178},[129],[15,57180,48],{"className":57181,"style":170},[129,133],[15,57183,57185],{"className":57184},[137],[15,57186,57188,57208],{"className":57187},[141,142],[15,57189,57191,57205],{"className":57190},[146],[15,57192,57194],{"className":57193,"style":1981},[150],[15,57195,57196,57199],{"style":48293},[15,57197],{"className":57198,"style":159},[158],[15,57200,57202],{"className":57201},[163,164,165,166],[15,57203,1459],{"className":57204},[129,133,166],[15,57206,175],{"className":57207},[174],[15,57209,57211],{"className":57210},[146],[15,57212,57214],{"className":57213,"style":182},[150],[15,57215],{},[15,57217,57219],{"className":57218},[195],[15,57220,59],{"className":57221},[228,229],[15,57223],{"className":57224,"style":199},[93],[15,57226,7473],{"className":57227},[203],[15,57229],{"className":57230,"style":199},[93],[15,57232,57234,57238,57300,57306,57309,57312,57315],{"className":57233},[120],[15,57235],{"className":57236,"style":57237},[124],"height:2.0074em;vertical-align:-0.686em;",[15,57239,57241,57244,57297],{"className":57240},[129],[15,57242],{"className":57243},[188,1803],[15,57245,57247],{"className":57246},[1443],[15,57248,57250,57289],{"className":57249},[141,142],[15,57251,57253,57286],{"className":57252},[146],[15,57254,57256,57267,57275],{"className":57255,"style":1816},[150],[15,57257,57258,57261],{"style":1819},[15,57259],{"className":57260,"style":1823},[158],[15,57262,57264],{"className":57263},[129],[15,57265,310],{"className":57266},[129],[15,57268,57269,57272],{"style":1832},[15,57270],{"className":57271,"style":1823},[158],[15,57273],{"className":57274,"style":1840},[1839],[15,57276,57277,57280],{"style":1843},[15,57278],{"className":57279,"style":1823},[158],[15,57281,57283],{"className":57282},[129],[15,57284,318],{"className":57285},[129],[15,57287,175],{"className":57288},[174],[15,57290,57292],{"className":57291},[146],[15,57293,57295],{"className":57294,"style":1862},[150],[15,57296],{},[15,57298],{"className":57299},[195,1803],[15,57301,57303],{"className":57302},[188],[15,57304,53],{"className":57305},[228,24461],[15,57307,318],{"className":57308},[129],[15,57310],{"className":57311,"style":3277},[93],[15,57313,315],{"className":57314},[2436],[15,57316],{"className":57317,"style":3277},[93],[15,57319,57321,57324,57361,57418,57421,57424],{"className":57320},[120],[15,57322],{"className":57323,"style":213},[124],[15,57325,57327],{"className":57326},[188],[15,57328,57330],{"className":57329},[228,4433],[15,57331,57333,57353],{"className":57332},[141,142],[15,57334,57336,57350],{"className":57335},[146],[15,57337,57339],{"className":57338,"style":4443},[150],[15,57340,57341,57344],{"style":4446},[15,57342],{"className":57343,"style":4450},[158],[15,57345,57346],{"style":53819},[4455,57347,57348],{"xmlns":4457,"width":53822,"height":73,"viewBox":53823},[4461,57349],{"d":53826},[15,57351,175],{"className":57352},[174],[15,57354,57356],{"className":57355},[146],[15,57357,57359],{"className":57358,"style":4473},[150],[15,57360],{},[15,57362,57364,57367],{"className":57363},[129],[15,57365,1388],{"className":57366,"style":12449},[129,133],[15,57368,57370],{"className":57369},[137],[15,57371,57373,57410],{"className":57372},[141,142],[15,57374,57376,57407],{"className":57375},[146],[15,57377,57379,57390],{"className":57378,"style":49566},[150],[15,57380,57381,57384],{"style":53858},[15,57382],{"className":57383,"style":159},[158],[15,57385,57387],{"className":57386},[163,164,165,166],[15,57388,541],{"className":57389},[129,166],[15,57391,57392,57395],{"style":1272},[15,57393],{"className":57394,"style":159},[158],[15,57396,57398],{"className":57397},[163,164,165,166],[15,57399,57401,57404],{"className":57400},[129,166],[15,57402,49367],{"className":57403},[129,166],[15,57405,39326],{"className":57406},[129,133,166],[15,57408,175],{"className":57409},[174],[15,57411,57413],{"className":57412},[146],[15,57414,57416],{"className":57415,"style":53894},[150],[15,57417],{},[15,57419],{"className":57420,"style":3277},[93],[15,57422,315],{"className":57423},[2436],[15,57425],{"className":57426,"style":3277},[93],[15,57428,57430,57433,57490,57570,57576],{"className":57429},[120],[15,57431],{"className":57432,"style":45833},[124],[15,57434,57436,57439],{"className":57435},[129],[15,57437,1388],{"className":57438,"style":12449},[129,133],[15,57440,57442],{"className":57441},[137],[15,57443,57445,57482],{"className":57444},[141,142],[15,57446,57448,57479],{"className":57447},[146],[15,57449,57451,57462],{"className":57450,"style":49566},[150],[15,57452,57453,57456],{"style":53858},[15,57454],{"className":57455,"style":159},[158],[15,57457,57459],{"className":57458},[163,164,165,166],[15,57460,318],{"className":57461},[129,166],[15,57463,57464,57467],{"style":1272},[15,57465],{"className":57466,"style":159},[158],[15,57468,57470],{"className":57469},[163,164,165,166],[15,57471,57473,57476],{"className":57472},[129,166],[15,57474,49367],{"className":57475},[129,166],[15,57477,39326],{"className":57478},[129,133,166],[15,57480,175],{"className":57481},[174],[15,57483,57485],{"className":57484},[146],[15,57486,57488],{"className":57487,"style":53894},[150],[15,57489],{},[15,57491,57493,57530],{"className":57492},[195],[15,57494,57496],{"className":57495},[195],[15,57497,57499],{"className":57498},[228,4433],[15,57500,57502,57522],{"className":57501},[141,142],[15,57503,57505,57519],{"className":57504},[146],[15,57506,57508],{"className":57507,"style":4443},[150],[15,57509,57510,57513],{"style":4446},[15,57511],{"className":57512,"style":4450},[158],[15,57514,57515],{"style":53819},[4455,57516,57517],{"xmlns":4457,"width":53822,"height":73,"viewBox":53823},[4461,57518],{"d":53826},[15,57520,175],{"className":57521},[174],[15,57523,57525],{"className":57524},[146],[15,57526,57528],{"className":57527,"style":4473},[150],[15,57529],{},[15,57531,57533],{"className":57532},[137],[15,57534,57536,57562],{"className":57535},[141,142],[15,57537,57539,57559],{"className":57538},[146],[15,57540,57542],{"className":57541,"style":54021},[150],[15,57543,57544,57547],{"style":54024},[15,57545],{"className":57546,"style":159},[158],[15,57548,57550],{"className":57549},[163,164,165,166],[15,57551,57553],{"className":57552},[129,166],[15,57554,57556],{"className":57555},[129,166],[15,57557,54040],{"className":57558,"style":2246},[129,221,166],[15,57560,175],{"className":57561},[174],[15,57563,57565],{"className":57564},[146],[15,57566,57568],{"className":57567,"style":54050},[150],[15,57569],{},[15,57571,57573],{"className":57572},[195],[15,57574,59],{"className":57575},[228,24461],[15,57577,107],{"className":57578},[129],[11,57580,57581,57582,57617,57618,57751,57752,57784],{},"这个式子的意思是：",[698,57583,57584,57585,1313],{},"两个分布的总变差越小，任何检验的错误率就越接近 ",[15,57586,57588,57605],{"className":57587,"translate":19},[23],[15,57589,57591],{"className":57590},[27],[29,57592,57593],{"xmlns":31},[34,57594,57595,57603],{},[37,57596,57597,57599,57601],{},[308,57598,318],{},[43,57600,544],{"mathvariant":65},[308,57602,310],{},[109,57604,56735],{"encoding":111},[15,57606,57608],{"className":57607,"ariaHidden":72},[116],[15,57609,57611,57614],{"className":57610},[120],[15,57612],{"className":57613,"style":125},[124],[15,57615,56735],{"className":57616},[129]," 当 ",[15,57619,57621,57647],{"className":57620,"translate":19},[23],[15,57622,57624],{"className":57623},[27],[29,57625,57626],{"xmlns":31},[34,57627,57628,57644],{},[37,57629,57630,57636,57638],{},[40,57631,57632,57634],{},[43,57633,1388],{},[308,57635,541],{},[50,57637,62],{},[40,57639,57640,57642],{},[43,57641,1388],{},[308,57643,318],{},[109,57645,57646],{"encoding":111},"P_0=P_1",[15,57648,57650,57705],{"className":57649,"ariaHidden":72},[116],[15,57651,57653,57656,57696,57699,57702],{"className":57652},[120],[15,57654],{"className":57655,"style":20229},[124],[15,57657,57659,57662],{"className":57658},[129],[15,57660,1388],{"className":57661,"style":12449},[129,133],[15,57663,57665],{"className":57664},[137],[15,57666,57668,57688],{"className":57667},[141,142],[15,57669,57671,57685],{"className":57670},[146],[15,57672,57674],{"className":57673,"style":1051},[150],[15,57675,57676,57679],{"style":34239},[15,57677],{"className":57678,"style":159},[158],[15,57680,57682],{"className":57681},[163,164,165,166],[15,57683,541],{"className":57684},[129,166],[15,57686,175],{"className":57687},[174],[15,57689,57691],{"className":57690},[146],[15,57692,57694],{"className":57693,"style":182},[150],[15,57695],{},[15,57697],{"className":57698,"style":199},[93],[15,57700,62],{"className":57701},[203],[15,57703],{"className":57704,"style":199},[93],[15,57706,57708,57711],{"className":57707},[120],[15,57709],{"className":57710,"style":20229},[124],[15,57712,57714,57717],{"className":57713},[129],[15,57715,1388],{"className":57716,"style":12449},[129,133],[15,57718,57720],{"className":57719},[137],[15,57721,57723,57743],{"className":57722},[141,142],[15,57724,57726,57740],{"className":57725},[146],[15,57727,57729],{"className":57728,"style":1051},[150],[15,57730,57731,57734],{"style":34239},[15,57732],{"className":57733,"style":159},[158],[15,57735,57737],{"className":57736},[163,164,165,166],[15,57738,318],{"className":57739},[129,166],[15,57741,175],{"className":57742},[174],[15,57744,57746],{"className":57745},[146],[15,57747,57749],{"className":57748,"style":182},[150],[15,57750],{}," 时，错误率至少是 ",[15,57753,57755,57772],{"className":57754,"translate":19},[23],[15,57756,57758],{"className":57757},[27],[29,57759,57760],{"xmlns":31},[34,57761,57762,57770],{},[37,57763,57764,57766,57768],{},[308,57765,318],{},[43,57767,544],{"mathvariant":65},[308,57769,310],{},[109,57771,56735],{"encoding":111},[15,57773,57775],{"className":57774,"ariaHidden":72},[116],[15,57776,57778,57781],{"className":57777},[120],[15,57779],{"className":57780,"style":125},[124],[15,57782,56735],{"className":57783},[129],"，这当然——数据里根本没有信息。",[11,57786,57787,57788,57816,57817,22169],{},"定理 3.1 要求 ",[15,57789,57791,57804],{"className":57790,"translate":19},[23],[15,57792,57794],{"className":57793},[27],[29,57795,57796],{"xmlns":31},[34,57797,57798,57802],{},[37,57799,57800],{},[43,57801,27529],{"mathvariant":65},[109,57803,53208],{"encoding":111},[15,57805,57807],{"className":57806,"ariaHidden":72},[116],[15,57808,57810,57813],{"className":57809},[120],[15,57811],{"className":57812,"style":522},[124],[15,57814,27529],{"className":57815},[129]," 满足三角不等式，平方损失并不满足。对平方损失，有一个常数稍有不同的版本。设 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",[15,59408,59410,59464],{"className":59409,"translate":19},[23],[15,59411,59413],{"className":59412},[27],[29,59414,59415],{"xmlns":31},[34,59416,59417,59461],{},[37,59418,59419,59421,59423,59425,59427,59429,59431,59437,59439,59441,59443,59445,59447,59449,59455,59457,59459],{},[43,59420,48354],{"mathvariant":65},[6856,59422,15788],{},[43,59424,26574],{"mathvariant":103},[50,59426,1407],{"stretchy":52},[43,59428,15763],{},[50,59430,53],{"stretchy":52},[305,59432,59433,59435],{},[43,59434,597],{},[43,59436,39326],{},[50,59438,59],{"stretchy":52},[50,59440,53],{"stretchy":52},[308,59442,318],{},[50,59444,315],{},[43,59446,15763],{},[50,59448,53],{"stretchy":52},[305,59450,59451,59453],{},[43,59452,597],{},[43,59454,39326],{},[50,59456,59],{"stretchy":52},[50,59458,59],{"stretchy":52},[50,59460,1423],{"stretchy":52},[109,59462,59463],{"encoding":111},"\\Delta\\,\\mathbb E[\\pi(X^n)(1-\\pi(X^n))]",[15,59465,59467,59538],{"className":59466,"ariaHidden":72},[116],[15,59468,59470,59473,59476,59479,59482,59485,59488,59491,59520,59523,59526,59529,59532,59535],{"className":59469},[120],[15,59471],{"className":59472,"style":125},[124],[15,59474,48354],{"className":59475},[129],[15,59477],{"className":59478,"style":262},[93],[15,59480,26574],{"className":59481},[129,285],[15,59483,1407],{"className":59484},[188],[15,59486,15763],{"className":59487,"style":887},[129,133],[15,59489,53],{"className":59490},[188],[15,59492,59494,59497],{"className":59493},[129],[15,59495,597],{"className":59496,"style":47970},[129,133],[15,59498,59500],{"className":59499},[137],[15,59501,59503],{"className":59502},[141],[15,59504,59506],{"className":59505},[146],[15,59507,59509],{"className":59508,"style":6377},[150],[15,59510,59511,59514],{"style":383},[15,59512],{"className":59513,"style":159},[158],[15,59515,59517],{"className":59516},[163,164,165,166],[15,59518,39326],{"className":59519},[129,133,166],[15,59521,59],{"className":59522},[195],[15,59524,53],{"className":59525},[188],[15,59527,318],{"className":59528},[129],[15,59530],{"className":59531,"style":3277},[93],[15,59533,315],{"className":59534},[2436],[15,59536],{"className":59537,"style":3277},[93],[15,59539,59541,59544,59547,59550,59579],{"className":59540},[120],[15,59542],{"className":59543,"style":125},[124],[15,59545,15763],{"className":59546,"style":887},[129,133],[15,59548,53],{"className":59549},[188],[15,59551,59553,59556],{"className":59552},[129],[15,59554,597],{"className":59555,"style":47970},[129,133],[15,59557,59559],{"className":59558},[137],[15,59560,59562],{"className":59561},[141],[15,59563,59565],{"className":59564},[146],[15,59566,59568],{"className":59567,"style":6377},[150],[15,59569,59570,59573],{"style":383},[15,59571],{"className":59572,"style":159},[158],[15,59574,59576],{"className":59575},[163,164,165,166],[15,59577,39326],{"className":59578},[129,133,166],[15,59580,59582],{"className":59581},[195],"))]","。由 ",[15,59585,59587,59637],{"className":59586,"translate":19},[23],[15,59588,59590],{"className":59589},[27],[29,59591,59592],{"xmlns":31},[34,59593,59594,59634],{},[37,59595,59596,59598,59600,59602,59604,59606,59608,59610,59616,59618,59620,59622,59624,59626,59628,59630,59632],{},[43,59597,15763],{},[50,59599,53],{"stretchy":52},[308,59601,318],{},[50,59603,315],{},[43,59605,15763],{},[50,59607,59],{"stretchy":52},[50,59609,7473],{},[1443,59611,59612,59614],{},[308,59613,318],{},[308,59615,310],{},[43,59617,2146],{},[50,59619,69],{},[50,59621,53],{"stretchy":52},[43,59623,15763],{},[50,59625,91],{"separator":72},[308,59627,318],{},[50,59629,315],{},[43,59631,15763],{},[50,59633,59],{"stretchy":52},[109,59635,59636],{"encoding":111},"\\pi(1-\\pi)\\ge\\frac12\\min(\\pi,1-\\pi)",[15,59638,59640,59664,59685,59793],{"className":59639,"ariaHidden":72},[116],[15,59641,59643,59646,59649,59652,59655,59658,59661],{"className":59642},[120],[15,59644],{"className":59645,"style":125},[124],[15,59647,15763],{"className":59648,"style":887},[129,133],[15,59650,53],{"className":59651},[188],[15,59653,318],{"className":59654},[129],[15,59656],{"className":59657,"style":3277},[93],[15,59659,315],{"className":59660},[2436],[15,59662],{"className":59663,"style":3277},[93],[15,59665,59667,59670,59673,59676,59679,59682],{"className":59666},[120],[15,59668],{"className":59669,"style":125},[124],[15,59671,15763],{"className":59672,"style":887},[129,133],[15,59674,59],{"className":59675},[195],[15,59677],{"className":59678,"style":199},[93],[15,59680,7473],{"className":59681},[203],[15,59683],{"className":59684,"style":199},[93],[15,59686,59688,59692,59763,59766,59769,59772,59775,59778,59781,59784,59787,59790],{"className":59687},[120],[15,59689],{"className":59690,"style":59691},[124],"height:1.1901em;vertical-align:-0.345em;",[15,59693,59695,59698,59760],{"className":59694},[129],[15,59696],{"className":59697},[188,1803],[15,59699,59701],{"className":59700},[1443],[15,59702,59704,59752],{"className":59703},[141,142],[15,59705,59707,59749],{"className":59706},[146],[15,59708,59711,59726,59734],{"className":59709,"style":59710},[150],"height:0.8451em;",[15,59712,59714,59717],{"style":59713},"top:-2.655em;",[15,59715],{"className":59716,"style":1823},[158],[15,59718,59720],{"className":59719},[163,164,165,166],[15,59721,59723],{"className":59722},[129,166],[15,59724,310],{"className":59725},[129,166],[15,59727,59728,59731],{"style":1832},[15,59729],{"className":59730,"style":1823},[158],[15,59732],{"className":59733,"style":1840},[1839],[15,59735,59737,59740],{"style":59736},"top:-3.394em;",[15,59738],{"className":59739,"style":1823},[158],[15,59741,59743],{"className":59742},[163,164,165,166],[15,59744,59746],{"className":59745},[129,166],[15,59747,318],{"className":59748},[129,166],[15,59750,175],{"className":59751},[174],[15,59753,59755],{"className":59754},[146],[15,59756,59758],{"className":59757,"style":23490},[150],[15,59759],{},[15,59761],{"className":59762},[195,1803],[15,59764],{"className":59765,"style":262},[93],[15,59767,2146],{"className":59768},[217],[15,59770,53],{"className":59771},[188],[15,59773,15763],{"className":59774,"style":887},[129,133],[15,59776,91],{"className":59777},[254],[15,59779],{"className":59780,"style":262},[93],[15,59782,318],{"className":59783},[129],[15,59785],{"className":59786,"style":3277},[93],[15,59788,315],{"className":59789},[2436],[15,59791],{"className":59792,"style":3277},[93],[15,59794,59796,59799,59802],{"className":59795},[120],[15,59797],{"className":59798,"style":125},[124],[15,59800,15763],{"className":59801,"style":887},[129,133],[15,59803,59],{"className":59804},[195]," 与均匀先验下贝叶斯检验误差的下界 ",[15,59807,59809,59867],{"className":59808,"translate":19},[23],[15,59810,59812],{"className":59811},[27],[29,59813,59814],{"xmlns":31},[34,59815,59816,59864],{},[37,59817,59818,59820,59822,59824,59826,59828,59830,59832,59834,59836,59838,59840,59842,59844,59850,59852,59854,59856,59862],{},[43,59819,26574],{"mathvariant":103},[50,59821,1407],{"stretchy":52},[43,59823,2146],{},[50,59825,69],{},[50,59827,53],{"stretchy":52},[43,59829,15763],{},[50,59831,91],{"separator":72},[308,59833,318],{},[50,59835,315],{},[43,59837,15763],{},[50,59839,59],{"stretchy":52},[50,59841,1423],{"stretchy":52},[50,59843,7473],{},[1443,59845,59846,59848],{},[308,59847,318],{},[308,59849,310],{},[50,59851,53],{"stretchy":52},[308,59853,318],{},[50,59855,315],{},[37,59857,59858,59860],{},[43,59859,53409],{"mathvariant":65},[43,59861,53412],{"mathvariant":65},[50,59863,59],{"stretchy":52},[109,59865,59866],{"encoding":111},"\\mathbb E[\\min(\\pi,1-\\pi)]\\ge\\frac12(1-\\mathrm{TV})",[15,59868,59870,59909,59931,60020],{"className":59869,"ariaHidden":72},[116],[15,59871,59873,59876,59879,59882,59885,59888,59891,59894,59897,59900,59903,59906],{"className":59872},[120],[15,59874],{"className":59875,"style":125},[124],[15,59877,26574],{"className":59878},[129,285],[15,59880,1407],{"className":59881},[188],[15,59883,2146],{"className":59884},[217],[15,59886,53],{"className":59887},[188],[15,59889,15763],{"className":59890,"style":887},[129,133],[15,59892,91],{"className":59893},[254],[15,59895],{"className":59896,"style":262},[93],[15,59898,318],{"className":59899},[129],[15,59901],{"className":59902,"style":3277},[93],[15,59904,315],{"className":59905},[2436],[15,59907],{"className":59908,"style":3277},[93],[15,59910,59912,59915,59918,59922,59925,59928],{"className":59911},[120],[15,59913],{"className":59914,"style":125},[124],[15,59916,15763],{"className":59917,"style":887},[129,133],[15,59919,59921],{"className":59920},[195],")]",[15,59923],{"className":59924,"style":199},[93],[15,59926,7473],{"className":59927},[203],[15,59929],{"className":59930,"style":199},[93],[15,59932,59934,59937,60005,60008,60011,60014,60017],{"className":59933},[120],[15,59935],{"className":59936,"style":59691},[124],[15,59938,59940,59943,60002],{"className":59939},[129],[15,59941],{"className":59942},[188,1803],[15,59944,59946],{"className":59945},[1443],[15,59947,59949,59994],{"className":59948},[141,142],[15,59950,59952,59991],{"className":59951},[146],[15,59953,59955,59969,59977],{"className":59954,"style":59710},[150],[15,59956,59957,59960],{"style":59713},[15,59958],{"className":59959,"style":1823},[158],[15,59961,59963],{"className":59962},[163,164,165,166],[15,59964,59966],{"className":59965},[129,166],[15,59967,310],{"className":59968},[129,166],[15,59970,59971,59974],{"style":1832},[15,59972],{"className":59973,"style":1823},[158],[15,59975],{"className":59976,"style":1840},[1839],[15,59978,59979,59982],{"style":59736},[15,59980],{"className":59981,"style":1823},[158],[15,59983,59985],{"className":59984},[163,164,165,166],[15,59986,59988],{"className":59987},[129,166],[15,59989,318],{"className":59990},[129,166],[15,59992,175],{"className":59993},[174],[15,59995,59997],{"className":59996},[146],[15,59998,60000],{"className":59999,"style":23490},[150],[15,60001],{},[15,60003],{"className":60004},[195,1803],[15,60006,53],{"className":60007},[188],[15,60009,318],{"className":60010},[129],[15,60012],{"className":60013,"style":3277},[93],[15,60015,315],{"className":60016},[2436],[15,60018],{"className":60019,"style":3277},[93],[15,60021,60023,60026,60032],{"className":60022},[120],[15,60024],{"className":60025,"style":125},[124],[15,60027,60029],{"className":60028},[129],[15,60030,54040],{"className":60031,"style":2246},[129,221],[15,60033,59],{"className":60034},[195],"，风险至少是 ",[15,60037,60039,60071],{"className":60038,"translate":19},[23],[15,60040,60042],{"className":60041},[27],[29,60043,60044],{"xmlns":31},[34,60045,60046,60068],{},[37,60047,60048,60054,60056,60058,60060,60066],{},[1443,60049,60050,60052],{},[43,60051,48354],{"mathvariant":65},[308,60053,14508],{},[50,60055,53],{"stretchy":52},[308,60057,318],{},[50,60059,315],{},[37,60061,60062,60064],{},[43,60063,53409],{"mathvariant":65},[43,60065,53412],{"mathvariant":65},[50,60067,59],{"stretchy":52},[109,60069,60070],{"encoding":111},"\\frac{\\Delta}{4}(1-\\mathrm{TV})",[15,60072,60074,60165],{"className":60073,"ariaHidden":72},[116],[15,60075,60077,60081,60150,60153,60156,60159,60162],{"className":60076},[120],[15,60078],{"className":60079,"style":60080},[124],"height:1.2173em;vertical-align:-0.345em;",[15,60082,60084,60087,60147],{"className":60083},[129],[15,60085],{"className":60086},[188,1803],[15,60088,60090],{"className":60089},[1443],[15,60091,60093,60139],{"className":60092},[141,142],[15,60094,60096,60136],{"className":60095},[146],[15,60097,60100,60114,60122],{"className":60098,"style":60099},[150],"height:0.8723em;",[15,60101,60102,60105],{"style":59713},[15,60103],{"className":60104,"style":1823},[158],[15,60106,60108],{"className":60107},[163,164,165,166],[15,60109,60111],{"className":60110},[129,166],[15,60112,14508],{"className":60113},[129,166],[15,60115,60116,60119],{"style":1832},[15,60117],{"className":60118,"style":1823},[158],[15,60120],{"className":60121,"style":1840},[1839],[15,60123,60124,60127],{"style":59736},[15,60125],{"className":60126,"style":1823},[158],[15,60128,60130],{"className":60129},[163,164,165,166],[15,60131,60133],{"className":60132},[129,166],[15,60134,48354],{"className":60135},[129,166],[15,60137,175],{"className":60138},[174],[15,60140,60142],{"className":60141},[146],[15,60143,60145],{"className":60144,"style":23490},[150],[15,60146],{},[15,60148],{"className":60149},[195,1803],[15,60151,53],{"className":60152},[188],[15,60154,318],{"className":60155},[129],[15,60157],{"className":60158,"style":3277},[93],[15,60160,315],{"className":60161},[2436],[15,60163],{"className":60164,"style":3277},[93],[15,60166,60168,60171,60177],{"className":60167},[120],[15,60169],{"className":60170,"style":125},[124],[15,60172,60174],{"className":60173},[129],[15,60175,54040],{"className":60176,"style":2246},[129,221],[15,60178,59],{"className":60179},[195],"。极小极大风险不小于贝叶斯风险，得证。",[575,60182,60184],{"id":60183},"_32-从-tv-到-kl","3.2 从 TV 到 KL",[11,60186,60187],{},"总变差通常不好直接算，尤其对乘积分布。Pinsker 不等式把它换成 KL 散度：",[15,60189,60191],{"className":60190,"translate":19},[18],[15,60192,60194,60258],{"className":60193,"translate":19},[23],[15,60195,60197],{"className":60196},[27],[29,60198,60199],{"xmlns":31,"display":32},[34,60200,60201,60255],{},[37,60202,60203,60205,60207,60209,60211,60221,60223,60253],{},[50,60204,33823],{"stretchy":52},[43,60206,1388],{},[50,60208,315],{},[43,60210,34354],{},[40,60212,60213,60215],{},[50,60214,33823],{"stretchy":52},[37,60216,60217,60219],{},[43,60218,53409],{"mathvariant":65},[43,60220,53412],{"mathvariant":65},[50,60222,2717],{},[29015,60224,60225],{},[37,60226,60227,60233,60243,60245,60247,60249,60251],{},[1443,60228,60229,60231],{},[308,60230,318],{},[308,60232,310],{},[40,60234,60235,60237],{},[43,60236,906],{},[37,60238,60239,60241],{},[43,60240,34524],{"mathvariant":65},[43,60242,1373],{"mathvariant":65},[50,60244,53],{"stretchy":52},[43,60246,1388],{},[43,60248,33823],{"mathvariant":65},[43,60250,34354],{},[50,60252,59],{"stretchy":52},[43,60254,107],{"mathvariant":65},[109,60256,60257],{"encoding":111},"\\lVert P-Q\\rVert_{\\mathrm{TV}}\\le\\sqrt{\\frac12 D_{\\mathrm{KL}}(P\\Vert Q)}.",[15,60259,60261,60282,60346],{"className":60260,"ariaHidden":72},[116],[15,60262,60264,60267,60270,60273,60276,60279],{"className":60263},[120],[15,60265],{"className":60266,"style":125},[124],[15,60268,33823],{"className":60269},[188],[15,60271,1388],{"className":60272,"style":12449},[129,133],[15,60274],{"className":60275,"style":3277},[93],[15,60277,315],{"className":60278},[2436],[15,60280],{"className":60281,"style":3277},[93],[15,60283,60285,60288,60291,60337,60340,60343],{"className":60284},[120],[15,60286],{"className":60287,"style":125},[124],[15,60289,34354],{"className":60290},[129,133],[15,60292,60294,60297],{"className":60293},[195],[15,60295,33823],{"className":60296},[195],[15,60298,60300],{"className":60299},[137],[15,60301,60303,60329],{"className":60302},[141,142],[15,60304,60306,60326],{"className":60305},[146],[15,60307,60309],{"className":60308,"style":1525},[150],[15,60310,60311,60314],{"style":154},[15,60312],{"className":60313,"style":159},[158],[15,60315,60317],{"className":60316},[163,164,165,166],[15,60318,60320],{"className":60319},[129,166],[15,60321,60323],{"className":60322},[129,166],[15,60324,54040],{"className":60325,"style":2246},[129,221,166],[15,60327,175],{"className":60328},[174],[15,60330,60332],{"className":60331},[146],[15,60333,60335],{"className":60334,"style":182},[150],[15,60336],{},[15,60338],{"className":60339,"style":199},[93],[15,60341,2717],{"className":60342},[203],[15,60344],{"className":60345,"style":199},[93],[15,60347,60349,60353,60523],{"className":60348},[120],[15,60350],{"className":60351,"style":60352},[124],"height:2.44em;vertical-align:-0.7884em;",[15,60354,60356],{"className":60355},[129,29123],[15,60357,60359,60514],{"className":60358},[141,142],[15,60360,60362,60511],{"className":60361},[146],[15,60363,60366,60498],{"className":60364,"style":60365},[150],"height:1.6516em;",[15,60367,60369,60372],{"className":60368,"style":29138},[29137],[15,60370],{"className":60371,"style":29142},[158],[15,60373,60375,60437,60483,60486,60489,60492,60495],{"className":60374,"style":29146},[129],[15,60376,60378,60381,60434],{"className":60377},[129],[15,60379],{"className":60380},[188,1803],[15,60382,60384],{"className":60383},[1443],[15,60385,60387,60426],{"className":60386},[141,142],[15,60388,60390,60423],{"className":60389},[146],[15,60391,60393,60404,60412],{"className":60392,"style":1816},[150],[15,60394,60395,60398],{"style":1819},[15,60396],{"className":60397,"style":1823},[158],[15,60399,60401],{"className":60400},[129],[15,60402,310],{"className":60403},[129],[15,60405,60406,60409],{"style":1832},[15,60407],{"className":60408,"style":1823},[158],[15,60410],{"className":60411,"style":1840},[1839],[15,60413,60414,60417],{"style":1843},[15,60415],{"className":60416,"style":1823},[158],[15,60418,60420],{"className":60419},[129],[15,60421,318],{"className":60422},[129],[15,60424,175],{"className":60425},[174],[15,60427,60429],{"className":60428},[146],[15,60430,60432],{"className":60431,"style":1862},[150],[15,60433],{},[15,60435],{"className":60436},[195,1803],[15,60438,60440,60443],{"className":60439},[129],[15,60441,906],{"className":60442,"style":170},[129,133],[15,60444,60446],{"className":60445},[137],[15,60447,60449,60475],{"className":60448},[141,142],[15,60450,60452,60472],{"className":60451},[146],[15,60453,60455],{"className":60454,"style":1525},[150],[15,60456,60457,60460],{"style":48293},[15,60458],{"className":60459,"style":159},[158],[15,60461,60463],{"className":60462},[163,164,165,166],[15,60464,60466],{"className":60465},[129,166],[15,60467,60469],{"className":60468},[129,166],[15,60470,34928],{"className":60471},[129,221,166],[15,60473,175],{"className":60474},[174],[15,60476,60478],{"className":60477},[146],[15,60479,60481],{"className":60480,"style":182},[150],[15,60482],{},[15,60484,53],{"className":60485},[188],[15,60487,1388],{"className":60488,"style":12449},[129,133],[15,60490,33823],{"className":60491},[129],[15,60493,34354],{"className":60494},[129,133],[15,60496,59],{"className":60497},[195],[15,60499,60501,60504],{"style":60500},"top:-3.6116em;",[15,60502],{"className":60503,"style":29142},[158],[15,60505,60507],{"className":60506,"style":29283},[29282],[4455,60508,60509],{"xmlns":4457,"width":29286,"height":29287,"viewBox":29288,"preserveAspectRatio":29289},[4461,60510],{"d":29292},[15,60512,175],{"className":60513},[174],[15,60515,60517],{"className":60516},[146],[15,60518,60521],{"className":60519,"style":60520},[150],"height:0.7884em;",[15,60522],{},[15,60524,107],{"className":60525},[129],[11,60527,60528],{},"而 KL 对乘积分布是可加的：",[15,60530,60532],{"className":60531,"translate":19},[18],[15,60533,60535,60623],{"className":60534,"translate":19},[23],[15,60536,60538],{"className":60537},[27],[29,60539,60540],{"xmlns":31,"display":32},[34,60541,60542,60620],{},[37,60543,60544,60554,60556,60568,60570,60582,60584,60586,60588,60590,60600,60602,60608,60610,60616,60618],{},[40,60545,60546,60548],{},[43,60547,906],{},[37,60549,60550,60552],{},[43,60551,34524],{"mathvariant":65},[43,60553,1373],{"mathvariant":65},[50,60555,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[23160,60557,60558,60560,60562],{},[43,60559,1388],{},[308,60561,541],{},[37,60563,60564,60566],{},[50,60565,49367],{},[43,60567,39326],{},[43,60569,33823],{"mathvariant":65},[23160,60571,60572,60574,60576],{},[43,60573,1388],{},[308,60575,318],{},[37,60577,60578,60580],{},[50,60579,49367],{},[43,60581,39326],{},[50,60583,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,60585,62],{},[43,60587,39326],{},[6856,60589,15788],{},[40,60591,60592,60594],{},[43,60593,906],{},[37,60595,60596,60598],{},[43,60597,34524],{"mathvariant":65},[43,60599,1373],{"mathvariant":65},[50,60601,53],{"stretchy":52},[40,60603,60604,60606],{},[43,60605,1388],{},[308,60607,541],{},[43,60609,33823],{"mathvariant":65},[40,60611,60612,60614],{},[43,60613,1388],{},[308,60615,318],{},[50,60617,59],{"stretchy":52},[43,60619,107],{"mathvariant":65},[109,60621,60622],{"encoding":111},"D_{\\mathrm{KL}}\\bigl(P_0^{\\otimes n}\\Vert P_1^{\\otimes n}\\bigr)=n\\,D_{\\mathrm{KL}}(P_0\\Vert 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,61696,61699],{"className":61697,"style":61698},[150],"height:0.8953em;",[15,61700],{},[15,61702,61704],{"className":61703,"style":5052},[195,5051],[15,61705,59],{"className":61706},[228,165],[15,61708],{"className":61709,"style":262},[93],[15,61711,107],{"className":61712},[129],[11,61714,61715,61716,804,61719,61748,61749,804,61752,61837,61838,62016],{},"这个式子把下界拆成了两个因子：",[698,61717,61718],{},"损失间隔",[15,61720,61722,61736],{"className":61721,"translate":19},[23],[15,61723,61725],{"className":61724},[27],[29,61726,61727],{"xmlns":31},[34,61728,61729,61733],{},[37,61730,61731],{},[43,61732,29575],{},[109,61734,61735],{"encoding":111},"\\delta",[15,61737,61739],{"className":61738,"ariaHidden":72},[116],[15,61740,61742,61745],{"className":61741},[120],[15,61743],{"className":61744,"style":522},[124],[15,61746,29575],{"className":61747,"style":29856},[129,133],"，和",[698,61750,61751],{},"信息量",[15,61753,61755,61779],{"className":61754,"translate":19},[23],[15,61756,61758],{"className":61757},[27],[29,61759,61760],{"xmlns":31},[34,61761,61762,61776],{},[37,61763,61764,61766],{},[43,61765,39326],{},[40,61767,61768,61770],{},[43,61769,906],{},[37,61771,61772,61774],{},[43,61773,34524],{"mathvariant":65},[43,61775,1373],{"mathvariant":65},[109,61777,61778],{"encoding":111},"nD_{\\mathrm{KL}}",[15,61780,61782],{"className":61781,"ariaHidden":72},[116],[15,61783,61785,61788,61791],{"className":61784},[120],[15,61786],{"className":61787,"style":20229},[124],[15,61789,39326],{"className":61790},[129,133],[15,61792,61794,61797],{"className":61793},[129],[15,61795,906],{"className":61796,"style":170},[129,133],[15,61798,61800],{"className":61799},[137],[15,61801,61803,61829],{"className":61802},[141,142],[15,61804,61806,61826],{"className":61805},[146],[15,61807,61809],{"className":61808,"style":1525},[150],[15,61810,61811,61814],{"style":48293},[15,61812],{"className":61813,"style":159},[158],[15,61815,61817],{"className":61816},[163,164,165,166],[15,61818,61820],{"className":61819},[129,166],[15,61821,61823],{"className":61822},[129,166],[15,61824,34928],{"className":61825},[129,221,166],[15,61827,175],{"className":61828},[174],[15,61830,61832],{"className":61831},[146],[15,61833,61835],{"className":61834,"style":182},[150],[15,61836],{},"。要让下界为正，必须 ",[15,61839,61841,61873],{"className":61840,"translate":19},[23],[15,61842,61844],{"className":61843},[27],[29,61845,61846],{"xmlns":31},[34,61847,61848,61870],{},[37,61849,61850,61856,61866,61868],{},[1443,61851,61852,61854],{},[43,61853,39326],{},[308,61855,310],{},[40,61857,61858,61860],{},[43,61859,906],{},[37,61861,61862,61864],{},[43,61863,34524],{"mathvariant":65},[43,61865,1373],{"mathvariant":65},[50,61867,7910],{},[308,61869,318],{},[109,61871,61872],{"encoding":111},"\\frac n2D_{\\mathrm{KL}}\u003C1",[15,61874,61876,62007],{"className":61875,"ariaHidden":72},[116],[15,61877,61879,61883,61952,61998,62001,62004],{"className":61878},[120],[15,61880],{"className":61881,"style":61882},[124],"height:1.0404em;vertical-align:-0.345em;",[15,61884,61886,61889,61949],{"className":61885},[129],[15,61887],{"className":61888},[188,1803],[15,61890,61892],{"className":61891},[1443],[15,61893,61895,61941],{"className":61894},[141,142],[15,61896,61898,61938],{"className":61897},[146],[15,61899,61902,61916,61924],{"className":61900,"style":61901},[150],"height:0.6954em;",[15,61903,61904,61907],{"style":59713},[15,61905],{"className":61906,"style":1823},[158],[15,61908,61910],{"className":61909},[163,164,165,166],[15,61911,61913],{"className":61912},[129,166],[15,61914,310],{"className":61915},[129,166],[15,61917,61918,61921],{"style":1832},[15,61919],{"className":61920,"style":1823},[158],[15,61922],{"className":61923,"style":1840},[1839],[15,61925,61926,61929],{"style":59736},[15,61927],{"className":61928,"style":1823},[158],[15,61930,61932],{"className":61931},[163,164,165,166],[15,61933,61935],{"className":61934},[129,166],[15,61936,39326],{"className":61937},[129,133,166],[15,61939,175],{"className":61940},[174],[15,61942,61944],{"className":61943},[146],[15,61945,61947],{"className":61946,"style":23490},[150],[15,61948],{},[15,61950],{"className":61951},[195,1803],[15,61953,61955,61958],{"className":61954},[129],[15,61956,906],{"className":61957,"style":170},[129,133],[15,61959,61961],{"className":61960},[137],[15,61962,61964,61990],{"className":61963},[141,142],[15,61965,61967,61987],{"className":61966},[146],[15,61968,61970],{"className":61969,"style":1525},[150],[15,61971,61972,61975],{"style":48293},[15,61973],{"className":61974,"style":159},[158],[15,61976,61978],{"className":61977},[163,164,165,166],[15,61979,61981],{"className":61980},[129,166],[15,61982,61984],{"className":61983},[129,166],[15,61985,34928],{"className":61986},[129,221,166],[15,61988,175],{"className":61989},[174],[15,61991,61993],{"className":61992},[146],[15,61994,61996],{"className":61995,"style":182},[150],[15,61997],{},[15,61999],{"className":62000,"style":199},[93],[15,62002,7910],{"className":62003},[203],[15,62005],{"className":62006,"style":199},[93],[15,62008,62010,62013],{"className":62009},[120],[15,62011],{"className":62012,"style":4886},[124],[15,62014,318],{"className":62015},[129],"；换句话说，如果两个分布在这个样本量下已经能被可靠地区分，两点方法就给不出非平凡的下界了。",[11,62018,62019,62020,62054],{},"这也解释了为什么下界里的样本量总以「",[15,62021,62023,62039],{"className":62022,"translate":19},[23],[15,62024,62026],{"className":62025},[27],[29,62027,62028],{"xmlns":31},[34,62029,62030,62036],{},[37,62031,62032,62034],{},[43,62033,39326],{},[50,62035,11422],{},[109,62037,62038],{"encoding":111},"n\\times",[15,62040,62042],{"className":62041,"ariaHidden":72},[116],[15,62043,62045,62048,62051],{"className":62044},[120],[15,62046],{"className":62047,"style":8518},[124],[15,62049,39326],{"className":62050},[129,133],[15,62052,11422],{"className":62053},[129]," 单样本信息」的形式出现：独立样本的信息是线性累加的。",[575,62056,62058],{"id":62057},"_33-一个例子bernoulli-均值","3.3 一个例子：Bernoulli 均值",[11,62060,21938,62061,7610,62179,62283],{},[15,62062,62064,62099],{"className":62063,"translate":19},[23],[15,62065,62067],{"className":62066},[27],[29,62068,62069],{"xmlns":31},[34,62070,62071,62096],{},[37,62072,62073,62079,62081,62090,62092,62094],{},[40,62074,62075,62077],{},[43,62076,597],{},[43,62078,1459],{},[50,62080,991],{},[37,62082,62083,62085,62087],{},[43,62084,33895],{"mathvariant":65},[43,62086,10699],{"mathvariant":65},[43,62088,62089],{"mathvariant":65},"r",[50,62091,53],{"stretchy":52},[43,62093,11],{},[50,62095,59],{"stretchy":52},[109,62097,62098],{"encoding":111},"X_i\\sim\\mathrm{Ber}(p)",[15,62100,62102,62157],{"className":62101,"ariaHidden":72},[116],[15,62103,62105,62108,62148,62151,62154],{"className":62104},[120],[15,62106],{"className":62107,"style":20229},[124],[15,62109,62111,62114],{"className":62110},[129],[15,62112,597],{"className":62113,"style":47970},[129,133],[15,62115,62117],{"className":62116},[137],[15,62118,62120,62140],{"className":62119},[141,142],[15,62121,62123,62137],{"className":62122},[146],[15,62124,62126],{"className":62125,"style":1981},[150],[15,62127,62128,62131],{"style":47985},[15,62129],{"className":62130,"style":159},[158],[15,62132,62134],{"className":62133},[163,164,165,166],[15,62135,1459],{"className":62136},[129,133,166],[15,62138,175],{"className":62139},[174],[15,62141,62143],{"className":62142},[146],[15,62144,62146],{"className":62145,"style":182},[150],[15,62147],{},[15,62149],{"className":62150,"style":199},[93],[15,62152,991],{"className":62153},[203],[15,62155],{"className":62156,"style":199},[93],[15,62158,62160,62163,62170,62173,62176],{"className":62159},[120],[15,62161],{"className":62162,"style":125},[124],[15,62164,62166],{"className":62165},[129],[15,62167,62169],{"className":62168},[129,221],"Ber",[15,62171,53],{"className":62172},[188],[15,62174,11],{"className":62175},[129,133],[15,62177,59],{"className":62178},[195],[15,62180,62182,62220],{"className":62181,"translate":19},[23],[15,62183,62185],{"className":62184},[27],[29,62186,62187],{"xmlns":31},[34,62188,62189,62217],{},[37,62190,62191,62193,62195,62197,62199,62201,62203,62205,62207,62209,62211,62213,62215],{},[43,62192,11],{},[50,62194,100],{},[50,62196,636],{"stretchy":52},[308,62198,318],{},[43,62200,544],{"mathvariant":65},[308,62202,310],{},[50,62204,91],{"separator":72},[308,62206,318],{},[43,62208,544],{"mathvariant":65},[308,62210,310],{},[50,62212,2682],{},[43,62214,48354],{"mathvariant":65},[50,62216,645],{"stretchy":52},[109,62218,62219],{"encoding":111},"p\\in\\{1\u002F2,1\u002F2+\\Delta\\}",[15,62221,62223,62241,62271],{"className":62222,"ariaHidden":72},[116],[15,62224,62226,62229,62232,62235,62238],{"className":62225},[120],[15,62227],{"className":62228,"style":10959},[124],[15,62230,11],{"className":62231},[129,133],[15,62233],{"className":62234,"style":199},[93],[15,62236,100],{"className":62237},[203],[15,62239],{"className":62240,"style":199},[93],[15,62242,62244,62247,62250,62253,62256,62259,62262,62265,62268],{"className":62243},[120],[15,62245],{"className":62246,"style":125},[124],[15,62248,636],{"className":62249},[188],[15,62251,56735],{"className":62252},[129],[15,62254,91],{"className":62255},[254],[15,62257],{"className":62258,"style":262},[93],[15,62260,56735],{"className":62261},[129],[15,62263],{"className":62264,"style":3277},[93],[15,62266,2682],{"className":62267},[2436],[15,62269],{"className":62270,"style":3277},[93],[15,62272,62274,62277,62280],{"className":62273},[120],[15,62275],{"className":62276,"style":125},[124],[15,62278,48354],{"className":62279},[129],[15,62281,645],{"className":62282},[195],"，损失为平方损失。两个分布的单样本 KL 是",[15,62285,62287],{"className":62286,"translate":19},[18],[15,62288,62290,62418],{"className":62289,"translate":19},[23],[15,62291,62293],{"className":62292},[27],[29,62294,62295],{"xmlns":31,"display":32},[34,62296,62297,62415],{},[37,62298,62299,62309,62311,62319,62321,62329,62331,62333,62341,62343,62351,62353,62355,62357,62359,62361,62363,62365,62371,62373,62375,62377,62379,62381,62383,62389,62391,62393,62395,62401,62403,62405,62407,62409,62411,62413],{},[40,62300,62301,62303],{},[43,62302,906],{},[37,62304,62305,62307],{},[43,62306,34524],{"mathvariant":65},[43,62308,1373],{"mathvariant":65},[50,62310,53],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[37,62312,62313,62315,62317],{},[43,62314,33895],{"mathvariant":65},[43,62316,10699],{"mathvariant":65},[43,62318,62089],{"mathvariant":65},[50,62320,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[2612,62322,62323],{"scriptlevel":541,"displaystyle":52},[1443,62324,62325,62327],{},[308,62326,318],{},[308,62328,310],{},[50,62330,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,62332,33823],{"fence":52,"stretchy":72,"minsize":45705,"maxsize":45705},[37,62334,62335,62337,62339],{},[43,62336,33895],{"mathvariant":65},[43,62338,10699],{"mathvariant":65},[43,62340,62089],{"mathvariant":65},[50,62342,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[2612,62344,62345],{"scriptlevel":541,"displaystyle":52},[1443,62346,62347,62349],{},[308,62348,318],{},[308,62350,310],{},[50,62352,2682],{},[43,62354,48354],{"mathvariant":65},[50,62356,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,62358,59],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[93,62360],{"linebreak":24721},[50,62362,62],{},[50,62364,315],{},[1443,62366,62367,62369],{},[308,62368,318],{},[308,62370,310],{},[43,62372,14727],{},[50,62374,69],{},[50,62376,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[308,62378,318],{},[50,62380,315],{},[308,62382,14508],{},[305,62384,62385,62387],{},[43,62386,48354],{"mathvariant":65},[308,62388,310],{},[50,62390,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,62392,32405],{},[308,62394,310],{},[305,62396,62397,62399],{},[43,62398,48354],{"mathvariant":65},[308,62400,310],{},[93,62402],{"width":95},[50,62404,53],{"stretchy":52},[43,62406,48354],{"mathvariant":65},[50,62408,723],{},[308,62410,541],{},[50,62412,59],{"stretchy":52},[43,62414,107],{"mathvariant":65},[109,62416,62417],{"encoding":111},"D_{\\mathrm{KL}}\\Bigl(\\mathrm{Ber}\\bigl(\\tfrac12\\bigr)\\Big\\Vert\\mathrm{Ber}\\bigl(\\tfrac12+\\Delta\\bigr)\\Bigr) \\\\\n=-\\frac12\\log\\bigl(1-4\\Delta^2\\bigr)\n\\approx 2\\Delta^2\n\\qquad(\\Delta\\to0).",[15,62419,62421,62698,62719,62722,62734,62831,62885,62941],{"className":62420,"ariaHidden":72},[116],[15,62422,62424,62427,62473,62479,62485,62491,62559,62565,62609,62615,62621,62689,62692,62695],{"className":62423},[120],[15,62425],{"className":62426,"style":45833},[124],[15,62428,62430,62433],{"className":62429},[129],[15,62431,906],{"className":62432,"style":170},[129,133],[15,62434,62436],{"className":62435},[137],[15,62437,62439,62465],{"className":62438},[141,142],[15,62440,62442,62462],{"className":62441},[146],[15,62443,62445],{"className":62444,"style":1525},[150],[15,62446,62447,62450],{"style":48293},[15,62448],{"className":62449,"style":159},[158],[15,62451,62453],{"className":62452},[163,164,165,166],[15,62454,62456],{"className":62455},[129,166],[15,62457,62459],{"className":62458},[129,166],[15,62460,34928],{"className":62461},[129,221,166],[15,62463,175],{"className":62464},[174],[15,62466,62468],{"className":62467},[146],[15,62469,62471],{"className":62470,"style":182},[150],[15,62472],{},[15,62474,62476],{"className":62475},[188],[15,62477,53],{"className":62478},[228,24461],[15,62480,62482],{"className":62481},[129],[15,62483,62169],{"className":62484},[129,221],[15,62486,62488],{"className":62487},[188],[15,62489,53],{"className":62490},[228,229],[15,62492,62494,62497,62556],{"className":62493},[129],[15,62495],{"className":62496},[188,1803],[15,62498,62500],{"className":62499},[1443],[15,62501,62503,62548],{"className":62502},[141,142],[15,62504,62506,62545],{"className":62505},[146],[15,62507,62509,62523,62531],{"className":62508,"style":59710},[150],[15,62510,62511,62514],{"style":59713},[15,62512],{"className":62513,"style":1823},[158],[15,62515,62517],{"className":62516},[163,164,165,166],[15,62518,62520],{"className":62519},[129,166],[15,62521,310],{"className":62522},[129,166],[15,62524,62525,62528],{"style":1832},[15,62526],{"className":62527,"style":1823},[158],[15,62529],{"className":62530,"style":1840},[1839],[15,62532,62533,62536],{"style":59736},[15,62534],{"className":62535,"style":1823},[158],[15,62537,62539],{"className":62538},[163,164,165,166],[15,62540,62542],{"className":62541},[129,166],[15,62543,318],{"className":62544},[129,166],[15,62546,175],{"className":62547},[174],[15,62549,62551],{"className":62550},[146],[15,62552,62554],{"className":62553,"style":23490},[150],[15,62555],{},[15,62557],{"className":62558},[195,1803],[15,62560,62562],{"className":62561},[195],[15,62563,59],{"className":62564},[228,229],[15,62566,62568],{"className":62567},[129],[15,62569,62571],{"className":62570},[228,4433],[15,62572,62574,62600],{"className":62573},[141,142],[15,62575,62577,62597],{"className":62576},[146],[15,62578,62581],{"className":62579,"style":62580},[150],"height:1.15em;",[15,62582,62584,62588],{"style":62583},"top:-3.15em;",[15,62585],{"className":62586,"style":62587},[158],"height:3.8em;",[15,62589,62591],{"style":62590},"width:0.556em;height:1.8em;",[4455,62592,62594],{"xmlns":4457,"width":53822,"height":45705,"viewBox":62593},"0 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E\\bigl[(p_i-\\hat p)^2\\bigr]\n\\ge\n\\frac{\\Delta^2}{4}\\left(1-\\frac12\\right)\n=\\frac{1}{32n}.",[15,63387,63389,63505,63598,63795],{"className":63388,"ariaHidden":72},[116],[15,63390,63392,63396,63441,63444,63447,63453,63456,63496,63499,63502],{"className":63391},[120],[15,63393],{"className":63394,"style":63395},[124],"height:1.5777em;vertical-align:-0.7277em;",[15,63397,63399],{"className":63398},[217,1874],[15,63400,63402,63432],{"className":63401},[141,142],[15,63403,63405,63429],{"className":63404},[146],[15,63406,63408,63419],{"className":63407,"style":4383},[150],[15,63409,63410,63413],{"style":53444},[15,63411],{"className":63412,"style":1823},[158],[15,63414,63416],{"className":63415},[163,164,165,166],[15,63417,1459],{"className":63418},[129,133,166],[15,63420,63421,63424],{"style":2287},[15,63422],{"className":63423,"style":1823},[158],[15,63425,63426],{},[15,63427,19705],{"className":63428},[217],[15,63430,175],{"className":63431},[174],[15,63433,63435],{"className":63434},[146],[15,63436,63439],{"className":63437,"style":63438},[150],"height:0.7277em;",[15,63440],{},[15,63442],{"className":63443,"style":262},[93],[15,63445,26574],{"className":63446},[129,285],[15,63448,63450],{"className":63449},[188],[15,63451,1407],{"className":63452},[228,229],[15,63454,53],{"className":63455},[188],[15,63457,63459,63462],{"className":63458},[129],[15,63460,11],{"className":63461},[129,133],[15,63463,63465],{"className":63464},[137],[15,63466,63468,63488],{"className":63467},[141,142],[15,63469,63471,63485],{"className":63470},[146],[15,63472,63474],{"className":63473,"style":1981},[150],[15,63475,63476,63479],{"style":154},[15,63477],{"className":63478,"style":159},[158],[15,63480,63482],{"className":63481},[163,164,165,166],[15,63483,1459],{"className":63484},[129,133,166],[15,63486,175],{"className":63487},[174],[15,63489,63491],{"className":63490},[146],[15,63492,63494],{"className":63493,"style":182},[150],[15,63495],{},[15,63497],{"className":63498,"style":3277},[93],[15,63500,315],{"className":63501},[2436],[15,63503],{"className":63504,"style":3277},[93],[15,63506,63508,63511,63554,63583,63589,63592,63595],{"className":63507},[120],[15,63509],{"className":63510,"style":62837},[124],[15,63512,63514],{"className":63513},[129,48500],[15,63515,63517,63545],{"className":63516},[141,142],[15,63518,63520,63542],{"className":63519},[146],[15,63521,63523,63531],{"className":63522,"style":522},[150],[15,63524,63525,63528],{"style":2287},[15,63526],{"className":63527,"style":1823},[158],[15,63529,11],{"className":63530},[129,133],[15,63532,63533,63536],{"style":2287},[15,63534],{"className":63535,"style":1823},[158],[15,63537,63539],{"className":63538,"style":48528},[48527],[15,63540,48483],{"className":63541},[129],[15,63543,175],{"className":63544},[174],[15,63546,63548],{"className":63547},[146],[15,63549,63552],{"className":63550,"style":63551},[150],"height:0.1944em;",[15,63553],{},[15,63555,63557,63560],{"className":63556},[195],[15,63558,59],{"className":63559},[195],[15,63561,63563],{"className":63562},[137],[15,63564,63566],{"className":63565},[141],[15,63567,63569],{"className":63568},[146],[15,63570,63572],{"className":63571,"style":40991},[150],[15,63573,63574,63577],{"style":1272},[15,63575],{"className":63576,"style":159},[158],[15,63578,63580],{"className":63579},[163,164,165,166],[15,63581,310],{"className":63582},[129,166],[15,63584,63586],{"className":63585},[195],[15,63587,1423],{"className":63588},[228,229],[15,63590],{"className":63591,"style":199},[93],[15,63593,7473],{"className":63594},[203],[15,63596],{"className":63597,"style":199},[93],[15,63599,63601,63605,63694,63697,63786,63789,63792],{"className":63600},[120],[15,63602],{"className":63603,"style":63604},[124],"height:2.4411em;vertical-align:-0.95em;",[15,63606,63608,63611,63691],{"className":63607},[129],[15,63609],{"className":63610},[188,1803],[15,63612,63614],{"className":63613},[1443],[15,63615,63617,63683],{"className":63616},[141,142],[15,63618,63620,63680],{"className":63619},[146],[15,63621,63624,63635,63643],{"className":63622,"style":63623},[150],"height:1.4911em;",[15,63625,63626,63629],{"style":1819},[15,63627],{"className":63628,"style":1823},[158],[15,63630,63632],{"className":63631},[129],[15,63633,14508],{"className":63634},[129],[15,63636,63637,63640],{"style":1832},[15,63638],{"className":63639,"style":1823},[158],[15,63641],{"className":63642,"style":1840},[1839],[15,63644,63645,63648],{"style":1843},[15,63646],{"className":63647,"style":1823},[158],[15,63649,63651],{"className":63650},[129],[15,63652,63654,63657],{"className":63653},[129],[15,63655,48354],{"className":63656},[129],[15,63658,63660],{"className":63659},[137],[15,63661,63663],{"className":63662},[141],[15,63664,63666],{"className":63665},[146],[15,63667,63669],{"className":63668,"style":380},[150],[15,63670,63671,63674],{"style":383},[15,63672],{"className":63673,"style":159},[158],[15,63675,63677],{"className":63676},[163,164,165,166],[15,63678,310],{"className":63679},[129,166],[15,63681,175],{"className":63682},[174],[15,63684,63686],{"className":63685},[146],[15,63687,63689],{"className":63688,"style":1862},[150],[15,63690],{},[15,63692],{"className":63693},[195,1803],[15,63695],{"className":63696,"style":262},[93],[15,63698,63700,63706,63709,63712,63715,63718,63780],{"className":63699},[449],[15,63701,63703],{"className":63702,"style":5052},[188,5051],[15,63704,53],{"className":63705},[228,165],[15,63707,318],{"className":63708},[129],[15,63710],{"className":63711,"style":3277},[93],[15,63713,315],{"className":63714},[2436],[15,63716],{"className":63717,"style":3277},[93],[15,63719,63721,63724,63777],{"className":63720},[129],[15,63722],{"className":63723},[188,1803],[15,63725,63727],{"className":63726},[1443],[15,63728,63730,63769],{"className":63729},[141,142],[15,63731,63733,63766],{"className":63732},[146],[15,63734,63736,63747,63755],{"className":63735,"style":1816},[150],[15,63737,63738,63741],{"style":1819},[15,63739],{"className":63740,"style":1823},[158],[15,63742,63744],{"className":63743},[129],[15,63745,310],{"className":63746},[129],[15,63748,63749,63752],{"style":1832},[15,63750],{"className":63751,"style":1823},[158],[15,63753],{"className":63754,"style":1840},[1839],[15,63756,63757,63760],{"style":1843},[15,63758],{"className":63759,"style":1823},[158],[15,63761,63763],{"className":63762},[129],[15,63764,318],{"className":63765},[129],[15,63767,175],{"className":63768},[174],[15,63770,63772],{"className":63771},[146],[15,63773,63775],{"className":63774,"style":1862},[150],[15,63776],{},[15,63778],{"className":63779},[195,1803],[15,63781,63783],{"className":63782,"style":5052},[195,5051],[15,63784,59],{"className":63785},[228,165],[15,63787],{"className":63788,"style":199},[93],[15,63790,62],{"className":63791},[203],[15,63793],{"className":63794,"style":199},[93],[15,63796,63798,63801,63866],{"className":63797},[120],[15,63799],{"className":63800,"style":57237},[124],[15,63802,63804,63807,63863],{"className":63803},[129],[15,63805],{"className":63806},[188,1803],[15,63808,63810],{"className":63809},[1443],[15,63811,63813,63855],{"className":63812},[141,142],[15,63814,63816,63852],{"className":63815},[146],[15,63817,63819,63833,63841],{"className":63818,"style":1816},[150],[15,63820,63821,63824],{"style":1819},[15,63822],{"className":63823,"style":1823},[158],[15,63825,63827,63830],{"className":63826},[129],[15,63828,63378],{"className":63829},[129],[15,63831,39326],{"className":63832},[129,133],[15,63834,63835,63838],{"style":1832},[15,63836],{"className":63837,"style":1823},[158],[15,63839],{"className":63840,"style":1840},[1839],[15,63842,63843,63846],{"style":1843},[15,63844],{"className":63845,"style":1823},[158],[15,63847,63849],{"className":63848},[129],[15,63850,318],{"className":63851},[129],[15,63853,175],{"className":63854},[174],[15,63856,63858],{"className":63857},[146],[15,63859,63861],{"className":63860,"style":1862},[150],[15,63862],{},[15,63864],{"className":63865},[195,1803],[15,63867,107],{"className":63868},[129],[11,63870,63871,63872,63907,63908],{},"所以 Bernoulli 均值的极小极大速率是 ",[15,63873,63875,63892],{"className":63874,"translate":19},[23],[15,63876,63878],{"className":63877},[27],[29,63879,63880],{"xmlns":31},[34,63881,63882,63890],{},[37,63883,63884,63886,63888],{},[308,63885,318],{},[43,63887,544],{"mathvariant":65},[43,63889,39326],{},[109,63891,48150],{"encoding":111},[15,63893,63895],{"className":63894,"ariaHidden":72},[116],[15,63896,63898,63901,63904],{"className":63897},[120],[15,63899],{"className":63900,"style":125},[124],[15,63902,29852],{"className":63903},[129],[15,63905,39326],{"className":63906},[129,133],"。这个结论不稀奇，但推导方式值得记住：",[698,63909,63910,63911,63946],{},"我们只用了两个点，就把「",[15,63912,63914,63931],{"className":63913,"translate":19},[23],[15,63915,63917],{"className":63916},[27],[29,63918,63919],{"xmlns":31},[34,63920,63921,63929],{},[37,63922,63923,63925,63927],{},[308,63924,318],{},[43,63926,544],{"mathvariant":65},[43,63928,39326],{},[109,63930,48150],{"encoding":111},[15,63932,63934],{"className":63933,"ariaHidden":72},[116],[15,63935,63937,63940,63943],{"className":63936},[120],[15,63938],{"className":63939,"style":125},[124],[15,63941,29852],{"className":63942},[129],[15,63944,39326],{"className":63945},[129,133]," 不可能被突破」这件事证明完了。",[575,63948,63950],{"id":63949},"_34-两点方法的局限","3.4 两点方法的局限",[11,63952,63953,63954,63982],{},"两点方法简单、常数清楚，但它只能看见一个方向。考虑 ",[15,63955,63957,63970],{"className":63956,"translate":19},[23],[15,63958,63960],{"className":63959},[27],[29,63961,63962],{"xmlns":31},[34,63963,63964,63968],{},[37,63965,63966],{},[43,63967,10376],{},[109,63969,10376],{"encoding":111},[15,63971,63973],{"className":63972,"ariaHidden":72},[116],[15,63974,63976,63979],{"className":63975},[120],[15,63977],{"className":63978,"style":522},[124],[15,63980,10376],{"className":63981},[129,133]," 维高斯均值：",[15,63984,63986],{"className":63985,"translate":19},[18],[15,63987,63989,64060],{"className":63988,"translate":19},[23],[15,63990,63992],{"className":63991},[27],[29,63993,63994],{"xmlns":31,"display":32},[34,63995,63996,64057],{},[37,63997,63998,64004,64006,64008,64010,64016,64018,64020,64022,64024,64026,64032,64039,64041,64043,64045,64047,64049,64055],{},[40,63999,64000,64002],{},[43,64001,597],{},[308,64003,318],{},[50,64005,91],{"separator":72},[50,64007,335],{},[50,64009,91],{"separator":72},[40,64011,64012,64014],{},[43,64013,597],{},[43,64015,39326],{},[50,64017,991],{},[43,64019,12561],{},[50,64021,53],{"stretchy":52},[43,64023,48],{},[50,64025,91],{"separator":72},[305,64027,64028,64030],{},[43,64029,27294],{},[308,64031,310],{},[40,64033,64034,64037],{},[43,64035,64036],{},"I",[43,64038,10376],{},[50,64040,59],{"stretchy":52},[50,64042,91],{"separator":72},[93,64044],{"width":95},[43,64046,48],{},[50,64048,100],{},[305,64050,64051,64053],{},[43,64052,104],{"mathvariant":103},[43,64054,10376],{},[43,64056,107],{"mathvariant":65},[109,64058,64059],{"encoding":111},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d.",[15,64061,64063,64176,64290],{"className":64062,"ariaHidden":72},[116],[15,64064,64066,64069,64109,64112,64115,64118,64121,64124,64127,64167,64170,64173],{"className":64065},[120],[15,64067],{"className":64068,"style":34366},[124],[15,64070,64072,64075],{"className":64071},[129],[15,64073,597],{"className":64074,"style":47970},[129,133],[15,64076,64078],{"className":64077},[137],[15,64079,64081,64101],{"className":64080},[141,142],[15,64082,64084,64098],{"className":64083},[146],[15,64085,64087],{"className":64086,"style":1051},[150],[15,64088,64089,64092],{"style":47985},[15,64090],{"className":64091,"style":159},[158],[15,64093,64095],{"className":64094},[163,164,165,166],[15,64096,318],{"className":64097},[129,166],[15,64099,175],{"className":64100},[174],[15,64102,64104],{"className":64103},[146],[15,64105,64107],{"className":64106,"style":182},[150],[15,64108],{},[15,64110,91],{"className":64111},[254],[15,64113],{"className":64114,"style":262},[93],[15,64116,335],{"className":64117},[449],[15,64119],{"className":64120,"style":262},[93],[15,64122,91],{"className":64123},[254],[15,64125],{"className":64126,"style":262},[93],[15,64128,64130,64133],{"className":64129},[129],[15,64131,597],{"className":64132,"style":47970},[129,133],[15,64134,64136],{"className":64135},[137],[15,64137,64139,64159],{"className":64138},[141,142],[15,64140,64142,64156],{"className":64141},[146],[15,64143,64145],{"className":64144,"style":1163},[150],[15,64146,64147,64150],{"style":47985},[15,64148],{"className":64149,"style":159},[158],[15,64151,64153],{"className":64152},[163,164,165,166],[15,64154,39326],{"className":64155},[129,133,166],[15,64157,175],{"className":64158},[174],[15,64160,64162],{"className":64161},[146],[15,64163,64165],{"className":64164,"style":182},[150],[15,64166],{},[15,64168],{"className":64169,"style":199},[93],[15,64171,991],{"className":64172},[203],[15,64174],{"className":64175,"style":199},[93],[15,64177,64179,64182,64185,64188,64191,64194,64197,64226,64266,64269,64272,64275,64278,64281,64284,64287],{"className":64178},[120],[15,64180],{"className":64181,"style":40920},[124],[15,64183,12561],{"className":64184,"style":12597},[129,133],[15,64186,53],{"className":64187},[188],[15,64189,48],{"className":64190,"style":170},[129,133],[15,64192,91],{"className":64193},[254],[15,64195],{"className":64196,"style":262},[93],[15,64198,64200,64203],{"className":64199},[129],[15,64201,27294],{"className":64202,"style":887},[129,133],[15,64204,64206],{"className":64205},[137],[15,64207,64209],{"className":64208},[141],[15,64210,64212],{"className":64211},[146],[15,64213,64215],{"className":64214,"style":40991},[150],[15,64216,64217,64220],{"style":1272},[15,64218],{"className":64219,"style":159},[158],[15,64221,64223],{"className":64222},[163,164,165,166],[15,64224,310],{"className":64225},[129,166],[15,64227,64229,64232],{"className":64228},[129],[15,64230,64036],{"className":64231,"style":47970},[129,133],[15,64233,64235],{"className":64234},[137],[15,64236,64238,64258],{"className":64237},[141,142],[15,64239,64241,64255],{"className":64240},[146],[15,64242,64244],{"className":64243,"style":151},[150],[15,64245,64246,64249],{"style":47985},[15,64247],{"className":64248,"style":159},[158],[15,64250,64252],{"className":64251},[163,164,165,166],[15,64253,10376],{"className":64254},[129,133,166],[15,64256,175],{"className":64257},[174],[15,64259,64261],{"className":64260},[146],[15,64262,64264],{"className":64263,"style":182},[150],[15,64265],{},[15,64267,59],{"className":64268},[195],[15,64270,91],{"className":64271},[254],[15,64273],{"className":64274,"style":258},[93],[15,64276],{"className":64277,"style":262},[93],[15,64279,48],{"className":64280,"style":170},[129,133],[15,64282],{"className":64283,"style":199},[93],[15,64285,100],{"className":64286},[203],[15,64288],{"className":64289,"style":199},[93],[15,64291,64293,64297,64326],{"className":64292},[120],[15,64294],{"className":64295,"style":64296},[124],"height:0.8991em;",[15,64298,64300,64303],{"className":64299},[129],[15,64301,104],{"className":64302},[129,285],[15,64304,64306],{"className":64305},[137],[15,64307,64309],{"className":64308},[141],[15,64310,64312],{"className":64311},[146],[15,64313,64315],{"className":64314,"style":64296},[150],[15,64316,64317,64320],{"style":1272},[15,64318],{"className":64319,"style":159},[158],[15,64321,64323],{"className":64322},[163,164,165,166],[15,64324,10376],{"className":64325},[129,133,166],[15,64327,107],{"className":64328},[129],[11,64330,64331,64332,33872,64423,64561,64562,64681,64682,64749,64750,64856,64857,64892,64893,64967,64968,1313],{},"如果只取 ",[15,64333,64335,64356],{"className":64334,"translate":19},[23],[15,64336,64338],{"className":64337},[27],[29,64339,64340],{"xmlns":31},[34,64341,64342,64354],{},[37,64343,64344,64350,64352],{},[40,64345,64346,64348],{},[43,64347,48],{},[308,64349,541],{},[50,64351,62],{},[308,64353,541],{},[109,64355,48263],{"encoding":111},[15,64357,64359,64414],{"className":64358,"ariaHidden":72},[116],[15,64360,64362,64365,64405,64408,64411],{"className":64361},[120],[15,64363],{"className":64364,"style":2183},[124],[15,64366,64368,64371],{"className":64367},[129],[15,64369,48],{"className":64370,"style":170},[129,133],[15,64372,64374],{"className":64373},[137],[15,64375,64377,64397],{"className":64376},[141,142],[15,64378,64380,64394],{"className":64379},[146],[15,64381,64383],{"className":64382,"style":1051},[150],[15,64384,64385,64388],{"style":48293},[15,64386],{"className":64387,"style":159},[158],[15,64389,64391],{"className":64390},[163,164,165,166],[15,64392,541],{"className":64393},[129,166],[15,64395,175],{"className":64396},[174],[15,64398,64400],{"className":64399},[146],[15,64401,64403],{"className":64402,"style":182},[150],[15,64404],{},[15,64406],{"className":64407,"style":199},[93],[15,64409,62],{"className":64410},[203],[15,64412],{"className":64413,"style":199},[93],[15,64415,64417,64420],{"className":64416},[120],[15,64418],{"className":64419,"style":4886},[124],[15,64421,541],{"className":64422},[129],[15,64424,64426,64454],{"className":64425,"translate":19},[23],[15,64427,64429],{"className":64428},[27],[29,64430,64431],{"xmlns":31},[34,64432,64433,64451],{},[37,64434,64435,64441,64443,64445],{},[40,64436,64437,64439],{},[43,64438,48],{},[308,64440,318],{},[50,64442,62],{},[43,64444,8690],{},[40,64446,64447,64449],{},[43,64448,10699],{},[308,64450,318],{},[109,64452,64453],{"encoding":111},"\\theta_1=a e_1",[15,64455,64457,64512],{"className":64456,"ariaHidden":72},[116],[15,64458,64460,64463,64503,64506,64509],{"className":64459},[120],[15,64461],{"className":64462,"style":2183},[124],[15,64464,64466,64469],{"className":64465},[129],[15,64467,48],{"className":64468,"style":170},[129,133],[15,64470,64472],{"className":64471},[137],[15,64473,64475,64495],{"className":64474},[141,142],[15,64476,64478,64492],{"className":64477},[146],[15,64479,64481],{"className":64480,"style":1051},[150],[15,64482,64483,64486],{"style":48293},[15,64484],{"className":64485,"style":159},[158],[15,64487,64489],{"className":64488},[163,164,165,166],[15,64490,318],{"className":64491},[129,166],[15,64493,175],{"className":64494},[174],[15,64496,64498],{"className":64497},[146],[15,64499,64501],{"className":64500,"style":182},[150],[15,64502],{},[15,64504],{"className":64505,"style":199},[93],[15,64507,62],{"className":64508},[203],[15,64510],{"className":64511,"style":199},[93],[15,64513,64515,64518,64521],{"className":64514},[120],[15,64516],{"className":64517,"style":7985},[124],[15,64519,8690],{"className":64520},[129,133],[15,64522,64524,64527],{"className":64523},[129],[15,64525,10699],{"className":64526},[129,133],[15,64528,64530],{"className":64529},[137],[15,64531,64533,64553],{"className":64532},[141,142],[15,64534,64536,64550],{"className":64535},[146],[15,64537,64539],{"className":64538,"style":1051},[150],[15,64540,64541,64544],{"style":154},[15,64542],{"className":64543,"style":159},[158],[15,64545,64547],{"className":64546},[163,164,165,166],[15,64548,318],{"className":64549},[129,166],[15,64551,175],{"className":64552},[174],[15,64554,64556],{"className":64555},[146],[15,64557,64559],{"className":64558,"style":182},[150],[15,64560],{},"，那么 KL 是 ",[15,64563,64565,64599],{"className":64564,"translate":19},[23],[15,64566,64568],{"className":64567},[27],[29,64569,64570],{"xmlns":31},[34,64571,64572,64596],{},[37,64573,64574,64576,64582,64584,64586,64588,64594],{},[43,64575,39326],{},[305,64577,64578,64580],{},[43,64579,8690],{},[308,64581,310],{},[43,64583,544],{"mathvariant":65},[50,64585,53],{"stretchy":52},[308,64587,310],{},[305,64589,64590,64592],{},[43,64591,27294],{},[308,64593,310],{},[50,64595,59],{"stretchy":52},[109,64597,64598],{"encoding":111},"n a^2\u002F(2\\sigma^2)",[15,64600,64602],{"className":64601,"ariaHidden":72},[116],[15,64603,64605,64608,64611,64640,64643,64646,64649,64678],{"className":64604},[120],[15,64606],{"className":64607,"style":34009},[124],[15,64609,39326],{"className":64610},[129,133],[15,64612,64614,64617],{"className":64613},[129],[15,64615,8690],{"className":64616},[129,133],[15,64618,64620],{"className":64619},[137],[15,64621,64623],{"className":64622},[141],[15,64624,64626],{"className":64625},[146],[15,64627,64629],{"className":64628,"style":380},[150],[15,64630,64631,64634],{"style":383},[15,64632],{"className":64633,"style":159},[158],[15,64635,64637],{"className":64636},[163,164,165,166],[15,64638,310],{"className":64639},[129,166],[15,64641,544],{"className":64642},[129],[15,64644,53],{"className":64645},[188],[15,64647,310],{"className":64648},[129],[15,64650,64652,64655],{"className":64651},[129],[15,64653,27294],{"className":64654,"style":887},[129,133],[15,64656,64658],{"className":64657},[137],[15,64659,64661],{"className":64660},[141],[15,64662,64664],{"className":64663},[146],[15,64665,64667],{"className":64666,"style":380},[150],[15,64668,64669,64672],{"style":383},[15,64670],{"className":64671,"style":159},[158],[15,64673,64675],{"className":64674},[163,164,165,166],[15,64676,310],{"className":64677},[129,166],[15,64679,59],{"className":64680},[195],"，Le Cam 给出的下界是 ",[15,64683,64685,64707],{"className":64684,"translate":19},[23],[15,64686,64688],{"className":64687},[27],[29,64689,64690],{"xmlns":31},[34,64691,64692,64704],{},[37,64693,64694,64700,64702],{},[305,64695,64696,64698],{},[43,64697,8690],{},[308,64699,310],{},[43,64701,544],{"mathvariant":65},[308,64703,14508],{},[109,64705,64706],{"encoding":111},"a^2\u002F4",[15,64708,64710],{"className":64709,"ariaHidden":72},[116],[15,64711,64713,64716,64745],{"className":64712},[120],[15,64714],{"className":64715,"style":34009},[124],[15,64717,64719,64722],{"className":64718},[129],[15,64720,8690],{"className":64721},[129,133],[15,64723,64725],{"className":64724},[137],[15,64726,64728],{"className":64727},[141],[15,64729,64731],{"className":64730},[146],[15,64732,64734],{"className":64733,"style":380},[150],[15,64735,64736,64739],{"style":383},[15,64737],{"className":64738,"style":159},[158],[15,64740,64742],{"className":64741},[163,164,165,166],[15,64743,310],{"className":64744},[129,166],[15,64746,64748],{"className":64747},[129],"\u002F4"," 的量级，取 ",[15,64751,64753,64778],{"className":64752,"translate":19},[23],[15,64754,64756],{"className":64755},[27],[29,64757,64758],{"xmlns":31},[34,64759,64760,64775],{},[37,64761,64762,64764,64767,64769,64771],{},[43,64763,8690],{},[50,64765,64766],{},"≍",[43,64768,27294],{},[43,64770,544],{"mathvariant":65},[29015,64772,64773],{},[43,64774,39326],{},[109,64776,64777],{"encoding":111},"a\\asymp\\sigma\u002F\\sqrt n",[15,64779,64781,64800],{"className":64780,"ariaHidden":72},[116],[15,64782,64784,64788,64791,64794,64797],{"className":64783},[120],[15,64785],{"className":64786,"style":64787},[124],"height:0.4637em;",[15,64789,8690],{"className":64790},[129,133],[15,64792],{"className":64793,"style":199},[93],[15,64795,64766],{"className":64796},[203],[15,64798],{"className":64799,"style":199},[93],[15,64801,64803,64806,64809,64812],{"className":64802},[120],[15,64804],{"className":64805,"style":30447},[124],[15,64807,27294],{"className":64808,"style":887},[129,133],[15,64810,544],{"className":64811},[129],[15,64813,64815],{"className":64814},[129,29123],[15,64816,64818,64848],{"className":64817},[141,142],[15,64819,64821,64845],{"className":64820},[146],[15,64822,64824,64833],{"className":64823,"style":30463},[150],[15,64825,64827,64830],{"className":64826,"style":2287},[29137],[15,64828],{"className":64829,"style":1823},[158],[15,64831,39326],{"className":64832,"style":30473},[129,133],[15,64834,64835,64838],{"style":30476},[15,64836],{"className":64837,"style":1823},[158],[15,64839,64841],{"className":64840,"style":30483},[29282],[4455,64842,64843],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},[4461,64844],{"d":30490},[15,64846,175],{"className":64847},[174],[15,64849,64851],{"className":64850},[146],[15,64852,64854],{"className":64853,"style":30500},[150],[15,64855],{}," 后只有 ",[15,64858,64860,64877],{"className":64859,"translate":19},[23],[15,64861,64863],{"className":64862},[27],[29,64864,64865],{"xmlns":31},[34,64866,64867,64875],{},[37,64868,64869,64871,64873],{},[308,64870,318],{},[43,64872,544],{"mathvariant":65},[43,64874,39326],{},[109,64876,48150],{"encoding":111},[15,64878,64880],{"className":64879,"ariaHidden":72},[116],[15,64881,64883,64886,64889],{"className":64882},[120],[15,64884],{"className":64885,"style":125},[124],[15,64887,29852],{"className":64888},[129],[15,64890,39326],{"className":64891},[129,133],"。但真实的极小极大风险是 ",[15,64894,64896,64920],{"className":64895,"translate":19},[23],[15,64897,64899],{"className":64898},[27],[29,64900,64901],{"xmlns":31},[34,64902,64903,64917],{},[37,64904,64905,64911,64913,64915],{},[305,64906,64907,64909],{},[43,64908,27294],{},[308,64910,310],{},[43,64912,10376],{},[43,64914,544],{"mathvariant":65},[43,64916,39326],{},[109,64918,64919],{"encoding":111},"\\sigma^2d\u002Fn",[15,64921,64923],{"className":64922,"ariaHidden":72},[116],[15,64924,64926,64929,64958,64961,64964],{"className":64925},[120],[15,64927],{"className":64928,"style":34009},[124],[15,64930,64932,64935],{"className":64931},[129],[15,64933,27294],{"className":64934,"style":887},[129,133],[15,64936,64938],{"className":64937},[137],[15,64939,64941],{"className":64940},[141],[15,64942,64944],{"className":64943},[146],[15,64945,64947],{"className":64946,"style":380},[150],[15,64948,64949,64952],{"style":383},[15,64950],{"className":64951,"style":159},[158],[15,64953,64955],{"className":64954},[163,164,165,166],[15,64956,310],{"className":64957},[129,166],[15,64959,10376],{"className":64960},[129,133],[15,64962,544],{"className":64963},[129],[15,64965,39326],{"className":64966},[129,133],"——差了整整一个 ",[15,64969,64971,64984],{"className":64970,"translate":19},[23],[15,64972,64974],{"className":64973},[27],[29,64975,64976],{"xmlns":31},[34,64977,64978,64982],{},[37,64979,64980],{},[43,64981,10376],{},[109,64983,10376],{"encoding":111},[15,64985,64987],{"className":64986,"ariaHidden":72},[116],[15,64988,64990,64993],{"className":64989},[120],[15,64991],{"className":64992,"style":522},[124],[15,64994,10376],{"className":64995},[129,133],[11,64997,64998,64999,65027,65028,65056,65057,65085],{},"原因很直接：两点方法只探测了一个坐标方向，而 ",[15,65000,65002,65015],{"className":65001,"translate":19},[23],[15,65003,65005],{"className":65004},[27],[29,65006,65007],{"xmlns":31},[34,65008,65009,65013],{},[37,65010,65011],{},[43,65012,10376],{},[109,65014,10376],{"encoding":111},[15,65016,65018],{"className":65017,"ariaHidden":72},[116],[15,65019,65021,65024],{"className":65020},[120],[15,65022],{"className":65023,"style":522},[124],[15,65025,10376],{"className":65026},[129,133]," 维问题的困难来自 ",[15,65029,65031,65044],{"className":65030,"translate":19},[23],[15,65032,65034],{"className":65033},[27],[29,65035,65036],{"xmlns":31},[34,65037,65038,65042],{},[37,65039,65040],{},[43,65041,10376],{},[109,65043,10376],{"encoding":111},[15,65045,65047],{"className":65046,"ariaHidden":72},[116],[15,65048,65050,65053],{"className":65049},[120],[15,65051],{"className":65052,"style":522},[124],[15,65054,10376],{"className":65055},[129,133]," 个方向同时存在的不确定性。要捕捉这一点，需要让假设集合随着 ",[15,65058,65060,65073],{"className":65059,"translate":19},[23],[15,65061,65063],{"className":65062},[27],[29,65064,65065],{"xmlns":31},[34,65066,65067,65071],{},[37,65068,65069],{},[43,65070,10376],{},[109,65072,10376],{"encoding":111},[15,65074,65076],{"className":65075,"ariaHidden":72},[116],[15,65077,65079,65082],{"className":65078},[120],[15,65080],{"className":65081,"style":522},[124],[15,65083,10376],{"className":65084},[129,133]," 指数增长。这就是 Fano 不等式和 Assouad 引理的用武之地。",[570,65087,65089],{"id":65088},"四fano-不等式","四、Fano 不等式",[575,65091,65093],{"id":65092},"_41-信息论形式","4.1 信息论形式",[6559,65095,65096,65586,65846],{},[11,65097,65098,52879,65101,5831,65129,65197,65198,65226,65227,7610,65255,65389,65390,22169],{},[698,65099,65100],{},"定理 4.1（Fano 不等式）",[15,65102,65104,65117],{"className":65103,"translate":19},[23],[15,65105,65107],{"className":65106},[27],[29,65108,65109],{"xmlns":31},[34,65110,65111,65115],{},[37,65112,65113],{},[43,65114,34524],{},[109,65116,34524],{"encoding":111},[15,65118,65120],{"className":65119,"ariaHidden":72},[116],[15,65121,65123,65126],{"className":65122},[120],[15,65124],{"className":65125,"style":610},[124],[15,65127,34524],{"className":65128,"style":5665},[129,133],[15,65130,65132,65158],{"className":65131,"translate":19},[23],[15,65133,65135],{"className":65134},[27],[29,65136,65137],{"xmlns":31},[34,65138,65139,65155],{},[37,65140,65141,65143,65145,65147,65149,65151,65153],{},[50,65142,636],{"stretchy":52},[308,65144,318],{},[50,65146,91],{"separator":72},[50,65148,335],{},[50,65150,91],{"separator":72},[43,65152,43246],{},[50,65154,645],{"stretchy":52},[109,65156,65157],{"encoding":111},"\\{1,\\dots,M\\}",[15,65159,65161],{"className":65160,"ariaHidden":72},[116],[15,65162,65164,65167,65170,65173,65176,65179,65182,65185,65188,65191,65194],{"className":65163},[120],[15,65165],{"className":65166,"style":125},[124],[15,65168,636],{"className":65169},[188],[15,65171,318],{"className":65172},[129],[15,65174,91],{"className":65175},[254],[15,65177],{"className":65178,"style":262},[93],[15,65180,335],{"className":65181},[449],[15,65183],{"className":65184,"style":262},[93],[15,65186,91],{"className":65187},[254],[15,65189],{"className":65190,"style":262},[93],[15,65192,43246],{"className":65193,"style":12597},[129,133],[15,65195,645],{"className":65196},[195]," 上均匀分布，",[15,65199,65201,65214],{"className":65200,"translate":19},[23],[15,65202,65204],{"className":65203},[27],[29,65205,65206],{"xmlns":31},[34,65207,65208,65212],{},[37,65209,65210],{},[43,65211,597],{},[109,65213,597],{"encoding":111},[15,65215,65217],{"className":65216,"ariaHidden":72},[116],[15,65218,65220,65223],{"className":65219},[120],[15,65221],{"className":65222,"style":610},[124],[15,65224,597],{"className":65225,"style":47970},[129,133]," 的分布依赖于 ",[15,65228,65230,65243],{"className":65229,"translate":19},[23],[15,65231,65233],{"className":65232},[27],[29,65234,65235],{"xmlns":31},[34,65236,65237,65241],{},[37,65238,65239],{},[43,65240,34524],{},[109,65242,34524],{"encoding":111},[15,65244,65246],{"className":65245,"ariaHidden":72},[116],[15,65247,65249,65252],{"className":65248},[120],[15,65250],{"className":65251,"style":610},[124],[15,65253,34524],{"className":65254,"style":5665},[129,133],[15,65256,65258,65290],{"className":65257,"translate":19},[23],[15,65259,65261],{"className":65260},[27],[29,65262,65263],{"xmlns":31},[34,65264,65265,65287],{},[37,65266,65267,65273,65275,65281,65283,65285],{},[46847,65268,65269,65271],{"accent":72},[43,65270,34524],{},[50,65272,48483],{},[50,65274,62],{},[46847,65276,65277,65279],{"accent":72},[43,65278,34524],{},[50,65280,48483],{},[50,65282,53],{"stretchy":52},[43,65284,597],{},[50,65286,59],{"stretchy":52},[109,65288,65289],{"encoding":111},"\\hat K=\\hat K(X)",[15,65291,65293,65342],{"className":65292,"ariaHidden":72},[116],[15,65294,65296,65300,65333,65336,65339],{"className":65295},[120],[15,65297],{"className":65298,"style":65299},[124],"height:0.9468em;",[15,65301,65303],{"className":65302},[129,48500],[15,65304,65306],{"className":65305},[141],[15,65307,65309],{"className":65308},[146],[15,65310,65312,65320],{"className":65311,"style":65299},[150],[15,65313,65314,65317],{"style":2287},[15,65315],{"className":65316,"style":1823},[158],[15,65318,34524],{"className":65319,"style":5665},[129,133],[15,65321,65323,65326],{"style":65322},"top:-3.2523em;",[15,65324],{"className":65325,"style":1823},[158],[15,65327,65330],{"className":65328,"style":65329},[48527],"left:-0.1944em;",[15,65331,48483],{"className":65332},[129],[15,65334],{"className":65335,"style":199},[93],[15,65337,62],{"className":65338},[203],[15,65340],{"className":65341,"style":199},[93],[15,65343,65345,65349,65380,65383,65386],{"className":65344},[120],[15,65346],{"className":65347,"style":65348},[124],"height:1.1968em;vertical-align:-0.25em;",[15,65350,65352],{"className":65351},[129,48500],[15,65353,65355],{"className":65354},[141],[15,65356,65358],{"className":65357},[146],[15,65359,65361,65369],{"className":65360,"style":65299},[150],[15,65362,65363,65366],{"style":2287},[15,65364],{"className":65365,"style":1823},[158],[15,65367,34524],{"className":65368,"style":5665},[129,133],[15,65370,65371,65374],{"style":65322},[15,65372],{"className":65373,"style":1823},[158],[15,65375,65377],{"className":65376,"style":65329},[48527],[15,65378,48483],{"className":65379},[129],[15,65381,53],{"className":65382},[188],[15,65384,597],{"className":65385,"style":47970},[129,133],[15,65387,59],{"className":65388},[195]," 是任意估计。记 ",[15,65391,65393,65431],{"className":65392,"translate":19},[23],[15,65394,65396],{"className":65395},[27],[29,65397,65398],{"xmlns":31},[34,65399,65400,65428],{},[37,65401,65402,65408,65410,65412,65414,65416,65422,65424,65426],{},[40,65403,65404,65406],{},[43,65405,1388],{},[43,65407,10699],{},[50,65409,62],{},[43,65411,4961],{},[50,65413,69],{},[50,65415,53],{"stretchy":52},[46847,65417,65418,65420],{"accent":72},[43,65419,34524],{},[50,65421,48483],{},[50,65423,1418],{"mathvariant":65},[43,65425,34524],{},[50,65427,59],{"stretchy":52},[109,65429,65430],{"encoding":111},"P_e=\\Pr(\\hat K\\neq 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1-\\frac{I(K;X)+\\log 2}{\\log 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是互信息。",[11,65907,65908],{},"证明分两步。第一步是条件熵的界：",[15,65910,65912],{"className":65911,"translate":19},[18],[15,65913,65915,66003],{"className":65914,"translate":19},[23],[15,65916,65918],{"className":65917},[27],[29,65919,65920],{"xmlns":31,"display":32},[34,65921,65922,66000],{},[37,65923,65924,65926,65928,65930,65932,65938,65940,65942,65944,65946,65952,65954,65956,65962,65964,65966,65968,65970,65972,65974,65976,65978,65980,65982,65984,65986,65992,65994,65996,65998],{},[43,65925,819],{},[50,65927,53],{"stretchy":52},[43,65929,34524],{},[50,65931,2945],{},[46847,65933,65934,65936],{"accent":72},[43,65935,34524],{},[50,65937,48483],{},[50,65939,59],{"stretchy":52},[50,65941,2717],{},[43,65943,45],{},[50,65945,53],{"stretchy":52},[40,65947,65948,65950],{},[43,65949,1388],{},[43,65951,10699],{},[50,65953,59],{"stretchy":52},[50,65955,2682],{},[40,65957,65958,65960],{},[43,65959,1388],{},[43,65961,10699],{},[43,65963,14727],{},[50,65965,69],{},[50,65967,53],{"stretchy":52},[43,65969,43246],{},[50,65971,315],{},[308,65973,318],{},[50,65975,59],{"stretchy":52},[50,65977,2717],{},[43,65979,14727],{},[50,65981,69],{},[308,65983,310],{},[50,65985,2682],{},[40,65987,65988,65990],{},[43,65989,1388],{},[43,65991,10699],{},[43,65993,14727],{},[50,65995,69],{},[43,65997,43246],{},[50,65999,91],{"separator":72},[109,66001,66002],{"encoding":111},"H(K\\mid 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2",[15,66361,66363,66427],{"className":66362,"ariaHidden":72},[116],[15,66364,66366,66369,66372,66375,66415,66418,66421,66424],{"className":66365},[120],[15,66367],{"className":66368,"style":125},[124],[15,66370,45],{"className":66371},[129,133],[15,66373,53],{"className":66374},[188],[15,66376,66378,66381],{"className":66377},[129],[15,66379,1388],{"className":66380,"style":12449},[129,133],[15,66382,66384],{"className":66383},[137],[15,66385,66387,66407],{"className":66386},[141,142],[15,66388,66390,66404],{"className":66389},[146],[15,66391,66393],{"className":66392,"style":1163},[150],[15,66394,66395,66398],{"style":34239},[15,66396],{"className":66397,"style":159},[158],[15,66399,66401],{"className":66400},[163,164,165,166],[15,66402,10699],{"className":66403},[129,133,166],[15,66405,175],{"className":66406},[174],[15,66408,66410],{"className":66409},[146],[15,66411,66413],{"className":66412,"style":182},[150],[15,66414],{},[15,66416,59],{"className":66417},[195],[15,66419],{"className":66420,"style":199},[93],[15,66422,2717],{"className":66423},[203],[15,66425],{"className":66426,"style":199},[93],[15,66428,66430,66433,66438,66441],{"className":66429},[120],[15,66431],{"className":66432,"style":1682},[124],[15,66434,14777,66436],{"className":66435},[217],[15,66437,2247],{"style":2246},[15,66439],{"className":66440,"style":262},[93],[15,66442,310],{"className":66443},[129]," 是二元熵的最大值。第二步是数据处理不等式：",[15,66446,66448],{"className":66447,"translate":19},[18],[15,66449,66451,66541],{"className":66450,"translate":19},[23],[15,66452,66454],{"className":66453},[27],[29,66455,66456],{"xmlns":31,"display":32},[34,66457,66458,66538],{},[37,66459,66460,66462,66464,66466,66468,66474,66476,66478,66480,66482,66484,66486,66488,66490,66492,66494,66496,66498,66500,66502,66504,66506,66508,66510,66512,66514,66516,66518,66520,66522,66524,66526,66528,66530,66532,66534,66536],{},[43,66461,819],{},[50,66463,53],{"stretchy":52},[43,66465,34524],{},[50,66467,2945],{},[46847,66469,66470,66472],{"accent":72},[43,66471,34524],{},[50,66473,48483],{},[50,66475,59],{"stretchy":52},[50,66477,7473],{},[43,66479,819],{},[50,66481,53],{"stretchy":52},[43,66483,34524],{},[50,66485,2945],{},[43,66487,597],{},[50,66489,59],{"stretchy":52},[50,66491,62],{},[43,66493,819],{},[50,66495,53],{"stretchy":52},[43,66497,34524],{},[50,66499,59],{"stretchy":52},[50,66501,315],{},[43,66503,64036],{},[50,66505,53],{"stretchy":52},[43,66507,34524],{},[50,66509,65625],{"separator":72},[43,66511,597],{},[50,66513,59],{"stretchy":52},[50,66515,62],{},[43,66517,14727],{},[50,66519,69],{},[43,66521,43246],{},[50,66523,315],{},[43,66525,64036],{},[50,66527,53],{"stretchy":52},[43,66529,34524],{},[50,66531,65625],{"separator":72},[43,66533,597],{},[50,66535,59],{"stretchy":52},[43,66537,107],{"mathvariant":65},[109,66539,66540],{"encoding":111},"H(K\\mid \\hat K)\\ge H(K\\mid X)=H(K)-I(K;X)=\\log M-I(K;X).",[15,66542,66544,66568,66617,66641,66662,66689,66725,66751],{"className":66543,"ariaHidden":72},[116],[15,66545,66547,66550,66553,66556,66559,66562,66565],{"className":66546},[120],[15,66548],{"className":66549,"style":125},[124],[15,66551,819],{"className":66552,"style":66015},[129,133],[15,66554,53],{"className":66555},[188],[15,66557,34524],{"className":66558,"style":5665},[129,133],[15,66560],{"className":66561,"style":199},[93],[15,66563,2945],{"className":66564},[203],[15,66566],{"className":66567,"style":199},[93],[15,66569,66571,66574,66605,66608,66611,66614],{"className":66570},[120],[15,66572],{"className":66573,"style":65348},[124],[15,66575,66577],{"className":66576},[129,48500],[15,66578,66580],{"className":66579},[141],[15,66581,66583],{"className":66582},[146],[15,66584,66586,66594],{"className":66585,"style":65299},[150],[15,66587,66588,66591],{"style":2287},[15,66589],{"className":66590,"style":1823},[158],[15,66592,34524],{"className":66593,"style":5665},[129,133],[15,66595,66596,66599],{"style":65322},[15,66597],{"className":66598,"style":1823},[158],[15,66600,66602],{"className":66601,"style":65329},[48527],[15,66603,48483],{"className":66604},[129],[15,66606,59],{"className":66607},[195],[15,66609],{"className":66610,"style":199},[93],[15,66612,7473],{"className":66613},[203],[15,66615],{"className":66616,"style":199},[93],[15,66618,66620,66623,66626,66629,66632,66635,66638],{"className":66619},[120],[15,66621],{"className":66622,"style":125},[124],[15,66624,819],{"className":66625,"style":66015},[129,133],[15,66627,53],{"className":66628},[188],[15,66630,34524],{"className":66631,"style":5665},[129,133],[15,66633],{"className":66634,"style":199},[93],[15,66636,2945],{"className":66637},[203],[15,66639],{"className":66640,"style":199},[93],[15,66642,66644,66647,66650,66653,66656,66659],{"className":66643},[120],[15,66645],{"className":66646,"style":125},[124],[15,66648,597],{"className":66649,"style":47970},[129,133],[15,66651,59],{"className":66652},[195],[15,66654],{"className":66655,"style":199},[93],[15,66657,62],{"className":66658},[203],[15,66660],{"className":66661,"style":199},[93],[15,66663,66665,66668,66671,66674,66677,66680,66683,66686],{"className":66664},[120],[15,66666],{"className":66667,"style":125},[124],[15,66669,819],{"className":66670,"style":66015},[129,133],[15,66672,53],{"className":66673},[188],[15,66675,34524],{"className":66676,"style":5665},[129,133],[15,66678,59],{"className":66679},[195],[15,66681],{"className":66682,"style":3277},[93],[15,66684,315],{"className":66685},[2436],[15,66687],{"className":66688,"style":3277},[93],[15,66690,66692,66695,66698,66701,66704,66707,66710,66713,66716,66719,66722],{"className":66691},[120],[15,66693],{"className":66694,"style":125},[124],[15,66696,64036],{"className":66697,"style":47970},[129,133],[15,66699,53],{"className":66700},[188],[15,66702,34524],{"className":66703,"style":5665},[129,133],[15,66705,65625],{"className":66706},[254],[15,66708],{"className":66709,"style":262},[93],[15,66711,597],{"className":66712,"style":47970},[129,133],[15,66714,59],{"className":66715},[195],[15,66717],{"className":66718,"style":199},[93],[15,66720,62],{"className":66721},[203],[15,66723],{"className":66724,"style":199},[93],[15,66726,66728,66731,66736,66739,66742,66745,66748],{"className":66727},[120],[15,66729],{"className":66730,"style":1682},[124],[15,66732,14777,66734],{"className":66733},[217],[15,66735,2247],{"style":2246},[15,66737],{"className":66738,"style":262},[93],[15,66740,43246],{"className":66741,"style":12597},[129,133],[15,66743],{"className":66744,"style":3277},[93],[15,66746,315],{"className":66747},[2436],[15,66749],{"className":66750,"style":3277},[93],[15,66752,66754,66757,66760,66763,66766,66769,66772,66775,66778],{"className":66753},[120],[15,66755],{"className":66756,"style":125},[124],[15,66758,64036],{"className":66759,"style":47970},[129,133],[15,66761,53],{"className":66762},[188],[15,66764,34524],{"className":66765,"style":5665},[129,133],[15,66767,65625],{"className":66768},[254],[15,66770],{"className":66771,"style":262},[93],[15,66773,597],{"className":66774,"style":47970},[129,133],[15,66776,59],{"className":66777},[195],[15,66779,107],{"className":66780},[129],[11,66782,66783],{},"两式合并，整理即得。",[11,66785,66786,66787,67044,67045,67086,67087,67127,67128,67156],{},"这个不等式有一个很直观的读法：",[698,66788,66789,66790,66818,66819,66847,66848,66899,66900,4890,66986,67014,67015,67043],{},"如果 ",[15,66791,66793,66806],{"className":66792,"translate":19},[23],[15,66794,66796],{"className":66795},[27],[29,66797,66798],{"xmlns":31},[34,66799,66800,66804],{},[37,66801,66802],{},[43,66803,597],{},[109,66805,597],{"encoding":111},[15,66807,66809],{"className":66808,"ariaHidden":72},[116],[15,66810,66812,66815],{"className":66811},[120],[15,66813],{"className":66814,"style":610},[124],[15,66816,597],{"className":66817,"style":47970},[129,133]," 几乎不包含关于 ",[15,66820,66822,66835],{"className":66821,"translate":19},[23],[15,66823,66825],{"className":66824},[27],[29,66826,66827],{"xmlns":31},[34,66828,66829,66833],{},[37,66830,66831],{},[43,66832,34524],{},[109,66834,34524],{"encoding":111},[15,66836,66838],{"className":66837,"ariaHidden":72},[116],[15,66839,66841,66844],{"className":66840},[120],[15,66842],{"className":66843,"style":610},[124],[15,66845,34524],{"className":66846,"style":5665},[129,133]," 的信息（",[15,66849,66851,66869],{"className":66850,"translate":19},[23],[15,66852,66854],{"className":66853},[27],[29,66855,66856],{"xmlns":31},[34,66857,66858,66866],{},[37,66859,66860,66862,66864],{},[43,66861,64036],{},[50,66863,32405],{},[308,66865,541],{},[109,66867,66868],{"encoding":111},"I\\approx0",[15,66870,66872,66890],{"className":66871,"ariaHidden":72},[116],[15,66873,66875,66878,66881,66884,66887],{"className":66874},[120],[15,66876],{"className":66877,"style":610},[124],[15,66879,64036],{"className":66880,"style":47970},[129,133],[15,66882],{"className":66883,"style":199},[93],[15,66885,32405],{"className":66886},[203],[15,66888],{"className":66889,"style":199},[93],[15,66891,66893,66896],{"className":66892},[120],[15,66894],{"className":66895,"style":4886},[124],[15,66897,541],{"className":66898},[129],"），那么任何猜测的错误率至少是 ",[15,66901,66903,66933],{"className":66902,"translate":19},[23],[15,66904,66906],{"className":66905},[27],[29,66907,66908],{"xmlns":31},[34,66909,66910,66930],{},[37,66911,66912,66914,66916,66918,66920,66922,66924,66926,66928],{},[308,66913,318],{},[50,66915,315],{},[43,66917,14727],{},[50,66919,69],{},[308,66921,310],{},[43,66923,544],{"mathvariant":65},[43,66925,14727],{},[50,66927,69],{},[43,66929,43246],{},[109,66931,66932],{"encoding":111},"1-\\log 2\u002F\\log M",[15,66934,66936,66954],{"className":66935,"ariaHidden":72},[116],[15,66937,66939,66942,66945,66948,66951],{"className":66938},[120],[15,66940],{"className":66941,"style":16484},[124],[15,66943,318],{"className":66944},[129],[15,66946],{"className":66947,"style":3277},[93],[15,66949,315],{"className":66950},[2436],[15,66952],{"className":66953,"style":3277},[93],[15,66955,66957,66960,66965,66968,66972,66975,66980,66983],{"className":66956},[120],[15,66958],{"className":66959,"style":125},[124],[15,66961,14777,66963],{"className":66962},[217],[15,66964,2247],{"style":2246},[15,66966],{"className":66967,"style":262},[93],[15,66969,66971],{"className":66970},[129],"2\u002F",[15,66973],{"className":66974,"style":262},[93],[15,66976,14777,66978],{"className":66977},[217],[15,66979,2247],{"style":2246},[15,66981],{"className":66982,"style":262},[93],[15,66984,43246],{"className":66985,"style":12597},[129,133],[15,66987,66989,67002],{"className":66988,"translate":19},[23],[15,66990,66992],{"className":66991},[27],[29,66993,66994],{"xmlns":31},[34,66995,66996,67000],{},[37,66997,66998],{},[43,66999,43246],{},[109,67001,43246],{"encoding":111},[15,67003,67005],{"className":67004,"ariaHidden":72},[116],[15,67006,67008,67011],{"className":67007},[120],[15,67009],{"className":67010,"style":610},[124],[15,67012,43246],{"className":67013,"style":12597},[129,133]," 很大时接近 ",[15,67016,67018,67031],{"className":67017,"translate":19},[23],[15,67019,67021],{"className":67020},[27],[29,67022,67023],{"xmlns":31},[34,67024,67025,67029],{},[37,67026,67027],{},[308,67028,318],{},[109,67030,318],{"encoding":111},[15,67032,67034],{"className":67033,"ariaHidden":72},[116],[15,67035,67037,67040],{"className":67036},[120],[15,67038],{"className":67039,"style":4886},[124],[15,67041,318],{"className":67042},[129],"，也就是接近「瞎猜」。"," 反过来，要让错误率小，互信息必须达到 ",[15,67046,67048,67066],{"className":67047,"translate":19},[23],[15,67049,67051],{"className":67050},[27],[29,67052,67053],{"xmlns":31},[34,67054,67055,67063],{},[37,67056,67057,67059,67061],{},[43,67058,14727],{},[50,67060,69],{},[43,67062,43246],{},[109,67064,67065],{"encoding":111},"\\log M",[15,67067,67069],{"className":67068,"ariaHidden":72},[116],[15,67070,67072,67075,67080,67083],{"className":67071},[120],[15,67073],{"className":67074,"style":1682},[124],[15,67076,14777,67078],{"className":67077},[217],[15,67079,2247],{"style":2246},[15,67081],{"className":67082,"style":262},[93],[15,67084,43246],{"className":67085,"style":12597},[129,133]," 的量级——你需要至少 ",[15,67088,67090,67107],{"className":67089,"translate":19},[23],[15,67091,67093],{"className":67092},[27],[29,67094,67095],{"xmlns":31},[34,67096,67097,67105],{},[37,67098,67099,67101,67103],{},[43,67100,14727],{},[50,67102,69],{},[43,67104,43246],{},[109,67106,67065],{"encoding":111},[15,67108,67110],{"className":67109,"ariaHidden":72},[116],[15,67111,67113,67116,67121,67124],{"className":67112},[120],[15,67114],{"className":67115,"style":1682},[124],[15,67117,14777,67119],{"className":67118},[217],[15,67120,2247],{"style":2246},[15,67122],{"className":67123,"style":262},[93],[15,67125,43246],{"className":67126,"style":12597},[129,133]," 奈特的信息才能区分 ",[15,67129,67131,67144],{"className":67130,"translate":19},[23],[15,67132,67134],{"className":67133},[27],[29,67135,67136],{"xmlns":31},[34,67137,67138,67142],{},[37,67139,67140],{},[43,67141,43246],{},[109,67143,43246],{"encoding":111},[15,67145,67147],{"className":67146,"ariaHidden":72},[116],[15,67148,67150,67153],{"className":67149},[120],[15,67151],{"className":67152,"style":610},[124],[15,67154,43246],{"className":67155,"style":12597},[129,133]," 个等可能的选择。",[575,67158,67160],{"id":67159},"_42-从检验到估计","4.2 从检验到估计",[11,67162,67163],{},"把 Fano 用在极小极大下界上，需要一个把「损失」翻译成「猜错」的分离条件。",[6559,67165,67166,67391,67641,67952,68555,68637],{},[11,67167,67168,52879,67171,13763,67199,67227,67228,67390],{},[698,67169,67170],{},"定理 4.2（Fano 方法，度量形式）",[15,67172,67174,67187],{"className":67173,"translate":19},[23],[15,67175,67177],{"className":67176},[27],[29,67178,67179],{"xmlns":31},[34,67180,67181,67185],{},[37,67182,67183],{},[43,67184,10376],{},[109,67186,10376],{"encoding":111},[15,67188,67190],{"className":67189,"ariaHidden":72},[116],[15,67191,67193,67196],{"className":67192},[120],[15,67194],{"className":67195,"style":522},[124],[15,67197,10376],{"className":67198},[129,133],[15,67200,67202,67215],{"className":67201,"translate":19},[23],[15,67203,67205],{"className":67204},[27],[29,67206,67207],{"xmlns":31},[34,67208,67209,67213],{},[37,67210,67211],{},[43,67212,30875],{"mathvariant":65},[109,67214,31304],{"encoding":111},[15,67216,67218],{"className":67217,"ariaHidden":72},[116],[15,67219,67221,67224],{"className":67220},[120],[15,67222],{"className":67223,"style":610},[124],[15,67225,30875],{"className":67226},[129]," 上的度量，",[15,67229,67231,67265],{"className":67230,"translate":19},[23],[15,67232,67234],{"className":67233},[27],[29,67235,67236],{"xmlns":31},[34,67237,67238,67262],{},[37,67239,67240,67246,67248,67250,67252,67258,67260],{},[40,67241,67242,67244],{},[43,67243,48],{},[308,67245,318],{},[50,67247,91],{"separator":72},[50,67249,335],{},[50,67251,91],{"separator":72},[40,67253,67254,67256],{},[43,67255,48],{},[43,67257,43246],{},[50,67259,100],{},[43,67261,30875],{"mathvariant":65},[109,67263,67264],{"encoding":111},"\\theta_1,\\dots,\\theta_M\\in\\Theta",[15,67266,67268,67381],{"className":67267,"ariaHidden":72},[116],[15,67269,67271,67274,67314,67317,67320,67323,67326,67329,67332,67372,67375,67378],{"className":67270},[120],[15,67272],{"className":67273,"style":1682},[124],[15,67275,67277,67280],{"className":67276},[129],[15,67278,48],{"className":67279,"style":170},[129,133],[15,67281,67283],{"className":67282},[137],[15,67284,67286,67306],{"className":67285},[141,142],[15,67287,67289,67303],{"className":67288},[146],[15,67290,67292],{"className":67291,"style":1051},[150],[15,67293,67294,67297],{"style":48293},[15,67295],{"className":67296,"style":159},[158],[15,67298,67300],{"className":67299},[163,164,165,166],[15,67301,318],{"className":67302},[129,166],[15,67304,175],{"className":67305},[174],[15,67307,67309],{"className":67308},[146],[15,67310,67312],{"className":67311,"style":182},[150],[15,67313],{},[15,67315,91],{"className":67316},[254],[15,67318],{"className":67319,"style":262},[93],[15,67321,335],{"className":67322},[449],[15,67324],{"className":67325,"style":262},[93],[15,67327,91],{"className":67328},[254],[15,67330],{"className":67331,"style":262},[93],[15,67333,67335,67338],{"className":67334},[129],[15,67336,48],{"className":67337,"style":170},[129,133],[15,67339,67341],{"className":67340},[137],[15,67342,67344,67364],{"className":67343},[141,142],[15,67345,67347,67361],{"className":67346},[146],[15,67348,67350],{"className":67349,"style":1525},[150],[15,67351,67352,67355],{"style":48293},[15,67353],{"className":67354,"style":159},[158],[15,67356,67358],{"className":67357},[163,164,165,166],[15,67359,43246],{"className":67360,"style":12597},[129,133,166],[15,67362,175],{"className":67363},[174],[15,67365,67367],{"className":67366},[146],[15,67368,67370],{"className":67369,"style":182},[150],[15,67371],{},[15,67373],{"className":67374,"style":199},[93],[15,67376,100],{"className":67377},[203],[15,67379],{"className":67380,"style":199},[93],[15,67382,67384,67387],{"className":67383},[120],[15,67385],{"className":67386,"style":610},[124],[15,67388,30875],{"className":67389},[129]," 满足",[15,67392,67394],{"className":67393,"translate":19},[18],[15,67395,67397,67449],{"className":67396,"translate":19},[23],[15,67398,67400],{"className":67399},[27],[29,67401,67402],{"xmlns":31,"display":32},[34,67403,67404,67446],{},[37,67405,67406,67408,67410,67416,67418,67424,67426,67428,67430,67432,67434,67436,67438,67440,67442,67444],{},[43,67407,10376],{},[50,67409,53],{"stretchy":52},[40,67411,67412,67414],{},[43,67413,48],{},[43,67415,1459],{},[50,67417,91],{"separator":72},[40,67419,67420,67422],{},[43,67421,48],{},[43,67423,13955],{},[50,67425,59],{"stretchy":52},[50,67427,7473],{},[308,67429,310],{},[43,67431,29575],{},[93,67433],{"width":95},[50,67435,53],{"stretchy":52},[43,67437,1459],{},[50,67439,1418],{"mathvariant":65},[43,67441,13955],{},[50,67443,59],{"stretchy":52},[43,67445,107],{"mathvariant":65},[109,67447,67448],{"encoding":111},"d(\\theta_i,\\theta_j)\\ge 2\\delta\\qquad(i\\neq 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d(\\theta_K,\\theta_j)-d(\\theta_K,\\hat\\theta)>2\\delta-\\delta=\\delta>d(\\theta_K,\\hat\\theta),",[15,69742,69744,69846,69956,70057,70078,70096,70114],{"className":69743,"ariaHidden":72},[116],[15,69745,69747,69751,69754,69757,69788,69791,69794,69834,69837,69840,69843],{"className":69746},[120],[15,69748],{"className":69749,"style":69750},[124],"height:1.244em;vertical-align:-0.2861em;",[15,69752,10376],{"className":69753},[129,133],[15,69755,53],{"className":69756},[188],[15,69758,69760],{"className":69759},[129,48500],[15,69761,69763],{"className":69762},[141],[15,69764,69766],{"className":69765},[146],[15,69767,69769,69777],{"className":69768,"style":48496},[150],[15,69770,69771,69774],{"style":2287},[15,69772],{"className":69773,"style":1823},[158],[15,69775,48],{"className":69776,"style":170},[129,133],[15,69778,69779,69782],{"style":48520},[15,69780],{"className":69781,"style":1823},[158],[15,69783,69785],{"className":69784,"style":48528},[48527],[15,69786,48483],{"className":69787},[129],[15,69789,91],{"className":69790},[254],[15,69792],{"className":69793,"style":262},[93],[15,69795,69797,69800],{"className":69796},[129],[15,69798,48],{"className":69799,"style":170},[129,133],[15,69801,69803],{"className":69802},[137],[15,69804,69806,69826],{"className":69805},[141,142],[15,69807,69809,69823],{"className":69808},[146],[15,69810,69812],{"className":69811,"style":1981},[150],[15,69813,69814,69817],{"style":48293},[15,69815],{"className":69816,"style":159},[158],[15,69818,69820],{"className":69819},[163,164,165,166],[15,69821,13955],{"className":69822,"style":13970},[129,133,166],[15,69824,175],{"className":69825},[174],[15,69827,69829],{"className":69828},[146],[15,69830,69832],{"className":69831,"style":14041},[150],[15,69833],{},[15,69835,59],{"className":69836},[195],[15,69838],{"className":69839,"style":199},[93],[15,69841,7473],{"className":69842},[203],[15,69844],{"className":69845,"style":199},[93],[15,69847,69849,69852,69855,69858,69898,69901,69904,69944,69947,69950,69953],{"className":69848},[120],[15,69850],{"className":69851,"style":67458},[124],[15,69853,10376],{"className":69854},[129,133],[15,69856,53],{"className":69857},[188],[15,69859,69861,69864],{"className":69860},[129],[15,69862,48],{"className":69863,"style":170},[129,133],[15,69865,69867],{"className":69866},[137],[15,69868,69870,69890],{"className":69869},[141,142],[15,69871,69873,69887],{"className":69872},[146],[15,69874,69876],{"className":69875,"style":1525},[150],[15,69877,69878,69881],{"style":48293},[15,69879],{"className":69880,"style":159},[158],[15,69882,69884],{"className":69883},[163,164,165,166],[15,69885,34524],{"className":69886,"style":5665},[129,133,166],[15,69888,175],{"className":69889},[174],[15,69891,69893],{"className":69892},[146],[15,69894,69896],{"className":69895,"style":182},[150],[15,69897],{},[15,69899,91],{"className":69900},[254],[15,69902],{"className":69903,"style":262},[93],[15,69905,69907,69910],{"className":69906},[129],[15,69908,48],{"className":69909,"style":170},[129,133],[15,69911,69913],{"className":69912},[137],[15,69914,69916,69936],{"className":69915},[141,142],[15,69917,69919,69933],{"className":69918},[146],[15,69920,69922],{"className":69921,"style":1981},[150],[15,69923,69924,69927],{"style":48293},[15,69925],{"className":69926,"style":159},[158],[15,69928,69930],{"className":69929},[163,164,165,166],[15,69931,13955],{"className":69932,"style":13970},[129,133,166],[15,69934,175],{"className":69935},[174],[15,69937,69939],{"className":69938},[146],[15,69940,69942],{"className":69941,"style":14041},[150],[15,69943],{},[15,69945,59],{"className":69946},[195],[15,69948],{"className":69949,"style":3277},[93],[15,69951,315],{"className":69952},[2436],[15,69954],{"className":69955,"style":3277},[93],[15,69957,69959,69962,69965,69968,70008,70011,70014,70045,70048,70051,70054],{"className":69958},[120],[15,69960],{"className":69961,"style":50032},[124],[15,69963,10376],{"className":69964},[129,133],[15,69966,53],{"className":69967},[188],[15,69969,69971,69974],{"className":69970},[129],[15,69972,48],{"className":69973,"style":170},[129,133],[15,69975,69977],{"className":69976},[137],[15,69978,69980,70000],{"className":69979},[141,142],[15,69981,69983,69997],{"className":69982},[146],[15,69984,69986],{"className":69985,"style":1525},[150],[15,69987,69988,69991],{"style":48293},[15,69989],{"className":69990,"style":159},[158],[15,69992,69994],{"className":69993},[163,164,165,166],[15,69995,34524],{"className":69996,"style":5665},[129,133,166],[15,69998,175],{"className":69999},[174],[15,70001,70003],{"className":70002},[146],[15,70004,70006],{"className":70005,"style":182},[150],[15,70007],{},[15,70009,91],{"className":70010},[254],[15,70012],{"className":70013,"style":262},[93],[15,70015,70017],{"className":70016},[129,48500],[15,70018,70020],{"className":70019},[141],[15,70021,70023],{"className":70022},[146],[15,70024,70026,70034],{"className":70025,"style":48496},[150],[15,70027,7002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",[15,70213,70215,70228],{"className":70214,"translate":19},[23],[15,70216,70218],{"className":70217},[27],[29,70219,70220],{"xmlns":31},[34,70221,70222,70226],{},[37,70223,70224],{},[43,70225,34524],{},[109,70227,34524],{"encoding":111},[15,70229,70231],{"className":70230,"ariaHidden":72},[116],[15,70232,70234,70237],{"className":70233},[120],[15,70235],{"className":70236,"style":610},[124],[15,70238,34524],{"className":70239,"style":5665},[129,133]," 是唯一的最小值点，",[15,70242,70244,70266],{"className":70243,"translate":19},[23],[15,70245,70247],{"className":70246},[27],[29,70248,70249],{"xmlns":31},[34,70250,70251,70263],{},[37,70252,70253,70259,70261],{},[46847,70254,70255,70257],{"accent":72},[43,70256,34524],{},[50,70258,48483],{},[50,70260,62],{},[43,70262,34524],{},[109,70264,70265],{"encoding":111},"\\hat K=K",[15,70267,70269,70315],{"className":70268,"ariaHidden":72},[116],[15,70270,70272,70275,70306,70309,70312],{"className":70271},[120],[15,70273],{"className":70274,"style":65299},[124],[15,70276,70278],{"className":70277},[129,48500],[15,70279,70281],{"className":70280},[141],[15,70282,70284],{"className":70283},[146],[15,70285,70287,70295],{"className":70286,"style":65299},[150],[15,70288,70289,70292],{"style":2287},[15,70290],{"className":70291,"style":1823},[158],[15,70293,34524],{"className":70294,"style":5665},[129,133],[15,70296,70297,70300],{"style":65322},[15,70298],{"className":70299,"style":1823},[158],[15,70301,70303],{"className":70302,"style":65329},[48527],[15,70304,48483],{"className":70305},[129],[15,70307],{"className":70308,"style":199},[93],[15,70310,62],{"className":70311},[203],[15,70313],{"className":70314,"style":199},[93],[15,70316,70318,70321],{"className":70317},[120],[15,70319],{"className":70320,"style":610},[124],[15,70322,34524],{"className":70323,"style":5665},[129,133],"。因此",[15,70326,70328],{"className":70327,"translate":19},[18],[15,70329,70331,70389],{"className":70330,"translate":19},[23],[15,70332,70334],{"className":70333},[27],[29,70335,70336],{"xmlns":31,"display":32},[34,70337,70338,70386],{},[37,70339,70340,70342,70344,70346,70352,70354,70360,70362,70364,70366,70368,70370,70372,70378,70380,70382,70384],{},[50,70341,636],{"stretchy":52},[43,70343,10376],{},[50,70345,53],{"stretchy":52},[40,70347,70348,70350],{},[43,70349,48],{},[43,70351,34524],{},[50,70353,91],{"separator":72},[46847,70355,70356,70358],{"accent":72},[43,70357,48],{},[50,70359,48483],{},[50,70361,59],{"stretchy":52},[50,70363,7910],{},[43,70365,29575],{},[50,70367,645],{"stretchy":52},[50,70369,5647],{},[50,70371,636],{"stretchy":52},[46847,70373,70374,70376],{"accent":72},[43,70375,34524],{},[50,70377,48483],{},[50,70379,62],{},[43,70381,34524],{},[50,70383,645],{"stretchy":52},[50,70385,91],{"separator":72},[109,70387,70388],{"encoding":111},"\\{d(\\theta_K,\\hat\\theta)\u003C\\delta\\}\\subseteq\\{\\hat 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",[15,71310,71312,71325],{"className":71311,"translate":19},[23],[15,71313,71315],{"className":71314},[27],[29,71316,71317],{"xmlns":31},[34,71318,71319,71323],{},[37,71320,71321],{},[43,71322,43246],{},[109,71324,43246],{"encoding":111},[15,71326,71328],{"className":71327,"ariaHidden":72},[116],[15,71329,71331,71334],{"className":71330},[120],[15,71332],{"className":71333,"style":610},[124],[15,71335,43246],{"className":71336,"style":12597},[129,133]," 个风险的平均，平均不超过最大，定理得证。",[11,71339,71340,71341,71422,71423,30212,71451,71530,71531,71560,71561,71772,71773,71878],{},"对平方损失 ",[15,71342,71344,71366],{"className":71343,"translate":19},[23],[15,71345,71347],{"className":71346},[27],[29,71348,71349],{"xmlns":31},[34,71350,71351,71363],{},[37,71352,71353,71355,71357],{},[43,71354,27529],{"mathvariant":65},[50,71356,62],{},[305,71358,71359,71361],{},[43,71360,10376],{},[308,71362,310],{},[109,71364,71365],{"encoding":111},"\\ell=d^2",[15,71367,71369,71387],{"className":71368,"ariaHidden":72},[116],[15,71370,71372,71375,71378,71381,71384],{"className":71371},[120],[15,71373],{"className":71374,"style":522},[124],[15,71376,27529],{"className":71377},[129],[15,71379],{"className":71380,"style":199},[93],[15,71382,62],{"className":71383},[203],[15,71385],{"className":71386,"style":199},[93],[15,71388,71390,71393],{"className":71389},[120],[15,71391],{"className":71392,"style":380},[124],[15,71394,71396,71399],{"className":71395},[129],[15,71397,10376],{"className":71398},[129,133],[15,71400,71402],{"className":71401},[137],[15,71403,71405],{"className":71404},[141],[15,71406,71408],{"className":71407},[146],[15,71409,71411],{"className":71410,"style":380},[150],[15,71412,71413,71416],{"style":383},[15,71414],{"className":71415,"style":159},[158],[15,71417,71419],{"className":71418},[163,164,165,166],[15,71420,310],{"className":71421},[129,166],"，同样的解码论证把 ",[15,71424,71426,71439],{"className":71425,"translate":19},[23],[15,71427,71429],{"className":71428},[27],[29,71430,71431],{"xmlns":31},[34,71432,71433,71437],{},[37,71434,71435],{},[43,71436,10376],{},[109,71438,10376],{"encoding":111},[15,71440,71442],{"className":71441,"ariaHidden":72},[116],[15,71443,71445,71448],{"className":71444},[120],[15,71446],{"className":71447,"style":522},[124],[15,71449,10376],{"className":71450},[129,133],[15,71452,71454,71470],{"className":71453,"translate":19},[23],[15,71455,71457],{"className":71456},[27],[29,71458,71459],{"xmlns":31},[34,71460,71461,71467],{},[37,71462,71463],{},[29015,71464,71465],{},[43,71466,27529],{"mathvariant":65},[109,71468,71469],{"encoding":111},"\\sqrt{\\ell}",[15,71471,71473],{"className":71472,"ariaHidden":72},[116],[15,71474,71476,71480],{"className":71475},[120],[15,71477],{"className":71478,"style":71479},[124],"height:1.04em;vertical-align:-0.1078em;",[15,71481,71483],{"className":71482},[129,29123],[15,71484,71486,71521],{"className":71485},[141,142],[15,71487,71489,71518],{"className":71488},[146],[15,71490,71493,71505],{"className":71491,"style":71492},[150],"height:0.9322em;",[15,71494,71496,71499],{"className":71495,"style":2287},[29137],[15,71497],{"className":71498,"style":1823},[158],[15,71500,71502],{"className":71501,"style":30473},[129],[15,71503,27529],{"className":71504},[129],[15,71506,71508,71511],{"style":71507},"top:-2.8922em;",[15,71509],{"className":71510,"style":1823},[158],[15,71512,71514],{"className":71513,"style":30483},[29282],[4455,71515,71516],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},[4461,71517],{"d":30490},[15,71519,175],{"className":71520},[174],[15,71522,71524],{"className":71523},[146],[15,71525,71528],{"className":71526,"style":71527},[150],"height:0.1078em;",[15,71529],{},"。若平方间隔为 ",[15,71532,71534,71548],{"className":71533,"translate":19},[23],[15,71535,71537],{"className":71536},[27],[29,71538,71539],{"xmlns":31},[34,71540,71541,71545],{},[37,71542,71543],{},[43,71544,48354],{"mathvariant":65},[109,71546,71547],{"encoding":111},"\\Delta",[15,71549,71551],{"className":71550,"ariaHidden":72},[116],[15,71552,71554,71557],{"className":71553},[120],[15,71555],{"className":71556,"style":610},[124],[15,71558,48354],{"className":71559},[129],"，即 ",[15,71562,71564,71602],{"className":71563,"translate":19},[23],[15,71565,71567],{"className":71566},[27],[29,71568,71569],{"xmlns":31},[34,71570,71571,71599],{},[37,71572,71573,71575,71577,71583,71585,71591,71593,71595],{},[43,71574,10376],{},[50,71576,53],{"stretchy":52},[40,71578,71579,71581],{},[43,71580,48],{},[43,71582,1459],{},[50,71584,91],{"separator":72},[40,71586,71587,71589],{},[43,71588,48],{},[43,71590,13955],{},[50,71592,59],{"stretchy":52},[50,71594,7473],{},[29015,71596,71597],{},[43,71598,48354],{"mathvariant":65},[109,71600,71601],{"encoding":111},"d(\\theta_i,\\theta_j)\\ge\\sqrt{\\Delta}",[15,71603,71605,71715],{"className":71604,"ariaHidden":72},[116],[15,71606,71608,71611,71614,71617,71657,71660,71663,71703,71706,71709,71712],{"className":71607},[120],[15,71609],{"className":71610,"style":67458},[124],[15,71612,10376],{"className":71613},[129,133],[15,71615,53],{"className":71616},[188],[15,71618,71620,71623],{"className":71619},[129],[15,71621,48],{"className":71622,"style":170},[129,133],[15,71624,71626],{"className":71625},[137],[15,71627,71629,71649],{"className":71628},[141,142],[15,71630,71632,71646],{"className":71631},[146],[15,71633,71635],{"className":71634,"style":1981},[150],[15,71636,71637,71640],{"style":48293},[15,71638],{"className":71639,"style":159},[158],[15,71641,71643],{"className":71642},[163,164,165,166],[15,71644,1459],{"className":71645},[129,133,166],[15,71647,175],{"className":71648},[174],[15,71650,71652],{"className":71651},[146],[15,71653,71655],{"className":71654,"style":182},[150],[15,71656],{},[15,71658,91],{"className":71659},[254],[15,71661],{"className":71662,"style":262},[93],[15,71664,71666,71669],{"className":71665},[129],[15,71667,48],{"className":71668,"style":170},[129,133],[15,71670,71672],{"className":71671},[137],[15,71673,71675,71695],{"className":71674},[141,142],[15,71676,71678,71692],{"className":71677},[146],[15,71679,71681],{"className":71680,"style":1981},[150],[15,71682,71683,71686],{"style":48293},[15,71684],{"className":71685,"style":159},[158],[15,71687,71689],{"className":71688},[163,164,165,166],[15,71690,13955],{"className":71691,"style":13970},[129,133,166],[15,71693,175],{"className":71694},[174],[15,71696,71698],{"className":71697},[146],[15,71699,71701],{"className":71700,"style":14041},[150],[15,71702],{},[15,71704,59],{"className":71705},[195],[15,71707],{"className":71708,"style":199},[93],[15,71710,7473],{"className":71711},[203],[15,71713],{"className":71714,"style":199},[93],[15,71716,71718,71722],{"className":71717},[120],[15,71719],{"className":71720,"style":71721},[124],"height:1.04em;vertical-align:-0.1133em;",[15,71723,71725],{"className":71724},[129,29123],[15,71726,71728,71763],{"className":71727},[141,142],[15,71729,71731,71760],{"className":71730},[146],[15,71732,71735,71747],{"className":71733,"style":71734},[150],"height:0.9267em;",[15,71736,71738,71741],{"className":71737,"style":2287},[29137],[15,71739],{"className":71740,"style":1823},[158],[15,71742,71744],{"className":71743,"style":30473},[129],[15,71745,48354],{"className":71746},[129],[15,71748,71750,71753],{"style":71749},"top:-2.8867em;",[15,71751],{"className":71752,"style":1823},[158],[15,71754,71756],{"className":71755,"style":30483},[29282],[4455,71757,71758],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},[4461,71759],{"d":30490},[15,71761,175],{"className":71762},[174],[15,71764,71766],{"className":71765},[146],[15,71767,71770],{"className":71768,"style":71769},[150],"height:0.1133em;",[15,71771],{},"，则取 ",[15,71774,71776,71800],{"className":71775,"translate":19},[23],[15,71777,71779],{"className":71778},[27],[29,71780,71781],{"xmlns":31},[34,71782,71783,71797],{},[37,71784,71785,71787,71789,71793,71795],{},[43,71786,29575],{},[50,71788,62],{},[29015,71790,71791],{},[43,71792,48354],{"mathvariant":65},[43,71794,544],{"mathvariant":65},[308,71796,310],{},[109,71798,71799],{"encoding":111},"\\delta=\\sqrt{\\Delta}\u002F2",[15,71801,71803,71821],{"className":71802,"ariaHidden":72},[116],[15,71804,71806,71809,71812,71815,71818],{"className":71805},[120],[15,71807],{"className":71808,"style":522},[124],[15,71810,29575],{"className":71811,"style":29856},[129,133],[15,71813],{"className":71814,"style":199},[93],[15,71816,62],{"className":71817},[203],[15,71819],{"className":71820,"style":199},[93],[15,71822,71824,71828,71875],{"className":71823},[120],[15,71825],{"className":71826,"style":71827},[124],"height:1.1767em;vertical-align:-0.25em;",[15,71829,71831],{"className":71830},[129,29123],[15,71832,71834,71867],{"className":71833},[141,142],[15,71835,71837,71864],{"className":71836},[146],[15,71838,71840,71852],{"className":71839,"style":71734},[150],[15,71841,71843,71846],{"className":71842,"style":2287},[29137],[15,71844],{"className":71845,"style":1823},[158],[15,71847,71849],{"className":71848,"style":30473},[129],[15,71850,48354],{"className":71851},[129],[15,71853,71854,71857],{"style":71749},[15,71855],{"className":71856,"style":1823},[158],[15,71858,71860],{"className":71859,"style":30483},[29282],[4455,71861,71862],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},[4461,71863],{"d":30490},[15,71865,175],{"className":71866},[174],[15,71868,71870],{"className":71869},[146],[15,71871,71873],{"className":71872,"style":71769},[150],[15,71874],{},[15,71876,56618],{"className":71877},[129],"，得到",[15,71880,71882],{"className":71881,"translate":19},[18],[15,71883,71885,71987],{"className":71884,"translate":19},[23],[15,71886,71888],{"className":71887},[27],[29,71889,71890],{"xmlns":31,"display":32},[34,71891,71892,71984],{},[37,71893,71894,71900,71906,71908,71910,71912,71914,71920,71922,71928,71930,71932,71934,71940,71982],{},[1443,71895,71896,71898],{},[308,71897,318],{},[43,71899,43246],{},[2140,71901,71902,71904],{},[50,71903,1454],{},[43,71905,1459],{},[43,71907,26574],{"mathvariant":103},[50,71909,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[43,71911,27529],{"mathvariant":65},[50,71913,53],{"stretchy":52},[40,71915,71916,71918],{},[43,71917,48],{},[43,71919,1459],{},[50,71921,91],{"separator":72},[46847,71923,71924,71926],{"accent":72},[43,71925,48],{},[50,71927,48483],{},[50,71929,59],{"stretchy":52},[50,71931,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,71933,7473],{},[1443,71935,71936,71938],{},[43,71937,48354],{"mathvariant":65},[308,71939,14508],{},[37,71941,71942,71944,71946,71948,71980],{},[50,71943,53],{"fence":72},[308,71945,318],{},[50,71947,315],{},[1443,71949,71950,71972],{},[37,71951,71952,71954,71956,71958,71960,71962,71964,71966,71968,71970],{},[43,71953,64036],{},[50,71955,53],{"stretchy":52},[43,71957,34524],{},[50,71959,65625],{"separator":72},[43,71961,597],{},[50,71963,59],{"stretchy":52},[50,71965,2682],{},[43,71967,14727],{},[50,71969,69],{},[308,71971,310],{},[37,71973,71974,71976,71978],{},[43,71975,14727],{},[50,71977,69],{},[43,71979,43246],{},[50,71981,59],{"fence":72},[43,71983,107],{"mathvariant":65},[109,71985,71986],{"encoding":111},"\\frac1M\\sum_i\\mathbb E\\bigl[\\ell(\\theta_i,\\hat\\theta)\\bigr]\n\\ge\n\\frac{\\Delta}{4}\\left(1-\\frac{I(K;X)+\\log 2}{\\log M}\\right).",[15,71988,71990,72219],{"className":71989,"ariaHidden":72},[116],[15,71991,71993,71997,72059,72062,72106,72109,72112,72118,72121,72124,72164,72167,72170,72201,72204,72210,72213,72216],{"className":71992},[120],[15,71994],{"className":71995,"style":71996},[124],"height:2.5991em;vertical-align:-1.2777em;",[15,71998,72000,72003,72056],{"className":71999},[129],[15,72001],{"className":72002},[188,1803],[15,72004,72006],{"className":72005},[1443],[15,72007,72009,72048],{"className":72008},[141,142],[15,72010,72012,72045],{"className":72011},[146],[15,72013,72015,72026,72034],{"className":72014,"style":1816},[150],[15,72016,72017,72020],{"style":1819},[15,72018],{"className":72019,"style":1823},[158],[15,72021,72023],{"className":72022},[129],[15,72024,43246],{"className":72025,"style":12597},[129,133],[15,72027,72028,72031],{"style":1832},[15,72029],{"className":72030,"style":1823},[158],[15,72032],{"className":72033,"style":1840},[1839],[15,72035,72036,72039],{"style":1843},[15,72037],{"className":72038,"style":1823},[158],[15,72040,72042],{"className":72041},[129],[15,72043,318],{"className":72044},[129],[15,72046,175],{"className":72047},[174],[15,72049,72051],{"className":72050},[146],[15,72052,72054],{"className":72053,"style":1862},[150],[15,72055],{},[15,72057],{"className":72058},[195,1803],[15,72060],{"className":72061,"style":262},[93],[15,72063,72065],{"className":72064},[217,1874],[15,72066,72068,72098],{"className":72067},[141,142],[15,72069,72071,72095],{"className":72070},[146],[15,72072,72074,72085],{"className":72073,"style":12149},[150],[15,72075,72076,72079],{"style":1887},[15,72077],{"className":72078,"style":1891},[158],[15,72080,72082],{"className":72081},[163,164,165,166],[15,72083,1459],{"className":72084},[129,133,166],[15,72086,72087,72090],{"style":1909},[15,72088],{"className":72089,"style":1891},[158],[15,72091,72092],{},[15,72093,1454],{"className":72094},[217,1918,1919],[15,72096,175],{"className":72097},[174],[15,72099,72101],{"className":72100},[146],[15,72102,72104],{"className":72103,"style":1941},[150],[15,72105],{},[15,72107],{"className":72108,"style":262},[93],[15,72110,26574],{"className":72111},[129,285],[15,72113,72115],{"className":72114},[188],[15,72116,1407],{"className":72117},[228,229],[15,72119,27529],{"className":72120},[129],[15,72122,53],{"className":72123},[188],[15,72125,72127,72130],{"className":72126},[129],[15,72128,48],{"className":72129,"style":170},[129,133],[15,72131,72133],{"className":72132},[137],[15,72134,72136,72156],{"className":72135},[141,142],[15,72137,72139,72153],{"className":72138},[146],[15,72140,72142],{"className":72141,"style":1981},[150],[15,72143,72144,72147],{"style":48293},[15,72145],{"className":72146,"style":159},[158],[15,72148,72150],{"className":72149},[163,164,165,166],[15,72151,1459],{"className":72152},[129,133,166],[15,72154,175],{"className":72155},[174],[15,72157,72159],{"className":72158},[146],[15,72160,72162],{"className":72161,"style":182},[150],[15,72163],{},[15,72165,91],{"className":72166},[254],[15,72168],{"className":72169,"style":262},[93],[15,72171,72173],{"className":72172},[129,48500],[15,72174,72176],{"className":72175},[141],[15,72177,72179],{"className":72178},[146],[15,72180,72182,72190],{"className":72181,"style":48496},[150],[15,72183,72184,72187],{"style":2287},[15,72185],{"className":72186,"style":1823},[158],[15,72188,48],{"className":72189,"style":170},[129,133],[15,72191,72192,72195],{"style":48520},[15,72193],{"className":72194,"style":1823},[158],[15,72196,72198],{"className":72197,"style":48528},[48527],[15,72199,48483],{"className":72200},[129],[15,72202,59],{"className":72203},[195],[15,72205,72207],{"className":72206},[195],[15,72208,1423],{"className":72209},[228,229],[15,72211],{"className":72212,"style":199},[93],[15,72214,7473],{"className":72215},[203],[15,72217],{"className":72218,"style":199},[93],[15,72220,72222,72225,72287,72290,72425,72428],{"className":72221},[120],[15,72223],{"className":72224,"style":17014},[124],[15,72226,72228,72231,72284],{"className":72227},[129],[15,72229],{"className":72230},[188,1803],[15,72232,72234],{"className":72233},[1443],[15,72235,72237,72276],{"className":72236},[141,142],[15,72238,72240,72273],{"className":72239},[146],[15,72241,72243,72254,72262],{"className":72242,"style":58514},[150],[15,72244,72245,72248],{"style":1819},[15,72246],{"className":72247,"style":1823},[158],[15,72249,72251],{"className":72250},[129],[15,72252,14508],{"className":72253},[129],[15,72255,72256,72259],{"style":1832},[15,72257],{"className":72258,"style":1823},[158],[15,72260],{"className":72261,"style":1840},[1839],[15,72263,72264,72267],{"style":1843},[15,72265],{"className":72266,"style":1823},[158],[15,72268,72270],{"className":72269},[129],[15,72271,48354],{"className":72272},[129],[15,72274,175],{"className":72275},[174],[15,72277,72279],{"className":72278},[146],[15,72280,72282],{"className":72281,"style":1862},[150],[15,72283],{},[15,72285],{"className":72286},[195,1803],[15,72288],{"className":72289,"style":262},[93],[15,72291,72293,72299,72302,72305,72308,72311,72419],{"className":72292},[449],[15,72294,72296],{"className":72295,"style":5052},[188,5051],[15,72297,53],{"className":72298},[228,165],[15,72300,318],{"className":72301},[129],[15,72303],{"className":72304,"style":3277},[93],[15,72306,315],{"className":72307},[2436],[15,72309],{"className":72310,"style":3277},[93],[15,72312,72314,72317,72416],{"className":72313},[129],[15,72315],{"className":72316},[188,1803],[15,72318,72320],{"className":72319},[1443],[15,72321,72323,72408],{"className":72322},[141,142],[15,72324,72326,72405],{"className":72325},[146],[15,72327,72329,72348,72356],{"className":72328,"style":29165},[150],[15,72330,72331,72334],{"style":1819},[15,72332],{"className":72333,"style":1823},[158],[15,72335,72337,72342,72345],{"className":72336},[129],[15,72338,14777,72340],{"className":72339},[217],[15,72341,2247],{"style":2246},[15,72343],{"className":72344,"style":262},[93],[15,72346,43246],{"className":72347,"style":12597},[129,133],[15,72349,72350,72353],{"style":1832},[15,72351],{"className":72352,"style":1823},[158],[15,72354],{"className":72355,"style":1840},[1839],[15,72357,72358,72361],{"style":1843},[15,72359],{"className":72360,"style":1823},[158],[15,72362,72364,72367,72370,72373,72376,72379,72382,72385,72388,72391,72394,72399,72402],{"className":72363},[129],[15,72365,64036],{"className":72366,"style":47970},[129,133],[15,72368,53],{"className":72369},[188],[15,72371,34524],{"className":72372,"style":5665},[129,133],[15,72374,65625],{"className":72375},[254],[15,72377],{"className":72378,"style":262},[93],[15,72380,597],{"className":72381,"style":47970},[129,133],[15,72383,59],{"className":72384},[195],[15,72386],{"className":72387,"style":3277},[93],[15,72389,2682],{"className":72390},[2436],[15,72392],{"className":72393,"style":3277},[93],[15,72395,14777,72397],{"className":72396},[217],[15,72398,2247],{"style":2246},[15,72400],{"className":72401,"style":262},[93],[15,72403,310],{"className":72404},[129],[15,72406,175],{"className":72407},[174],[15,72409,72411],{"className":72410},[146],[15,72412,72414],{"className":72413,"style":65837},[150],[15,72415],{},[15,72417],{"className":72418},[195,1803],[15,72420,72422],{"className":72421,"style":5052},[195,5051],[15,72423,59],{"className":72424},[228,165],[15,72426],{"className":72427,"style":262},[93],[15,72429,107],{"className":72430},[129],[11,72432,72433,72434,72495,72496,1313,72768,72805],{},"这里用了 ",[15,72435,72437,72459],{"className":72436,"translate":19},[23],[15,72438,72440],{"className":72439},[27],[29,72441,72442],{"xmlns":31},[34,72443,72444,72456],{},[37,72445,72446,72448,72450,72452,72454],{},[50,72447,636],{"stretchy":52},[43,72449,10376],{},[50,72451,7910],{},[43,72453,29575],{},[50,72455,645],{"stretchy":52},[109,72457,72458],{"encoding":111},"\\{d\u003C\\delta\\}",[15,72460,72462,72483],{"className":72461,"ariaHidden":72},[116],[15,72463,72465,72468,72471,72474,72477,72480],{"className":72464},[120],[15,72466],{"className":72467,"style":125},[124],[15,72469,636],{"className":72470},[188],[15,72472,10376],{"className":72473},[129,133],[15,72475],{"className":72476,"style":199},[93],[15,72478,7910],{"className":72479},[203],[15,72481],{"className":72482,"style":199},[93],[15,72484,72486,72489,72492],{"className":72485},[120],[15,72487],{"className":72488,"style":125},[124],[15,72490,29575],{"className":72491,"style":29856},[129,133],[15,72493,645],{"className":72494},[195]," 蕴含解码正确，因此 ",[15,72497,72499,72559],{"className":72498,"translate":19},[23],[15,72500,72502],{"className":72501},[27],[29,72503,72504],{"xmlns":31},[34,72505,72506,72556],{},[37,72507,72508,72510,72512,72518,72520,72522,72528,72530,72532,72534,72536,72538,72540,72542,72544,72550],{},[43,72509,26574],{"mathvariant":103},[50,72511,1407],{"stretchy":52},[305,72513,72514,72516],{},[43,72515,10376],{},[308,72517,310],{},[50,72519,1423],{"stretchy":52},[50,72521,7473],{},[305,72523,72524,72526],{},[43,72525,29575],{},[308,72527,310],{},[43,72529,4961],{},[50,72531,69],{},[50,72533,53],{"stretchy":52},[43,72535,10376],{},[50,72537,7473],{},[43,72539,29575],{},[50,72541,59],{"stretchy":52},[50,72543,7473],{},[305,72545,72546,72548],{},[43,72547,29575],{},[308,72549,310],{},[40,72551,72552,72554],{},[43,72553,1388],{},[43,72555,10699],{},[109,72557,72558],{"encoding":111},"\\mathbb E[d^2]\\ge\\delta^2\\Pr(d\\ge\\delta)\\ge\\delta^2P_e",[15,72560,72562,72615,72671,72692],{"className":72561,"ariaHidden":72},[116],[15,72563,72565,72568,72571,72574,72603,72606,72609,72612],{"className":72564},[120],[15,72566],{"className":72567,"style":34009},[124],[15,72569,26574],{"className":72570},[129,285],[15,72572,1407],{"className":72573},[188],[15,72575,72577,72580],{"className":72576},[129],[15,72578,10376],{"className":72579},[129,133],[15,72581,72583],{"className":72582},[137],[15,72584,72586],{"className":72585},[141],[15,72587,72589],{"className":72588},[146],[15,72590,72592],{"className":72591,"style":380},[150],[15,72593,72594,72597],{"style":383},[15,72595],{"className":72596,"style":159},[158],[15,72598,72600],{"className":72599},[163,164,165,166],[15,72601,310],{"className":72602},[129,166],[15,72604,1423],{"className":72605},[195],[15,72607],{"className":72608,"style":199},[93],[15,72610,7473],{"className":72611},[203],[15,72613],{"className":72614,"style":199},[93],[15,72616,72618,72621,72650,72653,72656,72659,72662,72665,72668],{"className":72617},[120],[15,72619],{"className":72620,"style":34009},[124],[15,72622,72624,72627],{"className":72623},[129],[15,72625,29575],{"className":72626,"style":29856},[129,133],[15,72628,72630],{"className":72629},[137],[15,72631,72633],{"className":72632},[141],[15,72634,72636],{"className":72635},[146],[15,72637,72639],{"className":72638,"style":380},[150],[15,72640,72641,72644],{"style":383},[15,72642],{"className":72643,"style":159},[158],[15,72645,72647],{"className":72646},[163,164,165,166],[15,72648,310],{"className":72649},[129,166],[15,72651],{"className":72652,"style":262},[93],[15,72654,4961],{"className":72655},[217],[15,72657,53],{"className":72658},[188],[15,72660,10376],{"className":72661},[129,133],[15,72663],{"className":72664,"style":199},[93],[15,72666,7473],{"className":72667},[203],[15,72669],{"className":72670,"style":199},[93],[15,72672,72674,72677,72680,72683,72686,72689],{"className":72673},[120],[15,72675],{"className":72676,"style":125},[124],[15,72678,29575],{"className":72679,"style":29856},[129,133],[15,72681,59],{"className":72682},[195],[15,72684],{"className":72685,"style":199},[93],[15,72687,7473],{"className":72688},[203],[15,72690],{"className":72691,"style":199},[93],[15,72693,72695,72699,72728],{"className":72694},[120],[15,72696],{"className":72697,"style":72698},[124],"height:0.9641em;vertical-align:-0.15em;",[15,72700,72702,72705],{"className":72701},[129],[15,72703,29575],{"className":72704,"style":29856},[129,133],[15,72706,72708],{"className":72707},[137],[15,72709,72711],{"className":72710},[141],[15,72712,72714],{"className":72713},[146],[15,72715,72717],{"className":72716,"style":380},[150],[15,72718,72719,72722],{"style":383},[15,72720],{"className":72721,"style":159},[158],[15,72723,72725],{"className":72724},[163,164,165,166],[15,72726,310],{"className":72727},[129,166],[15,72729,72731,72734],{"className":72730},[129],[15,72732,1388],{"className":72733,"style":12449},[129,133],[15,72735,72737],{"className":72736},[137],[15,72738,72740,72760],{"className":72739},[141,142],[15,72741,72743,72757],{"className":72742},[146],[15,72744,72746],{"className":72745,"style":1163},[150],[15,72747,72748,72751],{"style":34239},[15,72749],{"className":72750,"style":159},[158],[15,72752,72754],{"className":72753},[163,164,165,166],[15,72755,10699],{"className":72756},[129,133,166],[15,72758,175],{"className":72759},[174],[15,72761,72763],{"className":72762},[146],[15,72764,72766],{"className":72765,"style":182},[150],[15,72767],{},[698,72769,72770,72771],{},"平方损失的常数是 ",[15,72772,72774,72792],{"className":72773,"translate":19},[23],[15,72775,72777],{"className":72776},[27],[29,72778,72779],{"xmlns":31},[34,72780,72781,72789],{},[37,72782,72783,72785,72787],{},[43,72784,48354],{"mathvariant":65},[43,72786,544],{"mathvariant":65},[308,72788,14508],{},[109,72790,72791],{"encoding":111},"\\Delta\u002F4",[15,72793,72795],{"className":72794,"ariaHidden":72},[116],[15,72796,72798,72801],{"className":72797},[120],[15,72799],{"className":72800,"style":125},[124],[15,72802,72804],{"className":72803},[129],"Δ\u002F4","，与 Le Cam 的平方损失版本一致。本文后面的例子只关心速率，常数不影响结论。",[11,72807,72808],{},"剩下的任务是把互信息控制住。由定义，",[15,72810,72812],{"className":72811,"translate":19},[18],[15,72813,72815,72930],{"className":72814,"translate":19},[23],[15,72816,72818],{"className":72817},[27],[29,72819,72820],{"xmlns":31,"display":32},[34,72821,72822,72927],{},[37,72823,72824,72826,72828,72830,72832,72834,72836,72838,72844,72858,72868,72870,72876,72878,72885,72887,72889,72891,72897,72899,72905,72919,72925],{},[43,72825,64036],{},[50,72827,53],{"stretchy":52},[43,72829,34524],{},[50,72831,65625],{"separator":72},[43,72833,597],{},[50,72835,59],{"stretchy":52},[50,72837,62],{},[1443,72839,72840,72842],{},[308,72841,318],{},[43,72843,43246],{},[1450,72845,72846,72848,72856],{},[50,72847,1454],{},[37,72849,72850,72852,72854],{},[43,72851,1459],{},[50,72853,62],{},[308,72855,318],{},[43,72857,43246],{},[40,72859,72860,72862],{},[43,72861,906],{},[37,72863,72864,72866],{},[43,72865,34524],{"mathvariant":65},[43,72867,1373],{"mathvariant":65},[50,72869,53],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,72871,72872,72874],{},[43,72873,1388],{},[43,72875,1459],{},[43,72877,33823],{"mathvariant":65},[46847,72879,72880,72882],{"accent":72},[43,72881,1388],{},[50,72883,72884],{},"ˉ",[50,72886,59],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,72888,91],{"separator":72},[93,72890],{"width":95},[46847,72892,72893,72895],{"accent":72},[43,72894,1388],{},[50,72896,72884],{},[50,72898,62],{},[1443,72900,72901,72903],{},[308,72902,318],{},[43,72904,43246],{},[1450,72906,72907,72909,72917],{},[50,72908,1454],{},[37,72910,72911,72913,72915],{},[43,72912,1459],{},[50,72914,62],{},[308,72916,318],{},[43,72918,43246],{},[40,72920,72921,72923],{},[43,72922,1388],{},[43,72924,1459],{},[43,72926,107],{"mathvariant":65},[109,72928,72929],{"encoding":111},"I(K;X)=\\frac1M\\sum_{i=1}^M D_{\\mathrm{KL}}\\bigl(P_i\\Vert \\bar P\\bigr),\n\\qquad \\bar P=\\frac1M\\sum_{i=1}^M P_i.",[15,72931,72933,72969,73289],{"className":72932,"ariaHidden":72},[116],[15,72934,72936,72939,72942,72945,72948,72951,72954,72957,72960,72963,72966],{"className":72935},[120],[15,72937],{"className":72938,"style":125},[124],[15,72940,64036],{"className":72941,"style":47970},[129,133],[15,72943,53],{"className":72944},[188],[15,72946,34524],{"className":72947,"style":5665},[129,133],[15,72949,65625],{"className":72950},[254],[15,72952],{"className":72953,"style":262},[93],[15,72955,597],{"className":72956,"style":47970},[129,133],[15,72958,59],{"className":72959},[195],[15,72961],{"className":72962,"style":199},[93],[15,72964,62],{"className":72965},[203],[15,72967],{"className":72968,"style":199},[93],[15,72970,72972,72975,73037,73040,73104,73107,73153,73159,73199,73202,73234,73240,73243,73246,73249,73280,73283,73286],{"className":72971},[120],[15,72973],{"className":72974,"style":68081},[124],[15,72976,72978,72981,73034],{"className":72977},[129],[15,72979],{"className":72980},[188,1803],[15,72982,72984],{"className":72983},[1443],[15,72985,72987,73026],{"className":72986},[141,142],[15,72988,72990,73023],{"className":72989},[146],[15,72991,72993,73004,73012],{"className":72992,"style":1816},[150],[15,72994,72995,72998],{"style":1819},[15,72996],{"className":72997,"style":1823},[158],[15,72999,73001],{"className":73000},[129],[15,73002,43246],{"className":73003,"style":12597},[129,133],[15,73005,73006,73009],{"style":1832},[15,73007],{"className":73008,"style":1823},[158],[15,73010],{"className":73011,"style":1840},[1839],[15,73013,73014,73017],{"style":1843},[15,73015],{"className":73016,"style":1823},[158],[15,73018,73020],{"className":73019},[129],[15,73021,318],{"className":73022},[129],[15,73024,175],{"className":73025},[174],[15,73027,73029],{"className":73028},[146],[15,73030,73032],{"className":73031,"style":1862},[150],[15,73033],{},[15,73035],{"className":73036},[195,1803],[15,73038],{"className":73039,"style":262},[93],[15,73041,73043],{"className":73042},[217,1874],[15,73044,73046,73096],{"className":73045},[141,142],[15,73047,73049,73093],{"className":73048},[146],[15,73050,73052,73072,73082],{"className":73051,"style":68159},[150],[15,73053,73054,73057],{"style":1887},[15,73055],{"className":73056,"style":1891},[158],[15,73058,73060],{"className":73059},[163,164,165,166],[15,73061,73063,73066,73069],{"className":73062},[129,166],[15,73064,1459],{"className":73065},[129,133,166],[15,73067,62],{"className":73068},[203,166],[15,73070,318],{"className":73071},[129,166],[15,73073,73074,73077],{"style":1909},[15,73075],{"className":73076,"style":1891},[158],[15,73078,73079],{},[15,73080,1454],{"className":73081},[217,1918,1919],[15,73083,73084,73087],{"style":1922},[15,73085],{"className":73086,"style":1891},[158],[15,73088,73090],{"className":73089},[163,164,165,166],[15,73091,43246],{"className":73092,"style":12597},[129,133,166],[15,73094,175],{"className":73095},[174],[15,73097,73099],{"className":73098},[146],[15,73100,73102],{"className":73101,"style":1941},[150],[15,73103],{},[15,73105],{"className":73106,"style":262},[93],[15,73108,73110,73113],{"className":73109},[129],[15,73111,906],{"className":73112,"style":170},[129,133],[15,73114,73116],{"className":73115},[137],[15,73117,73119,73145],{"className":73118},[141,142],[15,73120,73122,73142],{"className":73121},[146],[15,73123,73125],{"className":73124,"style":1525},[150],[15,73126,73127,73130],{"style":48293},[15,73128],{"className":73129,"style":159},[158],[15,73131,73133],{"className":73132},[163,164,165,166],[15,73134,73136],{"className":73135},[129,166],[15,73137,73139],{"className":73138},[129,166],[15,73140,34928],{"className":73141},[129,221,166],[15,73143,175],{"className":73144},[174],[15,73146,73148],{"className":73147},[146],[15,73149,73151],{"className":73150,"style":182},[150],[15,73152],{},[15,73154,73156],{"className":73155},[188],[15,73157,53],{"className":73158},[228,229],[15,73160,73162,73165],{"className":73161},[129],[15,73163,1388],{"className":73164,"style":12449},[129,133],[15,73166,73168],{"className":73167},[137],[15,73169,73171,73191],{"className":73170},[141,142],[15,73172,73174,73188],{"className":73173},[146],[15,73175,73177],{"className":73176,"style":1981},[150],[15,73178,73179,73182],{"style":34239},[15,73180],{"className":73181,"style":159},[158],[15,73183,73185],{"className":73184},[163,164,165,166],[15,73186,1459],{"className":73187},[129,133,166],[15,73189,175],{"className":73190},[174],[15,73192,73194],{"className":73193},[146],[15,73195,73197],{"className":73196,"style":182},[150],[15,73198],{},[15,73200,33823],{"className":73201},[129],[15,73203,73205],{"className":73204},[129,48500],[15,73206,73208],{"className":73207},[141],[15,73209,73211],{"className":73210},[146],[15,73212,73215,73223],{"className":73213,"style":73214},[150],"height:0.8201em;",[15,73216,73217,73220],{"style":2287},[15,73218],{"className":73219,"style":1823},[158],[15,73221,1388],{"className":73222,"style":12449},[129,133],[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KL 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\\frac1{M^2}\\sum_{i,j=1}^M D_{\\mathrm{KL}}(P_i\\Vert P_j)\n\\le \\max_{i,j}D_{\\mathrm{KL}}(P_i\\Vert P_j).",[15,73610,73612,73648,73963],{"className":73611,"ariaHidden":72},[116],[15,73613,73615,73618,73621,73624,73627,73630,73633,73636,73639,73642,73645],{"className":73614},[120],[15,73616],{"className":73617,"style":125},[124],[15,73619,64036],{"className":73620,"style":47970},[129,133],[15,73622,53],{"className":73623},[188],[15,73625,34524],{"className":73626,"style":5665},[129,133],[15,73628,65625],{"className":73629},[254],[15,73631],{"className":73632,"style":262},[93],[15,73634,597],{"className":73635,"style":47970},[129,133],[15,73637,59],{"className":73638},[195],[15,73640],{"className":73641,"style":199},[93],[15,73643,2717],{"className":73644},[203],[15,73646],{"className":73647,"style":199},[93],[15,73649,73651,73655,73743,73746,73816,73819,73865,73868,73908,73911,73951,73954,73957,73960],{"className":73650},[120],[15,73652],{"className":73653,"style":73654},[124],"height:3.2421em;vertical-align:-1.4138em;",[15,73656,73658,73661,73740],{"className":73657},[129],[15,73659],{"className":73660},[188,1803],[15,73662,73664],{"className":73663},[1443],[15,73665,73667,73732],{"className":73666},[141,142],[15,73668,73670,73729],{"className":73669},[146],[15,73671,73673,73710,73718],{"className":73672,"style":1816},[150],[15,73674,73675,73678],{"style":1819},[15,73676],{"className":73677,"style":1823},[158],[15,73679,73681],{"className":73680},[129],[15,73682,73684,73687],{"className":73683},[129],[15,73685,43246],{"className":73686,"style":12597},[129,133],[15,73688,73690],{"className":73689},[137],[15,73691,73693],{"className":73692},[141],[15,73694,73696],{"className":73695},[146],[15,73697,73699],{"className":73698,"style":31001},[150],[15,73700,73701,73704],{"style":17201},[15,73702],{"className":73703,"style":159},[158],[15,73705,73707],{"className":73706},[163,164,165,166],[15,73708,310],{"className":73709},[129,166],[15,73711,73712,73715],{"style":1832},[15,73713],{"className":73714,"style":1823},[158],[15,73716],{"className":73717,"style":1840},[1839],[15,73719,73720,73723],{"style":1843},[15,73721],{"className":73722,"style":1823},[158],[15,73724,73726],{"className":73725},[129],[15,73727,318],{"className":73728},[129],[15,73730,175],{"className":73731},[174],[15,73733,73735],{"className":73734},[146],[15,73736,73738],{"className":73737,"style":1862},[150],[15,73739],{},[15,73741],{"className":73742},[195,1803],[15,73744],{"className":73745,"style":262},[93],[15,73747,73749],{"className":73748},[217,1874],[15,73750,73752,73808],{"className":73751},[141,142],[15,73753,73755,73805],{"className":73754},[146],[15,73756,73758,73784,73794],{"className":73757,"style":68159},[150],[15,73759,73760,73763],{"style":1887},[15,73761],{"className":73762,"style":1891},[158],[15,73764,73766],{"className":73765},[163,164,165,166],[15,73767,73769,73772,73775,73778,73781],{"className":73768},[129,166],[15,73770,1459],{"className":73771},[129,133,166],[15,73773,91],{"className":73774},[254,166],[15,73776,13955],{"className":73777,"style":13970},[129,133,166],[15,73779,62],{"className":73780},[203,166],[15,73782,318],{"className":73783},[129,166],[15,73785,73786,73789],{"style":1909},[15,73787],{"className":73788,"style":1891},[158],[15,73790,73791],{},[15,73792,1454],{"className":73793},[217,1918,1919],[15,73795,73796,73799],{"style":1922},[15,73797],{"className":73798,"style":1891},[158],[15,73800,73802],{"className":73801},[163,164,165,166],[15,73803,43246],{"className":73804,"style":12597},[129,133,166],[15,73806,175],{"className":73807},[174],[15,73809,73811],{"className":73810},[146],[15,73812,73814],{"className":73813,"style":16035},[150],[15,73815],{},[15,73817],{"className":73818,"style":262},[93],[15,73820,73822,73825],{"className":73821},[129],[15,73823,906],{"className":73824,"style":170},[129,133],[15,73826,73828],{"className":73827},[137],[15,73829,73831,73857],{"className":73830},[141,142],[15,73832,73834,73854],{"className":73833},[146],[15,73835,73837],{"className":73836,"style":1525},[150],[15,73838,73839,73842],{"style":48293},[15,73840],{"className":73841,"style":159},[158],[15,73843,73845],{"className":73844},[163,164,165,166],[15,73846,73848],{"className":73847},[129,166],[15,73849,73851],{"className":73850},[129,166],[15,73852,34928],{"className":73853},[129,221,166],[15,73855,175],{"className":73856},[174],[15,73858,73860],{"className":73859},[146],[15,73861,73863],{"className":73862,"style":182},[150],[15,73864],{},[15,73866,53],{"className":73867},[188],[15,73869,73871,73874],{"className":73870},[129],[15,73872,1388],{"className":73873,"style":12449},[129,133],[15,73875,73877],{"className":73876},[137],[15,73878,73880,73900],{"className":73879},[141,142],[15,73881,73883,73897],{"className":73882},[146],[15,73884,73886],{"className":73885,"style":1981},[150],[15,73887,73888,73891],{"style":34239},[15,73889],{"className":73890,"style":159},[158],[15,73892,73894],{"className":73893},[163,164,165,166],[15,73895,1459],{"className":73896},[129,133,166],[15,73898,175],{"className":73899},[174],[15,73901,73903],{"className":73902},[146],[15,73904,73906],{"className":73905,"style":182},[150],[15,73907],{},[15,73909,33823],{"className":73910},[129],[15,73912,73914,73917],{"className":73913},[129],[15,73915,1388],{"className":73916,"style":12449},[129,133],[15,73918,73920],{"className":73919},[137],[15,73921,73923,73943],{"className":73922},[141,142],[15,73924,73926,73940],{"className":73925},[146],[15,73927,73929],{"className":73928,"style":1981},[150],[15,73930,73931,73934],{"style":34239},[15,73932],{"className":73933,"style":159},[158],[15,73935,73937],{"className":73936},[163,164,165,166],[15,73938,13955],{"className":73939,"style":13970},[129,133,166],[15,73941,175],{"className":73942},[174],[15,73944,73946],{"className":73945},[146],[15,73947,73949],{"className":73948,"style":14041},[150],[15,73950],{},[15,73952,59],{"className":73953},[195],[15,73955],{"className":73956,"style":199},[93],[15,73958,2717],{"className":73959},[203],[15,73961],{"className":73962,"style":199},[93],[15,73964,73966,73969,74022,74025,74071,74074,74114,74117,74157,74160],{"className":73965},[120],[15,73967],{"className":73968,"style":58179},[124],[15,73970,73972],{"className":73971},[217,1874],[15,73973,73975,74014],{"className":73974},[141,142],[15,73976,73978,74011],{"className":73977},[146],[15,73979,73981,74001],{"className":73980,"style":4383},[150],[15,73982,73983,73986],{"style":53444},[15,73984],{"className":73985,"style":1823},[158],[15,73987,73989],{"className":73988},[163,164,165,166],[15,73990,73992,73995,73998],{"className":73991},[129,166],[15,73993,1459],{"className":73994},[129,133,166],[15,73996,91],{"className":73997},[254,166],[15,73999,13955],{"className":74000,"style":13970},[129,133,166],[15,74002,74003,74006],{"style":2287},[15,74004],{"className":74005,"style":1823},[158],[15,74007,74008],{},[15,74009,19705],{"className":74010},[217],[15,74012,175],{"className":74013},[174],[15,74015,74017],{"className":74016},[146],[15,74018,74020],{"className":74019,"style":53488},[150],[15,74021],{},[15,74023],{"className":74024,"style":262},[93],[15,74026,74028,74031],{"className":74027},[129],[15,74029,906],{"className":74030,"style":170},[129,133],[15,74032,74034],{"className":74033},[137],[15,74035,74037,74063],{"className":74036},[141,142],[15,74038,74040,74060],{"className":74039},[146],[15,74041,74043],{"className":74042,"style":1525},[150],[15,74044,74045,74048],{"style":48293},[15,74046],{"className":74047,"style":159},[158],[15,74049,74051],{"className":74050},[163,164,165,166],[15,74052,74054],{"className":74053},[129,166],[15,74055,74057],{"className":74056},[129,166],[15,74058,34928],{"className":74059},[129,221,166],[15,74061,175],{"className":74062},[174],[15,74064,74066],{"className":74065},[146],[15,74067,74069],{"className":74068,"style":182},[150],[15,74070],{},[15,74072,53],{"className":74073},[188],[15,74075,74077,74080],{"className":74076},[129],[15,74078,1388],{"className":74079,"style":12449},[129,133],[15,74081,74083],{"className":74082},[137],[15,74084,74086,74106],{"className":74085},[141,142],[15,74087,74089,74103],{"className":74088},[146],[15,74090,74092],{"className":74091,"style":1981},[150],[15,74093,74094,74097],{"style":34239},[15,74095],{"className":74096,"style":159},[158],[15,74098,74100],{"className":74099},[163,164,165,166],[15,74101,1459],{"className":74102},[129,133,166],[15,74104,175],{"className":74105},[174],[15,74107,74109],{"className":74108},[146],[15,74110,74112],{"className":74111,"style":182},[150],[15,74113],{},[15,74115,33823],{"className":74116},[129],[15,74118,74120,74123],{"className":74119},[129],[15,74121,1388],{"className":74122,"style":12449},[129,133],[15,74124,74126],{"className":74125},[137],[15,74127,74129,74149],{"className":74128},[141,142],[15,74130,74132,74146],{"className":74131},[146],[15,74133,74135],{"className":74134,"style":1981},[150],[15,74136,74137,74140],{"style":34239},[15,74138],{"className":74139,"style":159},[158],[15,74141,74143],{"className":74142},[163,164,165,166],[15,74144,13955],{"className":74145,"style":13970},[129,133,166],[15,74147,175],{"className":74148},[174],[15,74150,74152],{"className":74151},[146],[15,74153,74155],{"className":74154,"style":14041},[150],[15,74156],{},[15,74158,59],{"className":74159},[195],[15,74161,107],{"className":74162},[129],[11,74164,74165,74166,74236,74237,74265,74266,74294,74295,10179,74335,74368],{},"如果每个 ",[15,74167,74169,74187],{"className":74168,"translate":19},[23],[15,74170,74172],{"className":74171},[27],[29,74173,74174],{"xmlns":31},[34,74175,74176,74184],{},[37,74177,74178],{},[40,74179,74180,74182],{},[43,74181,1388],{},[43,74183,1459],{},[109,74185,74186],{"encoding":111},"P_i",[15,74188,74190],{"className":74189,"ariaHidden":72},[116],[15,74191,74193,74196],{"className":74192},[120],[15,74194],{"className":74195,"style":20229},[124],[15,74197,74199,74202],{"className":74198},[129],[15,74200,1388],{"className":74201,"style":12449},[129,133],[15,74203,74205],{"className":74204},[137],[15,74206,74208,74228],{"className":74207},[141,142],[15,74209,74211,74225],{"className":74210},[146],[15,74212,74214],{"className":74213,"style":1981},[150],[15,74215,74216,74219],{"style":34239},[15,74217],{"className":74218,"style":159},[158],[15,74220,74222],{"className":74221},[163,164,165,166],[15,74223,1459],{"className":74224},[129,133,166],[15,74226,175],{"className":74227},[174],[15,74229,74231],{"className":74230},[146],[15,74232,74234],{"className":74233,"style":182},[150],[15,74235],{}," 都是 ",[15,74238,74240,74253],{"className":74239,"translate":19},[23],[15,74241,74243],{"className":74242},[27],[29,74244,74245],{"xmlns":31},[34,74246,74247,74251],{},[37,74248,74249],{},[43,74250,39326],{},[109,74252,39326],{"encoding":111},[15,74254,74256],{"className":74255,"ariaHidden":72},[116],[15,74257,74259,74262],{"className":74258},[120],[15,74260],{"className":74261,"style":4383},[124],[15,74263,39326],{"className":74264},[129,133]," 个独立样本的分布，那么成对 KL 是单样本 KL 的 ",[15,74267,74269,74282],{"className":74268,"translate":19},[23],[15,74270,74272],{"className":74271},[27],[29,74273,74274],{"xmlns":31},[34,74275,74276,74280],{},[37,74277,74278],{},[43,74279,39326],{},[109,74281,39326],{"encoding":111},[15,74283,74285],{"className":74284,"ariaHidden":72},[116],[15,74286,74288,74291],{"className":74287},[120],[15,74289],{"className":74290,"style":4383},[124],[15,74292,39326],{"className":74293},[129,133]," 倍。这样，Fano 下界就变成了「",[15,74296,74298,74315],{"className":74297,"translate":19},[23],[15,74299,74301],{"className":74300},[27],[29,74302,74303],{"xmlns":31},[34,74304,74305,74313],{},[37,74306,74307,74309,74311],{},[43,74308,14727],{},[50,74310,69],{},[43,74312,43246],{},[109,74314,67065],{"encoding":111},[15,74316,74318],{"className":74317,"ariaHidden":72},[116],[15,74319,74321,74324,74329,74332],{"className":74320},[120],[15,74322],{"className":74323,"style":1682},[124],[15,74325,14777,74327],{"className":74326},[217],[15,74328,2247],{"style":2246},[15,74330],{"className":74331,"style":262},[93],[15,74333,43246],{"className":74334,"style":12597},[129,133],[15,74336,74338,74353],{"className":74337,"translate":19},[23],[15,74339,74341],{"className":74340},[27],[29,74342,74343],{"xmlns":31},[34,74344,74345,74351],{},[37,74346,74347,74349],{},[43,74348,39326],{},[50,74350,11422],{},[109,74352,62038],{"encoding":111},[15,74354,74356],{"className":74355,"ariaHidden":72},[116],[15,74357,74359,74362,74365],{"className":74358},[120],[15,74360],{"className":74361,"style":8518},[124],[15,74363,39326],{"className":74364},[129,133],[15,74366,11422],{"className":74367},[129],"KL 的竞争」。",[575,74370,74372],{"id":74371},"_43-广义-fano-与局部版本","4.3 广义 Fano 与局部版本",[11,74374,74375,74376,74379],{},"定理 4.2 常被称为",[698,74377,74378],{},"广义 Fano 不等式","。它的使用可以总结成一句口诀：",[15,74381,74383],{"className":74382,"translate":19},[18],[15,74384,74386,74494],{"className":74385,"translate":19},[23],[15,74387,74389],{"className":74388},[27],[29,74390,74391],{"xmlns":31,"display":32},[34,74392,74393,74491],{},[37,74394,74395],{},[74396,74397,74399],"menclose",{"notation":74398},"box",[2612,74400,74401],{"scriptlevel":541,"displaystyle":52},[2612,74402,74403],{"scriptlevel":541,"displaystyle":52},[2612,74404,74405],{"scriptlevel":541,"displaystyle":72},[37,74406,74407,74410,74412,74414,74416,74418,74421,74423,74425,74427,74443,74453,74455,74461,74463,74469,74471,74473,74476,74482,74485,74487,74489],{},[6856,74408,74409],{}," 只要 ",[43,74411,14727],{},[50,74413,69],{},[43,74415,43246],{},[6856,74417,6858],{},[50,74419,74420],{},"≫",[6856,74422,6858],{},[43,74424,39326],{},[50,74426,10553],{},[2140,74428,74429,74435],{},[37,74430,74431,74433],{},[43,74432,19705],{},[50,74434,69],{},[37,74436,74437,74439,74441],{},[43,74438,1459],{},[50,74440,91],{"separator":72},[43,74442,13955],{},[40,74444,74445,74447],{},[43,74446,906],{},[37,74448,74449,74451],{},[43,74450,34524],{"mathvariant":65},[43,74452,1373],{"mathvariant":65},[50,74454,53],{"stretchy":52},[40,74456,74457,74459],{},[43,74458,1388],{},[43,74460,1459],{},[43,74462,33823],{"mathvariant":65},[40,74464,74465,74467],{},[43,74466,1388],{},[43,74468,13955],{},[50,74470,59],{"stretchy":52},[50,74472,91],{"separator":72},[6856,74474,74475],{}," 就有 ",[40,74477,74478,74480],{},[43,74479,43246],{"mathvariant":28910},[43,74481,39326],{},[50,74483,74484],{},"≳",[43,74486,29575],{},[43,74488,107],{"mathvariant":65},[6856,74490,6858],{},[109,74492,74493],{"encoding":111},"\\boxed{\\ \\text{只要}\\ \\log M\\ \\gg\\ n\\cdot \\max_{i,j}D_{\\mathrm{KL}}(P_i\\Vert P_j),\\ \\text{就有}\\ \\mathfrak M_n\\gtrsim\\delta.\\ }",[15,74495,74497],{"className":74496,"ariaHidden":72},[116],[15,74498,74500,74504],{"className":74499},[120],[15,74501],{"className":74502,"style":74503},[124],"height:2.2938em;vertical-align:-1.2038em;",[15,74505,74507],{"className":74506},[129],[15,74508,74510,74871],{"className":74509},[141,142],[15,74511,74513,74868],{"className":74512},[146],[15,74514,74517,74856],{"className":74515,"style":74516},[150],"height:1.09em;",[15,74518,74520,74524],{"style":74519},"top:-4.2938em;",[15,74521],{"className":74522,"style":74523},[158],"height:4.2938em;",[15,74525,74528],{"className":74526},[74527],"boxpad",[15,74529,74531],{"className":74530},[129],[15,74532,74534,74537,74544,74547,74550,74555,74558,74561,74564,74567,74570,74573,74576,74579,74582,74585,74588,74641,74644,74690,74693,74733,74736,74776,74779,74782,74785,74788,74795,74798,74838,74841,74844,74847,74850,74853],{"className":74533},[129],[15,74535,6858],{"className":74536},[93],[15,74538,74540],{"className":74539},[129,37682],[15,74541,74543],{"className":74542},[129,37690],"只要",[15,74545,6858],{"className":74546},[93],[15,74548],{"className":74549,"style":262},[93],[15,74551,14777,74553],{"className":74552},[217],[15,74554,2247],{"style":2246},[15,74556],{"className":74557,"style":262},[93],[15,74559,43246],{"className":74560,"style":12597},[129,133],[15,74562,6858],{"className":74563},[93],[15,74565],{"className":74566,"style":199},[93],[15,74568,74420],{"className":74569},[203],[15,74571,6858],{"className":74572},[93],[15,74574],{"className":74575,"style":199},[93],[15,74577,39326],{"className":74578},[129,133],[15,74580],{"className":74581,"style":3277},[93],[15,74583,10553],{"className":74584},[2436],[15,74586],{"className":74587,"style":3277},[93],[15,74589,74591],{"className":74590},[217,1874],[15,74592,74594,74633],{"className":74593},[141,142],[15,74595,74597,74630],{"className":74596},[146],[15,74598,74600,74620],{"className":74599,"style":4383},[150],[15,74601,74602,74605],{"style":53444},[15,74603],{"className":74604,"style":1823},[158],[15,74606,74608],{"className":74607},[163,164,165,166],[15,74609,74611,74614,74617],{"className":74610},[129,166],[15,74612,1459],{"className":74613},[129,133,166],[15,74615,91],{"className":74616},[254,166],[15,74618,13955],{"className":74619,"style":13970},[129,133,166],[15,74621,74622,74625],{"style":2287},[15,74623],{"className":74624,"style":1823},[158],[15,74626,74627],{},[15,74628,19705],{"className":74629},[217],[15,74631,175],{"className":74632},[174],[15,74634,74636],{"className":74635},[146],[15,74637,74639],{"className":74638,"style":53488},[150],[15,74640],{},[15,74642],{"className":74643,"style":262},[93],[15,74645,74647,74650],{"className":74646},[129],[15,74648,906],{"className":74649,"style":170},[129,133],[15,74651,74653],{"className":74652},[137],[15,74654,74656,74682],{"className":74655},[141,142],[15,74657,74659,74679],{"className":74658},[146],[15,74660,74662],{"className":74661,"style":1525},[150],[15,74663,74664,74667],{"style":48293},[15,74665],{"className":74666,"style":159},[158],[15,74668,74670],{"className":74669},[163,164,165,166],[15,74671,74673],{"className":74672},[129,166],[15,74674,74676],{"className":74675},[129,166],[15,74677,34928],{"className":74678},[129,221,166],[15,74680,175],{"className":74681},[174],[15,74683,74685],{"className":74684},[146],[15,74686,74688],{"className":74687,"style":182},[150],[15,74689],{},[15,74691,53],{"className":74692},[188],[15,74694,74696,74699],{"className":74695},[129],[15,74697,1388],{"className":74698,"style":12449},[129,133],[15,74700,74702],{"className":74701},[137],[15,74703,74705,74725],{"className":74704},[141,142],[15,74706,74708,74722],{"className":74707},[146],[15,74709,74711],{"className":74710,"style":1981},[150],[15,74712,74713,74716],{"style":34239},[15,74714],{"className":74715,"style":159},[158],[15,74717,74719],{"className":74718},[163,164,165,166],[15,74720,1459],{"className":74721},[129,133,166],[15,74723,175],{"className":74724},[174],[15,74726,74728],{"className":74727},[146],[15,74729,74731],{"className":74730,"style":182},[150],[15,74732],{},[15,74734,33823],{"className":74735},[129],[15,74737,74739,74742],{"className":74738},[129],[15,74740,1388],{"className":74741,"style":12449},[129,133],[15,74743,74745],{"className":74744},[137],[15,74746,74748,74768],{"className":74747},[141,142],[15,74749,74751,74765],{"className":74750},[146],[15,74752,74754],{"className":74753,"style":1981},[150],[15,74755,74756,74759],{"style":34239},[15,74757],{"className":74758,"style":159},[158],[15,74760,74762],{"className":74761},[163,164,165,166],[15,74763,13955],{"className":74764,"style":13970},[129,133,166],[15,74766,175],{"className":74767},[174],[15,74769,74771],{"className":74770},[146],[15,74772,74774],{"className":74773,"style":14041},[150],[15,74775],{},[15,74777,59],{"className":74778},[195],[15,74780,91],{"className":74781},[254],[15,74783,6858],{"className":74784},[93],[15,74786],{"className":74787,"style":262},[93],[15,74789,74791],{"className":74790},[129,37682],[15,74792,74794],{"className":74793},[129,37690],"就有",[15,74796,6858],{"className":74797},[93],[15,74799,74801,74804],{"className":74800},[129],[15,74802,43246],{"className":74803},[129,28937],[15,74805,74807],{"className":74806},[137],[15,74808,74810,74830],{"className":74809},[141,142],[15,74811,74813,74827],{"className":74812},[146],[15,74814,74816],{"className":74815,"style":1163},[150],[15,74817,74818,74821],{"style":154},[15,74819],{"className":74820,"style":159},[158],[15,74822,74824],{"className":74823},[163,164,165,166],[15,74825,39326],{"className":74826},[129,133,166],[15,74828,175],{"className":74829},[174],[15,74831,74833],{"className":74832},[146],[15,74834,74836],{"className":74835,"style":182},[150],[15,74837],{},[15,74839],{"className":74840,"style":199},[93],[15,74842,74484],{"className":74843},[203,13316],[15,74845],{"className":74846,"style":199},[93],[15,74848,29575],{"className":74849,"style":29856},[129,133],[15,74851,107],{"className":74852},[129],[15,74854,6858],{"className":74855},[93],[15,74857,74859,74862],{"style":74858},"top:-3.09em;",[15,74860],{"className":74861,"style":74523},[158],[15,74863],{"className":74864,"style":74867},[74865,74866],"stretchy","fbox","height:2.2938em;border-style:solid;border-width:0.04em;",[15,74869,175],{"className":74870},[174],[15,74872,74874],{"className":74873},[146],[15,74875,74878],{"className":74876,"style":74877},[150],"height:1.2038em;",[15,74879],{},[11,74881,74882,74883,74911,74912,74940,74941,74944,74945,74973],{},"这里 ",[15,74884,74886,74899],{"className":74885,"translate":19},[23],[15,74887,74889],{"className":74888},[27],[29,74890,74891],{"xmlns":31},[34,74892,74893,74897],{},[37,74894,74895],{},[43,74896,43246],{},[109,74898,43246],{"encoding":111},[15,74900,74902],{"className":74901,"ariaHidden":72},[116],[15,74903,74905,74908],{"className":74904},[120],[15,74906],{"className":74907,"style":610},[124],[15,74909,43246],{"className":74910,"style":12597},[129,133]," 是「用 ",[15,74913,74915,74928],{"className":74914,"translate":19},[23],[15,74916,74918],{"className":74917},[27],[29,74919,74920],{"xmlns":31},[34,74921,74922,74926],{},[37,74923,74924],{},[43,74925,29575],{},[109,74927,61735],{"encoding":111},[15,74929,74931],{"className":74930,"ariaHidden":72},[116],[15,74932,74934,74937],{"className":74933},[120],[15,74935],{"className":74936,"style":522},[124],[15,74938,29575],{"className":74939,"style":29856},[129,133]," 分开的假设个数」，也就是损失空间里的",[698,74942,74943],{},"打包数","（packing number）。于是下界问题变成了一个几何问题：在参数空间里能放多少个半径为 ",[15,74946,74948,74961],{"className":74947,"translate":19},[23],[15,74949,74951],{"className":74950},[27],[29,74952,74953],{"xmlns":31},[34,74954,74955,74959],{},[37,74956,74957],{},[43,74958,29575],{},[109,74960,61735],{"encoding":111},[15,74962,74964],{"className":74963,"ariaHidden":72},[116],[15,74965,74967,74970],{"className":74966},[120],[15,74968],{"className":74969,"style":522},[124],[15,74971,29575],{"className":74972,"style":29856},[129,133]," 的小球，同时又让它们在 KL 意义下足够接近？",[11,74975,74976,74977,75332,75333,75365],{},"Yang 与 Barron 的版本直接用 ",[15,74978,74980,75046],{"className":74979,"translate":19},[23],[15,74981,74983],{"className":74982},[27],[29,74984,74985],{"xmlns":31},[34,74986,74987,75043],{},[37,74988,74989,74991,74993,74995,74997,74999,75001,75003,75009,75015,75025,75027,75033,75035,75041],{},[43,74990,64036],{},[50,74992,53],{"stretchy":52},[43,74994,34524],{},[50,74996,65625],{"separator":72},[43,74998,597],{},[50,75000,59],{"stretchy":52},[50,75002,62],{},[1443,75004,75005,75007],{},[308,75006,318],{},[43,75008,43246],{},[40,75010,75011,75013],{},[50,75012,1454],{},[43,75014,1459],{},[40,75016,75017,75019],{},[43,75018,906],{},[37,75020,75021,75023],{},[43,75022,34524],{"mathvariant":65},[43,75024,1373],{"mathvariant":65},[50,75026,53],{"stretchy":52},[40,75028,75029,75031],{},[43,75030,1388],{},[43,75032,1459],{},[43,75034,33823],{"mathvariant":65},[46847,75036,75037,75039],{"accent":72},[43,75038,1388],{},[50,75040,72884],{},[50,75042,59],{"stretchy":52},[109,75044,75045],{"encoding":111},"I(K;X)=\\frac1M\\sum_i D_{\\mathrm{KL}}(P_i\\Vert\\bar P)",[15,75047,75049,75085],{"className":75048,"ariaHidden":72},[116],[15,75050,75052,75055,75058,75061,75064,75067,75070,75073,75076,75079,75082],{"className":75051},[120],[15,75053],{"className":75054,"style":125},[124],[15,75056,64036],{"className":75057,"style":47970},[129,133],[15,75059,53],{"className":75060},[188],[15,75062,34524],{"className":75063,"style":5665},[129,133],[15,75065,65625],{"className":75066},[254],[15,75068],{"className":75069,"style":262},[93],[15,75071,597],{"className":75072,"style":47970},[129,133],[15,75074,59],{"className":75075},[195],[15,75077],{"className":75078,"style":199},[93],[15,75080,62],{"className":75081},[203],[15,75083],{"className":75084,"style":199},[93],[15,75086,75088,75091,75159,75162,75203,75206,75252,75255,75295,75298,75329],{"className":75087},[120],[15,75089],{"className":75090,"style":59691},[124],[15,75092,75094,75097,75156],{"className":75093},[129],[15,75095],{"className":75096},[188,1803],[15,75098,75100],{"className":75099},[1443],[15,75101,75103,75148],{"className":75102},[141,142],[15,75104,75106,75145],{"className":75105},[146],[15,75107,75109,75123,75131],{"className":75108,"style":59710},[150],[15,75110,75111,75114],{"style":59713},[15,75112],{"className":75113,"style":1823},[158],[15,75115,75117],{"className":75116},[163,164,165,166],[15,75118,75120],{"className":75119},[129,166],[15,75121,43246],{"className":75122,"style":12597},[129,133,166],[15,75124,75125,75128],{"style":1832},[15,75126],{"className":75127,"style":1823},[158],[15,75129],{"className":75130,"style":1840},[1839],[15,75132,75133,75136],{"style":59736},[15,75134],{"className":75135,"style":1823},[158],[15,75137,75139],{"className":75138},[163,164,165,166],[15,75140,75142],{"className":75141},[129,166],[15,75143,318],{"className":75144},[129,166],[15,75146,175],{"className":75147},[174],[15,75149,75151],{"className":75150},[146],[15,75152,75154],{"className":75153,"style":23490},[150],[15,75155],{},[15,75157],{"className":75158},[195,1803],[15,75160],{"className":75161,"style":262},[93],[15,75163,75165,75168],{"className":75164},[217],[15,75166,1454],{"className":75167,"style":23352},[217,1918,23351],[15,75169,75171],{"className":75170},[137],[15,75172,75174,75195],{"className":75173},[141,142],[15,75175,75177,75192],{"className":75176},[146],[15,75178,75181],{"className":75179,"style":75180},[150],"height:0.162em;",[15,75182,75183,75186],{"style":23368},[15,75184],{"className":75185,"style":159},[158],[15,75187,75189],{"className":75188},[163,164,165,166],[15,75190,1459],{"className":75191},[129,133,166],[15,75193,175],{"className":75194},[174],[15,75196,75198],{"className":75197},[146],[15,75199,75201],{"className":75200,"style":23417},[150],[15,75202],{},[15,75204],{"className":75205,"style":262},[93],[15,75207,75209,75212],{"className":75208},[129],[15,75210,906],{"className":75211,"style":170},[129,133],[15,75213,75215],{"className":75214},[137],[15,75216,75218,75244],{"className":75217},[141,142],[15,75219,75221,75241],{"className":75220},[146],[15,75222,75224],{"className":75223,"style":1525},[150],[15,75225,75226,75229],{"style":48293},[15,75227],{"className":75228,"style":159},[158],[15,75230,75232],{"className":75231},[163,164,165,166],[15,75233,75235],{"className":75234},[129,166],[15,75236,75238],{"className":75237},[129,166],[15,75239,34928],{"className":75240},[129,221,166],[15,75242,175],{"className":75243},[174],[15,75245,75247],{"className":75246},[146],[15,75248,75250],{"className":75249,"style":182},[150],[15,75251],{},[15,75253,53],{"className":75254},[188],[15,75256,75258,75261],{"className":75257},[129],[15,75259,1388],{"className":75260,"style":12449},[129,133],[15,75262,75264],{"className":75263},[137],[15,75265,75267,75287],{"className":75266},[141,142],[15,75268,75270,75284],{"className":75269},[146],[15,75271,75273],{"className":75272,"style":1981},[150],[15,75274,75275,75278],{"style":34239},[15,75276],{"className":75277,"style":159},[158],[15,75279,75281],{"className":75280},[163,164,165,166],[15,75282,1459],{"className":75283},[129,133,166],[15,75285,175],{"className":75286},[174],[15,75288,75290],{"className":75289},[146],[15,75291,75293],{"className":75292,"style":182},[150],[15,75294],{},[15,75296,33823],{"className":75297},[129],[15,75299,75301],{"className":75300},[129,48500],[15,75302,75304],{"className":75303},[141],[15,75305,75307],{"className":75306},[146],[15,75308,75310,75318],{"className":75309,"style":73214},[150],[15,75311,75312,75315],{"style":2287},[15,75313],{"className":75314,"style":1823},[158],[15,75316,1388],{"className":75317,"style":12449},[129,133],[15,75319,75320,75323],{"style":65322},[15,75321],{"className":75322,"style":1823},[158],[15,75324,75326],{"className":75325,"style":48528},[48527],[15,75327,72884],{"className":75328},[129],[15,75330,59],{"className":75331},[195],"，而不是把它放缩成最大成对 KL。这在局部问题里更精确，常常能去掉多余的 ",[15,75334,75336,75351],{"className":75335,"translate":19},[23],[15,75337,75339],{"className":75338},[27],[29,75340,75341],{"xmlns":31},[34,75342,75343,75349],{},[37,75344,75345,75347],{},[43,75346,14727],{},[50,75348,69],{},[109,75350,30130],{"encoding":111},[15,75352,75354],{"className":75353,"ariaHidden":72},[116],[15,75355,75357,75360],{"className":75356},[120],[15,75358],{"className":75359,"style":1682},[124],[15,75361,14777,75363],{"className":75362},[217],[15,75364,2247],{"style":2246}," 因子。其思想是：不要要求所有假设两两都接近，只要它们整体上离它们的「重心」接近就够了。",[11,75367,75368,75369,75372,75373,75443,75444,75472,75473,75582,75583,75611,75612,75640,75641,1313],{},"Fano 还有一个",[698,75370,75371],{},"局部版本","：不要求 ",[15,75374,75376,75394],{"className":75375,"translate":19},[23],[15,75377,75379],{"className":75378},[27],[29,75380,75381],{"xmlns":31},[34,75382,75383,75391],{},[37,75384,75385],{},[40,75386,75387,75389],{},[43,75388,48],{},[43,75390,1459],{},[109,75392,75393],{"encoding":111},"\\theta_i",[15,75395,75397],{"className":75396,"ariaHidden":72},[116],[15,75398,75400,75403],{"className":75399},[120],[15,75401],{"className":75402,"style":2183},[124],[15,75404,75406,75409],{"className":75405},[129],[15,75407,48],{"className":75408,"style":170},[129,133],[15,75410,75412],{"className":75411},[137],[15,75413,75415,75435],{"className":75414},[141,142],[15,75416,75418,75432],{"className":75417},[146],[15,75419,75421],{"className":75420,"style":1981},[150],[15,75422,75423,75426],{"style":48293},[15,75424],{"className":75425,"style":159},[158],[15,75427,75429],{"className":75428},[163,164,165,166],[15,75430,1459],{"className":75431},[129,133,166],[15,75433,175],{"className":75434},[174],[15,75436,75438],{"className":75437},[146],[15,75439,75441],{"className":75440,"style":182},[150],[15,75442],{}," 覆盖整个参数空间，只要求它们落在一个半径 ",[15,75445,75447,75460],{"className":75446,"translate":19},[23],[15,75448,75450],{"className":75449},[27],[29,75451,75452],{"xmlns":31},[34,75453,75454,75458],{},[37,75455,75456],{},[43,75457,62089],{},[109,75459,62089],{"encoding":111},[15,75461,75463],{"className":75462,"ariaHidden":72},[116],[15,75464,75466,75469],{"className":75465},[120],[15,75467],{"className":75468,"style":4383},[124],[15,75470,62089],{"className":75471,"style":170},[129,133]," 的小球里。非参数速率 ",[15,75474,75476,75514],{"className":75475,"translate":19},[23],[15,75477,75479],{"className":75478},[27],[29,75480,75481],{"xmlns":31},[34,75482,75483,75511],{},[37,75484,75485],{},[305,75486,75487,75489],{},[43,75488,39326],{},[37,75490,75491,75493,75495,75497,75499,75501,75503,75505,75507,75509],{},[50,75492,315],{},[308,75494,310],{},[43,75496,36277],{},[43,75498,544],{"mathvariant":65},[50,75500,53],{"stretchy":52},[308,75502,310],{},[43,75504,36277],{},[50,75506,2682],{},[43,75508,10376],{},[50,75510,59],{"stretchy":52},[109,75512,75513],{"encoding":111},"n^{-2\\beta\u002F(2\\beta+d)}",[15,75515,75517],{"className":75516,"ariaHidden":72},[116],[15,75518,75520,75523],{"className":75519},[120],[15,75521],{"className":75522,"style":14948},[124],[15,75524,75526,75529],{"className":75525},[129],[15,75527,39326],{"className":75528},[129,133],[15,75530,75532],{"className":75531},[137],[15,75533,75535],{"className":75534},[141],[15,75536,75538],{"className":75537},[146],[15,75539,75541],{"className":75540,"style":14948},[150],[15,75542,75543,75546],{"style":383},[15,75544],{"className":75545,"style":159},[158],[15,75547,75549],{"className":75548},[163,164,165,166],[15,75550,75552,75555,75558,75561,75564,75567,75570,75573,75576,75579],{"className":75551},[129,166],[15,75553,315],{"className":75554},[129,166],[15,75556,310],{"className":75557},[129,166],[15,75559,36277],{"className":75560,"style":36293},[129,133,166],[15,75562,544],{"className":75563},[129,166],[15,75565,53],{"className":75566},[188,166],[15,75568,310],{"className":75569},[129,166],[15,75571,36277],{"className":75572,"style":36293},[129,133,166],[15,75574,2682],{"className":75575},[2436,166],[15,75577,10376],{"className":75578},[129,133,166],[15,75580,59],{"className":75581},[195,166]," 正是这样来的——局部的打包数随 ",[15,75584,75586,75599],{"className":75585,"translate":19},[23],[15,75587,75589],{"className":75588},[27],[29,75590,75591],{"xmlns":31},[34,75592,75593,75597],{},[37,75594,75595],{},[43,75596,62089],{},[109,75598,62089],{"encoding":111},[15,75600,75602],{"className":75601,"ariaHidden":72},[116],[15,75603,75605,75608],{"className":75604},[120],[15,75606],{"className":75607,"style":4383},[124],[15,75609,62089],{"className":75610,"style":170},[129,133]," 增长，而 KL 随 ",[15,75613,75615,75628],{"className":75614,"translate":19},[23],[15,75616,75618],{"className":75617},[27],[29,75619,75620],{"xmlns":31},[34,75621,75622,75626],{},[37,75623,75624],{},[43,75625,62089],{},[109,75627,62089],{"encoding":111},[15,75629,75631],{"className":75630,"ariaHidden":72},[116],[15,75632,75634,75637],{"className":75633},[120],[15,75635],{"className":75636,"style":4383},[124],[15,75638,62089],{"className":75639,"style":170},[129,133]," 增长，两者平衡给出最优的 ",[15,75642,75644,75657],{"className":75643,"translate":19},[23],[15,75645,75647],{"className":75646},[27],[29,75648,75649],{"xmlns":31},[34,75650,75651,75655],{},[37,75652,75653],{},[43,75654,62089],{},[109,75656,62089],{"encoding":111},[15,75658,75660],{"className":75659,"ariaHidden":72},[116],[15,75661,75663,75666],{"className":75662},[120],[15,75664],{"className":75665,"style":4383},[124],[15,75667,62089],{"className":75668,"style":170},[129,133],[570,75670,75672],{"id":75671},"五assouad-引理","五、Assouad 引理",[575,75674,75676],{"id":75675},"_51-超立方体","5.1 超立方体",[11,75678,75679,75680,75708],{},"Fano 处理的是「",[15,75681,75683,75696],{"className":75682,"translate":19},[23],[15,75684,75686],{"className":75685},[27],[29,75687,75688],{"xmlns":31},[34,75689,75690,75694],{},[37,75691,75692],{},[43,75693,43246],{},[109,75695,43246],{"encoding":111},[15,75697,75699],{"className":75698,"ariaHidden":72},[116],[15,75700,75702,75705],{"className":75701},[120],[15,75703],{"className":75704,"style":610},[124],[15,75706,43246],{"className":75707,"style":12597},[129,133]," 个任意假设」，Assouad 处理的是一种特殊但极有用的结构：超立方体。",[6559,75710,75711,75919,76688,76691],{},[11,75712,75713,52879,75716,7610,75820,75890,75891,15741],{},[698,75714,75715],{},"定理 5.1（Assouad 引理，平方损失）",[15,75717,75719,75749],{"className":75718,"translate":19},[23],[15,75720,75722],{"className":75721},[27],[29,75723,75724],{"xmlns":31},[34,75725,75726,75746],{},[37,75727,75728,75730,75732,75734,75736,75738,75740],{},[43,75729,30875],{"mathvariant":65},[50,75731,62],{},[50,75733,636],{"stretchy":52},[308,75735,541],{},[50,75737,91],{"separator":72},[308,75739,318],{},[305,75741,75742,75744],{},[50,75743,645],{"stretchy":52},[43,75745,348],{},[109,75747,75748],{"encoding":111},"\\Theta=\\{0,1\\}^m",[15,75750,75752,75770],{"className":75751,"ariaHidden":72},[116],[15,75753,75755,75758,75761,75764,75767],{"className":75754},[120],[15,75756],{"className":75757,"style":610},[124],[15,75759,30875],{"className":75760},[129],[15,75762],{"className":75763,"style":199},[93],[15,75765,62],{"className":75766},[203],[15,75768],{"className":75769,"style":199},[93],[15,75771,75773,75776,75779,75782,75785,75788,75791],{"className":75772},[120],[15,75774],{"className":75775,"style":125},[124],[15,75777,636],{"className":75778},[188],[15,75780,541],{"className":75781},[129],[15,75783,91],{"className":75784},[254],[15,75786],{"className":75787,"style":262},[93],[15,75789,318],{"className":75790},[129],[15,75792,75794,75797],{"className":75793},[195],[15,75795,645],{"className":75796},[195],[15,75798,75800],{"className":75799},[137],[15,75801,75803],{"className":75802},[141],[15,75804,75806],{"className":75805},[146],[15,75807,75809],{"className":75808,"style":6377},[150],[15,75810,75811,75814],{"style":383},[15,75812],{"className":75813,"style":159},[158],[15,75815,75817],{"className":75816},[163,164,165,166],[15,75818,348],{"className":75819},[129,133,166],[15,75821,75823,75841],{"className":75822,"translate":19},[23],[15,75824,75826],{"className":75825},[27],[29,75827,75828],{"xmlns":31},[34,75829,75830,75838],{},[37,75831,75832],{},[40,75833,75834,75836],{},[43,75835,1388],{},[43,75837,48],{},[109,75839,75840],{"encoding":111},"P_\\theta",[15,75842,75844],{"className":75843,"ariaHidden":72},[116],[15,75845,75847,75850],{"className":75846},[120],[15,75848],{"className":75849,"style":20229},[124],[15,75851,75853,75856],{"className":75852},[129],[15,75854,1388],{"className":75855,"style":12449},[129,133],[15,75857,75859],{"className":75858},[137],[15,75860,75862,75882],{"className":75861},[141,142],[15,75863,75865,75879],{"className":75864},[146],[15,75866,75868],{"className":75867,"style":151},[150],[15,75869,75870,75873],{"style":34239},[15,75871],{"className":75872,"style":159},[158],[15,75874,75876],{"className":75875},[163,164,165,166],[15,75877,48],{"className":75878,"style":170},[129,133,166],[15,75880,175],{"className":75881},[174],[15,75883,75885],{"className":75884},[146],[15,75886,75888],{"className":75887,"style":182},[150],[15,75889],{}," 是观测分布，损失为平方欧氏距离。对每个坐标 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X]",[15,77572,77574,77666,77690],{"className":77573,"ariaHidden":72},[116],[15,77575,77577,77580,77648,77651,77654,77657,77660,77663],{"className":77576},[120],[15,77578],{"className":77579,"style":50032},[124],[15,77581,77583,77614],{"className":77582},[129],[15,77584,77586],{"className":77585},[129,48500],[15,77587,77589],{"className":77588},[141],[15,77590,77592],{"className":77591},[146],[15,77593,77595,77603],{"className":77594,"style":48496},[150],[15,77596,77597,77600],{"style":2287},[15,77598],{"className":77599,"style":1823},[158],[15,77601,48],{"className":77602,"style":170},[129,133],[15,77604,77605,77608],{"style":48520},[15,77606],{"className":77607,"style":1823},[158],[15,77609,77611],{"className":77610,"style":48528},[48527],[15,77612,48483],{"className":77613},[129],[15,77615,77617],{"className":77616},[137],[15,77618,77620,77640],{"className":77619},[141,142],[15,77621,77623,77637],{"className":77622},[146],[15,77624,77626],{"className":77625,"style":1525},[150],[15,77627,77628,77631],{"style":48293},[15,77629],{"className":77630,"style":159},[158],[15,77632,77634],{"className":77633},[163,164,165,166],[15,77635,33895],{"className":77636,"style":33935},[129,133,166],[15,77638,175],{"className":77639},[174],[15,77641,77643],{"className":77642},[146],[15,77644,77646],{"className":77645,"style":182},[150],[15,77647],{},[15,77649,53],{"className":77650},[188],[15,77652,597],{"className":77653,"style":47970},[129,133],[15,77655,59],{"className":77656},[195],[15,77658],{"className":77659,"style":199},[93],[15,77661,62],{"className":77662},[203],[15,77664],{"className":77665,"style":199},[93],[15,77667,77669,77672,77675,77678,77681,77684,77687],{"className":77668},[120],[15,77670],{"className":77671,"style":125},[124],[15,77673,26574],{"className":77674},[129,285],[15,77676,1407],{"className":77677},[188],[15,77679,48],{"className":77680,"style":170},[129,133],[15,77682],{"className":77683,"style":199},[93],[15,77685,2945],{"className":77686},[203],[15,77688],{"className":77689,"style":199},[93],[15,77691,77693,77696,77699],{"className":77692},[120],[15,77694],{"className":77695,"style":125},[124],[15,77697,597],{"className":77698,"style":47970},[129,133],[15,77700,1423],{"className":77701},[195],"，其贝叶斯风险为",[15,77704,77706],{"className":77705,"translate":19},[18],[15,77707,77709,77815],{"className":77708,"translate":19},[23],[15,77710,77712],{"className":77711},[27],[29,77713,77714],{"xmlns":31,"display":32},[34,77715,77716,77812],{},[37,77717,77718,77732,77734,77736,77742,77744,77746,77748,77750,77752,77754,77760,77762,77764,77766,77768,77770,77772,77774,77780,77782,77784,77786,77788,77790,77792,77794,77800,77802,77804,77806,77808,77810],{},[1450,77719,77720,77722,77730],{},[50,77721,1454],{},[37,77723,77724,77726,77728],{},[43,77725,13955],{},[50,77727,62],{},[308,77729,318],{},[43,77731,348],{},[43,77733,26574],{"mathvariant":103},[50,77735,1407],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[40,77737,77738,77740],{},[43,77739,15763],{},[43,77741,13955],{},[50,77743,53],{"stretchy":52},[43,77745,597],{},[50,77747,59],{"stretchy":52},[50,77749,53],{"stretchy":52},[308,77751,318],{},[50,77753,315],{},[40,77755,77756,77758],{},[43,77757,15763],{},[43,77759,13955],{},[50,77761,53],{"stretchy":52},[43,77763,597],{},[50,77765,59],{"stretchy":52},[50,77767,59],{"stretchy":52},[50,77769,1423],{"fence":72,"stretchy":72,"minsize":73,"maxsize":73},[50,77771,91],{"separator":72},[93,77773],{"width":95},[40,77775,77776,77778],{},[43,77777,15763],{},[43,77779,13955],{},[50,77781,53],{"stretchy":52},[43,77783,597],{},[50,77785,59],{"stretchy":52},[50,77787,62],{},[43,77789,4961],{},[50,77791,69],{},[50,77793,53],{"stretchy":52},[40,77795,77796,77798],{},[43,77797,48],{},[43,77799,13955],{},[50,77801,62],{},[308,77803,318],{},[50,77805,2945],{},[43,77807,597],{},[50,77809,59],{"stretchy":52},[43,77811,107],{"mathvariant":65},[109,77813,77814],{"encoding":111},"\\sum_{j=1}^m\\mathbb E\\bigl[\\pi_j(X)(1-\\pi_j(X))\\bigr],\n\\qquad\n\\pi_j(X)=\\Pr(\\theta_j=1\\mid X).",[15,77816,77818,77965,78093,78154,78172],{"className":77817,"ariaHidden":72},[116],[15,77819,77821,77825,77889,77892,77895,77901,77941,77944,77947,77950,77953,77956,77959,77962],{"className":77820},[120],[15,77822],{"className":77823,"style":77824},[124],"height:3.0652em;vertical-align:-1.4138em;",[15,77826,77828],{"className":77827},[217,1874],[15,77829,77831,77881],{"className":77830},[141,142],[15,77832,77834,77878],{"className":77833},[146],[15,77835,77837,77857,77867],{"className":77836,"style":1884},[150],[15,77838,77839,77842],{"style":1887},[15,77840],{"className":77841,"style":1891},[158],[15,77843,77845],{"className":77844},[163,164,165,166],[15,77846,77848,77851,77854],{"className":77847},[129,166],[15,77849,13955],{"className":77850,"style":13970},[129,133,166],[15,77852,62],{"className":77853},[203,166],[15,77855,318],{"className":77856},[129,166],[15,77858,77859,77862],{"style":1909},[15,77860],{"className":77861,"style":1891},[158],[15,77863,77864],{},[15,77865,1454],{"className":77866},[217,1918,1919],[15,77868,77869,77872],{"style":1922},[15,77870],{"className":77871,"style":1891},[158],[15,77873,77875],{"className":77874},[163,164,165,166],[15,77876,348],{"className":77877},[129,133,166],[15,77879,175],{"className":77880},[174],[15,77882,77884],{"className":77883},[146],[15,77885,77887],{"className":77886,"style":16035},[150],[15,77888],{},[15,77890],{"className":77891,"style":262},[93],[15,77893,26574],{"className":77894},[129,285],[15,77896,77898],{"className":77897},[188],[15,77899,1407],{"className":77900},[228,229],[15,77902,77904,77907],{"className":77903},[129],[15,77905,15763],{"className":77906,"style":887},[129,133],[15,77908,77910],{"className":77909},[137],[15,77911,77913,77933],{"className":77912},[141,142],[15,77914,77916,77930],{"className":77915},[146],[15,77917,77919],{"className":77918,"style":1981},[150],[15,77920,77921,77924],{"style":1100},[15,77922],{"className":77923,"style":159},[158],[15,77925,77927],{"className":77926},[163,164,165,166],[15,77928,13955],{"className":77929,"style":13970},[129,133,166],[15,77931,175],{"className":77932},[174],[15,77934,77936],{"className":77935},[146],[15,77937,77939],{"className":77938,"style":14041},[150],[15,77940],{},[15,77942,53],{"className":77943},[188],[15,77945,597],{"className":77946,"style":47970},[129,133],[15,77948,59],{"className":77949},[195],[15,77951,53],{"className":77952},[188],[15,77954,318],{"className":77955},[129],[15,77957],{"className":77958,"style":3277},[93],[15,77960,315],{"className":77961},[2436],[15,77963],{"className":77964,"style":3277},[93],[15,77966,77968,77971,78011,78014,78017,78020,78026,78029,78032,78035,78075,78078,78081,78084,78087,78090],{"className":77967},[120],[15,77969],{"className":77970,"style":213},[124],[15,77972,77974,77977],{"className":77973},[129],[15,77975,15763],{"className":77976,"style":887},[129,133],[15,77978,77980],{"className":77979},[137],[15,77981,77983,78003],{"className":77982},[141,142],[15,77984,77986,78000],{"className":77985},[146],[15,77987,77989],{"className":77988,"style":1981},[150],[15,77990,77991,77994],{"style":1100},[15,77992],{"className":77993,"style":159},[158],[15,77995,77997],{"className":77996},[163,164,165,166],[15,77998,13955],{"className":77999,"style":13970},[129,133,166],[15,78001,175],{"className":78002},[174],[15,78004,78006],{"className":78005},[146],[15,78007,78009],{"className":78008,"style":14041},[150],[15,78010],{},[15,78012,53],{"className":78013},[188],[15,78015,597],{"className":78016,"style":47970},[129,133],[15,78018,6149],{"className":78019},[195],[15,78021,78023],{"className":78022},[195],[15,78024,1423],{"className":78025},[228,229],[15,78027,91],{"className":78028},[254],[15,78030],{"className":78031,"style":258},[93],[15,78033],{"className":78034,"style":262},[93],[15,78036,78038,78041],{"className":78037},[129],[15,78039,15763],{"className":78040,"style":887},[129,133],[15,78042,78044],{"className":78043},[137],[15,78045,78047,78067],{"className":78046},[141,142],[15,78048,78050,78064],{"className":78049},[146],[15,78051,78053],{"className":78052,"style":1981},[150],[15,78054,78055,78058],{"style":1100},[15,78056],{"className":78057,"style":159},[158],[15,78059,78061],{"className":78060},[163,164,165,166],[15,78062,13955],{"className":78063,"style":13970},[129,133,166],[15,78065,175],{"className":78066},[174],[15,78068,78070],{"className":78069},[146],[15,78071,78073],{"className":78072,"style":14041},[150],[15,78074],{},[15,78076,53],{"className":78077},[188],[15,78079,597],{"className":78080,"style":47970},[129,133],[15,78082,59],{"className":78083},[195],[15,78085],{"className":78086,"style":199},[93],[15,78088,62],{"className":78089},[203],[15,78091],{"className":78092,"style":199},[93],[15,78094,78096,78099,78102,78105,78145,78148,78151],{"className":78095},[120],[15,78097],{"className":78098,"style":67458},[124],[15,78100,4961],{"className":78101},[217],[15,78103,53],{"className":78104},[188],[15,78106,78108,78111],{"className":78107},[129],[15,78109,48],{"className":78110,"style":170},[129,133],[15,78112,78114],{"className":78113},[137],[15,78115,78117,78137],{"className":78116},[141,142],[15,78118,78120,78134],{"className":78119},[146],[15,78121,78123],{"className":78122,"style":1981},[150],[15,78124,78125,78128],{"style":48293},[15,78126],{"className":78127,"style":159},[158],[15,78129,78131],{"className":78130},[163,164,165,166],[15,78132,13955],{"className":78133,"style":13970},[129,133,166],[15,78135,175],{"className":78136},[174],[15,78138,78140],{"className":78139},[146],[15,78141,78143],{"className":78142,"style":14041},[150],[15,78144],{},[15,78146],{"className":78147,"style":199},[93],[15,78149,62],{"className":78150},[203],[15,78152],{"className":78153,"style":199},[93],[15,78155,78157,78160,78163,78166,78169],{"className":78156},[120],[15,78158],{"className":78159,"style":125},[124],[15,78161,318],{"className":78162},[129],[15,78164],{"className":78165,"style":199},[93],[15,78167,2945],{"className":78168},[203],[15,78170],{"className":78171,"style":199},[93],[15,78173,78175,78178,78181,78184],{"className":78174},[120],[15,78176],{"className":78177,"style":125},[124],[15,78179,597],{"className":78180,"style":47970},[129,133],[15,78182,59],{"className":78183},[195],[15,78185,107],{"className":78186},[129],[11,78188,78189,78190,78218,78219,78289,78290,10179,78375,78460,78461,7610],{},"固定 ",[15,78191,78193,78206],{"className":78192,"translate":19},[23],[15,78194,78196],{"className":78195},[27],[29,78197,78198],{"xmlns":31},[34,78199,78200,78204],{},[37,78201,78202],{},[43,78203,13955],{},[109,78205,13955],{"encoding":111},[15,78207,78209],{"className":78208,"ariaHidden":72},[116],[15,78210,78212,78215],{"className":78211},[120],[15,78213],{"className":78214,"style":11191},[124],[15,78216,13955],{"className":78217,"style":13970},[129,133],"。边际后验 ",[15,78220,78222,78240],{"className":78221,"translate":19},[23],[15,78223,78225],{"className":78224},[27],[29,78226,78227],{"xmlns":31},[34,78228,78229,78237],{},[37,78230,78231],{},[40,78232,78233,78235],{},[43,78234,15763],{},[43,78236,13955],{},[109,78238,78239],{"encoding":111},"\\pi_j",[15,78241,78243],{"className":78242,"ariaHidden":72},[116],[15,78244,78246,78249],{"className":78245},[120],[15,78247],{"className":78248,"style":14002},[124],[15,78250,78252,78255],{"className":78251},[129],[15,78253,15763],{"className":78254,"style":887},[129,133],[15,78256,78258],{"className":78257},[137],[15,78259,78261,78281],{"className":78260},[141,142],[15,78262,78264,78278],{"className":78263},[146],[15,78265,78267],{"className":78266,"style":1981},[150],[15,78268,78269,78272],{"style":1100},[15,78270],{"className":78271,"style":159},[158],[15,78273,78275],{"className":78274},[163,164,165,166],[15,78276,13955],{"className":78277,"style":13970},[129,133,166],[15,78279,175],{"className":78280},[174],[15,78282,78284],{"className":78283},[146],[15,78285,78287],{"className":78286,"style":14041},[150],[15,78288],{}," 正是 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在均匀先验下的后验概率。对任意 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3.1 的检验形式。于是每个坐标贡献至少 ",[15,79269,79271,79307],{"className":79270,"translate":19},[23],[15,79272,79274],{"className":79273},[27],[29,79275,79276],{"xmlns":31},[34,79277,79278,79304],{},[37,79279,79280,79286,79288,79290,79292,79302],{},[1443,79281,79282,79284],{},[308,79283,318],{},[308,79285,14508],{},[50,79287,53],{"stretchy":52},[308,79289,318],{},[50,79291,315],{},[40,79293,79294,79300],{},[37,79295,79296,79298],{},[43,79297,53409],{"mathvariant":65},[43,79299,53412],{"mathvariant":65},[43,79301,13955],{},[50,79303,59],{"stretchy":52},[109,79305,79306],{"encoding":111},"\\frac14(1-\\mathrm{TV}_j)",[15,79308,79310,79399],{"className":79309,"ariaHidden":72},[116],[15,79311,79313,79316,79384,79387,79390,79393,79396],{"className":79312},[120],[15,79314],{"className":79315,"style":59691},[124],[15,79317,79319,79322,79381],{"className":79318},[129],[15,79320],{"className":79321},[188,1803],[15,79323,79325],{"className":79324},[1443],[15,79326,79328,79373],{"className":79327},[141,142],[15,79329,79331,79370],{"className":79330},[146],[15,79332,79334,79348,79356],{"className":79333,"style":59710},[150],[15,79335,79336,79339],{"style":59713},[15,79337],{"className":79338,"style":1823},[158],[15,79340,79342],{"className":79341},[163,164,165,166],[15,79343,79345],{"className":79344},[129,166],[15,79346,14508],{"className":79347},[129,166],[15,79349,79350,79353],{"style":1832},[15,79351],{"className":79352,"style":1823},[158],[15,79354],{"className":79355,"style":1840},[1839],[15,79357,79358,79361],{"style":59736},[15,79359],{"className":79360,"style":1823},[158],[15,79362,79364],{"className":79363},[163,164,165,166],[15,79365,79367],{"className":79366},[129,166],[15,79368,318],{"className":79369},[129,166],[15,79371,175],{"className":79372},[174],[15,79374,79376],{"className":79375},[146],[15,79377,79379],{"className":79378,"style":23490},[150],[15,79380],{},[15,79382],{"className":79383},[195,1803],[15,79385,53],{"className":79386},[188],[15,79388,318],{"className":79389},[129],[15,79391],{"className":79392,"style":3277},[93],[15,79394,315],{"className":79395},[2436],[15,79397],{"className":79398,"style":3277},[93],[15,79400,79402,79405,79449],{"className":79401},[120],[15,79403],{"className":79404,"style":67458},[124],[15,79406,79408,79414],{"className":79407},[129],[15,79409,79411],{"className":79410},[129],[15,79412,54040],{"className":79413,"style":2246},[129,221],[15,79415,79417],{"className":79416},[137],[15,79418,79420,79441],{"className":79419},[141,142],[15,79421,79423,79438],{"className":79422},[146],[15,79424,79426],{"className":79425,"style":1981},[150],[15,79427,79429,79432],{"style":79428},"top:-2.55em;margin-right:0.05em;",[15,79430],{"className":79431,"style":159},[158],[15,79433,79435],{"className":79434},[163,164,165,166],[15,79436,13955],{"className":79437,"style":13970},[129,133,166],[15,79439,175],{"className":79440},[174],[15,79442,79444],{"className":79443},[146],[15,79445,79447],{"className":79446,"style":14041},[150],[15,79448],{},[15,79450,59],{"className":79451},[195],"，求和得",[15,79454,79456],{"className":79455,"translate":19},[18],[15,79457,79459,79545],{"className":79458,"translate":19},[23],[15,79460,79462],{"className":79461},[27],[29,79463,79464],{"xmlns":31,"display":32},[34,79465,79466,79542],{},[37,79467,79468,79474,79476,79482,79540],{},[40,79469,79470,79472],{},[43,79471,33895],{"mathvariant":28910},[43,79473,39326],{},[50,79475,7473],{},[1443,79477,79478,79480],{},[43,79479,348],{},[308,79481,14508],{},[37,79483,79484,79486,79488,79490,79500,79502,79514,79516,79528,79538],{},[50,79485,53],{"fence":72},[308,79487,318],{},[50,79489,315],{},[2140,79491,79492,79498],{},[37,79493,79494,79496],{},[43,79495,19705],{},[50,79497,69],{},[43,79499,13955],{},[50,79501,33823],{"stretchy":52},[40,79503,79504,79506],{},[43,79505,34354],{},[37,79507,79508,79510,79512],{},[43,79509,13955],{},[50,79511,91],{"separator":72},[308,79513,541],{},[50,79515,315],{},[40,79517,79518,79520],{},[43,79519,34354],{},[37,79521,79522,79524,79526],{},[43,79523,13955],{},[50,79525,91],{"separator":72},[308,79527,318],{},[40,79529,79530,79532],{},[50,79531,33823],{"stretchy":52},[37,79533,79534,79536],{},[43,79535,53409],{"mathvariant":65},[43,79537,53412],{"mathvariant":65},[50,79539,59],{"fence":72},[43,79541,107],{"mathvariant":65},[109,79543,79544],{"encoding":111},"\\mathfrak B_n\\ge\\frac m4\\left(1-\\max_j\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\\right).",[15,79546,79548,79604],{"className":79547,"ariaHidden":72},[116],[15,79549,79551,79555,79595,79598,79601],{"className":79550},[120],[15,79552],{"className":79553,"style":79554},[124],"height:0.8414em;vertical-align:-0.15em;",[15,79556,79558,79561],{"className":79557},[129],[15,79559,33895],{"className":79560},[129,28937],[15,79562,79564],{"className":79563},[137],[15,79565,79567,79587],{"className":79566},[141,142],[15,79568,79570,79584],{"className":79569},[146],[15,79571,79573],{"className":79572,"style":1163},[150],[15,79574,79575,79578],{"style":154},[15,79576],{"className":79577,"style":159},[158],[15,79579,79581],{"className":79580},[163,164,165,166],[15,79582,39326],{"className":79583},[129,133,166],[15,79585,175],{"className":79586},[174],[15,79588,79590],{"className":79589},[146],[15,79591,79593],{"className":79592,"style":182},[150],[15,79594],{},[15,79596],{"className":79597,"style":199},[93],[15,79599,7473],{"className":79600},[203],[15,79602],{"className":79603,"style":199},[93],[15,79605,79607,79610,79672,79675,79902,79905],{"className":79606},[120],[15,79608],{"className":79609,"style":17014},[124],[15,79611,79613,79616,79669],{"className":79612},[129],[15,79614],{"className":79615},[188,1803],[15,79617,79619],{"className":79618},[1443],[15,79620,79622,79661],{"className":79621},[141,142],[15,79623,79625,79658],{"className":79624},[146],[15,79626,79628,79639,79647],{"className":79627,"style":21221},[150],[15,79629,79630,79633],{"style":1819},[15,79631],{"className":79632,"style":1823},[158],[15,79634,79636],{"className":79635},[129],[15,79637,14508],{"className":79638},[129],[15,79640,79641,79644],{"style":1832},[15,79642],{"className":79643,"style":1823},[158],[15,79645],{"className":79646,"style":1840},[1839],[15,79648,79649,79652],{"style":1843},[15,79650],{"className":79651,"style":1823},[158],[15,79653,79655],{"className":79654},[129],[15,79656,348],{"className":79657},[129,133],[15,79659,175],{"className":79660},[174],[15,79662,79664],{"className":79663},[146],[15,79665,79667],{"className":79666,"style":1862},[150],[15,79668],{},[15,79670],{"className":79671},[195,1803],[15,79673],{"className":79674,"style":262},[93],[15,79676,79678,79684,79687,79690,79693,79696,79740,79743,79792,79795,79798,79801,79850,79896],{"className":79677},[449],[15,79679,79681],{"className":79680,"style":5052},[188,5051],[15,79682,53],{"className":79683},[228,165],[15,79685,318],{"className":79686},[129],[15,79688],{"className":79689,"style":3277},[93],[15,79691,315],{"className":79692},[2436],[15,79694],{"className":79695,"style":3277},[93],[15,79697,79699],{"className":79698},[217,1874],[15,79700,79702,79732],{"className":79701},[141,142],[15,79703,79705,79729],{"className":79704},[146],[15,79706,79708,79719],{"className":79707,"style":4383},[150],[15,79709,79710,79713],{"style":53444},[15,79711],{"className":79712,"style":1823},[158],[15,79714,79716],{"className":79715},[163,164,165,166],[15,79717,13955],{"className":79718,"style":13970},[129,133,166],[15,79720,79721,79724],{"style":2287},[15,79722],{"className":79723,"style":1823},[158],[15,79725,79726],{},[15,79727,19705],{"className":79728},[217],[15,79730,175],{"className":79731},[174],[15,79733,79735],{"className":79734},[146],[15,79736,79738],{"className":79737,"style":53488},[150],[15,79739],{},[15,79741,33823],{"className":79742},[188],[15,79744,79746,79749],{"className":79745},[129],[15,79747,34354],{"className":79748},[129,133],[15,79750,79752],{"className":79751},[137],[15,79753,79755,79784],{"className":79754},[141,142],[15,79756,79758,79781],{"className":79757},[146],[15,79759,79761],{"className":79760,"style":1981},[150],[15,79762,79763,79766],{"style":154},[15,79764],{"className":79765,"style":159},[158],[15,79767,79769],{"className":79768},[163,164,165,166],[15,79770,79772,79775,79778],{"className":79771},[129,166],[15,79773,13955],{"className":79774,"style":13970},[129,133,166],[15,79776,91],{"className":79777},[254,166],[15,79779,541],{"className":79780},[129,166],[15,79782,175],{"className":79783},[174],[15,79785,79787],{"className":79786},[146],[15,79788,79790],{"className":79789,"style":14041},[150],[15,79791],{},[15,79793],{"className":79794,"style":3277},[93],[15,79796,315],{"className":79797},[2436],[15,79799],{"className":79800,"style":3277},[93],[15,79802,79804,79807],{"className":79803},[129],[15,79805,34354],{"className":79806},[129,133],[15,79808,79810],{"className":79809},[137],[15,79811,79813,79842],{"className":79812},[141,142],[15,79814,79816,79839],{"className":79815},[146],[15,79817,79819],{"className":79818,"style":1981},[150],[15,79820,79821,79824],{"style":154},[15,79822],{"className":79823,"style":159},[158],[15,79825,79827],{"className":79826},[163,164,165,166],[15,79828,79830,79833,79836],{"className":79829},[129,166],[15,79831,13955],{"className":79832,"style":13970},[129,133,166],[15,79834,91],{"className":79835},[254,166],[15,79837,318],{"className":79838},[129,166],[15,79840,175],{"className":79841},[174],[15,79843,79845],{"className":79844},[146],[15,79846,79848],{"className":79847,"style":14041},[150],[15,79849],{},[15,79851,79853,79856],{"className":79852},[195],[15,79854,33823],{"className":79855},[195],[15,79857,79859],{"className":79858},[137],[15,79860,79862,79888],{"className":79861},[141,142],[15,79863,79865,79885],{"className":79864},[146],[15,79866,79868],{"className":79867,"style":1525},[150],[15,79869,79870,79873],{"style":154},[15,79871],{"className":79872,"style":159},[158],[15,79874,79876],{"className":79875},[163,164,165,166],[15,79877,79879],{"className":79878},[129,166],[15,79880,79882],{"className":79881},[129,166],[15,79883,54040],{"className":79884,"style":2246},[129,221,166],[15,79886,175],{"className":79887},[174],[15,79889,79891],{"className":79890},[146],[15,79892,79894],{"className":79893,"style":182},[150],[15,79895],{},[15,79897,79899],{"className":79898,"style":5052},[195,5051],[15,79900,59],{"className":79901},[228,165],[15,79903],{"className":79904,"style":262},[93],[15,79906,107],{"className":79907},[129],[11,79909,79910],{},"极小极大风险不小于贝叶斯风险，定理得证。",[575,79912,79914],{"id":79913},"_52-逐坐标下界","5.2 逐坐标下界",[11,79916,79917,79918,79946,79947,79975,79976,80004,80005,80033,80034,80062],{},"Assouad 的威力在于它把 ",[15,79919,79921,79934],{"className":79920,"translate":19},[23],[15,79922,79924],{"className":79923},[27],[29,79925,79926],{"xmlns":31},[34,79927,79928,79932],{},[37,79929,79930],{},[43,79931,348],{},[109,79933,348],{"encoding":111},[15,79935,79937],{"className":79936,"ariaHidden":72},[116],[15,79938,79940,79943],{"className":79939},[120],[15,79941],{"className":79942,"style":4383},[124],[15,79944,348],{"className":79945},[129,133]," 维问题拆成了 ",[15,79948,79950,79963],{"className":79949,"translate":19},[23],[15,79951,79953],{"className":79952},[27],[29,79954,79955],{"xmlns":31},[34,79956,79957,79961],{},[37,79958,79959],{},[43,79960,348],{},[109,79962,348],{"encoding":111},[15,79964,79966],{"className":79965,"ariaHidden":72},[116],[15,79967,79969,79972],{"className":79968},[120],[15,79970],{"className":79971,"style":4383},[124],[15,79973,348],{"className":79974},[129,133]," 个一维问题。只要每个坐标上的两个混合分布足够接近，",[15,79977,79979,79992],{"className":79978,"translate":19},[23],[15,79980,79982],{"className":79981},[27],[29,79983,79984],{"xmlns":31},[34,79985,79986,79990],{},[37,79987,79988],{},[43,79989,348],{},[109,79991,348],{"encoding":111},[15,79993,79995],{"className":79994,"ariaHidden":72},[116],[15,79996,79998,80001],{"className":79997},[120],[15,79999],{"className":80000,"style":4383},[124],[15,80002,348],{"className":80003},[129,133]," 个坐标就会贡献 ",[15,80006,80008,80021],{"className":80007,"translate":19},[23],[15,80009,80011],{"className":80010},[27],[29,80012,80013],{"xmlns":31},[34,80014,80015,80019],{},[37,80016,80017],{},[43,80018,348],{},[109,80020,348],{"encoding":111},[15,80022,80024],{"className":80023,"ariaHidden":72},[116],[15,80025,80027,80030],{"className":80026},[120],[15,80028],{"className":80029,"style":4383},[124],[15,80031,348],{"className":80032},[129,133]," 倍的下界——这正是高维速率里 ",[15,80035,80037,80050],{"className":80036,"translate":19},[23],[15,80038,80040],{"className":80039},[27],[29,80041,80042],{"xmlns":31},[34,80043,80044,80048],{},[37,80045,80046],{},[43,80047,10376],{},[109,80049,10376],{"encoding":111},[15,80051,80053],{"className":80052,"ariaHidden":72},[116],[15,80054,80056,80059],{"className":80055},[120],[15,80057],{"className":80058,"style":522},[124],[15,80060,10376],{"className":80061},[129,133]," 的来源。",[11,80064,80065,80066,80170,80171,16663,80260,80398],{},"对 ",[15,80067,80069,80099],{"className":80068,"translate":19},[23],[15,80070,80072],{"className":80071},[27],[29,80073,80074],{"xmlns":31},[34,80075,80076,80096],{},[37,80077,80078,80080,80082,80084,80086,80088,80090],{},[43,80079,30875],{"mathvariant":65},[50,80081,62],{},[50,80083,636],{"stretchy":52},[308,80085,541],{},[50,80087,91],{"separator":72},[43,80089,4849],{},[305,80091,80092,80094],{},[50,80093,645],{"stretchy":52},[43,80095,348],{},[109,80097,80098],{"encoding":111},"\\Theta=\\{0,\\varepsilon\\}^m",[15,80100,80102,80120],{"className":80101,"ariaHidden":72},[116],[15,80103,80105,80108,80111,80114,80117],{"className":80104},[120],[15,80106],{"className":80107,"style":610},[124],[15,80109,30875],{"className":80110},[129],[15,80112],{"className":80113,"style":199},[93],[15,80115,62],{"className":80116},[203],[15,80118],{"className":80119,"style":199},[93],[15,80121,80123,80126,80129,80132,80135,80138,80141],{"className":80122},[120],[15,80124],{"className":80125,"style":125},[124],[15,80127,636],{"className":80128},[188],[15,80130,541],{"className":80131},[129],[15,80133,91],{"className":80134},[254],[15,80136],{"className":80137,"style":262},[93],[15,80139,4849],{"className":80140},[129,133],[15,80142,80144,80147],{"className":80143},[195],[15,80145,645],{"className":80146},[195],[15,80148,80150],{"className":80149},[137],[15,80151,80153],{"className":80152},[141],[15,80154,80156],{"className":80155},[146],[15,80157,80159],{"className":80158,"style":6377},[150],[15,80160,80161,80164],{"style":383},[15,80162],{"className":80163,"style":159},[158],[15,80165,80167],{"className":80166},[163,164,165,166],[15,80168,348],{"className":80169},[129,133,166],"，令 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_2^2\n=\\varepsilon^2\\bigl\\lVert\\hat\\theta\u002F\\varepsilon-\\theta'\\bigr\\rVert_2^2,",[15,80476,80478,80527,80596,80714],{"className":80477,"ariaHidden":72},[116],[15,80479,80481,80484,80487,80518,80521,80524],{"className":80480},[120],[15,80482],{"className":80483,"style":50032},[124],[15,80485,33823],{"className":80486},[188],[15,80488,80490],{"className":80489},[129,48500],[15,80491,80493],{"className":80492},[141],[15,80494,80496],{"className":80495},[146],[15,80497,80499,80507],{"className":80498,"style":48496},[150],[15,80500,80501,80504],{"style":2287},[15,80502],{"className":80503,"style":1823},[158],[15,80505,48],{"className":80506,"style":170},[129,133],[15,80508,80509,80512],{"style":48520},[15,80510],{"className":80511,"style":1823},[158],[15,80513,80515],{"className":80514,"style":48528},[48527],[15,80516,48483],{"className":80517},[129],[15,80519],{"className":80520,"style":3277},[93],[15,80522,315],{"className":80523},[2436],[15,80525],{"className":80526,"style":3277},[93],[15,80528,80530,80533,80536,80587,80590,80593],{"className":80529},[120],[15,80531],{"className":80532,"style":40920},[124],[15,80534,48],{"className":80535,"style":170},[129,133],[15,80537,80539,80542],{"className":80538},[195],[15,80540,33823],{"className":80541},[195],[15,80543,80545],{"className":80544},[137],[15,80546,80548,80579],{"className":80547},[141,142],[15,80549,80551,80576],{"className":80550},[146],[15,80552,80554,80565],{"className":80553,"style":40991},[150],[15,80555,80556,80559],{"style":58448},[15,80557],{"className":80558,"style":159},[158],[15,80560,80562],{"className":80561},[163,164,165,166],[15,80563,310],{"className":80564},[129,166],[15,80566,80567,80570],{"style":1272},[15,80568],{"className":80569,"style":159},[158],[15,80571,80573],{"className":80572},[163,164,165,166],[15,80574,310],{"className":80575},[129,166],[15,80577,175],{"className":80578},[174],[15,80580,80582],{"className":80581},[146],[15,80583,80585],{"className":80584,"style":28173},[150],[15,80586],{},[15,80588],{"className":80589,"style":199},[93],[15,80591,62],{"className":80592},[203],[15,80594],{"className":80595,"style":199},[93],[15,80597,80599,80602,80631,80668,80699,80702,80705,80708,80711],{"className":80598},[120],[15,80600],{"className":80601,"style":70939},[124],[15,80603,80605,80608],{"className":80604},[129],[15,80606,4849],{"className":80607},[129,133],[15,80609,80611],{"className":80610},[137],[15,80612,80614],{"className":80613},[141],[15,80615,80617],{"className":80616},[146],[15,80618,80620],{"className":80619,"style":40991},[150],[15,80621,80622,80625],{"style":1272},[15,80623],{"className":80624,"style":159},[158],[15,80626,80628],{"className":80627},[163,164,165,166],[15,80629,310],{"className":80630},[129,166],[15,80632,80634],{"className":80633},[188],[15,80635,80637],{"className":80636},[228,4433],[15,80638,80640,80660],{"className":80639},[141,142],[15,80641,80643,80657],{"className":80642},[146],[15,80644,80646],{"className":80645,"style":4443},[150],[15,80647,80648,80651],{"style":4446},[15,80649],{"className":80650,"style":4450},[158],[15,80652,80653],{"style":53819},[4455,80654,80655],{"xmlns":4457,"width":53822,"height":73,"viewBox":53823},[4461,80656],{"d":53826},[15,80658,175],{"className":80659},[174],[15,80661,80663],{"className":80662},[146],[15,80664,80666],{"className":80665,"style":4473},[150],[15,80667],{},[15,80669,80671],{"className":80670},[129,48500],[15,80672,80674],{"className":80673},[141],[15,80675,80677],{"className":80676},[146],[15,80678,80680,80688],{"className":80679,"style":48496},[150],[15,80681,80682,80685],{"style":2287},[15,80683],{"className":80684,"style":1823},[158],[15,80686,48],{"className":80687,"style":170},[129,133],[15,80689,80690,80693],{"style":48520},[15,80691],{"className":80692,"style":1823},[158],[15,80694,80696],{"className":80695,"style":48528},[48527],[15,80697,48483],{"className":80698},[129],[15,80700,544],{"className":80701},[129],[15,80703,4849],{"className":80704},[129,133],[15,80706],{"className":80707,"style":3277},[93],[15,80709,315],{"className":80710},[2436],[15,80712],{"className":80713,"style":3277},[93],[15,80715,80717,80721,80753,80840],{"className":80716},[120],[15,80718],{"className":80719,"style":80720},[124],"height:1.4537em;vertical-align:-0.3997em;",[15,80722,80724,80727],{"className":80723},[129],[15,80725,48],{"className":80726,"style":170},[129,133],[15,80728,80730],{"className":80729},[137],[15,80731,80733],{"className":80732},[141],[15,80734,80736],{"className":80735},[146],[15,80737,80739],{"className":80738,"style":28137},[150],[15,80740,80741,80744],{"style":1272},[15,80742],{"className":80743,"style":159},[158],[15,80745,80747],{"className":80746},[163,164,165,166],[15,80748,80750],{"className":80749},[129,166],[15,80751,21510],{"className":80752},[129,166],[15,80754,80756,80793],{"className":80755},[195],[15,80757,80759],{"className":80758},[195],[15,80760,80762],{"className":80761},[228,4433],[15,80763,80765,80785],{"c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5.1 直接给出",[15,80847,80849],{"className":80848,"translate":19},[18],[15,80850,80852,81004],{"className":80851,"translate":19},[23],[15,80853,80855],{"className":80854},[27],[29,80856,80857],{"xmlns":31,"display":32},[34,80858,80859,81001],{},[37,80860,80861,80875,80903,80905,80907,80913,80915,80917,80925,80927,80941,80999],{},[2140,80862,80863,80869],{},[37,80864,80865,80867],{},[43,80866,48708],{},[50,80868,69],{},[46847,80870,80871,80873],{"accent":72},[43,80872,48],{},[50,80874,48483],{},[2140,80876,80877,80883],{},[37,80878,80879,80881],{},[43,80880,19705],{},[50,80882,69],{},[37,80884,80885,80887,80889,80891,80893,80895,80897],{},[43,80886,48],{},[50,80888,100],{},[50,80890,636],{"stretchy":52},[308,80892,541],{},[50,80894,91],{"separator":72},[43,80896,4849],{},[305,80898,80899,80901],{},[50,80900,645],{"stretchy":52},[43,80902,348],{},[43,80904,26574],{"mathvariant":103},[50,80906,33823],{"stretchy":52},[46847,80908,80909,80911],{"accent":72},[43,80910,48],{},[50,80912,48483],{},[50,80914,315],{},[43,80916,48],{},[23160,80918,80919,80921,80923],{},[50,80920,33823],{"stretchy":52},[308,80922,310],{},[308,80924,310],{},[50,80926,7473],{},[1443,80928,80929,80939],{},[37,80930,80931,80937],{},[305,80932,80933,80935],{},[43,80934,4849],{},[308,80936,310],{},[43,80938,348],{},[308,80940,14508],{},[37,80942,80943,80945,80947,80949,80959,80961,80973,80975,80987,80997],{},[50,80944,53],{"fence":72},[308,80946,318],{},[50,80948,315],{},[2140,80950,80951,80957],{},[37,80952,80953,80955],{},[43,80954,19705],{},[50,80956,69],{},[43,80958,13955],{},[50,80960,33823],{"stretchy":52},[40,80962,80963,80965],{},[43,80964,34354],{},[37,80966,80967,80969,80971],{},[43,80968,13955],{},[50,80970,91],{"separator":72},[308,80972,541],{},[50,80974,315],{},[40,80976,80977,80979],{},[43,80978,34354],{},[37,80980,80981,80983,80985],{},[43,80982,13955],{},[50,80984,91],{"separator":72},[308,80986,318],{},[40,80988,80989,80991],{},[50,80990,33823],{"stretchy":52},[37,80992,80993,80995],{},[43,80994,53409],{"mathvariant":65},[43,80996,53412],{"mathvariant":65},[50,80998,59],{"fence":72},[43,81000,107],{"mathvariant":65},[109,81002,81003],{"encoding":111},"\\inf_{\\hat\\theta}\\max_{\\theta\\in\\{0,\\varepsilon\\}^m}\\mathbb 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Hamming 距离，同样的逐坐标论证给出常数 ",[15,81641,81643,81660],{"className":81642,"translate":19},[23],[15,81644,81646],{"className":81645},[27],[29,81647,81648],{"xmlns":31},[34,81649,81650,81658],{},[37,81651,81652,81654,81656],{},[308,81653,318],{},[43,81655,544],{"mathvariant":65},[308,81657,310],{},[109,81659,56735],{"encoding":111},[15,81661,81663],{"className":81662,"ariaHidden":72},[116],[15,81664,81666,81669],{"className":81665},[120],[15,81667],{"className":81668,"style":125},[124],[15,81670,56735],{"className":81671},[129],"：把后验均值换成阈值检验即可。",[570,81674,81676],{"id":81675},"六两个完整的例子","六、两个完整的例子",[575,81678,81680,81681],{"id":81679},"_61-高斯均值σ2dnsigma2dnσ2dn","6.1 高斯均值：",[15,81682,81684,81707],{"className":81683,"translate":19},[23],[15,81685,81687],{"className":81686},[27],[29,81688,81689],{"xmlns":31},[34,81690,81691,81705],{},[37,81692,81693,81699,81701,81703],{},[305,81694,81695,81697],{},[43,81696,27294],{},[308,81698,310],{},[43,81700,10376],{},[43,81702,544],{"mathvariant":65},[43,81704,39326],{},[109,81706,64919],{"encoding":111},[15,81708,81710],{"className":81709,"ariaHidden":72},[116],[15,81711,81713,81716,81745,81748,81751],{"className":81712},[120],[15,81714],{"className":81715,"style":34009},[124],[15,81717,81719,81722],{"className":81718},[129],[15,81720,27294],{"className":81721,"style":887},[129,133],[15,81723,81725],{"className":81724},[137],[15,81726,81728],{"className":81727},[141],[15,81729,81731],{"className":81730},[146],[15,81732,81734],{"className":81733,"style":380},[150],[15,81735,81736,81739],{"style":383},[15,81737],{"className":81738,"style":159},[158],[15,81740,81742],{"className":81741},[163,164,165,166],[15,81743,310],{"className":81744},[129,166],[15,81746,10376],{"className":81747},[129,133],[15,81749,544],{"className":81750},[129],[15,81752,39326],{"className":81753},[129,133],[11,81755,81756],{},"模型：",[15,81758,81760],{"className":81759,"translate":19},[18],[15,81761,81763,81833],{"className":81762,"translate":19},[23],[15,81764,81766],{"className":81765},[27],[29,81767,81768],{"xmlns":31,"display":32},[34,81769,81770,81830],{},[37,81771,81772,81778,81780,81782,81784,81790,81792,81794,81796,81798,81800,81806,81812,81814,81816,81818,81820,81822,81828],{},[40,81773,81774,81776],{},[43,81775,597],{},[308,81777,318],{},[50,81779,91],{"separator":72},[50,81781,335],{},[50,81783,91],{"separator":72},[40,81785,81786,81788],{},[43,81787,597],{},[43,81789,39326],{},[50,81791,991],{},[43,81793,12561],{},[50,81795,53],{"stretchy":52},[43,81797,48],{},[50,81799,91],{"separator":72},[305,81801,81802,81804],{},[43,81803,27294],{},[308,81805,310],{},[40,81807,81808,81810],{},[43,81809,64036],{},[43,81811,10376],{},[50,81813,59],{"stretchy":52},[50,81815,91],{"separator":72},[93,81817],{"width":95},[43,81819,48],{},[50,81821,100],{},[305,81823,81824,81826],{},[43,81825,104],{"mathvariant":103},[43,81827,10376],{},[50,81829,91],{"separator":72},[109,81831,81832],{"encoding":111},"X_1,\\dots,X_n\\sim N(\\theta,\\sigma^2 I_d),\\qquad \\theta\\in\\mathbb R^d,",[15,81834,81836,81949,82063],{"className":81835,"ariaHidden":72},[116],[15,81837,81839,81842,81882,81885,81888,81891,81894,81897,81900,81940,81943,81946],{"className":81838},[120],[15,81840],{"className":81841,"style":34366},[124],[15,81843,81845,81848],{"className":81844},[129],[15,81846,597],{"className":81847,"style":47970},[129,133],[15,81849,81851],{"className":81850},[137],[15,81852,81854,81874],{"className":81853},[141,142],[15,81855,81857,81871],{"className":81856},[146],[15,81858,81860],{"className":81859,"style":1051},[150],[15,81861,81862,81865],{"style":47985},[15,81863],{"className":81864,"style":159},[158],[15,81866,81868],{"className":81867},[163,164,165,166],[15,81869,318],{"className":81870},[129,166],[15,81872,175],{"className":81873},[174],[15,81875,81877],{"className":81876},[146],[15,81878,81880],{"className":81879,"style":182},[150],[15,81881],{},[15,81883,91],{"className":81884},[254],[15,81886],{"className":81887,"style":262},[93],[15,81889,335],{"className":81890},[449],[15,81892],{"className":81893,"style":262},[93],[15,81895,91],{"className":81896},[254],[15,81898],{"className":81899,"style":262},[93],[15,81901,81903,81906],{"className":81902},[129],[15,81904,597],{"className":81905,"style":47970},[129,133],[15,81907,81909],{"className":81908},[137],[15,81910,81912,81932],{"className":81911},[141,142],[15,81913,81915,81929],{"className":81914},[146],[15,81916,81918],{"className":81917,"style":1163},[150],[15,81919,81920,81923],{"style":47985},[15,81921],{"className":81922,"style":159},[158],[15,81924,81926],{"className":81925},[163,164,165,166],[15,81927,39326],{"className":81928},[129,133,166],[15,81930,175],{"className":81931},[174],[15,81933,81935],{"className":81934},[146],[15,81936,81938],{"className":81937,"style":182},[150],[15,81939],{},[15,81941],{"className":81942,"style":199},[93],[15,81944,991],{"className":81945},[203],[15,81947],{"className":81948,"style":199},[93],[15,81950,81952,81955,81958,81961,81964,81967,81970,81999,82039,82042,82045,82048,82051,82054,82057,82060],{"className":81951},[120],[15,81953],{"className":81954,"style":40920},[124],[15,81956,12561],{"className":81957,"style":12597},[129,133],[15,81959,53],{"className":81960},[188],[15,81962,48],{"className":81963,"style":170},[129,133],[15,81965,91],{"className":81966},[254],[15,81968],{"className":81969,"style":262},[93],[15,81971,81973,81976],{"className":81972},[129],[15,81974,27294],{"className":81975,"style":887},[129,133],[15,81977,81979],{"className":81978},[137],[15,81980,81982],{"className":81981},[141],[15,81983,81985],{"className":81984},[146],[15,81986,81988],{"className":81987,"style":40991},[150],[15,81989,81990,81993],{"style":1272},[15,81991],{"className":81992,"style":159},[158],[15,81994,81996],{"className":81995},[163,164,165,166],[15,81997,310],{"className":81998},[129,166],[15,82000,82002,82005],{"className":82001},[129],[15,82003,64036],{"className":82004,"style":47970},[129,133],[15,82006,82008],{"className":82007},[137],[15,82009,82011,82031],{"className":82010},[141,142],[15,82012,82014,82028],{"className":82013},[146],[15,82015,82017],{"className":82016,"style":151},[150],[15,82018,82019,82022],{"style":47985},[15,82020],{"className":82021,"style":159},[158],[15,82023,82025],{"className":82024},[163,164,165,166],[15,82026,10376],{"className":82027},[129,133,166],[15,82029,175],{"className":82030},[174],[15,82032,82034],{"className":82033},[146],[15,82035,82037],{"className":82036,"style":182},[150],[15,82038],{},[15,82040,59],{"className":82041},[195],[15,82043,91],{"className":82044},[254],[15,82046],{"className":82047,"style":258},[93],[15,82049],{"className":82050,"style":262},[93],[15,82052,48],{"className":82053,"style":170},[129,133],[15,82055],{"className":82056,"style":199},[93],[15,82058,100],{"className":82059},[203],[15,82061],{"className":82062,"style":199},[93],[15,82064,82066,82070,82099],{"className":82065},[120],[15,82067],{"className":82068,"style":82069},[124],"height:1.0935em;vertical-align:-0.1944em;",[15,82071,82073,82076],{"className":82072},[129],[15,82074,104],{"className":82075},[129,285],[15,82077,82079],{"className":82078},[137],[15,82080,82082],{"className":82081},[141],[15,82083,82085],{"className":82084},[146],[15,82086,82088],{"className":82087,"style":64296},[150],[15,82089,82090,82093],{"style":1272},[15,82091],{"className":82092,"style":159},[158],[15,82094,82096],{"className":82095},[163,164,165,166],[15,82097,10376],{"className":82098},[129,133,166],[15,82100,91],{"className":82101},[254],[11,82103,82104,82105,1313],{},"损失为 ",[15,82106,82108,82140],{"className":82107,"translate":19},[23],[15,82109,82111],{"className":82110},[27],[29,82112,82113],{"xmlns":31},[34,82114,82115,82137],{},[37,82116,82117,82119,82125,82127,82129],{},[50,82118,33823],{"stretchy":52},[46847,82120,82121,82123],{"accent":72},[43,82122,48],{},[50,82124,48483],{},[50,82126,315],{},[43,82128,48],{},[23160,82130,82131,82133,82135],{},[50,82132,33823],{"stretchy":52},[308,82134,310],{},[308,82136,310],{},[109,82138,82139],{"encoding":111},"\\lVert\\hat\\theta-\\theta\\rVert_2^2",[15,82141,82143,82192],{"className":82142,"ariaHidden":72},[116],[15,82144,82146,82149,82152,82183,82186,82189],{"className":82145},[120],[15,82147],{"className":82148,"style":50032},[124],[15,82150,33823],{"className":82151},[188],[15,82153,82155],{"className":82154},[129,48500],[15,82156,82158],{"className":82157},[141],[15,82159,82161],{"className":82160},[146],[15,82162,82164,82172],{"className":82163,"style":48496},[150],[15,82165,82166,82169],{"style":2287},[15,82167],{"className":82168,"style":1823},[158],[15,82170,48],{"className":82171,"style":170},[129,133],[15,82173,82174,82177],{"style":48520},[15,82175],{"className":82176,"style":1823},[158],[15,82178,82180],{"className":82179,"style":48528},[48527],[15,82181,48483],{"className":82182},[129],[15,82184],{"className":82185,"style":3277},[93],[15,82187,315],{"className":82188},[2436],[15,82190],{"className":82191,"style":3277},[93],[15,82193,82195,82198,82201],{"className":82194},[120],[15,82196],{"className":82197,"style":34009},[124],[15,82199,48],{"className":82200,"style":170},[129,133],[15,82202,82204,82207],{"className":82203},[195],[15,82205,33823],{"className":82206},[195],[15,82208,82210],{"className":82209},[137],[15,82211,82213,82244],{"className":82212},[141,142],[15,82214,82216,82241],{"className":82215},[146],[15,82217,82219,82230],{"className":82218,"style":380},[150],[15,82220,82221,82224],{"style":50169},[15,82222],{"className":82223,"style":159},[158],[15,82225,82227],{"className":82226},[163,164,165,166],[15,82228,310],{"className":82229},[129,166],[15,82231,82232,82235],{"style":383},[15,82233],{"className":82234,"style":159},[158],[15,82236,82238],{"className":82237},[163,164,165,166],[15,82239,310],{"className":82240},[129,166],[15,82242,175],{"className":82243},[174],[15,82245,82247],{"className":82246},[146],[15,82248,82250],{"className":82249,"style":38948},[150],[15,82251],{},[11,82253,82254,82257,82258,82319],{},[698,82255,82256],{},"上界。"," 样本均值 ",[15,82259,82261,82279],{"className":82260,"translate":19},[23],[15,82262,82264],{"className":82263},[27],[29,82265,82266],{"xmlns":31},[34,82267,82268,82276],{},[37,82269,82270],{},[46847,82271,82272,82274],{"accent":72},[43,82273,597],{},[50,82275,72884],{},[109,82277,82278],{"encoding":111},"\\bar X",[15,82280,82282],{"className":82281,"ariaHidden":72},[116],[15,82283,82285,82288],{"className":82284},[120],[15,82286],{"className":82287,"style":73214},[124],[15,82289,82291],{"className":82290},[129,48500],[15,82292,82294],{"className":82293},[141],[15,82295,82297],{"className":82296},[146],[15,82298,82300,82308],{"className":82299,"style":73214},[150],[15,82301,82302,82305],{"style":2287},[15,82303],{"className":82304,"style":1823},[158],[15,82306,597],{"className":82307,"style":47970},[129,133],[15,82309,82310,82313],{"style":65322},[15,82311],{"className":82312,"style":1823},[158],[15,82314,82316],{"className":82315,"style":48528},[48527],[15,82317,72884],{"className":82318},[129]," 的风险是",[15,82321,82323],{"className":82322,"translate":19},[18],[15,82324,82326,82378],{"className":82325,"translate":19},[23],[15,82327,82329],{"className":82328},[27],[29,82330,82331],{"xmlns":31,"display":32},[34,82332,82333,82375],{},[37,82334,82335,82337,82339,82345,82347,82349,82357,82359,82373],{},[43,82336,26574],{"mathvariant":103},[50,82338,33823],{"stretchy":52},[46847,82340,82341,82343],{"accent":72},[43,82342,597],{},[50,82344,72884],{},[50,82346,315],{},[43,82348,48],{},[23160,82350,82351,82353,82355],{},[50,82352,33823],{"stretchy":52},[308,82354,310],{},[308,82356,310],{},[50,82358,62],{},[1443,82360,82361,82371],{},[37,82362,82363,82369],{},[305,82364,82365,82367],{},[43,82366,27294],{},[308,82368,310],{},[43,82370,10376],{},[43,82372,39326],{},[43,82374,107],{"mathvariant":65},[109,82376,82377],{"encoding":111},"\\mathbb E\\lVert\\bar X-\\theta\\rVert_2^2=\\frac{\\sigma^2d}{n}.",[15,82379,82381,82434,82503],{"className":82380,"ariaHidden":72},[116],[15,82382,82384,82388,82391,82394,82425,82428,82431],{"className":82383},[120],[15,82385],{"className":82386,"style":82387},[124],"height:1.0701em;vertical-align:-0.25em;",[15,82389,26574],{"className":82390},[129,285],[15,82392,33823],{"className":82393},[188],[15,82395,82397],{"className":82396},[129,48500],[15,82398,82400],{"className":82399},[141],[15,82401,82403],{"className":82402},[146],[15,82404,82406,82414],{"className":82405,"style":73214},[150],[15,82407,82408,82411],{"style":2287},[15,82409],{"className":82410,"style":1823},[158],[15,82412,597],{"className":82413,"style":47970},[129,133],[15,82415,82416,82419],{"style":65322},[15,82417],{"className":82418,"style":1823},[158],[15,82420,82422],{"className":82421,"style":48528},[48527],[15,82423,72884],{"className":82424},[129],[15,82426],{"className":82427,"style":3277},[93],[15,82429,315],{"className":82430},[2436],[15,82432],{"className":82433,"style":3277},[93],[15,82435,82437,82440,82443,82494,82497,82500],{"className":82436},[120],[15,82438],{"className":82439,"style":40920},[124],[15,82441,48],{"className":82442,"style":170},[129,133],[15,82444,82446,82449],{"className":82445},[195],[15,82447,33823],{"className":82448},[195],[15,82450,82452],{"className":82451},[137],[15,82453,82455,82486],{"className":82454},[141,142],[15,82456,82458,82483],{"className":82457},[146],[15,82459,82461,82472],{"className":82460,"style":40991},[150],[15,82462,82463,82466],{"style":58448},[15,82464],{"className":82465,"style":159},[158],[15,82467,82469],{"className":82468},[163,164,165,166],[15,82470,310],{"className":82471},[129,166],[15,82473,82474,82477],{"style":1272},[15,82475],{"className":82476,"style":159},[158],[15,82478,82480],{"className":82479},[163,164,165,166],[15,82481,310],{"className":82482},[129,166],[15,82484,175],{"className":82485},[174],[15,82487,82489],{"className":82488},[146],[15,82490,82492],{"className":82491,"style":28173},[150],[15,82493],{},[15,82495],{"className":82496,"style":199},[93],[15,82498,62],{"className":82499},[203],[15,82501],{"className":82502,"style":199},[93],[15,82504,82506,82510,82601],{"className":82505},[120],[15,82507],{"className":82508,"style":82509},[124],"height:2.1771em;vertical-align:-0.686em;",[15,82511,82513,82516,82598],{"className":82512},[129],[15,82514],{"className":82515},[188,1803],[15,82517,82519],{"className":82518},[1443],[15,82520,82522,82590],{"className":82521},[141,142],[15,82523,82525,82587],{"className":82524},[146],[15,82526,82528,82539,82547],{"className":82527,"style":63623},[150],[15,82529,82530,82533],{"style":1819},[15,82531],{"className":82532,"style":1823},[158],[15,82534,82536],{"className":82535},[129],[15,82537,39326],{"className":82538},[129,133],[15,82540,82541,82544],{"style":1832},[15,82542],{"className":82543,"style":1823},[158],[15,82545],{"className":82546,"style":1840},[1839],[15,82548,82549,82552],{"style":1843},[15,82550],{"className":82551,"style":1823},[158],[15,82553,82555,82584],{"className":82554},[129],[15,82556,82558,82561],{"className":82557},[129],[15,82559,27294],{"className":82560,"style":887},[129,133],[15,82562,82564],{"className":82563},[137],[15,82565,82567],{"className":82566},[141],[15,82568,82570],{"className":82569},[146],[15,82571,82573],{"className":82572,"style":380},[150],[15,82574,82575,82578],{"style":383},[15,82576],{"className":82577,"style":159},[158],[15,82579,82581],{"className":82580},[163,164,165,166],[15,82582,310],{"className":82583},[129,166],[15,82585,10376],{"className":82586},[129,133],[15,82588,175],{"className":82589},[174],[15,82591,82593],{"className":82592},[146],[15,82594,82596],{"className":82595,"style":1862},[150],[15,82597],{},[15,82599],{"className":82600},[195,1803],[15,82602,107],{"className":82603},[129],[11,82605,82606,82609,82610,82755,82756,82784,82785,10179,82869,82953,82954,82982],{},[698,82607,82608],{},"下界。"," 把参数限制到超立方体 ",[15,82611,82613,82647],{"className":82612,"translate":19},[23],[15,82614,82616],{"className":82615},[27],[29,82617,82618],{"xmlns":31},[34,82619,82620,82644],{},[37,82621,82622,82628,82630,82632,82634,82636,82638],{},[40,82623,82624,82626],{},[43,82625,30875],{"mathvariant":65},[308,82627,541],{},[50,82629,62],{},[50,82631,636],{"stretchy":52},[308,82633,541],{},[50,82635,91],{"separator":72},[43,82637,4849],{},[305,82639,82640,82642],{},[50,82641,645],{"stretchy":52},[43,82643,10376],{},[109,82645,82646],{"encoding":111},"\\Theta_0=\\{0,\\varepsilon\\}^d",[15,82648,82650,82705],{"className":82649,"ariaHidden":72},[116],[15,82651,82653,82656,82696,82699,82702],{"className":82652},[120],[15,82654],{"className":82655,"style":20229},[124],[15,82657,82659,82662],{"className":82658},[129],[15,82660,30875],{"className":82661},[129],[15,82663,82665],{"className":82664},[137],[15,82666,82668,82688],{"className":82667},[141,142],[15,82669,82671,82685],{"className":82670},[146],[15,82672,82674],{"className":82673,"style":1051},[150],[15,82675,82676,82679],{"style":154},[15,82677],{"className":82678,"style":159},[158],[15,82680,82682],{"className":82681},[163,164,165,166],[15,82683,541],{"className":82684},[129,166],[15,82686,175],{"className":82687},[174],[15,82689,82691],{"className":82690},[146],[15,82692,82694],{"className":82693,"style":182},[150],[15,82695],{},[15,82697],{"className":82698,"style":199},[93],[15,82700,62],{"className":82701},[203],[15,82703],{"className":82704,"style":199},[93],[15,82706,82708,82711,82714,82717,82720,82723,82726],{"className":82707},[120],[15,82709],{"className":82710,"style":10978},[124],[15,82712,636],{"className":82713},[188],[15,82715,541],{"className":82716},[129],[15,82718,91],{"className":82719},[254],[15,82721],{"className":82722,"style":262},[93],[15,82724,4849],{"className":82725},[129,133],[15,82727,82729,82732],{"className":82728},[195],[15,82730,645],{"className":82731},[195],[15,82733,82735],{"className":82734},[137],[15,82736,82738],{"className":82737},[141],[15,82739,82741],{"className":82740},[146],[15,82742,82744],{"className":82743,"style":10490},[150],[15,82745,82746,82749],{"style":383},[15,82747],{"className":82748,"style":159},[158],[15,82750,82752],{"className":82751},[163,164,165,166],[15,82753,10376],{"className":82754},[129,133,166],"。对坐标 ",[15,82757,82759,82772],{"className":82758,"translate":19},[23],[15,82760,82762],{"className":82761},[27],[29,82763,82764],{"xmlns":31},[34,82765,82766,82770],{},[37,82767,82768],{},[43,82769,13955],{},[109,82771,13955],{"encoding":111},[15,82773,82775],{"className":82774,"ariaHidden":72},[116],[15,82776,82778,82781],{"className":82777},[120],[15,82779],{"className":82780,"style":11191},[124],[15,82782,13955],{"className":82783,"style":13970},[129,133],"，混合分布 ",[15,82786,82788,82811],{"className":82787,"translate":19},[23],[15,82789,82791],{"className":82790},[27],[29,82792,82793],{"xmlns":31},[34,82794,82795,82809],{},[37,82796,82797],{},[40,82798,82799,82801],{},[43,82800,34354],{},[37,82802,82803,82805,82807],{},[43,82804,13955],{},[50,82806,91],{"separator":72},[308,82808,541],{},[109,82810,78316],{"encoding":111},[15,82812,82814],{"className":82813,"ariaHidden":72},[116],[15,82815,82817,82820],{"className":82816},[120],[15,82818],{"className":82819,"style":76064},[124],[15,82821,82823,82826],{"className":82822},[129],[15,82824,34354],{"className":82825},[129,133],[15,82827,82829],{"className":82828},[137],[15,82830,82832,82861],{"className":82831},[141,142],[15,82833,82835,82858],{"className":82834},[146],[15,82836,82838],{"className":82837,"style":1981},[150],[15,82839,82840,82843],{"style":154},[15,82841],{"className":82842,"style":159},[158],[15,82844,82846],{"className":82845},[163,164,165,166],[15,82847,82849,82852,82855],{"className":82848},[129,166],[15,82850,13955],{"className":82851,"style":13970},[129,133,166],[15,82853,91],{"className":82854},[254,166],[15,82856,541],{"className":82857},[129,166],[15,82859,175],{"className":82860},[174],[15,82862,82864],{"className":82863},[146],[15,82865,82867],{"className":82866,"style":14041},[150],[15,82868],{},[15,82870,82872,82895],{"className":82871,"translate":19},[23],[15,82873,82875],{"className":82874},[27],[29,82876,82877],{"xmlns":31},[34,82878,82879,82893],{},[37,82880,82881],{},[40,82882,82883,82885],{},[43,82884,34354],{},[37,82886,82887,82889,82891],{},[43,82888,13955],{},[50,82890,91],{"separator":72},[308,82892,318],{},[109,82894,78401],{"encoding":111},[15,82896,82898],{"className":82897,"ariaHidden":72},[116],[15,82899,82901,82904],{"className":82900},[120],[15,82902],{"className":82903,"style":76064},[124],[15,82905,82907,82910],{"className":82906},[129],[15,82908,34354],{"className":82909},[129,133],[15,82911,82913],{"className":82912},[137],[15,82914,82916,82945],{"className":82915},[141,142],[15,82917,82919,82942],{"className":82918},[146],[15,82920,82922],{"className":82921,"style":1981},[150],[15,82923,82924,82927],{"style":154},[15,82925],{"className":82926,"style":159},[158],[15,82928,82930],{"className":82929},[163,164,165,166],[15,82931,82933,82936,82939],{"className":82932},[129,166],[15,82934,13955],{"className":82935,"style":13970},[129,133,166],[15,82937,91],{"className":82938},[254,166],[15,82940,318],{"className":82941},[129,166],[15,82943,175],{"className":82944},[174],[15,82946,82948],{"className":82947},[146],[15,82949,82951],{"className":82950,"style":14041},[150],[15,82952],{}," 只在这一维的均值上相差 ",[15,82955,82957,82970],{"className":82956,"translate":19},[23],[15,82958,82960],{"className":82959},[27],[29,82961,82962],{"xmlns":31},[34,82963,82964,82968],{},[37,82965,82966],{},[43,82967,4849],{},[109,82969,34320],{"encoding":111},[15,82971,82973],{"className":82972,"ariaHidden":72},[116],[15,82974,82976,82979],{"className":82975},[120],[15,82977],{"className":82978,"style":4383},[124],[15,82980,4849],{"className":82981},[129,133],"。由 KL 的联合凸性，",[15,82984,82986],{"className":82985,"translate":19},[18],[15,82987,82989,83067],{"className":82988,"translate":19},[23],[15,82990,82992],{"className":82991},[27],[29,82993,82994],{"xmlns":31,"display":32},[34,82995,82996,83064],{},[37,82997,82998,83008,83010,83022,83024,83036,83038,83040,83062],{},[40,82999,83000,83002],{},[43,83001,906],{},[37,83003,83004,83006],{},[43,83005,34524],{"mathvariant":65},[43,83007,1373],{"mathvariant":65},[50,83009,53],{"stretchy":52},[40,83011,83012,83014],{},[43,83013,34354],{},[37,83015,83016,83018,83020],{},[43,83017,13955],{},[50,83019,91],{"separator":72},[308,83021,541],{},[43,83023,33823],{"mathvariant":65},[40,83025,83026,83028],{},[43,83027,34354],{},[37,83029,83030,83032,83034],{},[43,83031,13955],{},[50,83033,91],{"separator":72},[308,83035,318],{},[50,83037,59],{"stretchy":52},[50,83039,2717],{},[1443,83041,83042,83052],{},[37,83043,83044,83046],{},[43,83045,39326],{},[305,83047,83048,83050],{},[43,83049,4849],{},[308,83051,310],{},[37,83053,83054,83056],{},[308,83055,310],{},[305,83057,83058,83060],{},[43,83059,27294],{},[308,83061,310],{},[43,83063,107],{"mathvariant":65},[109,83065,83066],{"encoding":111},"D_{\\mathrm{KL}}(Q_{j,0}\\Vert Q_{j,1})\n\\le \\frac{n\\varepsilon^2}{2\\sigma^2}.",[15,83068,83070,83238],{"className":83069,"ariaHidden":72},[116],[15,83071,83073,83076,83122,83125,83174,83177,83226,83229,83232,83235],{"className":83072},[120],[15,83074],{"className":83075,"style":67458},[124],[15,83077,83079,83082],{"className":83078},[129],[15,83080,906],{"className":83081,"style":170},[129,133],[15,83083,83085],{"className":83084},[137],[15,83086,83088,83114],{"className":83087},[141,142],[15,83089,83091,83111],{"className":83090},[146],[15,83092,83094],{"className":83093,"style":1525},[150],[15,83095,83096,83099],{"style":48293},[15,83097],{"className":83098,"style":159},[158],[15,83100,83102],{"className":83101},[163,164,165,166],[15,83103,83105],{"className":83104},[129,166],[15,83106,83108],{"className":83107},[129,166],[15,83109,34928],{"className":83110},[129,221,166],[15,83112,175],{"className":83113},[174],[15,83115,83117],{"className":83116},[146],[15,83118,83120],{"className":83119,"style":182},[150],[15,83121],{},[15,83123,53],{"className":83124},[188],[15,83126,83128,83131],{"className":83127},[129],[15,83129,34354],{"className":83130},[129,133],[15,83132,83134],{"className":83133},[137],[15,83135,83137,83166],{"className":83136},[141,142],[15,83138,83140,83163],{"className":83139},[146],[15,83141,83143],{"className":83142,"style":1981},[150],[15,83144,83145,83148],{"style":154},[15,83146],{"className":83147,"style":159},[158],[15,83149,83151],{"className":83150},[163,164,165,166],[15,83152,83154,83157,83160],{"className":83153},[129,166],[15,83155,13955],{"className":83156,"style":13970},[129,133,166],[15,83158,91],{"className":83159},[254,166],[15,83161,541],{"className":83162},[129,166],[15,83164,175],{"className":83165},[174],[15,83167,83169],{"className":83168},[146],[15,83170,83172],{"className":83171,"style":14041},[150],[15,83173],{},[15,83175,33823],{"className":83176},[129],[15,83178,83180,83183],{"className":83179},[129],[15,83181,34354],{"className":83182},[129,133],[15,83184,83186],{"className":83185},[137],[15,83187,83189,83218],{"className":83188},[141,142],[15,83190,83192,83215],{"className":83191},[146],[15,83193,83195],{"className":83194,"style":1981},[150],[15,83196,83197,83200],{"style":154},[15,83198],{"className":83199,"style":159},[158],[15,83201,83203],{"className":83202},[163,164,165,166],[15,83204,83206,83209,83212],{"className":83205},[129,166],[15,83207,13955],{"className":83208,"style":13970},[129,133,166],[15,83210,91],{"className":83211},[254,166],[15,83213,318],{"className":83214},[129,166],[15,83216,175],{"className":83217},[174],[15,83219,83221],{"className":83220},[146],[15,83222,83224],{"className":83223,"style":14041},[150],[15,83225],{},[15,83227,59],{"className":83228},[195],[15,83230],{"className":83231,"style":199},[93],[15,83233,2717],{"className":83234},[203],[15,83236],{"className":83237,"style":199},[93],[15,83239,83241,83244,83364],{"className":83240},[120],[15,83242],{"className":83243,"style":82509},[124],[15,83245,83247,83250,83361],{"className":83246},[129],[15,83248],{"className":83249},[188,1803],[15,83251,83253],{"className":83252},[1443],[15,83254,83256,83353],{"className":83255},[141,142],[15,83257,83259,83350],{"className":83258},[146],[15,83260,83262,83302,83310],{"className":83261,"style":63623},[150],[15,83263,83264,83267],{"style":1819},[15,83265],{"className":83266,"style":1823},[158],[15,83268,83270,83273],{"className":83269},[129],[15,83271,310],{"className":83272},[129],[15,83274,83276,83279],{"className":83275},[129],[15,83277,27294],{"className":83278,"style":887},[129,133],[15,83280,83282],{"className":83281},[137],[15,83283,83285],{"className":83284},[141],[15,83286,83288],{"className":83287},[146],[15,83289,83291],{"className":83290,"style":31001},[150],[15,83292,83293,83296],{"style":17201},[15,83294],{"className":83295,"style":159},[158],[15,83297,83299],{"className":83298},[163,164,165,166],[15,83300,310],{"className":83301},[129,166],[15,83303,83304,83307],{"style":1832},[15,83305],{"className":83306,"style":1823},[158],[15,83308],{"className":83309,"style":1840},[1839],[15,83311,83312,83315],{"style":1843},[15,83313],{"className":83314,"style":1823},[158],[15,83316,83318,83321],{"className":83317},[129],[15,83319,39326],{"className":83320},[129,133],[15,83322,83324,83327],{"className":83323},[129],[15,83325,4849],{"className":83326},[129,133],[15,83328,83330],{"className":83329},[137],[15,83331,83333],{"className":83332},[141],[15,83334,83336],{"className":83335},[146],[15,83337,83339],{"className":83338,"style":380},[150],[15,83340,83341,83344],{"style":383},[15,83342],{"className":83343,"style":159},[158],[15,83345,83347],{"className":83346},[163,164,165,166],[15,83348,310],{"className":83349},[129,166],[15,83351,175],{"className":83352},[174],[15,83354,83356],{"className":83355},[146],[15,83357,83359],{"className":83358,"style":1862},[150],[15,83360],{},[15,83362],{"className":83363},[195,1803],[15,83365,107],{"className":83366},[129],[11,83368,83369],{},"Pinsker 给出",[15,83371,83373],{"className":83372,"translate":19},[18],[15,83374,83376,83446],{"className":83375,"translate":19},[23],[15,83377,83379],{"className":83378},[27],[29,83380,83381],{"xmlns":31,"display":32},[34,83382,83383,83443],{},[37,83384,83385,83387,83399,83401,83413,83423,83425,83441],{},[50,83386,33823],{"stretchy":52},[40,83388,83389,83391],{},[43,83390,34354],{},[37,83392,83393,83395,83397],{},[43,83394,13955],{},[50,83396,91],{"separator":72},[308,83398,541],{},[50,83400,315],{},[40,83402,83403,83405],{},[43,83404,34354],{},[37,83406,83407,83409,83411],{},[43,83408,13955],{},[50,83410,91],{"separator":72},[308,83412,318],{},[40,83414,83415,83417],{},[50,83416,33823],{"stretchy":52},[37,83418,83419,83421],{},[43,83420,53409],{"mathvariant":65},[43,83422,53412],{"mathvariant":65},[50,83424,2717],{},[1443,83426,83427,83435],{},[37,83428,83429,83431],{},[43,83430,4849],{},[29015,83432,83433],{},[43,83434,39326],{},[37,83436,83437,83439],{},[308,83438,310],{},[43,83440,27294],{},[43,83442,107],{"mathvariant":65},[109,83444,83445],{"encoding":111},"\\lVert Q_{j,0}-Q_{j,1}\\rVert_{\\mathrm{TV}}\n\\le \\frac{\\varepsilon\\sqrt n}{2\\sigma}.",[15,83447,83449,83516,83626],{"className":83448,"ariaHidden":72},[116],[15,83450,83452,83455,83458,83507,83510,83513],{"className":83451},[120],[15,83453],{"className":83454,"style":67458},[124],[15,83456,33823],{"className":83457},[188],[15,83459,83461,83464],{"className":83460},[129],[15,83462,34354],{"className":83463},[129,133],[15,83465,83467],{"className":83466},[137],[15,83468,83470,83499],{"className":83469},[141,142],[15,83471,83473,83496],{"className":83472},[146],[15,83474,83476],{"className":83475,"style":1981},[150],[15,83477,83478,83481],{"style":154},[15,83479],{"className":83480,"style":159},[158],[15,83482,83484],{"className":83483},[163,164,165,166],[15,83485,83487,83490,83493],{"className":83486},[129,166],[15,83488,13955],{"className":83489,"style":13970},[129,133,166],[15,83491,91],{"className":83492},[254,166],[15,83494,541],{"className":83495},[129,166],[15,83497,175],{"className":83498},[174],[15,83500,83502],{"className":83501},[146],[15,83503,83505],{"className":83504,"style":14041},[150],[15,83506],{},[15,83508],{"className":83509,"style":3277},[93],[15,83511,315],{"className":83512},[2436],[15,83514],{"className":83515,"style":3277},[93],[15,83517,83519,83522,83571,83617,83620,83623],{"className":83518},[120],[15,83520],{"className":83521,"style":67458},[124],[15,83523,83525,83528],{"className":83524},[129],[15,83526,34354],{"className":83527},[129,133],[15,83529,83531],{"className":83530},[137],[15,83532,83534,83563],{"className":83533},[141,142],[15,83535,83537,83560],{"className":83536},[146],[15,83538,83540],{"className":83539,"style":1981},[150],[15,83541,83542,83545],{"style":154},[15,83543],{"className":83544,"style":159},[158],[15,83546,83548],{"className":83547},[163,164,165,166],[15,83549,83551,83554,83557],{"className":83550},[129,166],[15,83552,13955],{"className":83553,"style":13970},[129,133,166],[15,83555,91],{"className":83556},[254,166],[15,83558,318],{"className":83559},[129,166],[15,83561,175],{"className":83562},[174],[15,83564,83566],{"className":83565},[146],[15,83567,83569],{"className":83568,"style":14041},[150],[15,83570],{},[15,83572,83574,83577],{"className":83573},[195],[15,83575,33823],{"className":83576},[195],[15,83578,83580],{"className":83579},[137],[15,83581,83583,83609],{"className":83582},[141,142],[15,83584,83586,83606],{"className":83585},[146],[15,83587,83589],{"className":83588,"style":1525},[150],[15,83590,83591,83594],{"style":154},[15,83592],{"className":83593,"style":159},[158],[15,83595,83597],{"className":83596},[163,164,165,166],[15,83598,83600],{"className":83599},[129,166],[15,83601,83603],{"className":83602},[129,166],[15,83604,54040],{"className":83605,"style":2246},[129,221,166],[15,83607,175],{"className":83608},[174],[15,83610,83612],{"className":83611},[146],[15,83613,83615],{"className":83614,"style":182},[150],[15,83616],{},[15,83618],{"className":83619,"style":199},[93],[15,83621,2717],{"className":83622},[203],[15,83624],{"className":83625,"style":199},[93],[15,83627,83629,83633,83743],{"className":83628},[120],[15,83630],{"className":83631,"style":83632},[124],"height:2.1633em;vertical-align:-0.686em;",[15,83634,83636,83639,83740],{"className":83635},[129],[15,83637],{"className":83638},[188,1803],[15,83640,83642],{"className":83641},[1443],[15,83643,83645,83732],{"className":83644},[141,142],[15,83646,83648,83729],{"className":83647},[146],[15,83649,83652,83666,83674],{"className":83650,"style":83651},[150],"height:1.4773em;",[15,83653,83654,83657],{"style":1819},[15,83655],{"className":83656,"style":1823},[158],[15,83658,83660,83663],{"className":83659},[129],[15,83661,310],{"className":83662},[129],[15,83664,27294],{"className":83665,"style":887},[129,133],[15,83667,83668,83671],{"style":1832},[15,83669],{"className":83670,"style":1823},[158],[15,83672],{"className":83673,"style":1840},[1839],[15,83675,83676,83679],{"style":1843},[15,83677],{"className":83678,"style":1823},[158],[15,83680,83682,83685],{"className":83681},[129],[15,83683,4849],{"className":83684},[129,133],[15,83686,83688],{"className":83687},[129,29123],[15,83689,83691,83721],{"className":83690},[141,142],[15,83692,83694,83718],{"className":83693},[146],[15,83695,83697,83706],{"className":83696,"style":30463},[150],[15,83698,83700,83703],{"className":83699,"style":2287},[29137],[15,83701],{"className":83702,"style":1823},[158],[15,83704,39326],{"className":83705,"style":30473},[129,133],[15,83707,83708,83711],{"style":30476},[15,83709],{"className":83710,"style":1823},[158],[15,83712,83714],{"className":83713,"style":30483},[29282],[4455,83715,83716],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},[4461,83717],{"d":30490},[15,83719,175],{"className":83720},[174],[15,83722,83724],{"className":83723},[146],[15,83725,83727],{"className":83726,"style":30500},[150],[15,83728],{},[15,83730,175],{"className":83731},[174],[15,83733,83735],{"className":83734},[146],[15,83736,83738],{"className":83737,"style":1862},[150],[15,83739],{},[15,83741],{"className":83742},[195,1803],[15,83744,107],{"className":83745},[129],[11,83747,7441,83748,83852,83853,83885],{},[15,83749,83751,83775],{"className":83750,"translate":19},[23],[15,83752,83754],{"className":83753},[27],[29,83755,83756],{"xmlns":31},[34,83757,83758,83772],{},[37,83759,83760,83762,83764,83766,83768],{},[43,83761,4849],{},[50,83763,62],{},[43,83765,27294],{},[43,83767,544],{"mathvariant":65},[29015,83769,83770],{},[43,83771,39326],{},[109,83773,83774],{"encoding":111},"\\varepsilon=\\sigma\u002F\\sqrt n",[15,83776,83778,83796],{"className":83777,"ariaHidden":72},[116],[15,83779,83781,83784,83787,83790,83793],{"className":83780},[120],[15,83782],{"className":83783,"style":4383},[124],[15,83785,4849],{"className":83786},[129,133],[15,83788],{"className":83789,"style":199},[93],[15,83791,62],{"className":83792},[203],[15,83794],{"className":83795,"style":199},[93],[15,83797,83799,83802,83805,83808],{"className":83798},[120],[15,83800],{"className":83801,"style":30447},[124],[15,83803,27294],{"className":83804,"style":887},[129,133],[15,83806,544],{"className":83807},[129],[15,83809,83811],{"className":83810},[129,29123],[15,83812,83814,83844],{"className":83813},[141,142],[15,83815,83817,83841],{"className":83816},[146],[15,83818,83820,83829],{"className":83819,"style":30463},[150],[15,83821,83823,83826],{"className":83822,"style":2287},[29137],[15,83824],{"className":83825,"style":1823},[158],[15,83827,39326],{"className":83828,"style":30473},[129,133],[15,83830,83831,83834],{"style":30476},[15,83832],{"className":83833,"style":1823},[158],[15,83835,83837],{"className":83836,"style":30483},[29282],[4455,83838,83839],{"xmlns":4457,"width":29286,"height":30486,"viewBox":30487,"preserveAspectRatio":29289},[4461,83840],{"d":30490},[15,83842,175],{"className":83843},[174],[15,83845,83847],{"className":83846},[146],[15,83848,83850],{"className":83849,"style":30500},[150],[15,83851],{},"，则 TV 至多 ",[15,83854,83856,83873],{"className":83855,"translate":19},[23],[15,83857,83859],{"className":83858},[27],[29,83860,83861],{"xmlns":31},[34,83862,83863,83871],{},[37,83864,83865,83867,83869],{},[308,83866,318],{},[43,83868,544],{"mathvariant":65},[308,83870,310],{},[109,83872,56735],{"encoding":111},[15,83874,83876],{"className":83875,"ariaHidden":72},[116],[15,83877,83879,83882],{"className":83878},[120],[15,83880],{"className":83881,"style":125},[124],[15,83883,56735],{"className":83884},[129],"。Assouad：",[15,83887,83889],{"className":83888,"translate":19},[18],[15,83890,83892,84000],{"className":83891,"translate":19},[23],[15,83893,83895],{"className":83894},[27],[29,83896,83897],{"xmlns":31,"display":32},[34,83898,83899,83997],{},[37,83900,83901,83921,83923,83925,83931,83933,83935,83943,83945,83959,83975,83977,83995],{},[2140,83902,83903,83909],{},[37,83904,83905,83907],{},[43,83906,19705],{},[50,83908,69],{},[37,83910,83911,83913,83915],{},[43,83912,48],{},[50,83914,100],{},[40,83916,83917,83919],{},[43,83918,30875],{"mathvariant":65},[308,83920,541],{},[43,83922,26574],{"mathvariant":103},[50,83924,33823],{"stretchy":52},[46847,83926,83927,83929],{"accent":72},[43,83928,48],{},[50,83930,48483],{},[50,83932,315],{},[43,83934,48],{},[23160,83936,83937,83939,83941],{},[50,83938,33823],{"stretchy":52},[308,83940,310],{},[308,83942,310],{},[50,83944,7473],{},[1443,83946,83947,83957],{},[37,83948,83949,83955],{},[305,83950,83951,83953],{},[43,83952,4849],{},[308,83954,310],{},[43,83956,10376],{},[308,83958,14508],{},[37,83960,83961,83963,83965,83967,83973],{},[50,83962,53],{"fence":72},[308,83964,318],{},[50,83966,315],{},[1443,83968,83969,83971],{},[308,83970,318],{},[308,83972,310],{},[50,83974,59],{"fence":72},[50,83976,62],{},[1443,83978,83979,83989],{},[37,83980,83981,83987],{},[305,83982,83983,83985],{},[43,83984,27294],{},[308,83986,310],{},[43,83988,10376],{},[37,83990,83991,83993],{},[308,83992,32915],{},[43,83994,39326],{},[43,83996,107],{"mathvariant":65},[109,83998,83999],{"encoding":111},"\\max_{\\theta\\in\\Theta_0}\\mathbb E\\lVert\\hat\\theta-\\theta\\rVert_2^2\n\\ge\n\\frac{\\varepsilon^2d}{4}\\left(1-\\frac12\\right)\n=\\frac{\\sigma^2d}{8n}.",[15,84001,84003,84151,84220,84418],{"className":84002,"ariaHidden":72},[116],[15,84004,84006,84010,84102,84105,84108,84111,84142,84145,84148],{"className":84005},[120],[15,84007],{"className":84008,"style":84009},[124],"height:1.8101em;vertical-align:-0.8522em;",[15,84011,84013],{"className":84012},[217,1874],[15,84014,84016,84093],{"className":84015},[141,142],[15,84017,84019,84090],{"className":84018},[146],[15,84020,84022,84080],{"className":84021,"style":4383},[150],[15,84023,84024,84027],{"style":2266},[15,84025],{"className":84026,"style":1823},[158],[15,84028,84030],{"className":84029},[163,164,165,166],[15,84031,84033,84036,84039],{"className":84032},[129,166],[15,84034,48],{"className":84035,"style":170},[129,133,166],[15,84037,100],{"className":84038},[203,166],[15,84040,84042,84045],{"className":84041},[129,166],[15,84043,30875],{"className":84044},[129,166],[15,84046,84048],{"className":84047},[137],[15,84049,84051,84072],{"className":84050},[141,142],[15,84052,84054,84069],{"className":84053},[146],[15,84055,84057],{"className":84056,"style":46995},[150],[15,84058,84060,84063],{"style":84059},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[15,84061],{"className":84062,"style":15987},[158],[15,84064,84066],{"className":84065},[163,15991,229,166],[15,84067,541],{"className":84068},[129,166],[15,84070,175],{"className":84071},[174],[15,84073,84075],{"className":84074},[146],[15,84076,84078],{"className":84077,"style":16004},[150],[15,84079],{},[15,84081,84082,84085],{"style":2287},[15,84083],{"className":84084,"style":1823},[158],[15,84086,84087],{},[15,84088,19705],{"className":84089},[217],[15,84091,175],{"className":84092},[174],[15,84094,84096],{"className":84095},[146],[15,84097,84100],{"className":84098,"style":84099},[150],"height:0.8522em;",[15,84101],{},[15,84103],{"className":84104,"style":262},[93],[15,84106,26574],{"className":84107},[129,285],[15,84109,33823],{"className":84110},[188],[15,84112,84114],{"className":84113},[129,48500],[15,84115,84117],{"className":84116},[141],[15,84118,84120],{"className":84119},[146],[15,84121,84123,84131],{"className":84122,"style":48496},[150],[15,84124,84125,84128],{"style":2287},[15,84126],{"className":84127,"style":1823},[158],[15,84129,48],{"className":84130,"style":170},[129,133],[15,84132,84133,84136],{"style":48520},[15,84134],{"className":84135,"style":1823},[158],[15,84137,84139],{"className":84138,"style":48528},[48527],[15,84140,48483],{"className":84141},[129],[15,84143],{"className":84144,"style":3277},[93],[15,84146,315],{"className":84147},[2436],[15,84149],{"className":84150,"style":3277},[93],[15,84152,84154,84157,84160,84211,84214,84217],{"className":84153},[120],[15,84155],{"className":84156,"style":40920},[124],[15,84158,48],{"className":84159,"style":170},[129,133],[15,84161,84163,84166],{"className":84162},[195],[15,84164,33823],{"className":84165},[195],[15,84167,84169],{"className":84168},[137],[15,84170,84172,84203],{"className":84171},[141,142],[15,84173,84175,84200],{"className":84174},[146],[15,84176,84178,84189],{"className":84177,"style":40991},[150],[15,84179,84180,84183],{"style":58448},[15,84181],{"className":84182,"style":159},[158],[15,84184,84186],{"className":84185},[163,164,165,166],[15,84187,310],{"className":84188},[129,166],[15,84190,84191,84194],{"style":1272},[15,84192],{"className":84193,"style":159},[158],[15,84195,84197],{"className":84196},[163,164,165,166],[15,84198,310],{"className":84199},[129,166],[15,84201,175],{"className":84202},[174],[15,84204,84206],{"className":84205},[146],[15,84207,84209],{"className":84208,"style":28173},[150],[15,84210],{},[15,84212],{"className":84213,"style":199},[93],[15,84215,7473],{"className":84216},[203],[15,84218],{"className":84219,"style":199},[93],[15,84221,84223,84226,84317,84320,84409,84412,84415],{"className":84222},[120],[15,84224],{"className":84225,"style":63604},[124],[15,84227,84229,84232,84314],{"className":84228},[129],[15,84230],{"className":84231},[188,1803],[15,84233,84235],{"className":84234},[1443],[15,84236,84238,84306],{"className":84237},[141,142],[15,84239,84241,84303],{"className":84240},[146],[15,84242,84244,84255,84263],{"className":84243,"style":63623},[150],[15,84245,84246,84249],{"style":1819},[15,84247],{"className":84248,"style":1823},[158],[15,84250,84252],{"className":84251},[129],[15,84253,14508],{"className":84254},[129],[15,84256,84257,84260],{"style":1832},[15,84258],{"className":84259,"style":1823},[158],[15,84261],{"className":84262,"style":1840},[1839],[15,84264,84265,84268],{"style":1843},[15,84266],{"className":84267,"style":1823},[158],[15,84269,84271,84300],{"className":84270},[129],[15,84272,84274,84277],{"className":84273},[129],[15,84275,4849],{"className":84276},[129,133],[15,84278,84280],{"className":84279},[137],[15,84281,84283],{"className":84282},[141],[15,84284,84286],{"className":84285},[146],[15,84287,84289],{"className":84288,"style":380},[150],[15,84290,84291,84294],{"style":383},[15,84292],{"className":84293,"style":159},[158],[15,84295,84297],{"className":84296},[163,164,165,166],[15,84298,310],{"className":84299},[129,166],[15,84301,10376],{"className":84302},[129,133],[15,84304,175],{"className":84305},[174],[15,84307,84309],{"className":84308},[146],[15,84310,84312],{"className":84311,"style":1862},[150],[15,84313],{},[15,84315],{"className":84316},[195,1803],[15,84318],{"className":84319,"style":262},[93],[15,84321,84323,84329,84332,84335,84338,84341,84403],{"className":84322},[449],[15,84324,84326],{"className":84325,"style":5052},[188,5051],[15,84327,53],{"className":84328},[228,165],[15,84330,318],{"className":84331},[129],[15,84333],{"className":84334,"style":3277},[93],[15,84336,315],{"className":84337},[2436],[15,84339],{"className":84340,"style":3277},[93],[15,84342,84344,84347,84400],{"className":84343},[129],[15,84345],{"className":84346},[188,1803],[15,84348,84350],{"className":84349},[1443],[15,84351,84353,84392],{"className":84352},[141,142],[15,84354,84356,84389],{"className":84355},[146],[15,84357,84359,84370,84378],{"className":84358,"style":1816},[150],[15,84360,84361,84364],{"style":1819},[15,84362],{"className":84363,"style":1823},[158],[15,84365,84367],{"className":84366},[129],[15,84368,310],{"className":84369},[129],[15,84371,84372,84375],{"style":1832},[15,84373],{"className":84374,"style":1823},[158],[15,84376],{"className":84377,"style":1840},[1839],[15,84379,84380,84383],{"style":1843},[15,84381],{"className":84382,"style":1823},[158],[15,84384,84386],{"className":84385},[129],[15,84387,318],{"className":84388},[129],[15,84390,175],{"className":84391},[174],[15,84393,84395],{"className":84394},[146],[15,84396,84398],{"className":84397,"style":1862},[150],[15,84399],{},[15,84401],{"className":84402},[195,1803],[15,84404,84406],{"className":84405,"style":5052},[195,5051],[15,84407,59],{"className":84408},[228,165],[15,84410],{"className":84411,"style":199},[93],[15,84413,62],{"className":84414},[203],[15,84416],{"className":84417,"style":199},[93],[15,84419,84421,84424,84518],{"className":84420},[120],[15,84422],{"className":84423,"style":82509},[124],[15,84425,84427,84430,84515],{"className":84426},[129],[15,84428],{"className":84429},[188,1803],[15,84431,84433],{"className":84432},[1443],[15,84434,84436,84507],{"className":84435},[141,142],[15,84437,84439,84504],{"className":84438},[146],[15,84440,84442,84456,84464],{"className":84441,"style":63623},[150],[15,84443,84444,84447],{"style":1819},[15,84445],{"className":84446,"style":1823},[158],[15,84448,84450,84453],{"className":84449},[129],[15,84451,32915],{"className":84452},[129],[15,84454,39326],{"className":84455},[129,133],[15,84457,84458,84461],{"style":1832},[15,84459],{"className":84460,"style":1823},[158],[15,84462],{"className":84463,"style":1840},[1839],[15,84465,84466,84469],{"style":1843},[15,84467],{"className":84468,"style":1823},[158],[15,84470,84472,84501],{"className":84471},[129],[15,84473,84475,84478],{"className":84474},[129],[15,84476,27294],{"className":84477,"style":887},[129,133],[15,84479,84481],{"className":84480},[137],[15,84482,84484],{"className":84483},[141],[15,84485,84487],{"className":84486},[146],[15,84488,84490],{"className":84489,"style":380},[150],[15,84491,84492,84495],{"style":383},[15,84493],{"className":84494,"style":159},[158],[15,84496,84498],{"className":84497},[163,164,165,166],[15,84499,310],{"className":84500},[129,166],[15,84502,10376],{"className":84503},[129,133],[15,84505,175],{"className":84506},[174],[15,84508,84510],{"className":84509},[146],[15,84511,84513],{"className":84512,"style":1862},[150],[15,84514],{},[15,84516],{"className":84517},[195,1803],[15,84519,107],{"className":84520},[129],[11,84522,84523,84524,84552,84553,84626],{},"上下界只差常数 ",[15,84525,84527,84540],{"className":84526,"translate":19},[23],[15,84528,84530],{"className":84529},[27],[29,84531,84532],{"xmlns":31},[34,84533,84534,84538],{},[37,84535,84536],{},[308,84537,32915],{},[109,84539,32915],{"encoding":111},[15,84541,84543],{"className":84542,"ariaHidden":72},[116],[15,84544,84546,84549],{"className":84545},[120],[15,84547],{"className":84548,"style":4886},[124],[15,84550,32915],{"className":84551},[129],"，速率 ",[15,84554,84556,84579],{"className":84555,"translate":19},[23],[15,84557,84559],{"className":84558},[27],[29,84560,84561],{"xmlns":31},[34,84562,84563,84577],{},[37,84564,84565,84571,84573,84575],{},[305,84566,84567,84569],{},[43,84568,27294],{},[308,84570,310],{},[43,84572,10376],{},[43,84574,544],{"mathvariant":65},[43,84576,39326],{},[109,84578,64919],{"encoding":111},[15,84580,84582],{"className":84581,"ariaHidden":72},[116],[15,84583,84585,84588,84617,84620,84623],{"className":84584},[120],[15,84586],{"className":84587,"style":34009},[124],[15,84589,84591,84594],{"className":84590},[129],[15,84592,27294],{"className":84593,"style":887},[129,133],[15,84595,84597],{"className":84596},[137],[15,84598,84600],{"className":84599},[141],[15,84601,84603],{"className":84602},[146],[15,84604,84606],{"className":84605,"style":380},[150],[15,84607,84608,84611],{"style":383},[15,84609],{"className":84610,"style":159},[158],[15,84612,84614],{"className":84613},[163,164,165,166],[15,84615,310],{"className":84616},[129,166],[15,84618,10376],{"className":84619},[129,133],[15,84621,544],{"className":84622},[129],[15,84624,39326],{"className":84625},[129,133]," 被夹住了。",[11,84628,84629,84630,84658],{},"这个例子也说明了 Fano 与 Assouad 的分工：Fano 需要构造一个 ",[15,84631,84633,84646],{"className":84632,"translate":19},[23],[15,84634,84636],{"className":84635},[27],[29,84637,84638],{"xmlns":31},[34,84639,84640,84644],{},[37,84641,84642],{},[43,84643,29575],{},[109,84645,61735],{"encoding":111},[15,84647,84649],{"className":84648,"ariaHidden":72},[116],[15,84650,84652,84655],{"className":84651},[120],[15,84653],{"className":84654,"style":522},[124],[15,84656,29575],{"className":84657,"style":29856},[129,133],"-分离的打包，Assouad 则直接利用乘积结构，把打包问题化成了逐坐标的 TV 计算。",[575,84660,84662,84663],{"id":84661},"_62-非参数回归n2β2βdn-2beta2betadn2β2βd","6.2 非参数回归：",[15,84664,84666,84703],{"className":84665,"translate":19},[23],[15,84667,84669],{"className":84668},[27],[29,84670,84671],{"xmlns":31},[34,84672,84673,84701],{},[37,84674,84675],{},[305,84676,84677,84679],{},[43,84678,39326],{},[37,84680,84681,84683,84685,84687,84689,84691,84693,84695,84697,84699],{},[50,84682,315],{},[308,84684,310],{},[43,84686,36277],{},[43,84688,544],{"mathvariant":65},[50,84690,53],{"stretchy":52},[308,84692,310],{},[43,84694,36277],{},[50,84696,2682],{},[43,84698,10376],{},[50,84700,59],{"stretchy":52},[109,84702,75513],{"encoding":111},[15,84704,84706],{"className":84705,"ariaHidden":72},[116],[15,84707,84709,84712],{"className":84708},[120],[15,84710],{"className":84711,"style":14948},[124],[15,84713,84715,84718],{"className":84714},[129],[15,84716,39326],{"className":84717},[129,133],[15,84719,84721],{"className":84720},[137],[15,84722,84724],{"className":84723},[141],[15,84725,84727],{"className":84726},[146],[15,84728,84730],{"className":84729,"style":14948},[150],[15,84731,84732,84735],{"style":383},[15,84733],{"className":84734,"style":159},[158],[15,84736,84738],{"className":84737},[163,164,165,166],[15,84739,84741,84744,84747,84750,84753,84756,84759,84762,84765,84768],{"className":84740},[129,166],[15,84742,315],{"className":84743},[129,166],[15,84745,310],{"className":84746},[129,166],[15,84748,36277],{"className":84749,"style":36293},[129,133,166],[15,84751,544],{"className":84752},[129,166],[15,84754,53],{"className":84755},[188,166],[15,84757,310],{"className":84758},[129,166],[15,84760,36277],{"className":84761,"style":36293},[129,133,166],[15,84763,2682],{"className":84764},[2436,166],[15,84766,10376],{"className":84767},[129,133,166],[15,84769,59],{"className":84770},[195,166],[11,84772,81756],{},[15,84774,84776],{"className":84775,"translate":19},[18],[15,84777,84779,84890],{"className":84778,"translate":19},[23],[15,84780,84782],{"className":84781},[27],[29,84783,84784],{"xmlns":31,"display":32},[34,84785,84786,84887],{},[37,84787,84788,84794,84796,84798,84800,84806,84808,84810,84817,84819,84821,84827,84829,84839,84841,84843,84845,84847,84849,84855,84857,84859,84861,84867,84869,84871,84873,84875,84877,84883,84885],{},[40,84789,84790,84792],{},[43,84791,631],{},[43,84793,1459],{},[50,84795,62],{},[43,84797,36569],{},[50,84799,53],{"stretchy":52},[40,84801,84802,84804],{},[43,84803,597],{},[43,84805,1459],{},[50,84807,59],{"stretchy":52},[50,84809,2682],{},[40,84811,84812,84815],{},[43,84813,84814],{},"ξ",[43,84816,1459],{},[50,84818,91],{"separator":72},[93,84820],{"width":95},[40,84822,84823,84825],{},[43,84824,597],{},[43,84826,1459],{},[50,84828,991],{},[37,84830,84831,84833,84835,84837],{},[43,84832,67663],{"mathvariant":65},[43,84834,39326],{"mathvariant":65},[43,84836,1459],{"mathvariant":65},[43,84838,36569],{"mathvariant":65},[50,84840,53],{"stretchy":52},[50,84842,1407],{"stretchy":52},[308,84844,541],{},[50,84846,91],{"separator":72},[308,84848,318],{},[305,84850,84851,84853],{},[50,84852,1423],{"stretchy":52},[43,84854,10376],{},[50,84856,59],{"stretchy":52},[50,84858,91],{"separator":72},[93,84860],{"width":95},[40,84862,84863,84865],{},[43,84864,84814],{},[43,84866,1459],{},[50,84868,991],{},[43,84870,12561],{},[50,84872,53],{"stretchy":52},[308,84874,541],{},[50,84876,91],{"separator":72},[305,84878,84879,84881],{},[43,84880,27294],{},[308,84882,310],{},[50,84884,59],{"stretchy":52},[50,84886,91],{"separator":72},[109,84888,84889],{"encoding":111},"Y_i=f(X_i)+\\xi_i,\\qquad X_i\\sim\\mathrm{Unif}([0,1]^d),\\qquad \\xi_i\\sim N(0,\\sigma^2),",[15,84891,84893,84949,85013,85119,85238],{"className":84892,"ariaHidden":72},[116],[15,84894,84896,84899,84940,84943,84946],{"className":84895},[120],[15,84897],{"className":84898,"style":20229},[124],[15,84900,84902,84905],{"className":84901},[129],[15,84903,631],{"className":84904,"style":3277},[129,133],[15,84906,84908],{"className":84907},[137],[15,84909,84911,84932],{"className":84910},[141,142],[15,84912,84914,84929],{"className":84913},[146],[15,84915,84917],{"className":84916,"style":1981},[150],[15,84918,84920,84923],{"style":84919},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[15,84921],{"className":84922,"style":159},[158],[15,84924,84926],{"className":84925},[163,164,165,166],[15,84927,1459],{"className":84928},[129,133,166],[15,84930,175],{"className":84931},[174],[15,84933,84935],{"className":84934},[146],[15,84936,84938],{"className":84937,"style":182},[150],[15,84939],{},[15,84941],{"className":84942,"style":199},[93],[15,84944,62],{"className":84945},[203],[15,84947],{"className":84948,"style":199},[93],[15,84950,84952,84955,84958,84961,85001,85004,85007,85010],{"className":84951},[120],[15,84953],{"className":84954,"style":125},[124],[15,84956,36569],{"className":84957,"style":36584},[129,133],[15,84959,53],{"className":84960},[188],[15,84962,84964,84967],{"className":84963},[129],[15,84965,597],{"className":84966,"style":47970},[129,133],[15,84968,84970],{"className":84969},[137],[15,84971,84973,84993],{"className":84972},[141,142],[15,84974,84976,84990],{"className":84975},[146],[15,84977,84979],{"className":84978,"style":1981},[150],[15,84980,84981,84984],{"style":47985},[15,84982],{"className":84983,"style":159},[158],[15,84985,84987],{"className":84986},[163,164,165,166],[15,84988,1459],{"className":84989},[129,133,166],[15,84991,175],{"className":84992},[174],[15,84994,84996],{"className":84995},[146],[15,84997,84999],{"className":84998,"style":182},[150],[15,85000],{},[15,85002,59],{"className":85003},[195],[15,85005],{"className":85006,"style":3277},[93],[15,85008,2682],{"className":85009},[2436],[15,85011],{"className":85012,"style":3277},[93],[15,85014,85016,85019,85061,85064,85067,85070,85110,85113,85116],{"className":85015},[120],[15,85017],{"className":85018,"style":1682},[124],[15,85020,85022,85026],{"className":85021},[129],[15,85023,84814],{"className":85024,"style":85025},[129,133],"margin-right:0.046em;",[15,85027,85029],{"className":85028},[137],[15,85030,85032,85053],{"className":85031},[141,142],[15,85033,85035,85050],{"className":85034},[146],[15,85036,85038],{"className":85037,"style":1981},[150],[15,85039,85041,85044],{"style":85040},"top:-2.55em;margin-left:-0.046em;margin-right:0.05em;",[15,85042],{"className":85043,"style":159},[158],[15,85045,85047],{"className":85046},[163,164,165,166],[15,85048,1459],{"className":85049},[129,133,166],[15,85051,175],{"className":85052},[174],[15,85054,85056],{"className":85055},[146],[15,85057,85059],{"className":85058,"style":182},[150],[15,85060],{},[15,85062,91],{"className":85063},[254],[15,85065],{"className":85066,"style":258},[93],[15,85068],{"className":85069,"style":262},[93],[15,85071,85073,85076],{"className":85072},[129],[15,85074,597],{"className":85075,"style":47970},[129,133],[15,85077,85079],{"className":85078},[137],[15,85080,85082,85102],{"className":85081},[141,142],[15,85083,85085,85099],{"className":85084},[146],[15,85086,85088],{"className":85087,"style":1981},[150],[15,85089,85090,85093],{"style":47985},[15,85091],{"className":85092,"style":159},[158],[15,85094,85096],{"className":85095},[163,164,165,166],[15,85097,1459],{"className":85098},[129,133,166],[15,85100,175],{"className":85101},[174],[15,85103,85105],{"className":85104},[146],[15,85106,85108],{"className":85107,"style":182},[150],[15,85109],{},[15,85111],{"className":85112,"style":199},[93],[15,85114,991],{"className":85115},[203],[15,85117],{"className":85118,"style":199},[93],[15,85120,85122,85126,85132,85136,85139,85142,85145,85148,85177,85180,85183,85186,85189,85229,85232,85235],{"className":85121},[120],[15,85123],{"className":85124,"style":85125},[124],"height:1.1491em;vertical-align:-0.25em;",[15,85127,85129],{"className":85128},[129],[15,85130,67720],{"className":85131,"style":48850},[129,221],[15,85133,85135],{"className":85134},[188],"([",[15,85137,541],{"className":85138},[129],[15,85140,91],{"className":85141},[254],[15,85143],{"className":85144,"style":262},[93],[15,85146,318],{"className":85147},[129],[15,85149,85151,85154],{"className":85150},[195],[15,85152,1423],{"className":85153},[195],[15,85155,85157],{"className":85156},[137],[15,85158,85160],{"className":85159},[141],[15,85161,85163],{"className":85162},[146],[15,85164,85166],{"className":85165,"style":64296},[150],[15,85167,85168,85171],{"style":1272},[15,85169],{"className":85170,"style":159},[158],[15,85172,85174],{"className":85173},[163,164,165,166],[15,85175,10376],{"className":85176},[129,133,166],[15,85178,59],{"className":85179},[195],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属于 ",[15,85326,85328,85341],{"className":85327,"translate":19},[23],[15,85329,85331],{"className":85330},[27],[29,85332,85333],{"xmlns":31},[34,85334,85335,85339],{},[37,85336,85337],{},[43,85338,36277],{},[109,85340,36280],{"encoding":111},[15,85342,85344],{"className":85343,"ariaHidden":72},[116],[15,85345,85347,85350],{"className":85346},[120],[15,85348],{"className":85349,"style":1682},[124],[15,85351,36277],{"className":85352,"style":36293},[129,133],"-Hölder 球 ",[15,85355,85357,85382],{"className":85356,"translate":19},[23],[15,85358,85360],{"className":85359},[27],[29,85361,85362],{"xmlns":31},[34,85363,85364,85379],{},[37,85365,85366,85369,85371,85373,85375,85377],{},[43,85367,85368],{"mathvariant":65},"Σ",[50,85370,53],{"stretchy":52},[43,85372,36277],{},[50,85374,91],{"separator":72},[43,85376,1373],{},[50,85378,59],{"stretchy":52},[109,85380,85381],{"encoding":111},"\\Sigma(\\beta,L)",[15,85383,85385],{"className":85384,"ariaHidden":72},[116],[15,85386,85388,85391,85394,85397,85400,85403,85406,85409],{"className":85387},[120],[15,85389],{"className":85390,"style":125},[124],[15,85392,85368],{"className":85393},[129],[15,85395,53],{"className":85396},[188],[15,85398,36277],{"className":85399,"style":36293},[129,133],[15,85401,91],{"className":85402},[254],[15,85404],{"className":85405,"style":262},[93],[15,85407,1373],{"className":85408},[129,133],[15,85410,59],{"className":85411},[195],"，损失为 ",[15,85414,85416,85452],{"className":85415,"translate":19},[23],[15,85417,85419],{"className":85418},[27],[29,85420,85421],{"xmlns":31},[34,85422,85423,85449],{},[37,85424,85425,85427,85433,85435,85437],{},[50,85426,33823],{"stretchy":52},[46847,85428,85429,85431],{"accent":72},[43,85430,36569],{},[50,85432,48483],{},[50,85434,315],{},[43,85436,36569],{},[23160,85438,85439,85441,85447],{},[50,85440,33823],{"stretchy":52},[305,85442,85443,85445],{},[43,85444,1373],{},[308,85446,310],{},[308,85448,310],{},[109,85450,85451],{"encoding":111},"\\lVert\\hat 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核估计或局部多项式估计可以达到",[15,85615,85617],{"className":85616,"translate":19},[18],[15,85618,85620,85658],{"className":85619,"translate":19},[23],[15,85621,85623],{"className":85622},[27],[29,85624,85625],{"xmlns":31,"display":32},[34,85626,85627,85655],{},[37,85628,85629],{},[305,85630,85631,85633],{},[43,85632,39326],{},[37,85634,85635,85637],{},[50,85636,315],{},[1443,85638,85639,85645],{},[37,85640,85641,85643],{},[308,85642,310],{},[43,85644,36277],{},[37,85646,85647,85649,85651,85653],{},[308,85648,310],{},[43,85650,36277],{},[50,85652,2682],{},[43,85654,10376],{},[109,85656,85657],{"encoding":111},"n^{-\\frac{2\\beta}{2\\beta+d}}",[15,85659,85661],{"className":85660,"ariaHidden":72},[116],[15,85662,85664,85668],{"className":85663},[120],[15,85665],{"className":85666,"style":85667},[124],"height:1.1167em;",[15,85669,85671,85674],{"className":85670},[129],[15,85672,39326],{"className":85673},[129,133],[15,85675,85677],{"className":85676},[137],[15,85678,85680],{"className":85679},[141],[15,85681,85683],{"className":85682},[146],[15,85684,85686],{"className":85685,"style":85667},[150],[15,85687,85689,85692],{"style":85688},"top:-3.4458em;margin-right:0.05em;",[15,85690],{"className":85691,"style":1823},[158],[15,85693,85695],{"className":85694},[163,164,165,166],[15,85696,85698,85701],{"className":85697},[129,166],[15,85699,315],{"className":85700},[129,166],[15,85702,85704,85708,85785],{"className":85703},[129,166],[15,85705],{"className":85706},[188,1803,163,15991,85707],"size6",[15,85709,85711],{"className":85710},[1443],[15,85712,85714,85776],{"className":85713},[141,142],[15,85715,85717,85773],{"className":85716},[146],[15,85718,85721,85745,85755],{"className":85719,"style":85720},[150],"height:0.9584em;",[15,85722,85724,85727],{"style":85723},"top:-2.656em;",[15,85725],{"className":85726,"style":1823},[158],[15,85728,85730],{"className":85729},[163,15991,229,166],[15,85731,85733,85736,85739,85742],{"className":85732},[129,166],[15,85734,310],{"className":85735},[129,166],[15,85737,36277],{"className":85738,"style":36293},[129,133,166],[15,85740,2682],{"className":85741},[2436,166],[15,85743,10376],{"className":85744},[129,133,166],[15,85746,85748,85751],{"style":85747},"top:-3.2255em;",[15,85749],{"className":85750,"style":1823},[158],[15,85752],{"className":85753,"style":85754},[1839,166],"border-bottom-width:0.049em;",[15,85756,85758,85761],{"style":85757},"top:-3.4624em;",[15,85759],{"className":85760,"style":1823},[158],[15,85762,85764],{"className":85763},[163,15991,229,166],[15,85765,85767,85770],{"className":85766},[129,166],[15,85768,310],{"className":85769},[129,166],[15,85771,36277],{"className":85772,"style":36293},[129,133,166],[15,85774,175],{"className":85775},[174],[15,85777,85779],{"className":85778},[146],[15,85780,85783],{"className":85781,"style":85782},[150],"height:0.4829em;",[15,85784],{},[15,85786],{"className":85787},[195,1803,163,15991,85707],[11,85789,85790,85791,85793],{},"（见 Tsybakov 第 1 章）。",[698,85792,10671],{},"用 Fano。",[11,85795,7441,85796,2455,85870,85952,85953,86044,86045,86073,86074,86104],{},[15,85797,85799,85825],{"className":85798,"translate":19},[23],[15,85800,85802],{"className":85801},[27],[29,85803,85804],{"xmlns":31},[34,85805,85806,85822],{},[37,85807,85808,85810,85812,85814,85816,85818,85820],{},[43,85809,45],{},[50,85811,100],{},[50,85813,53],{"stretchy":52},[308,85815,541],{},[50,85817,91],{"separator":72},[308,85819,318],{},[50,85821,59],{"stretchy":52},[109,85823,85824],{"encoding":111},"h\\in(0,1)",[15,85826,85828,85846],{"className":85827,"ariaHidden":72},[116],[15,85829,85831,85834,85837,85840,85843],{"className":85830},[120],[15,85832],{"className":85833,"style":2405},[124],[15,85835,45],{"className":85836},[129,133],[15,85838],{"className":85839,"style":199},[93],[15,85841,100],{"className":85842},[203],[15,85844],{"className":85845,"style":199},[93],[15,85847,85849,85852,85855,85858,85861,85864,85867],{"className":85848},[120],[15,85850],{"className":85851,"style":125},[124],[15,85853,53],{"className":85854},[188],[15,85856,541],{"className":85857},[129],[15,85859,91],{"className":85860},[254],[15,85862],{"className":85863,"style":262},[93],[15,85865,318],{"className":85866},[129],[15,85868,59],{"className":85869},[195],[15,85871,85873,85899],{"className":85872,"translate":19},[23],[15,85874,85876],{"className":85875},[27],[29,85877,85878],{"xmlns":31},[34,85879,85880,85896],{},[37,85881,85882,85884,85886,85888,85890],{},[50,85883,1407],{"stretchy":52},[308,85885,541],{},[50,85887,91],{"separator":72},[308,85889,318],{},[305,85891,85892,85894],{},[50,85893,1423],{"stretchy":52},[43,85895,10376],{},[109,85897,85898],{"encoding":111},"[0,1]^d",[15,85900,85902],{"className":85901,"ariaHidden":72},[116],[15,85903,85905,85908,85911,85914,85917,85920,85923],{"className":85904},[120],[15,85906],{"className":85907,"style":10978},[124],[15,85909,1407],{"className":85910},[188],[15,85912,541],{"className":85913},[129],[15,85915,91],{"className":85916},[254],[15,85918],{"className":85919,"style":262},[93],[15,85921,318],{"className":85922},[129],[15,85924,85926,85929],{"className":85925},[195],[15,85927,1423],{"className":85928},[195],[15,85930,85932],{"className":85931},[137],[15,85933,85935],{"className":85934},[141],[15,85936,85938],{"className":85937},[146],[15,85939,85941],{"className":85940,"style":10490},[150],[15,85942,85943,85946],{"style":383},[15,85944],{"className":85945,"style":159},[158],[15,85947,85949],{"className":85948},[163,164,165,166],[15,85950,10376],{"className":85951},[129,133,166]," 分成 ",[15,85954,85956,85982],{"className":85955,"translate":19},[23],[15,85957,85959],{"className":85958},[27],[29,85960,85961],{"xmlns":31},[34,85962,85963,85979],{},[37,85964,85965,85967,85969],{},[43,85966,348],{},[50,85968,62],{},[305,85970,85971,85973],{},[43,85972,45],{},[37,85974,85975,85977],{},[50,85976,315],{},[43,85978,10376],{},[109,85980,85981],{"encoding":111},"m=h^{-d}",[15,85983,85985,86003],{"className":85984,"ariaHidden":72},[116],[15,85986,85988,85991,85994,85997,86000],{"className":85987},[120],[15,85989],{"className":85990,"style":4383},[124],[15,85992,348],{"className":85993},[129,133],[15,85995],{"className":85996,"style":199},[93],[15,85998,62],{"className":85999},[203],[15,86001],{"className":86002,"style":199},[93],[15,86004,86006,86009],{"className":86005},[120],[15,86007],{"className":86008,"style":10490},[124],[15,86010,86012,86015],{"className":86011},[129],[15,86013,45],{"className":86014},[129,133],[15,86016,86018],{"className":86017},[137],[15,86019,86021],{"className":86020},[141],[15,86022,86024],{"className":86023},[146],[15,86025,86027],{"className":86026,"style":10490},[150],[15,86028,86029,86032],{"style":383},[15,86030],{"className":86031,"style":159},[158],[15,86033,86035],{"className":86034},[163,164,165,166],[15,86036,86038,86041],{"className":86037},[129,166],[15,86039,315],{"className":86040},[129,166],[15,86042,10376],{"className":86043},[129,133,166]," 个边长为 ",[15,86046,86048,86061],{"className":86047,"translate":19},[23],[15,86049,86051],{"className":86050},[27],[29,86052,86053],{"xmlns":31},[34,86054,86055,86059],{},[37,86056,86057],{},[43,86058,45],{},[109,86060,45],{"encoding":111},[15,86062,86064],{"className":86063,"ariaHidden":72},[116],[15,86065,86067,86070],{"className":86066},[120],[15,86068],{"className":86069,"style":522},[124],[15,86071,45],{"className":86072},[129,133]," 的小方块。取一个光滑的鼓包函数 ",[15,86075,86077,86092],{"className":86076,"translate":19},[23],[15,86078,86080],{"className":86079},[27],[29,86081,86082],{"xmlns":31},[34,86083,86084,86089],{},[37,86085,86086],{},[43,86087,86088],{},"φ",[109,86090,86091],{"encoding":111},"\\varphi",[15,86093,86095],{"className":86094,"ariaHidden":72},[116],[15,86096,86098,86101],{"className":86097},[120],[15,86099],{"className":86100,"style":5399},[124],[15,86102,86088],{"className":86103},[129,133],"，支撑在单位方块内，令",[15,86106,86108],{"className":86107,"translate":19},[18],[15,86109,86111,86161],{"className":86110,"translate":19},[23],[15,86112,86114],{"className":86113},[27],[29,86115,86116],{"xmlns":31,"display":32},[34,86117,86118,86158],{},[37,86119,86120,86126,86128,86130,86132,86134,86136,86138,86154,86156],{},[40,86121,86122,86124],{},[43,86123,86088],{},[43,86125,13955],{},[50,86127,53],{"stretchy":52},[43,86129,56],{},[50,86131,59],{"stretchy":52},[50,86133,62],{},[43,86135,86088],{},[50,86137,53],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[1443,86139,86140,86152],{},[37,86141,86142,86144,86146],{},[43,86143,56],{},[50,86145,315],{},[40,86147,86148,86150],{},[43,86149,56],{},[43,86151,13955],{},[43,86153,45],{},[50,86155,59],{"fence":72,"stretchy":72,"minsize":45705,"maxsize":45705},[50,86157,91],{"separator":72},[109,86159,86160],{"encoding":111},"\\varphi_j(x)=\\varphi\\Bigl(\\frac{x-x_j}{h}\\Bigr),",[15,86162,86164,86228],{"className":86163,"ariaHidden":72},[116],[15,86165,86167,86170,86210,86213,86216,86219,86222,86225],{"className":86166},[120],[15,86168],{"className":86169,"style":67458},[124],[15,86171,86173,86176],{"className":86172},[129],[15,86174,86088],{"className":86175},[129,133],[15,86177,86179],{"className":86178},[137],[15,86180,86182,86202],{"className":86181},[141,142],[15,86183,86185,86199],{"className":86184},[146],[15,86186,86188],{"className":86187,"style":1981},[150],[15,86189,86190,86193],{"style":154},[15,86191],{"className":86192,"style":159},[158],[15,86194,86196],{"className":86195},[163,164,165,166],[15,86197,13955],{"className":86198,"style":13970},[129,133,166],[15,86200,175],{"className":86201},[174],[15,86203,86205],{"className":86204},[146],[15,86206,86208],{"className":86207,"style":14041},[150],[15,86209],{},[15,86211,53],{"className":86212},[188],[15,86214,56],{"className":86215},[129,133],[15,86217,59],{"className":86218},[195],[15,86220],{"className":86221,"style":199},[93],[15,86223,62],{"className":86224},[203],[15,86226],{"className":86227,"style":199},[93],[15,86229,86231,86235,86238,86244,86356,86362],{"className":86230},[120],[15,86232],{"className":86233,"style":86234},[124],"height:1.9463em;vertical-align:-0.686em;",[15,86236,86088],{"className":86237},[129,133],[15,86239,86241],{"className":86240},[188],[15,86242,53],{"className":86243},[228,24461],[15,86245,86247,86250,86353],{"className":86246},[129],[15,86248],{"className":86249},[188,1803],[15,86251,86253],{"className":86252},[1443],[15,86254,86256,86345],{"className":86255},[141,142],[15,86257,86259,86342],{"className":86258},[146],[15,86260,86263,86274,86282],{"className":86261,"style":86262},[150],"height:1.2603em;",[15,86264,86265,86268],{"style":1819},[15,86266],{"className":86267,"style":1823},[158],[15,86269,86271],{"className":86270},[129],[15,86272,45],{"className":86273},[129,133],[15,86275,86276,86279],{"style":1832},[15,86277],{"className":86278,"style":1823},[158],[15,86280],{"className":86281,"style":1840},[1839],[15,86283,86284,86287],{"style":1843},[15,86285],{"className":86286,"style":1823},[158],[15,86288,86290,86293,86296,86299,86302],{"className":86289},[129],[15,86291,56],{"className":86292},[129,133],[15,86294],{"className":86295,"style":3277},[93],[15,86297,315],{"className":86298},[2436],[15,86300],{"className":86301,"style":3277},[93],[15,86303,86305,86308],{"className":86304},[129],[15,86306,56],{"className":86307},[129,133],[15,86309,86311],{"className":86310},[137],[15,86312,86314,86334],{"className":86313},[141,142],[15,86315,86317,86331],{"className":86316},[146],[15,86318,86320],{"className":86319,"style":1981},[150],[15,86321,86322,86325],{"style":154},[15,86323],{"className":86324,"style":159},[158],[15,86326,86328],{"className":86327},[163,164,165,166],[15,86329,13955],{"className":86330,"style":13970},[129,133,166],[15,86332,175],{"className":86333},[174],[15,86335,86337],{"className":86336},[146],[15,86338,86340],{"className":86339,"style":14041},[150],[15,86341],{},[15,86343,175],{"className":86344},[174],[15,86346,86348],{"className":86347},[146],[15,86349,86351],{"className":86350,"style":1862},[150],[15,86352],{},[15,86354],{"className":86355},[195,1803],[15,86357,86359],{"className":86358},[195],[15,86360,59],{"className":86361},[228,24461],[15,86363,91],{"className":86364},[254],[11,86366,28697,86367,86437,86438,86466,86467,86558],{},[15,86368,86370,86388],{"className":86369,"translate":19},[23],[15,86371,86373],{"className":86372},[27],[29,86374,86375],{"xmlns":31},[34,86376,86377,86385],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是第 ",[15,86439,86441,86454],{"className":86440,"translate":19},[23],[15,86442,86444],{"className":86443},[27],[29,86445,86446],{"xmlns":31},[34,86447,86448,86452],{},[37,86449,86450],{},[43,86451,13955],{},[109,86453,13955],{"encoding":111},[15,86455,86457],{"className":86456,"ariaHidden":72},[116],[15,86458,86460,86463],{"className":86459},[120],[15,86461],{"className":86462,"style":11191},[124],[15,86464,13955],{"className":86465,"style":13970},[129,133]," 个方块的中心。对任意子集 ",[15,86468,86470,86500],{"className":86469,"translate":19},[23],[15,86471,86473],{"className":86472},[27],[29,86474,86475],{"xmlns":31},[34,86476,86477,86497],{},[37,86478,86479,86481,86483,86485,86487,86489,86491,86493,86495],{},[43,86480,940],{},[50,86482,5647],{},[50,86484,636],{"stretchy":52},[308,86486,318],{},[50,86488,91],{"separator":72},[50,86490,335],{},[50,86492,91],{"separator":72},[43,86494,348],{},[50,86496,645],{"stretchy":52},[109,86498,86499],{"encoding":111},"S\\subseteq\\{1,\\dots,m\\}",[15,86501,86503,86522],{"className":86502,"ariaHidden":72},[116],[15,86504,86506,86510,86513,86516,86519],{"className":86505},[120],[15,86507],{"className":86508,"style":86509},[124],"height:0.8193em;vertical-align:-0.136em;",[15,86511,940],{"className":86512,"style":1013},[129,133],[15,86514],{"className":86515,"style":199},[93],[15,86517,5647],{"className":86518},[203],[15,86520],{"className":86521,"style":199},[93],[15,86523,86525,86528,86531,86534,86537,86540,86543,86546,86549,86552,86555],{"className":86524},[120],[15,86526],{"className":86527,"style":125},[124],[15,86529,636],{"className":86530},[188],[15,86532,318],{"className":86533},[129],[15,86535,91],{"className":86536},[254],[15,86538],{"className":86539,"style":262},[93],[15,86541,335],{"className":86542},[449],[15,86544],{"className":86545,"style":262},[93],[15,86547,91],{"className":86548},[254],[15,86550],{"className":86551,"style":262},[93],[15,86553,348],{"className":86554},[129,133],[15,86556,645],{"className":86557},[195],"，定义",[15,86560,86562],{"className":86561,"translate":19},[18],[15,86563,86565,86625],{"className":86564,"translate":19},[23],[15,86566,86568],{"className":86567},[27],[29,86569,86570],{"xmlns":31,"display":32},[34,86571,86572,86622],{},[37,86573,86574,86580,86582,86584,86596,86602,86604,86606,86608,86610,86612,86614,86620],{},[40,86575,86576,86578],{},[43,86577,36569],{},[43,86579,940],{},[50,86581,62],{},[43,86583,8690],{},[2140,86585,86586,86588],{},[50,86587,1454],{},[37,86589,86590,86592,86594],{},[43,86591,13955],{},[50,86593,100],{},[43,86595,940],{},[40,86597,86598,86600],{},[43,86599,86088],{},[43,86601,13955],{},[50,86603,91],{"separator":72},[93,86605],{"width":95},[43,86607,8690],{},[50,86609,62],{},[43,86611,5628],{},[6856,86613,15788],{},[305,86615,86616,86618],{},[43,86617,45],{},[43,86619,36277],{},[43,86621,107],{"mathvariant":65},[109,86623,86624],{"encoding":111},"f_S=a\\sum_{j\\in S}\\varphi_j,\\qquad a=c\\,h^\\beta.",[15,86626,86628,86683,86814],{"className":86627,"ariaHidden":72},[116],[15,86629,86631,86634,86674,86677,86680],{"className":86630},[120],[15,86632],{"className":86633,"style":1682},[124],[15,86635,86637,86640],{"className":86636},[129],[15,86638,36569],{"className":86639,"style":36584},[129,133],[15,86641,86643],{"className":86642},[137],[15,86644,86646,86666],{"className":86645},[141,142],[15,86647,86649,86663],{"className":86648},[146],[15,86650,86652],{"className":86651,"style":1525},[150],[15,86653,86654,86657],{"style":39822},[15,86655],{"className":86656,"style":159},[158],[15,86658,86660],{"className":86659},[163,164,165,166],[15,86661,940],{"className":86662,"style":1013},[129,133,166],[15,86664,175],{"className":86665},[174],[15,86667,86669],{"className":86668},[146],[15,86670,86672],{"className":86671,"style":182},[150],[15,86673],{},[15,86675],{"className":86676,"style":199},[93],[15,86678,62],{"className":86679},[203],[15,86681],{"className":86682,"style":199},[93],[15,86684,86686,86690,86693,86696,86750,86753,86793,86796,86799,86802,86805,86808,86811],{"className":86685},[120],[15,86687],{"className":86688,"style":86689},[124],"height:2.4804em;vertical-align:-1.4304em;",[15,86691,8690],{"className":86692},[129,133],[15,86694],{"className":86695,"style":262},[93],[15,86697,86699],{"className":86698},[217,1874],[15,86700,86702,86741],{"className":86701},[141,142],[15,86703,86705,86738],{"className":86704},[146],[15,86706,86708,86728],{"className":86707,"style":12149},[150],[15,86709,86710,86713],{"style":12796},[15,86711],{"className":86712,"style":1891},[158],[15,86714,86716],{"className":86715},[163,164,165,166],[15,86717,86719,86722,86725],{"className":86718},[129,166],[15,86720,13955],{"className":86721,"style":13970},[129,133,166],[15,86723,100],{"className":86724},[203,166],[15,86726,940],{"className":86727,"style":1013},[129,133,166],[15,86729,86730,86733],{"style":1909},[15,86731],{"className":86732,"style":1891},[158],[15,86734,86735],{},[15,86736,1454],{"className":86737},[217,1918,1919],[15,86739,175],{"className":86740},[174],[15,86742,86744],{"className":86743},[146],[15,86745,86748],{"className":86746,"style":86747},[150],"height:1.4304em;",[15,86749],{},[15,86751],{"className":86752,"style":262},[93],[15,86754,86756,86759],{"className":86755},[129],[15,86757,86088],{"className":86758},[129,133],[15,86760,86762],{"className":86761},[137],[15,86763,86765,86785],{"className":86764},[141,142],[15,86766,86768,86782],{"className":86767},[146],[15,86769,86771],{"className":86770,"style":1981},[150],[15,86772,86773,86776],{"style":154},[15,86774],{"className":86775,"style":159},[158],[15,86777,86779],{"className":86778},[163,164,165,166],[15,86780,13955],{"className":86781,"style":13970},[129,133,166],[15,86783,175],{"className":86784},[174],[15,86786,86788],{"className":86787},[146],[15,86789,86791],{"className":86790,"style":14041},[150],[15,86792],{},[15,86794,91],{"className":86795},[254],[15,86797],{"className":86798,"style":258},[93],[15,86800],{"className":86801,"style":262},[93],[15,86803,8690],{"className":86804},[129,133],[15,86806],{"className":86807,"style":199},[93],[15,86809,62],{"className":86810},[203],[15,86812],{"className":86813,"style":199},[93],[15,86815,86817,86820,86823,86826,86855],{"className":86816},[120],[15,86818],{"className":86819,"style":64296},[124],[15,86821,5628],{"className":86822},[129,133],[15,86824],{"className":86825,"style":262},[93],[15,86827,86829,86832],{"className":86828},[129],[15,86830,45],{"className":86831},[129,133],[15,86833,86835],{"className":86834},[137],[15,86836,86838],{"className":86837},[141],[15,86839,86841],{"className":86840},[146],[15,86842,86844],{"className":86843,"style":64296},[150],[15,86845,86846,86849],{"style":1272},[15,86847],{"className":86848,"style":159},[158],[15,86850,86852],{"className":86851},[163,164,165,166],[15,86853,36277],{"className":86854,"style":36293},[129,133,166],[15,86856,107],{"className":86857},[129],[11,86859,17335,86860,86888,86889,87009,87010,87069,87070,15075,87098,87126,87127,1313],{},[15,86861,86863,86876],{"className":86862,"translate":19},[23],[15,86864,86866],{"className":86865},[27],[29,86867,86868],{"xmlns":31},[34,86869,86870,86874],{},[37,86871,86872],{},[43,86873,5628],{},[109,86875,5628],{"encoding":111},[15,86877,86879],{"className":86878,"ariaHidden":72},[116],[15,86880,86882,86885],{"className":86881},[120],[15,86883],{"className":86884,"style":4383},[124],[15,86886,5628],{"className":86887},[129,133]," 足够小时，",[15,86890,86892,86924],{"className":86891,"translate":19},[23],[15,86893,86895],{"className":86894},[27],[29,86896,86897],{"xmlns":31},[34,86898,86899,86921],{},[37,86900,86901,86907,86909,86911,86913,86915,86917,86919],{},[40,86902,86903,86905],{},[43,86904,36569],{},[43,86906,940],{},[50,86908,100],{},[43,86910,85368],{"mathvariant":65},[50,86912,53],{"stretchy":52},[43,86914,36277],{},[50,86916,91],{"separator":72},[43,86918,1373],{},[50,86920,59],{"stretchy":52},[109,86922,86923],{"encoding":111},"f_S\\in\\Sigma(\\beta,L)",[15,86925,86927,86982],{"className":86926,"ariaHidden":72},[116],[15,86928,86930,86933,86973,86976,86979],{"className":86929},[120],[15,86931],{"className":86932,"style":1682},[124],[15,86934,86936,86939],{"className":86935},[129],[15,86937,36569],{"className":86938,"style":36584},[129,133],[15,86940,86942],{"className":86941},[137],[15,86943,86945,86965],{"className":86944},[141,142],[15,86946,86948,86962],{"className":86947},[146],[15,86949,86951],{"className":86950,"style":1525},[150],[15,86952,86953,86956],{"style":39822},[15,86954],{"className":86955,"style":159},[158],[15,86957,86959],{"className":86958},[163,164,165,166],[15,86960,940],{"className":86961,"style":1013},[129,133,166],[15,86963,175],{"className":86964},[174],[15,86966,86968],{"className":86967},[146],[15,86969,86971],{"className":86970,"style":182},[150],[15,86972],{},[15,86974],{"className":86975,"style":199},[93],[15,86977,100],{"className":86978},[203],[15,86980],{"className":86981,"style":199},[93],[15,86983,86985,86988,86991,86994,86997,87000,87003,87006],{"className":86984},[120],[15,86986],{"className":86987,"style":125},[124],[15,86989,85368],{"className":86990},[129],[15,86992,53],{"className":86993},[188],[15,86995,36277],{"className":86996,"style":36293},[129,133],[15,86998,91],{"className":86999},[254],[15,87001],{"className":87002,"style":262},[93],[15,87004,1373],{"className":87005},[129,133],[15,87007,59],{"className":87008},[195],"：鼓包的高度是 ",[15,87011,87013,87031],{"className":87012,"translate":19},[23],[15,87014,87016],{"className":87015},[27],[29,87017,87018],{"xmlns":31},[34,87019,87020,87028],{},[37,87021,87022],{},[305,87023,87024,87026],{},[43,87025,45],{},[43,87027,36277],{},[109,87029,87030],{"encoding":111},"h^\\beta",[15,87032,87034],{"className":87033,"ariaHidden":72},[116],[15,87035,87037,87040],{"className":87036},[120],[15,87038],{"className":87039,"style":10490},[124],[15,87041,87043,87046],{"className":87042},[129],[15,87044,45],{"className":87045},[129,133],[15,87047,87049],{"className":87048},[137],[15,87050,87052],{"className":87051},[141],[15,87053,87055],{"className":87054},[146],[15,87056,87058],{"className":87057,"style":10490},[150],[15,87059,87060,87063],{"style":383},[15,87061],{"className":87062,"style":159},[158],[15,87064,87066],{"className":87065},[163,164,165,166],[15,87067,36277],{"className":87068,"style":36293},[129,133,166],"、宽度是 ",[15,87071,87073,87086],{"className":87072,"translate":19},[23],[15,87074,87076],{"className":87075},[27],[29,87077,87078],{"xmlns":31},[34,87079,87080,87084],{},[37,87081,87082],{},[43,87083,45],{},[109,87085,45],{"encoding":111},[15,87087,87089],{"className":87088,"ariaHidden":72},[116],[15,87090,87092,87095],{"className":87091},[120],[15,87093],{"className":87094,"style":522},[124],[15,87096,45],{"className":87097},[129,133],[15,87099,87101,87114],{"className":87100,"translate":19},[23],[15,87102,87104],{"className":87103},[27],[29,87105,87106],{"xmlns":31},[34,87107,87108,87112],{},[37,87109,87110],{},[43,87111,36277],{},[109,87113,36280],{"encoding":111},[15,87115,87117],{"className":87116,"ariaHidden":72},[116],[15,87118,87120,87123],{"className":87119},[120],[15,87121],{"className":87122,"style":1682},[124],[15,87124,36277],{"className":87125,"style":36293},[129,133]," 阶导数的量级是 ",[15,87128,87130,87160],{"className":87129,"translate":19},[23],[15,87131,87133],{"className":87132},[27],[29,87134,87135],{"xmlns":31},[34,87136,87137,87157],{},[37,87138,87139,87145,87147,87153,87155],{},[305,87140,87141,87143],{},[43,87142,45],{},[43,87144,36277],{},[43,87146,544],{"mathvariant":65},[305,87148,87149,87151],{},[43,87150,45],{},[43,87152,36277],{},[50,87154,62],{},[308,87156,318],{},[109,87158,87159],{"encoding":111},"h^\\beta\u002Fh^\\beta=1",[15,87161,87163,87239],{"className":87162,"ariaHidden":72},[116],[15,87164,87166,87169,87198,87201,87230,87233,87236],{"className":87165},[120],[15,87167],{"className":87168,"style":10978},[124],[15,87170,87172,87175],{"className":87171},[129],[15,87173,45],{"className":87174},[129,133],[15,87176,87178],{"className":87177},[137],[15,87179,87181],{"className":87180},[141],[15,87182,87184],{"className":87183},[146],[15,87185,87187],{"className":87186,"style":10490},[150],[15,87188,87189,87192],{"style":383},[15,87190],{"className":87191,"style":159},[158],[15,87193,87195],{"className":87194},[163,164,165,166],[15,87196,36277],{"className":87197,"style":36293},[129,133,166],[15,87199,544],{"className":87200},[129],[15,87202,87204,87207],{"className":87203},[129],[15,87205,45],{"className":87206},[129,133],[15,87208,87210],{"className":87209},[137],[15,87211,87213],{"className":87212},[141],[15,87214,87216],{"className":87215},[146],[15,87217,87219],{"className":87218,"style":10490},[150],[15,87220,87221,87224],{"style":383},[15,87222],{"className":87223,"style":159},[158],[15,87225,87227],{"className":87226},[163,164,165,166],[15,87228,36277],{"className":87229,"style":36293},[129,133,166],[15,87231],{"className":87232,"style":199},[93],[15,87234,62],{"className":87235},[203],[15,87237],{"className":87238,"style":199},[93],[15,87240,87242,87245],{"className":87241},[120],[15,87243],{"className":87244,"style":4886},[124],[15,87246,318],{"className":87247},[129],[11,87249,87250,87251,6801,87411,87470,87471,22169],{},"两个函数 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距离只取决于它们不同的那些鼓包。若 ",[15,87472,87474,87505],{"className":87473,"translate":19},[23],[15,87475,87477],{"className":87476},[27],[29,87478,87479],{"xmlns":31},[34,87480,87481,87502],{},[37,87482,87483,87485,87487,87490,87496,87498,87500],{},[43,87484,2945],{"mathvariant":65},[43,87486,940],{},[43,87488,87489],{"mathvariant":65},"△",[305,87491,87492,87494],{},[43,87493,940],{},[50,87495,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[43,87497,2945],{"mathvariant":65},[50,87499,62],{},[43,87501,16562],{},[109,87503,87504],{"encoding":111},"|S\\triangle 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",[15,88349,88351,88381],{"className":88350,"translate":19},[23],[15,88352,88354],{"className":88353},[27],[29,88355,88356],{"xmlns":31},[34,88357,88358,88378],{},[37,88359,88360,88362,88364],{},[43,88361,43246],{},[50,88363,62],{},[305,88365,88366,88368],{},[308,88367,310],{},[37,88369,88370,88376],{},[40,88371,88372,88374],{},[43,88373,5628],{},[308,88375,541],{},[43,88377,348],{},[109,88379,88380],{"encoding":111},"M=2^{c_0m}",[15,88382,88384,88402],{"className":88383,"ariaHidden":72},[116],[15,88385,88387,88390,88393,88396,88399],{"className":88386},[120],[15,88388],{"className":88389,"style":610},[124],[15,88391,43246],{"className":88392,"style":12597},[129,133],[15,88394],{"className":88395,"style":199},[93],[15,88397,62],{"className":88398},[203],[15,88400],{"className":88401,"style":199},[93],[15,88403,88405,88408],{"className":88404},[120],[15,88406],{"className":88407,"style":6377},[124],[15,88409,88411,88414],{"className":88410},[129],[15,88412,310],{"className":88413},[129],[15,88415,88417],{"className":88416},[137],[15,88418,88420],{"className":88419},[141],[15,88421,88423],{"className":88422},[146],[15,88424,88426],{"className":88425,"style":6377},[150],[15,88427,88428,88431],{"style":383},[15,88429],{"className":88430,"style":159},[158],[15,88432,88434],{"className":88433},[163,164,165,166],[15,88435,88437,88477],{"className":88436},[129,166],[15,88438,88440,88443],{"className":88439},[129,166],[15,88441,5628],{"className":88442},[129,133,166],[15,88444,88446],{"className":88445},[137],[15,88447,88449,88469],{"className":88448},[141,142],[15,88450,88452,88466],{"className":88451},[146],[15,88453,88455],{"className":88454,"style":46995},[150],[15,88456,88457,88460],{"style":84059},[15,88458],{"className":88459,"style":15987},[158],[15,88461,88463],{"className":88462},[163,15991,229,166],[15,88464,541],{"className":88465},[129,166],[15,88467,175],{"className":88468},[174],[15,88470,88472],{"className":88471},[146],[15,88473,88475],{"className":88474,"style":16004},[150],[15,88476],{},[15,88478,348],{"className":88479},[129,133,166]," 个码字（",[15,88482,88484,88506],{"className":88483,"translate":19},[23],[15,88485,88487],{"className":88486},[27],[29,88488,88489],{"xmlns":31},[34,88490,88491,88503],{},[37,88492,88493,88499,88501],{},[40,88494,88495,88497],{},[43,88496,5628],{},[308,88498,541],{},[50,88500,4852],{},[308,88502,541],{},[109,88504,88505],{"encoding":111},"c_0>0",[15,88507,88509,88564],{"className":88508,"ariaHidden":72},[116],[15,88510,88512,88515,88555,88558,88561],{"className":88511},[120],[15,88513],{"className":88514,"style":7929},[124],[15,88516,88518,88521],{"className":88517},[129],[15,88519,5628],{"className":88520},[129,133],[15,88522,88524],{"className":88523},[137],[15,88525,88527,88547],{"className":88526},[141,142],[15,88528,88530,88544],{"className":88529},[146],[15,88531,88533],{"className":88532,"style":1051},[150],[15,88534,88535,88538],{"style":154},[15,88536],{"className":88537,"style":159},[158],[15,88539,88541],{"className":88540},[163,164,165,166],[15,88542,541],{"className":88543},[129,166],[15,88545,175],{"className":88546},[174],[15,88548,88550],{"className":88549},[146],[15,88551,88553],{"className":88552,"style":182},[150],[15,88554],{},[15,88556],{"className":88557,"style":199},[93],[15,88559,4852],{"className":88560},[203],[15,88562],{"className":88563,"style":199},[93],[15,88565,88567,88570],{"className":88566},[120],[15,88568],{"className":88569,"style":4886},[124],[15,88571,541],{"className":88572},[129]," 是常数），任意两个码字至少在 ",[15,88575,88577,88595],{"className":88576,"translate":19},[23],[15,88578,88580],{"className":88579},[27],[29,88581,88582],{"xmlns":31},[34,88583,88584,88592],{},[37,88585,88586,88588,88590],{},[43,88587,348],{},[43,88589,544],{"mathvariant":65},[308,88591,14508],{},[109,88593,88594],{"encoding":111},"m\u002F4",[15,88596,88598],{"className":88597,"ariaHidden":72},[116],[15,88599,88601,88604,88607],{"className":88600},[120],[15,88602],{"className":88603,"style":125},[124],[15,88605,348],{"className":88606},[129,133],[15,88608,64748],{"className":88609},[129]," 个位置不同。于是",[15,88612,88614],{"className":88613,"translate":19},[18],[15,88615,88617,88681],{"className":88616,"translate":19},[23],[15,88618,88620],{"className":88619},[27],[29,88621,88622],{"xmlns":31,"display":32},[34,88623,88624,88678],{},[37,88625,88626,88628,88634,88636,88646,88658,88660,88666,88676],{},[50,88627,33823],{"stretchy":52},[40,88629,88630,88632],{},[43,88631,36569],{},[43,88633,940],{},[50,88635,315],{},[40,88637,88638,88640],{},[43,88639,36569],{},[305,88641,88642,88644],{},[43,88643,940],{},[50,88645,21510],{"mathvariant":65,"lspace":2604,"rspace":2604},[23160,88647,88648,88650,88656],{},[50,88649,33823],{"stretchy":52},[305,88651,88652,88654],{},[43,88653,1373],{},[308,88655,310],{},[308,88657,310],{},[50,88659,7473],{},[40,88661,88662,88664],{},[43,88663,5628],{},[308,88665,318],{},[305,88667,88668,88670],{},[43,88669,45],{},[37,88671,88672,88674],{},[308,88673,310],{},[43,88675,36277],{},[50,88677,91],{"separator":72},[109,88679,88680],{"encoding":111},"\\lVert f_S-f_{S'}\\rVert_{L^2}^2\n\\ge c_1h^{2\\beta},",[15,88682,88684,88742,88909],{"className":88683,"ariaHidden":72},[116],[15,88685,88687,88690,88693,88733,88736,88739],{"className":88686},[120],[15,88688],{"className":88689,"style":125},[124],[15,88691,33823],{"className":88692},[188],[15,88694,88696,88699],{"className":88695},[129],[15,88697,36569],{"className":88698,"style":36584},[129,133],[15,88700,88702],{"className":88701},[137],[15,88703,88705,88725],{"className":88704},[141,142],[15,88706,88708,88722],{"className":88707},[146],[15,88709,88711],{"className":88710,"style":1525},[150],[15,88712,88713,88716],{"style":39822},[15,88714],{"className":88715,"style":159},[158],[15,88717,88719],{"className":88718},[163,164,165,166],[15,88720,940],{"className":88721,"style":1013},[129,133,166],[15,88723,175],{"className":88724},[174],[15,88726,88728],{"className":88727},[146],[15,88729,88731],{"className":88730,"style":182},[150],[15,88732],{},[15,88734],{"className":88735,"style":3277},[93],[15,88737,315],{"className":88738},[2436],[15,88740],{"className":88741,"style":3277},[93],[15,88743,88745,88748,88820,88900,88903,88906],{"className":88744},[120],[15,88746],{"className":88747,"style":87775},[124],[15,88749,88751,88754],{"className":88750},[129],[15,88752,36569],{"className":88753,"style":36584},[129,133],[15,88755,88757],{"className":88756},[137],[15,88758,88760,88812],{"className":88759},[141,142],[15,88761,88763,88809],{"className":88762},[146],[15,88764,88766],{"className":88765,"style":1525},[150],[15,88767,88768,88771],{"style":39822},[15,88769],{"className":88770,"style":159},[158],[15,88772,88774],{"className":88773},[163,164,165,166],[15,88775,88777],{"className":88776},[129,166],[15,88778,88780,88783],{"className":88779},[129,166],[15,88781,940],{"className":88782,"style":1013},[129,133,166],[15,88784,88786],{"className":88785},[137],[15,88787,88789],{"className":88788},[141],[15,88790,88792],{"className":88791},[146],[15,88793,88795],{"className":88794,"style":23275},[150],[15,88796,88797,88800],{"style":23278},[15,88798],{"className":88799,"style":15987},[158],[15,88801,88803],{"className":88802},[163,15991,229,166],[15,88804,88806],{"className":88805},[129,166],[15,88807,21510],{"className":88808},[129,166],[15,88810,175],{"className":88811},[174],[15,88813,88815],{"className":88814},[146],[15,88816,88818],{"className":88817,"style":182},[150],[15,88819],{},[15,88821,88823,88826],{"className":88822},[195],[15,88824,33823],{"className":88825},[195],[15,88827,88829],{"className":88828},[137],[15,88830,88832,88892],{"className":88831},[141,142],[15,88833,88835,88889],{"className":88834},[146],[15,88836,88838,88878],{"className":88837,"style":40991},[150],[15,88839,88840,88843],{"style":87868},[15,88841],{"className":88842,"style":159},[158],[15,88844,88846],{"className":88845},[163,164,165,166],[15,88847,88849],{"className":88848},[129,166],[15,88850,88852,88855],{"className":88851},[129,166],[15,88853,1373],{"className":88854},[129,133,166],[15,88856,88858],{"className":88857},[137],[15,88859,88861],{"className":88860},[141],[15,88862,88864],{"className":88863},[146],[15,88865,88867],{"className":88866,"style":85574},[150],[15,88868,88869,88872],{"style":23278},[15,88870],{"className":88871,"style":15987},[158],[15,88873,88875],{"className":88874},[163,15991,229,166],[15,88876,310],{"className":88877},[129,166],[15,88879,88880,88883],{"style":1272},[15,88881],{"className":88882,"style":159},[158],[15,88884,88886],{"className":88885},[163,164,165,166],[15,88887,310],{"className":88888},[129,166],[15,88890,175],{"className":88891},[174],[15,88893,88895],{"className":88894},[146],[15,88896,88898],{"className":88897,"style":87927},[150],[15,88899],{},[15,88901],{"className":88902,"style":199},[93],[15,88904,7473],{"className":88905},[203],[15,88907],{"className":88908,"style":199},[93],[15,88910,88912,88915,88955,88990],{"className":88911},[120],[15,88913],{"className":88914,"style":82069},[124],[15,88916,88918,88921],{"className":88917},[129],[15,88919,5628],{"className":88920},[129,133],[15,88922,88924],{"className":88923},[137],[15,88925,88927,88947],{"className":88926},[141,142],[15,88928,88930,88944],{"className":88929},[146],[15,88931,88933],{"className":88932,"style":1051},[150],[15,88934,88935,88938],{"style":154},[15,88936],{"className":88937,"style":159},[158],[15,88939,88941],{"className":88940},[163,164,165,166],[15,88942,318],{"className":88943},[129,166],[15,88945,175],{"className":88946},[174],[15,88948,88950],{"className":88949},[146],[15,88951,88953],{"className":88952,"style":182},[150],[15,88954],{},[15,88956,88958,88961],{"className":88957},[129],[15,88959,45],{"className":88960},[129,133],[15,88962,88964],{"className":88963},[137],[15,88965,88967],{"className":88966},[141],[15,88968,88970],{"className":88969},[146],[15,88971,88973],{"className":88972,"style":64296},[150],[15,88974,88975,88978],{"style":1272},[15,88976],{"className":88977,"style":159},[158],[15,88979,88981],{"className":88980},[163,164,165,166]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要求 ",[15,90094,90096,90128],{"className":90095,"translate":19},[23],[15,90097,90099],{"className":90098},[27],[29,90100,90101],{"xmlns":31},[34,90102,90103,90125],{},[37,90104,90105,90107,90109,90111,90113,90115],{},[43,90106,14727],{},[50,90108,69],{},[43,90110,43246],{},[50,90112,74420],{},[43,90114,39326],{},[305,90116,90117,90119],{},[43,90118,45],{},[37,90120,90121,90123],{},[308,90122,310],{},[43,90124,36277],{},[109,90126,90127],{"encoding":111},"\\log M\\gg n h^{2\\beta}",[15,90129,90131,90157],{"className":90130,"ariaHidden":72},[116],[15,90132,90134,90137,90142,90145,90148,90151,90154],{"className":90133},[120],[15,90135],{"className":90136,"style":1682},[124],[15,90138,14777,90140],{"className":90139},[217],[15,90141,2247],{"style":2246},[15,90143],{"className":90144,"style":262},[93],[15,90146,43246],{"className":90147,"style":12597},[129,133],[15,90149],{"className":90150,"style":199},[93],[15,90152,74420],{"className":90153},[203],[15,90155],{"className":90156,"style":199},[93],[15,90158,90160,90163,90166],{"className":90159},[120],[15,90161],{"className":90162,"style":10490},[124],[15,90164,39326],{"className":90165},[129,133],[15,90167,90169,90172],{"className":90168},[129],[15,90170,45],{"className":90171},[129,133],[15,90173,90175],{"className":90174},[137],[15,90176,90178],{"className":90177},[141],[15,90179,90181],{"className":90180},[146],[15,90182,90184],{"className":90183,"style":10490},[150],[15,90185,90186,90189],{"style":383},[15,90187],{"className":90188,"style":159},[158],[15,90190,90192],{"className":90191},[163,164,165,166],[15,90193,90195,90198],{"className":90194},[129,166],[15,90196,310],{"className":90197},[129,166],[15,90199,36277],{"className":90200,"style":36293},[129,133,166],"，即",[15,90203,90205],{"className":90204,"translate":19},[18],[15,90206,90208,90279],{"className":90207,"translate":19},[23],[15,90209,90211],{"className":90210},[27],[29,90212,90213],{"xmlns":31,"display":32},[34,90214,90215,90276],{},[37,90216,90217,90227,90229,90231,90241,90243,90246,90248,90250,90252,90274],{},[305,90218,90219,90221],{},[43,90220,45],{},[37,90222,90223,90225],{},[50,90224,315],{},[43,90226,10376],{},[50,90228,74484],{},[43,90230,39326],{},[305,90232,90233,90235],{},[43,90234,45],{},[37,90236,90237,90239],{},[308,90238,310],{},[43,90240,36277],{},[93,90242],{"width":16921},[50,90244,90245],{},"⟺",[93,90247],{"width":16921},[43,90249,45],{},[50,90251,26567],{},[305,90253,90254,90256],{},[43,90255,39326],{},[37,90257,90258,90260],{},[50,90259,315],{},[1443,90261,90262,90264],{},[308,90263,318],{},[37,90265,90266,90268,90270,90272],{},[308,90267,310],{},[43,90269,36277],{},[50,90271,2682],{},[43,90273,10376],{},[43,90275,107],{"mathvariant":65},[109,90277,90278],{"encoding":111},"h^{-d}\\gtrsim n h^{2\\beta}\n\\quad\\Longleftrightarrow\\quad\nh\\lesssim n^{-\\frac{1}{2\\beta+d}}.",[15,90280,90282,90333,90393,90412],{"className":90281,"ariaHidden":72},[116],[15,90283,90285,90289,90324,90327,90330],{"className":90284},[120],[15,90286],{"className":90287,"style":90288},[124],"height:1.1287em;vertical-align:-0.2296em;",[15,90290,90292,90295],{"className":90291},[129],[15,90293,45],{"className":90294},[129,133],[15,90296,90298],{"className":90297},[137],[15,90299,90301],{"className":90300},[141],[15,90302,90304],{"className":90303},[146],[15,90305,90307],{"className":90306,"style":64296},[150],[15,90308,90309,90312],{"style":1272},[15,90310],{"className":90311,"style":159},[158],[15,90313,90315],{"className":90314},[163,164,165,166],[15,90316,90318,90321],{"className":90317},[129,166],[15,90319,315],{"className":90320},[129,166],[15,90322,10376],{"className":90323},[129,133,166],[15,90325],{"className":90326,"style":199},[93],[15,90328,74484],{"className":90329},[203,13316],[15,90331],{"className":90332,"style":199},[93],[15,90334,90336,90340,90343,90378,90381,90384,90387,90390],{"className":90335},[120],[15,90337],{"className":90338,"style":90339},[124],"height:0.9231em;vertical-align:-0.024em;",[15,90341,39326],{"className":90342},[129,133],[15,90344,90346,90349],{"className":90345},[129],[15,90347,45],{"className":90348},[129,133],[15,90350,90352],{"className":90351},[137],[15,90353,90355],{"className":90354},[141],[15,90356,90358],{"className":90357},[146],[15,90359,90361],{"className":90360,"style":64296},[150],[15,90362,90363,90366],{"style":1272},[15,90364],{"className":90365,"style":159},[158],[15,90367,90369],{"className":90368},[163,164,165,166],[15,90370,90372,90375],{"className":90371},[129,166],[15,90373,310],{"className":90374},[129,166],[15,90376,36277],{"className":90377,"style":36293},[129,133,166],[15,90379],{"className":90380,"style":17294},[93],[15,90382],{"className":90383,"style":199},[93],[15,90385,90245],{"className":90386},[203],[15,90388],{"className":90389,"style":17294},[93],[15,90391],{"className":90392,"style":199},[93],[15,90394,90396,90400,90403,90406,90409],{"className":90395},[120],[15,90397],{"className":90398,"style":90399},[124],"height:0.9592em;vertical-align:-0.2296em;",[15,90401,45],{"className":90402},[129,133],[15,90404],{"className":90405,"style":199},[93],[15,90407,26567],{"className":90408},[203,13316],[15,90410],{"className":90411,"style":199},[93],[15,90413,90415,90419,90530],{"className":90414},[120],[15,90416],{"className":90417,"style":90418},[124],"height:1.0368em;",[15,90420,90422,90425],{"className":90421},[129],[15,90423,39326],{"className":90424},[129,133],[15,90426,90428],{"className":90427},[137],[15,90429,90431],{"className":90430},[141],[15,90432,90434],{"className":90433},[146],[15,90435,90437],{"className":90436,"style":90418},[150],[15,90438,90439,90442],{"style":85688},[15,90440],{"className":90441,"style":1823},[158],[15,90443,90445],{"className":90444},[163,164,165,166],[15,90446,90448,90451],{"className":90447},[129,166],[15,90449,315],{"className":90450},[129,166],[15,90452,90454,90457,90527],{"className":90453},[129,166],[15,90455],{"className":90456},[188,1803,163,15991,85707],[15,90458,90460],{"className":90459},[1443],[15,90461,90463,90519],{"className":90462},[141,142],[15,90464,90466,90516],{"className":90465},[146],[15,90467,90470,90493,90501],{"className":90468,"style":90469},[150],"height:0.8443em;",[15,90471,90472,90475],{"style":85723},[15,90473],{"className":90474,"style":1823},[158],[15,90476,90478],{"className":90477},[163,15991,229,166],[15,90479,90481,90484,90487,90490],{"className":90480},[129,166],[15,90482,310],{"className":90483},[129,166],[15,90485,36277],{"className":90486,"style":36293},[129,133,166],[15,90488,2682],{"className":90489},[2436,166],[15,90491,10376],{"className":90492},[129,133,166],[15,90494,90495,90498],{"style":85747},[15,90496],{"className":90497,"style":1823},[158],[15,90499],{"className":90500,"style":85754},[1839,166],[15,90502,90504,90507],{"style":90503},"top:-3.384em;",[15,90505],{"className":90506,"style":1823},[158],[15,90508,90510],{"className":90509},[163,15991,229,166],[15,90511,90513],{"className":90512},[129,166],[15,90514,318],{"className":90515},[129,166],[15,90517,175],{"className":90518},[174],[15,90520,90522],{"className":90521},[146],[15,90523,90525],{"className":90524,"style":85782},[150],[15,90526],{},[15,90528],{"className":90529},[195,1803,163,15991,85707],[15,90531,107],{"className":90532},[129],[11,90534,7441,90535,90658],{},[15,90536,90538,90578],{"className":90537,"translate":19},[23],[15,90539,90541],{"className":90540},[27],[29,90542,90543],{"xmlns":31},[34,90544,90545,90575],{},[37,90546,90547,90549,90551],{},[43,90548,45],{},[50,90550,64766],{},[305,90552,90553,90555],{},[43,90554,39326],{},[37,90556,90557,90559,90561,90563,90565,90567,90569,90571,90573],{},[50,90558,315],{},[308,90560,318],{},[43,90562,544],{"mathvariant":65},[50,90564,53],{"stretchy":52},[308,90566,310],{},[43,90568,36277],{},[50,90570,2682],{},[43,90572,1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n^{-1\u002F(2\\beta+d)}",[15,90579,90581,90599],{"className":90580,"ariaHidden":72},[116],[15,90582,90584,90587,90590,90593,90596],{"className":90583},[120],[15,90585],{"className":90586,"style":522},[124],[15,90588,45],{"className":90589},[129,133],[15,90591],{"className":90592,"style":199},[93],[15,90594,64766],{"className":90595},[203],[15,90597],{"className":90598,"style":199},[93],[15,90600,90602,90605],{"className":90601},[120],[15,90603],{"className":90604,"style":14948},[124],[15,90606,90608,90611],{"className":90607},[129],[15,90609,39326],{"className":90610},[129,133],[15,90612,90614],{"className":90613},[137],[15,90615,90617],{"className":90616},[141],[15,90618,90620],{"className":90619},[146],[15,90621,90623],{"className":90622,"style":14948},[150],[15,90624,90625,90628],{"style":383},[15,90626],{"className":90627,"style":159},[158],[15,90629,90631],{"className":90630},[163,164,165,166],[15,90632,90634,90637,90640,90643,90646,90649,90652,90655],{"className":90633},[129,166],[15,90635,315],{"className":90636},[129,166],[15,90638,29852],{"className":90639},[129,166],[15,90641,53],{"className":90642},[188,166],[15,90644,310],{"className":90645},[129,166],[15,90647,36277],{"className":90648,"style":36293},[129,133,166],[15,90650,2682],{"className":90651},[2436,166],[15,90653,10376],{"className":90654},[129,133,166],[15,90656,59],{"className":90657},[195,166],"，下界就是",[15,90660,90662],{"className":90661,"translate":19},[18],[15,90663,90665,90725],{"className":90664,"translate":19},[23],[15,90666,90668],{"className":90667},[27],[29,90669,90670],{"xmlns":31,"display":32},[34,90671,90672,90722],{},[37,90673,90674,90680,90682,90692,90694,90720],{},[40,90675,90676,90678],{},[43,90677,43246],{"mathvariant":28910},[43,90679,39326],{},[50,90681,74484],{},[305,90683,90684,90686],{},[43,90685,45],{},[37,90687,90688,90690],{},[308,90689,310],{},[43,90691,36277],{},[50,90693,64766],{},[305,90695,90696,90698],{},[43,90697,39326],{},[37,90699,90700,90702],{},[50,90701,315],{},[1443,90703,90704,90710],{},[37,90705,90706,90708],{},[308,90707,310],{},[43,90709,36277],{},[37,90711,90712,90714,90716,90718],{},[308,90713,310],{},[43,90715,36277],{},[50,90717,2682],{},[43,90719,10376],{},[50,90721,91],{"separator":72},[109,90723,90724],{"encoding":111},"\\mathfrak M_n\\gtrsim h^{2\\beta}\n\\asymp n^{-\\frac{2\\beta}{2\\beta+d}},",[15,90726,90728,90783,90833],{"className":90727,"ariaHidden":72},[116],[15,90729,90731,90734,90774,90777,90780],{"className":90730},[120],[15,90732],{"className":90733,"style":90399},[124],[15,90735,90737,90740],{"className":90736},[129],[15,90738,43246],{"className":90739},[129,28937],[15,90741,90743],{"className":90742},[137],[15,90744,90746,90766],{"className":90745},[141,142],[15,90747,90749,90763],{"className":90748},[146],[15,90750,90752],{"className":90751,"style":1163},[150],[15,90753,90754,90757],{"style":154},[15,90755],{"className":90756,"style":159},[158],[15,90758,90760],{"className":90759},[163,164,165,166],[15,90761,39326],{"className":90762},[129,133,166],[15,90764,175],{"className":90765},[174],[15,90767,90769],{"className":90768},[146],[15,90770,90772],{"className":90771,"style":182},[150],[15,90773],{},[15,90775],{"className":90776,"style":199},[93],[15,90778,74484],{"className":90779},[203,13316],[15,90781],{"className":90782,"style":199},[93],[15,90784,90786,90789,90824,90827,90830],{"className":90785},[120],[15,90787],{"className":90788,"style":64296},[124],[15,90790,90792,90795],{"className":90791},[129],[15,90793,45],{"className":90794},[129,133],[15,90796,90798],{"className":90797},[137],[15,90799,90801],{"className":90800},[141],[15,90802,90804],{"className":90803},[146],[15,90805,90807],{"className":90806,"style":64296},[150],[15,90808,90809,90812],{"style":1272},[15,90810],{"className":90811,"style":159},[158],[15,90813,90815],{"className":90814},[163,164,165,166],[15,90816,90818,90821],{"className":90817},[129,166],[15,90819,310],{"className":90820},[129,166],[15,90822,36277],{"className":90823,"style":36293},[129,133,166],[15,90825],{"className":90826,"style":199},[93],[15,90828,64766],{"className":90829},[203],[15,90831],{"className":90832,"style":199},[93],[15,90834,90836,90840,90952],{"className":90835},[120],[15,90837],{"className":90838,"style":90839},[124],"height:1.3111em;vertical-align:-0.1944em;",[15,90841,90843,90846],{"className":90842},[129],[15,90844,39326],{"className":90845},[129,133],[15,90847,90849],{"className":90848},[137],[15,90850,90852],{"className":90851},[141],[15,90853,90855],{"className":90854},[146],[15,90856,90858],{"className":90857,"style":85667},[150],[15,90859,90860,90863],{"style":85688},[15,90861],{"className":90862,"style":1823},[158],[15,90864,90866],{"className":90865},[163,164,165,166],[15,90867,90869,90872],{"className":90868},[129,166],[15,90870,315],{"className":90871},[129,166],[15,90873,90875,90878,90949],{"className":90874},[129,166],[15,90876],{"className":90877},[188,1803,163,15991,85707],[15,90879,90881],{"className":90880},[1443],[15,90882,90884,90941],{"className":90883},[141,142],[15,90885,90887,90938],{"className":90886},[146],[15,90888,90890,90913,90921],{"className":90889,"style":85720},[150],[15,90891,90892,90895],{"style":85723},[15,90893],{"className":90894,"style":1823},[158],[15,90896,90898],{"className":90897},[163,15991,229,166],[15,90899,90901,90904,90907,90910],{"className":90900},[129,166],[15,90902,310],{"className":90903},[129,166],[15,90905,36277],{"className":90906,"style":36293},[129,133,166],[15,90908,2682],{"className":90909},[2436,166],[15,90911,10376],{"className":90912},[129,133,166],[15,90914,90915,90918],{"style":85747},[15,90916],{"className":90917,"style":1823},[158],[15,90919],{"className":90920,"style":85754},[1839,166],[15,90922,90923,90926],{"style":85757},[15,90924],{"className":90925,"style":1823},[158],[15,90927,90929],{"className":90928},[163,15991,229,166],[15,90930,90932,90935],{"className":90931},[129,166],[15,90933,310],{"className":90934},[129,166],[15,90936,36277],{"className":90937,"style":36293},[129,133,166],[15,90939,175],{"className":90940},[174],[15,90942,90944],{"className":90943},[146],[15,90945,90947],{"className":90946,"style":85782},[150],[15,90948],{},[15,90950],{"className":90951},[195,1803,163,15991,85707],[15,90953,91],{"className":90954},[254],[11,90956,90957],{},"与上界同阶。证毕。",[11,90959,90960,90961,91208,91209,91237],{},"这个构造值得反复看：",[698,90962,90963,90964,90992,90993,91062,91063,91137,91138,91207],{},"鼓包的宽度 ",[15,90965,90967,90980],{"className":90966,"translate":19},[23],[15,90968,90970],{"className":90969},[27],[29,90971,90972],{"xmlns":31},[34,90973,90974,90978],{},[37,90975,90976],{},[43,90977,45],{},[109,90979,45],{"encoding":111},[15,90981,90983],{"className":90982,"ariaHidden":72},[116],[15,90984,90986,90989],{"className":90985},[120],[15,90987],{"className":90988,"style":522},[124],[15,90990,45],{"className":90991},[129,133]," 控制两件事——损失间隔（",[15,90994,90996,91018],{"className":90995,"translate":19},[23],[15,90997,90999],{"className":90998},[27],[29,91000,91001],{"xmlns":31},[34,91002,91003,91015],{},[37,91004,91005],{},[305,91006,91007,91009],{},[43,91008,45],{},[37,91010,91011,91013],{},[308,91012,310],{},[43,91014,36277],{},[109,91016,91017],{"encoding":111},"h^{2\\beta}",[15,91019,91021],{"className":91020,"ariaHidden":72},[116],[15,91022,91024,91027],{"className":91023},[120],[15,91025],{"className":91026,"style":10490},[124],[15,91028,91030,91033],{"className":91029},[129],[15,91031,45],{"className":91032},[129,133],[15,91034,91036],{"className":91035},[137],[15,91037,91039],{"className":91038},[141],[15,91040,91042],{"className":91041},[146],[15,91043,91045],{"className":91044,"style":10490},[150],[15,91046,91047,91050],{"style":383},[15,91048],{"className":91049,"style":159},[158],[15,91051,91053],{"className":91052},[163,164,165,166],[15,91054,91056,91059],{"className":91055},[129,166],[15,91057,310],{"className":91058},[129,166],[15,91060,36277],{"className":91061,"style":36293},[129,133,166],"）和 KL（",[15,91064,91066,91090],{"className":91065,"translate":19},[23],[15,91067,91069],{"className":91068},[27],[29,91070,91071],{"xmlns":31},[34,91072,91073,91087],{},[37,91074,91075,91077],{},[43,91076,39326],{},[305,91078,91079,91081],{},[43,91080,45],{},[37,91082,91083,91085],{},[308,91084,310],{},[43,91086,36277],{},[109,91088,91089],{"encoding":111},"n h^{2\\beta}",[15,91091,91093],{"className":91092,"ariaHidden":72},[116],[15,91094,91096,91099,91102],{"className":91095},[120],[15,91097],{"className":91098,"style":10490},[124],[15,91100,39326],{"className":91101},[129,133],[15,91103,91105,91108],{"className":91104},[129],[15,91106,45],{"className":91107},[129,133],[15,91109,91111],{"className":91110},[137],[15,91112,91114],{"className":91113},[141],[15,91115,91117],{"className":91116},[146],[15,91118,91120],{"className":91119,"style":10490},[150],[15,91121,91122,91125],{"style":383},[15,91123],{"className":91124,"style":159},[158],[15,91126,91128],{"className":91127},[163,164,165,166],[15,91129,91131,91134],{"className":91130},[129,166],[15,91132,310],{"className":91133},[129,166],[15,91135,36277],{"className":91136,"style":36293},[129,133,166],"），而鼓包的个数 ",[15,91139,91141,91163],{"className":91140,"translate":19},[23],[15,91142,91144],{"className":91143},[27],[29,91145,91146],{"xmlns":31},[34,91147,91148,91160],{},[37,91149,91150],{},[305,91151,91152,91154],{},[43,91153,45],{},[37,91155,91156,91158],{},[50,91157,315],{},[43,91159,10376],{},[109,91161,91162],{"encoding":111},"h^{-d}",[15,91164,91166],{"className":91165,"ariaHidden":72},[116],[15,91167,91169,91172],{"className":91168},[120],[15,91170],{"className":91171,"style":10490},[124],[15,91173,91175,91178],{"className":91174},[129],[15,91176,45],{"className":91177},[129,133],[15,91179,91181],{"className":91180},[137],[15,91182,91184],{"className":91183},[141],[15,91185,91187],{"className":91186},[146],[15,91188,91190],{"className":91189,"style":10490},[150],[15,91191,91192,91195],{"style":383},[15,91193],{"className":91194,"style":159},[158],[15,91196,91198],{"className":91197},[163,164,165,166],[15,91199,91201,91204],{"className":91200},[129,166],[15,91202,315],{"className":91203},[129,166],[15,91205,10376],{"className":91206},[129,133,166]," 控制假设的个数。"," 三者平衡，就得到非参数速率。维度 ",[15,91210,91212,91225],{"className":91211,"translate":19},[23],[15,91213,91215],{"className":91214},[27],[29,91216,91217],{"xmlns":31},[34,91218,91219,91223],{},[37,91220,91221],{},[43,91222,10376],{},[109,91224,10376],{"encoding":111},[15,91226,91228],{"className":91227,"ariaHidden":72},[116],[15,91229,91231,91234],{"className":91230},[120],[15,91232],{"className":91233,"style":522},[124],[15,91235,10376],{"className":91236},[129,133]," 出现在指数上，正是因为它决定了「同样宽度下能放多少个鼓包」。",[575,91239,91241],{"id":91240},"_63-一些经典速率","6.3 一些经典速率",[18040,91243,91244,91257],{},[18043,91245,91246],{},[18046,91247,91248,91251,91254],{},[18049,91249,91250],{},"问题",[18049,91252,91253],{},"参数类",[18049,91255,91256],{},"极小极大速率",[18057,91258,91259,91471,91645,91818,92196],{},[18046,91260,91261,91292,91396],{},[18062,91262,91263,91291],{},[15,91264,91266,91279],{"className":91265,"translate":19},[23],[15,91267,91269],{"className":91268},[27],[29,91270,91271],{"xmlns":31},[34,91272,91273,91277],{},[37,91274,91275],{},[43,91276,10376],{},[109,91278,10376],{"encoding":111},[15,91280,91282],{"className":91281,"ariaHidden":72},[116],[15,91283,91285,91288],{"className":91284},[120],[15,91286],{"className":91287,"style":522},[124],[15,91289,10376],{"className":91290},[129,133]," 维高斯均值",[18062,91293,91294],{},[15,91295,91297,91323],{"className":91296,"translate":19},[23],[15,91298,91300],{"className":91299},[27],[29,91301,91302],{"xmlns":31},[34,91303,91304,91320],{},[37,91305,91306,91308,91310,91316,91318],{},[50,91307,33823],{"stretchy":52},[43,91309,48],{},[40,91311,91312,91314],{},[50,91313,33823],{"stretchy":52},[308,91315,310],{},[50,91317,2717],{},[43,91319,104],{},[109,91321,91322],{"encoding":111},"\\lVert\\theta\\rVert_2\\le R",[15,91324,91326,91387],{"className":91325,"ariaHidden":72},[116],[15,91327,91329,91332,91335,91338,91378,91381,91384],{"className":91328},[120],[15,91330],{"className":91331,"style":125},[124],[15,91333,33823],{"className":91334},[188],[15,91336,48],{"className":91337,"style":170},[129,133],[15,91339,91341,91344],{"className":91340},[195],[15,91342,33823],{"className":91343},[195],[15,91345,91347],{"className":91346},[137],[15,91348,91350,91370],{"className":91349},[141,142],[15,91351,91353,91367],{"className":91352},[146],[15,91354,91356],{"className":91355,"style":1051},[150],[15,91357,91358,91361],{"style":154},[15,91359],{"className":91360,"style":159},[158],[15,91362,91364],{"className":91363},[163,164,165,166],[15,91365,310],{"className":91366},[129,166],[15,91368,175],{"className":91369},[174],[15,91371,91373],{"className":91372},[146],[15,91374,91376],{"className":91375,"style":182},[150],[15,91377],{},[15,91379],{"className":91380,"style":199},[93],[15,91382,2717],{"className":91383},[203],[15,91385],{"className":91386,"style":199},[93],[15,91388,91390,91393],{"className":91389},[120],[15,91391],{"className":91392,"style":610},[124],[15,91394,104],{"className":91395,"style":33871},[129,133],[18062,91397,91398],{},[15,91399,91401,91424],{"className":91400,"translate":19},[23],[15,91402,91404],{"className":91403},[27],[29,91405,91406],{"xmlns":31},[34,91407,91408,91422],{},[37,91409,91410,91416,91418,91420],{},[305,91411,91412,91414],{},[43,91413,27294],{},[308,91415,310],{},[43,91417,10376],{},[43,91419,544],{"mathvariant":65},[43,91421,39326],{},[109,91423,64919],{"encoding":111},[15,91425,91427],{"className":91426,"ariaHidden":72},[116],[15,91428,91430,91433,91462,91465,91468],{"className":91429},[120],[15,91431],{"className":91432,"style":34009},[124],[15,91434,91436,91439],{"className":91435},[129],[15,91437,27294],{"className":91438,"style":887},[129,133],[15,91440,91442],{"className":91441},[137],[15,91443,91445],{"className":91444},[141],[15,91446,91448],{"className":91447},[146],[15,91449,91451],{"className":91450,"style":380},[150],[15,91452,91453,91456],{"style":383},[15,91454],{"className":91455,"style":159},[158],[15,91457,91459],{"className":91458},[163,164,165,166],[15,91460,310],{"className":91461},[129,166],[15,91463,10376],{"className":91464},[129,133],[15,91466,544],{"className":91467},[129],[15,91469,39326],{"className":91470},[129,133],[18046,91472,91473,91476,91535],{},[18062,91474,91475],{},"非参数回归",[18062,91477,91478,91506,91507],{},[15,91479,91481,91494],{"className":91480,"translate":19},[23],[15,91482,91484],{"className":91483},[27],[29,91485,91486],{"xmlns":31},[34,91487,91488,91492],{},[37,91489,91490],{},[43,91491,36277],{},[109,91493,36280],{"encoding":111},[15,91495,91497],{"className":91496,"ariaHidden":72},[116],[15,91498,91500,91503],{"className":91499},[120],[15,91501],{"className":91502,"style":1682},[124],[15,91504,36277],{"className":91505,"style":36293},[129,133],"-Hölder，",[15,91508,91510,91523],{"className":91509,"translate":19},[23],[15,91511,91513],{"className":91512},[27],[29,91514,91515],{"xmlns":31},[34,91516,91517,91521],{},[37,91518,91519],{},[43,91520,1373],{},[109,91522,1373],{"encoding":111},[15,91524,91526],{"className":91525,"ariaHidden":72},[116],[15,91527,91529,91532],{"className":91528},[120],[15,91530],{"className":91531,"style":610},[124],[15,91533,1373],{"className":91534},[129,133],[18062,91536,91537],{},[15,91538,91540,91577],{"className":91539,"translate":19},[23],[15,91541,91543],{"className":91542},[27],[29,91544,91545],{"xmlns":31},[34,91546,91547,91575],{},[37,91548,91549],{},[305,91550,91551,91553],{},[43,91552,39326],{},[37,91554,91555,91557,91559,91561,91563,91565,91567,91569,91571,91573],{},[50,91556,315],{},[308,91558,310],{},[43,91560,36277],{},[43,91562,544],{"mathvariant":65},[50,91564,53],{"stretchy":52},[308,91566,310],{},[43,91568,36277],{},[50,91570,2682],{},[43,91572,10376],{},[50,91574,59],{"stretchy":52},[109,91576,75513],{"encoding":111},[15,91578,91580],{"className":91579,"ariaHidden":72},[116],[15,91581,91583,91586],{"className":91582},[120],[15,91584],{"className":91585,"style":14948},[124],[15,91587,91589,91592],{"className":91588},[129],[15,91590,39326],{"className":91591},[129,133],[15,91593,91595],{"className":91594},[137],[15,91596,91598],{"className":91597},[141],[15,91599,91601],{"className":91600},[146],[15,91602,91604],{"className":91603,"style":14948},[150],[15,91605,91606,91609],{"style":383},[15,91607],{"className":91608,"style":159},[158],[15,91610,91612],{"className":91611},[163,164,165,166],[15,91613,91615,91618,91621,91624,91627,91630,91633,91636,91639,91642],{"className":91614},[129,166],[15,91616,315],{"className":91617},[129,166],[15,91619,310],{"className":91620},[129,166],[15,91622,36277],{"className":91623,"style":36293},[129,133,166],[15,91625,544],{"className":91626},[129,166],[15,91628,53],{"className":91629},[188,166],[15,91631,310],{"className":91632},[129,166],[15,91634,36277],{"classNa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s\\ll d",[15,91888,91890,91951,91969],{"className":91889,"ariaHidden":72},[116],[15,91891,91893,91896,91899,91902,91942,91945,91948],{"className":91892},[120],[15,91894],{"className":91895,"style":125},[124],[15,91897,33823],{"className":91898},[188],[15,91900,48],{"className":91901,"style":170},[129,133],[15,91903,91905,91908],{"className":91904},[195],[15,91906,33823],{"className":91907},[195],[15,91909,91911],{"className":91910},[137],[15,91912,91914,91934],{"className":91913},[141,142],[15,91915,91917,91931],{"className":91916},[146],[15,91918,91920],{"className":91919,"style":1051},[150],[15,91921,91922,91925],{"style":154},[15,91923],{"className":91924,"style":159},[158],[15,91926,91928],{"className":91927},[163,164,165,166],[15,91929,541],{"className":91930},[129,166],[15,91932,175],{"className":91933},[174],[15,91935,91937],{"className":91936},[146],[15,91938,91940],{"className":91939,"style":182},[150],[15,91941],{},[15,91943],{"className":91944,"style":199},[93],[15,91946,2717],{"className":91947},[203],[15,91949],{"className":91950,"style":199},[93],[15,91952,91954,91957,91960,91963,91966],{"className":91953},[120],[15,91955],{"className":91956,"style":4867},[124],[15,91958,22154],{"className":91959},[129,133],[15,91961],{"className":91962,"style":199},[93],[15,91964,91881],{"className":91965},[203],[15,91967],{"className":91968,"style":199},[93],[15,91970,91972,91975],{"className":91971},[120],[15,91973],{"className":91974,"style":522},[124],[15,91976,10376],{"className":91977},[129,133],[18062,91979,91980],{},[15,91981,91983,92023],{"className":91982,"translate":19},[23],[15,91984,91986],{"className":91985},[27],[29,91987,91988],{"xmlns":31},[34,91989,91990,92020],{},[37,91991,91992,92008,92010,92012],{},[2612,91993,91994],{"scriptlevel":541,"displaystyle":72},[1443,91995,91996,92006],{},[37,91997,91998,92004],{},[305,91999,92000,92002],{},[43,92001,27294],{},[308,92003,310],{},[43,92005,22154],{},[43,92007,39326],{},[43,92009,14727],{},[50,92011,69],{},[2612,92013,92014],{"scriptlevel":541,"displaystyle":72},[1443,92015,92016,92018],{},[43,92017,10376],{},[43,92019,22154],{},[109,92021,92022],{"encoding":111},"\\dfrac{\\sigma^2s}{n}\\log\\dfrac ds",[15,92024,92026],{"className":92025,"ariaHidden":72},[116],[15,92027,92029,92032,92123,92126,92131,92134],{"className":92028},[120],[15,92030],{"className":92031,"style":82509},[124],[15,92033,92035,92038,92120],{"className":92034},[129],[15,92036],{"className":92037},[188,1803],[15,92039,92041],{"className":92040},[1443],[15,92042,92044,92112],{"className":92043},[141,142],[15,92045,92047,92109],{"className":92046},[146],[15,92048,92050,92061,92069],{"className":92049,"style":63623},[150],[15,92051,92052,92055],{"style":1819},[15,92053],{"className":92054,"style":1823},[158],[15,92056,92058],{"className":92057},[129],[15,92059,39326],{"className":92060},[129,133],[15,92062,92063,92066],{"style":1832},[15,92064],{"className":92065,"style":1823},[158],[15,92067],{"className":92068,"style":1840},[1839],[15,92070,92071,92074],{"style":1843},[15,92072],{"className":92073,"style":1823},[158],[15,92075,92077,92106],{"className":92076},[129],[15,92078,92080,92083],{"className":92079},[129],[15,92081,27294],{"className":92082,"style":887},[129,133],[15,92084,92086],{"className":92085},[137],[15,92087,92089],{"className":92088},[141],[15,92090,92092],{"className":92091},[146],[15,92093,92095],{"className":92094,"style":380},[150],[15,92096,92097,92100],{"style":383},[15,92098],{"className":92099,"style":159},[158],[15,92101,92103],{"className":92102},[163,164,165,166],[15,92104,310],{"className":92105},[129,166],[15,92107,22154],{"className":92108},[129,133],[15,92110,175],{"className":92111},[174],[15,92113,92115],{"className":92114},[146],[15,92116,92118],{"className":92117,"style":1862},[150],[15,92119],{},[15,92121],{"className":92122},[195,1803],[15,92124],{"className":92125,"style":262},[93],[15,92127,14777,92129],{"className":92128},[217],[15,92130,2247],{"style":2246},[15,92132],{"className":92133,"style":262},[93],[15,92135,92137,92140,92193],{"className":92136},[129],[15,92138],{"className":92139},[188,1803],[15,92141,92143],{"className":92142},[1443],[15,92144,92146,92185],{"className":92145},[141,142],[15,92147,92149,92182],{"className":92148},[146],[15,92150,92152,92163,92171],{"className":92151,"style":17171},[150],[15,92153,92154,92157],{"style":1819},[15,92155],{"className":92156,"style":1823},[158],[15,92158,92160],{"className":92159},[129],[15,92161,22154],{"className":92162},[129,133],[15,92164,92165,92168],{"style":1832},[15,92166],{"className":92167,"style":1823},[158],[15,92169],{"className":92170,"style":1840},[1839],[15,92172,92173,92176],{"style":1843},[15,92174],{"className":92175,"style":1823},[158],[15,92177,92179],{"className":92178},[129],[15,92180,10376],{"className":92181},[129,133],[15,92183,175],{"className":92184},[174],[15,92186,92188],{"className":92187},[146],[15,92189,92191],{"className":92190,"style":1862},[150],[15,92192],{},[15,92194],{"className":92195},[195,1803],[18046,92197,92198,92201,92204],{},[18062,92199,92200],{},"一维单调回归",[18062,92202,92203],{},"单调、有界",[18062,92205,92206],{},[15,92207,92209,92235],{"className":92208,"translate":19},[23],[15,92210,92212],{"className":92211},[27],[29,92213,92214],{"xmlns":31},[34,92215,92216,92232],{},[37,92217,92218],{},[305,92219,92220,92222],{},[43,92221,39326],{},[37,92223,92224,92226,92228,92230],{},[50,92225,315],{},[308,92227,310],{},[43,92229,544],{"mathvariant":65},[308,92231,9879],{},[109,92233,92234],{"encoding":111},"n^{-2\u002F3}",[15,92236,92238],{"className":92237,"ariaHidden":72},[116],[15,92239,92241,92244],{"className":92240},[120],[15,92242],{"className":92243,"style":14948},[124],[15,92245,92247,92250],{"className":92246},[129],[15,92248,39326],{"className":92249},[129,133],[15,92251,92253],{"className":92252},[137],[15,92254,92256],{"className":92255},[141],[15,92257,92259],{"className":92258},[146],[15,92260,92262],{"className":92261,"style":14948},[150],[15,92263,92264,92267],{"style":383},[15,92265],{"className":92266,"style":159},[158],[15,92268,92270],{"className":92269},[163,164,165,166],[15,92271,92273,92276],{"className":92272},[129,166],[15,92274,315],{"className":92275},[129,166],[15,92277,92279],{"className":92278},[129,166],"2\u002F3",[11,92281,92282],{},"这张表里的每一个速率，都可以用本文的三种方法之一推出来。稀疏均值的下界通常用 Fano 加一个随机支撑的打包；单调回归则用 Le Cam 加一个局部的「台阶」构造。",[570,92284,92286],{"id":92285},"七三种方法的比较","七、三种方法的比较",[18040,92288,92289,92308],{},[18043,92290,92291],{},[18046,92292,92293,92296,92299,92302,92305],{},[18049,92294,92295],{},"方法",[18049,92297,92298],{},"假设个数",[18049,92300,92301],{},"距离",[18049,92303,92304],{},"优势",[18049,92306,92307],{},"局限",[18057,92309,92310,92354,92398],{},[18046,92311,92312,92315,92345,92348,92351],{},[18062,92313,92314],{},"Le Cam",[18062,92316,92317],{},[15,92318,92320,92333],{"className":92319,"translate":19},[23],[15,92321,92323],{"className":92322},[27],[29,92324,92325],{"xmlns":31},[34,92326,92327,92331],{},[37,92328,92329],{},[308,92330,310],{},[109,92332,310],{"encoding":111},[15,92334,92336],{"className":92335,"ariaHidden":72},[116],[15,92337,92339,92342],{"className":92338},[120],[15,92340],{"className":92341,"style":4886},[124],[15,92343,310],{"className":92344},[129],[18062,92346,92347],{},"TV \u002F KL",[18062,92349,92350],{},"简单，常数清楚",[18062,92352,92353],{},"只探测一个方向，高维问题会松",[18046,92355,92356,92359,92389,92392,92395],{},[18062,92357,92358],{},"Fano",[18062,92360,92361],{},[15,92362,92364,92377],{"className":92363,"translate":19},[23],[15,92365,92367],{"className":92366},[27],[29,92368,92369],{"xmlns":31},[34,92370,92371,92375],{},[37,92372,92373],{},[43,92374,43246],{},[109,92376,43246],{"encoding":111},[15,92378,92380],{"className":92379,"ariaHidden":72},[116],[15,92381,92383,92386],{"className":92382},[120],[15,92384],{"className":92385,"style":610},[124],[15,92387,43246],{"className":92388,"style":12597},[129,133],[18062,92390,92391],{},"互信息 \u002F KL",[18062,92393,92394],{},"通用，适合高维与非参数",[18062,92396,92397],{},"需要构造打包，常数较松",[18046,92399,92400,92403,92463,92466,92469],{},[18062,92401,92402],{},"Assouad",[18062,92404,92405],{},[15,92406,92408,92425],{"className":92407,"translate":19},[23],[15,92409,92411],{"className":92410},[27],[29,92412,92413],{"xmlns":31},[34,92414,92415,92423],{},[37,92416,92417],{},[305,92418,92419,92421],{},[308,92420,310],{},[43,92422,348],{},[109,92424,6367],{"encoding":111},[15,92426,92428],{"className":92427,"ariaHidden":72},[116],[15,92429,92431,92434],{"className":92430},[120],[15,92432],{"className":92433,"style":6377},[124],[15,92435,92437,92440],{"className":92436},[129],[15,92438,310],{"className":92439},[129],[15,92441,92443],{"className":92442},[137],[15,92444,92446],{"className":92445},[141],[15,92447,92449],{"className":92448},[146],[15,92450,92452],{"className":92451,"style":6377},[150],[15,92453,92454,92457],{"style":383},[15,92455],{"className":92456,"style":159},[158],[15,92458,92460],{"className":92459},[163,164,165,166],[15,92461,348],{"className":92462},[129,133,166],[18062,92464,92465],{},"逐坐标 TV",[18062,92467,92468],{},"乘积结构下最锋利",[18062,92470,92471],{},"只适用于超立方体结构",[11,92473,92474],{},"三者的关系可以这样理解：",[21730,92476,92477,92483,92518],{},[21733,92478,92479,92482],{},[698,92480,92481],{},"Le Cam 是 Fano 的两点版本。"," 当只有两个假设时，Fano 退化为两点检验；Le Cam 用 TV 给出了更干净的常数。",[21733,92484,92485,92488,92489,92517],{},[698,92486,92487],{},"Fano 是 Le Cam 的多假设推广。"," 它把「两个点分不清」推广为「",[15,92490,92492,92505],{"className":92491,"translate":19},[23],[15,92493,92495],{"className":92494},[27],[29,92496,92497],{"xmlns":31},[34,92498,92499,92503],{},[37,92500,92501],{},[43,92502,43246],{},[109,92504,43246],{"encoding":111},[15,92506,92508],{"className":92507,"ariaHidden":72},[116],[15,92509,92511,92514],{"className":92510},[120],[15,92512],{"className":92513,"style":610},[124],[15,92515,43246],{"className":92516,"style":12597},[129,133]," 个点平均分不清」。",[21733,92519,92520,92523,92524,92582,92583,92611],{},[698,92521,92522],{},"Assouad 是 Fano 在乘积结构上的精细化。"," 它不直接对 ",[15,92525,92527,92544],{"className":92526,"translate":19},[23],[15,92528,92530],{"className":92529},[27],[29,92531,92532],{"xmlns":31},[34,92533,92534,92542],{},[37,92535,92536],{},[305,92537,92538,92540],{},[308,92539,310],{},[43,92541,348],{},[109,92543,6367],{"encoding":111},[15,92545,92547],{"className":92546,"ariaHidden":72},[116],[15,92548,92550,92553],{"className":92549},[120],[15,92551],{"className":92552,"style":6377},[124],[15,92554,92556,92559],{"className":92555},[129],[15,92557,310],{"className":92558},[129],[15,92560,92562],{"className":92561},[137],[15,92563,92565],{"className":92564},[141],[15,92566,92568],{"className":92567},[146],[15,92569,92571],{"className":92570,"style":6377},[150],[15,92572,92573,92576],{"style":383},[15,92574],{"className":92575,"style":159},[158],[15,92577,92579],{"className":92578},[163,164,165,166],[15,92580,348],{"className":92581},[129,133,166]," 个假设做 Fano，而是逐坐标做 Le Cam，再把 ",[15,92584,92586,92599],{"className":92585,"translate":19},[23],[15,92587,92589],{"className":92588},[27],[29,92590,92591],{"xmlns":31},[34,92592,92593,92597],{},[37,92594,92595],{},[43,92596,348],{},[109,92598,348],{"encoding":111},[15,92600,92602],{"className":92601,"ariaHidden":72},[116],[15,92603,92605,92608],{"className":92604},[120],[15,92606],{"className":92607,"style":4383},[124],[15,92609,348],{"className":92610},[129,133]," 个下界加起来。",[11,92613,92614],{},"实际使用时，选择哪种方法取决于参数空间的结构：",[21730,92616,92617,92620,92664],{},[21733,92618,92619],{},"一维、局部、两样本问题：Le Cam 通常足够，而且常数最干净。",[21733,92621,92622,92623,92663],{},"高维、非参数、需要 ",[15,92624,92626,92643],{"className":92625,"translate":19},[23],[15,92627,92629],{"className":92628},[27],[29,92630,92631],{"xmlns":31},[34,92632,92633,92641],{},[37,92634,92635,92637,92639],{},[43,92636,14727],{},[50,92638,69],{},[43,92640,43246],{},[109,92642,67065],{"encoding":111},[15,92644,92646],{"className":92645,"ariaHidden":72},[116],[15,92647,92649,92652,92657,92660],{"className":92648},[120],[15,92650],{"className":92651,"style":1682},[124],[15,92653,14777,92655],{"className":92654},[217],[15,92656,2247],{"style":2246},[15,92658],{"className":92659,"style":262},[93],[15,92661,43246],{"className":92662,"style":12597},[129,133]," 的地方：Fano 是标准工具。",[21733,92665,92666,92667,39046,92737,92807],{},"乘积结构、",[15,92668,92670,92688],{"className":92669,"translate":19},[23],[15,92671,92673],{"className":92672},[27],[29,92674,92675],{"xmlns":31},[34,92676,92677,92685],{},[37,92678,92679],{},[40,92680,92681,92683],{},[43,92682,27529],{"mathvariant":65},[308,92684,310],{},[109,92686,92687],{"encoding":111},"\\ell_2",[15,92689,92691],{"className":92690,"ariaHidden":72},[116],[15,92692,92694,92697],{"className":92693},[120],[15,92695],{"className":92696,"style":2183},[124],[15,92698,92700,92703],{"className":92699},[129],[15,92701,27529],{"className":92702},[129],[15,92704,92706],{"className":92705},[137],[15,92707,92709,92729],{"className":92708},[141,142],[15,92710,92712,92726],{"className":92711},[146],[15,92713,92715],{"className":92714,"style":1051},[150],[15,92716,92717,92720],{"style":154},[15,92718],{"className":92719,"style":159},[158],[15,92721,92723],{"className":92722},[163,164,165,166],[15,92724,310],{"className":92725},[129,166],[15,92727,175],{"className":92728},[174],[15,92730,92732],{"className":92731},[146],[15,92733,92735],{"className":92734,"style":182},[150],[15,92736],{},[15,92738,92740,92758],{"className":92739,"translate":19},[23],[15,92741,92743],{"className":92742},[27],[29,92744,92745],{"xmlns":31},[34,92746,92747,92755],{},[37,92748,92749],{},[40,92750,92751,92753],{},[43,92752,27529],{"mathvariant":65},[43,92754,4909],{"mathvariant":65},[109,92756,92757],{"encoding":111},"\\ell_\\infty",[15,92759,92761],{"className":92760,"ariaHidden":72},[116],[15,92762,92764,92767],{"className":92763},[120],[15,92765],{"className":92766,"style":2183},[124],[15,92768,92770,92773],{"className":92769},[129],[15,92771,27529],{"className":92772},[129],[15,92774,92776],{"className":92775},[137],[15,92777,92779,92799],{"className":92778},[141,142],[15,92780,92782,92796],{"className":92781},[146],[15,92783,92785],{"className":92784,"style":1163},[150],[15,92786,92787,92790],{"style":154},[15,92788],{"className":92789,"style":159},[158],[15,92791,92793],{"className":92792},[163,164,165,166],[15,92794,4909],{"className":92795},[129,166],[15,92797,175],{"className":92798},[174],[15,92800,92802],{"className":92801},[146],[15,92803,92805],{"className":92804,"style":182},[150],[15,92806],{}," 速率：Assouad 最方便。",[570,92809,36424],{"id":36424},[11,92811,92812,92813,92848,92849,92877],{},"回到开头的问题：为什么 ",[15,92814,92816,92833],{"className":92815,"translate":19},[23],[15,92817,92819],{"className":92818},[27],[29,92820,92821],{"xmlns":31},[34,92822,92823,92831],{},[37,92824,92825,92827,92829],{},[308,92826,318],{},[43,92828,544],{"mathvariant":65},[43,92830,39326],{},[109,92832,48150],{"encoding":111},[15,92834,92836],{"className":92835,"ariaHidden":72},[116],[15,92837,92839,92842,92845],{"className":92838},[120],[15,92840],{"className":92841,"style":125},[124],[15,92843,29852],{"className":92844},[129],[15,92846,39326],{"className":92847},[129,133]," 不能再小？因为两个足够接近的分布，在 ",[15,92850,92852,92865],{"className":92851,"translate":19},[23],[15,92853,92855],{"className":92854},[27],[29,92856,92857],{"xmlns":31},[34,92858,92859,92863],{},[37,92860,92861],{},[43,92862,39326],{},[109,92864,39326],{"encoding":111},[15,92866,92868],{"className":92867,"ariaHidden":72},[116],[15,92869,92871,92874],{"className":92870},[120],[15,92872],{"className":92873,"style":4383},[124],[15,92875,39326],{"className":92876},[129,133]," 个样本下仍然难以区分；而任何估计器都必须先区分它们。这个论证只需要两个点，所以叫两点方法。",[11,92879,92880,92881,92909,92910,92938,92939,92967],{},"但当困难来自许多方向时，两个点就不够了。Fano 把「两个点」换成「",[15,92882,92884,92897],{"className":92883,"translate":19},[23],[15,92885,92887],{"className":92886},[27],[29,92888,92889],{"xmlns":31},[34,92890,92891,92895],{},[37,92892,92893],{},[43,92894,43246],{},[109,92896,43246],{"encoding":111},[15,92898,92900],{"className":92899,"ariaHidden":72},[116],[15,92901,92903,92906],{"className":92902},[120],[15,92904],{"className":92905,"style":610},[124],[15,92907,43246],{"className":92908,"style":12597},[129,133]," 个点」，用互信息度量整体的不可区分性；Assouad 则把 ",[15,92911,92913,92926],{"className":92912,"translate":19},[23],[15,92914,92916],{"className":92915},[27],[29,92917,92918],{"xmlns":31},[34,92919,92920,92924],{},[37,92921,92922],{},[43,92923,348],{},[109,92925,348],{"encoding":111},[15,92927,92929],{"className":92928,"ariaHidden":72},[116],[15,92930,92932,92935],{"className":92931},[120],[15,92933],{"className":92934,"style":4383},[124],[15,92936,348],{"className":92937},[129,133]," 维问题拆成 ",[15,92940,92942,92955],{"className":92941,"translate":19},[23],[15,92943,92945],{"className":92944},[27],[29,92946,92947],{"xmlns":31},[34,92948,92949,92953],{},[37,92950,92951],{},[43,92952,348],{},[109,92954,348],{"encoding":111},[15,92956,92958],{"className":92957,"ariaHidden":72},[116],[15,92959,92961,92964],{"className":92960},[120],[15,92962],{"className":92963,"style":4383},[124],[15,92965,348],{"className":92966},[129,133]," 个一维问题，逐坐标地累加下界。",[11,92969,92970],{},"如果只记三句话：",[30672,92972,92973,92976,93067],{},[21733,92974,92975],{},"下界证明的第一步永远是归约：把估计问题换成一个有限检验问题。",[21733,92977,92978,92979,93007,93008,93066],{},"Le Cam 用两个点，Fano 用 ",[15,92980,92982,92995],{"className":92981,"translate":19},[23],[15,92983,92985],{"className":92984},[27],[29,92986,92987],{"xmlns":31},[34,92988,92989,92993],{},[37,92990,92991],{},[43,92992,43246],{},[109,92994,43246],{"encoding":111},[15,92996,92998],{"className":92997,"ariaHidden":72},[116],[15,92999,93001,93004],{"className":93000},[120],[15,93002],{"className":93003,"style":610},[124],[15,93005,43246],{"className":93006,"style":12597},[129,133]," 个点，Assouad 用 ",[15,93009,93011,93028],{"className":93010,"translate":19},[23],[15,93012,93014],{"className":93013},[27],[29,93015,93016],{"xmlns":31},[34,93017,93018,93026],{},[37,93019,93020],{},[305,93021,93022,93024],{},[308,93023,310],{},[43,93025,348],{},[109,93027,6367],{"encoding":111},[15,93029,93031],{"className":93030,"ariaHidden":72},[116],[15,93032,93034,93037],{"className":93033},[120],[15,93035],{"className":93036,"style":6377},[124],[15,93038,93040,93043],{"className":93039},[129],[15,93041,310],{"className":93042},[129],[15,93044,93046],{"className":93045},[137],[15,93047,93049],{"className":93048},[141],[15,93050,93052],{"className":93051},[146],[15,93053,93055],{"className":93054,"style":6377},[150],[15,93056,93057,93060],{"style":383},[15,93058],{"className":93059,"style":159},[158],[15,93061,93063],{"className":93062},[163,164,165,166],[15,93064,348],{"className":93065},[129,133,166]," 个点；工具在变，思想都是「构造一个在损失上离得远、在信息上离得近的假设集合」。",[21733,93068,93069],{},"信息论下界告诉你统计上不可能；它不告诉你计算上不可能。深度学习时代真正难的，往往是后一个问题。",{"title":36432,"searchDepth":36433,"depth":36433,"links":93071},[93072,93073,93078,93084,93089,93093,93100,93101],{"id":48460,"depth":36433,"text":48461},{"id":49090,"depth":36433,"text":49091,"children":93074},[93075,93076,93077],{"id":49094,"depth":36438,"text":49095},{"id":51622,"depth":36438,"text":51623},{"id":52219,"depth":36438,"text":52220},{"id":52863,"depth":36433,"text":52864,"children":93079},[93080,93081,93082,93083],{"id":52867,"depth":36438,"text":52868},{"id":60183,"depth":36438,"text":60184},{"id":62057,"depth":36438,"text":62058},{"id":63949,"depth":36438,"text":63950},{"id":65088,"depth":36433,"text":65089,"children":93085},[93086,93087,93088],{"id":65092,"depth":36438,"text":65093},{"id":67159,"depth":36438,"text":67160},{"id":74371,"depth":36438,"text":74372},{"id":75671,"depth":36433,"text":75672,"children":93090},[93091,93092],{"id":75675,"depth":36438,"text":75676},{"id":79913,"depth":36438,"text":79914},{"id":81675,"depth":36433,"text":81676,"children":93094},[93095,93097,93099],{"id":81679,"depth":36438,"text":93096},"6.1 高斯均值：σ2d\u002Fn\\sigma^2d\u002Fnσ2d\u002Fn",{"id":84661,"depth":36438,"text":93098},"6.2 非参数回归：n−2β\u002F(2β+d)n^{-2\\beta\u002F(2\\beta+d)}n−2β\u002F(2β+d)",{"id":91240,"depth":36438,"text":91241},{"id":92285,"depth":36433,"text":92286},{"id":36424,"depth":36433,"text":36424},{},"\u002Fblog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa",{"title":47900,"description":47905},"blog\u002F2025\u002F2025-09-27-fano-bu-deng-shi-yu-le-cam-fang-fa","极小极大下界的三种基本工具：Le Cam 两点方法、Fano 不等式与 Assouad 引理。本文从决策论框架出发，给出完整证明与常数，并用高斯均值、非参数回归等例子展示如何把信息论不等式变成速率下界。",[36476,36477],"2025-09-27T16:28:39+08:00","PCaKGvsKVNVeAyaMYmmlfttL41cAUYqcbixx-fGLAmI",1792493791684]