[{"data":1,"prerenderedAt":24610},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions":3},{"post":4,"nextPost":19494,"prevPost":19504},{"id":5,"title":6,"body":7,"description":13,"draft":19491,"enableComment":19492,"extension":19493,"image":19494,"important":19491,"location":19494,"meta":19495,"navigation":19492,"ogImage":19494,"onday":19496,"path":19497,"seo":19498,"stem":19499,"summary":19500,"tags":19501,"__hash__":19503},"blog\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions.md","高维非凸优化的「鞍点主导」",{"type":8,"value":9,"toc":19470},"minimark",[10,14,17,35,38,42,45,129,132,136,468,1400,1687,3053,3176,3841,3915,4412,4828,5069,5072,5488,5524,6843,6904,7185,7188,7191,7194,7768,8545,8611,8616,8619,8622,8625,8766,8928,9016,9300,9368,9667,10036,10039,10386,10388,10887,10890,11107,11168,11179,11182,11185,11188,11192,11230,11542,11574,11578,11735,11858,11861,11989,11992,12876,13013,13146,13149,13221,13642,13829,14025,14086,14569,14575,14579,14584,14779,15112,15173,15833,15948,16002,16315,16414,16489,16521,16524,16529,16533,16536,16671,16860,17021,17027,17314,17534,17537,17541,17548,18345,18348,18574,18577,18803,18916,18919,18922,19291,19298,19301,19304,19307,19318,19321,19324,19384,19445,19448,19453,19456,19460],[11,12,13],"p",{},"深度学习中，Roon 说过一句话，被很多人当作信条：最大的 whitepill 是：局部极小值在高维空间中很罕见。也就是说，你的模型很难卡在局部最优里。但它到底有没有道理？如果有道理，道理又在哪里？",[11,15,16],{},"这句观点并非对每个光滑函数都字面成立。比如凸函数可能只有一个全局极小值，压根没有鞍点。它的精确版本需要一个「景观的概率模型」，以及退化与非退化临界点之间的区分。这篇文章会把这句直觉，分三步严格化：",[18,19,20,24,32],"ol",{},[21,22,23],"li",{},"对 Morse 函数而言，临界点是孤立的，因此在所处区域中占据零体积。",[21,25,26,27,31],{},"在非退化临界点的条件下，局部极小值要求 Hessian ",[28,29,30],"strong",{},"正定","，即所有 N 个独立曲率方向都为正。",[21,33,34],{},"在 Hessian 表现得像大型中心化随机对称矩阵的随机景观模型中，正定是指数级罕见的事件，鞍点则压倒性地常见。",[11,36,37],{},"Kac–Rice 公式为这一现象在高斯随机场中提供了严格的计数框架，Morse 理论则解释了为什么「指数」是临界点的自然拓扑不变量。下面我们一步步把这条链路走完。",[39,40,41],"h2",{"id":41},"高维的地形景观",[11,43,44],{},"我们习惯把优化问题想象成一幅二维或三维的地形图：有山峰（极大值）、山谷（极小值）、还有山口（鞍点，像马鞍一样一边上一边下）。这种直觉在低维时很管用。",[11,46,47,48,96,97,125,126],{},"但当维度 ",[49,50,54,76],"span",{"className":51,"translate":53},[52],"katex","no",[49,55,58],{"className":56},[57],"katex-mathml",[59,60,62],"math",{"xmlns":61},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[63,64,65,72],"semantics",{},[66,67,68],"mrow",{},[69,70,71],"mi",{},"N",[73,74,71],"annotation",{"encoding":75},"application\u002Fx-tex",[49,77,81],{"className":78,"ariaHidden":80},[79],"katex-html","true",[49,82,85,90],{"className":83},[84],"base",[49,86],{"className":87,"style":89},[88],"strut","height:0.6833em;",[49,91,71],{"className":92,"style":95},[93,94],"mord","mathnormal","margin-right:0.109em;"," 变得很大——几千、几百万，甚至更多——这幅图景就开始「失真」。想象你站在高维空间的一个点上：此时脚下不再只有「前后左右」两个方向，而是有 ",[49,98,100,113],{"className":99,"translate":53},[52],[49,101,103],{"className":102},[57],[59,104,105],{"xmlns":61},[63,106,107,111],{},[66,108,109],{},[69,110,71],{},[73,112,71],{"encoding":75},[49,114,116],{"className":115,"ariaHidden":80},[79],[49,117,119,122],{"className":118},[84],[49,120],{"className":121,"style":89},[88],[49,123,71],{"className":124,"style":95},[93,94]," 个相互独立的方向可以移动。要说这个点是一个真正的谷底（局部极小值），你必须保证：",[28,127,128],{},"无论往哪个方向迈一小步，海拔都会升高。",[11,130,131],{},"这个直觉在数学上如何精确化？我们从最基础的定义开始。",[39,133,135],{"id":134},"临界点梯度与-hessian","临界点、梯度与 Hessian",[11,137,138,139,268,269,407,408,467],{},"设 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是二阶连续可微函数，记作 ",[49,270,272,311],{"className":271,"translate":53},[52],[49,273,275],{"className":274},[57],[59,276,277],{"xmlns":61},[63,278,279,308],{},[66,280,281,283,286,295,299,305],{},[69,282,153],{},[155,284,285],{},"∈",[159,287,288,291],{},[69,289,290],{},"C",[292,293,294],"mn",{},"2",[155,296,298],{"stretchy":297},"false","(",[159,300,301,303],{},[69,302,164],{"mathvariant":163},[69,304,71],{},[155,306,307],{"stretchy":297},")",[73,309,310],{"encoding":75},"f \\in 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假设保证了 Hessian 良定义且对称。",[469,470,471,749],"blockquote",{},[11,472,473,476,477,505,506,748],{},[28,474,475],{},"定义 2.1（梯度）","。函数 ",[49,478,480,493],{"className":479,"translate":53},[52],[49,481,483],{"className":482},[57],[59,484,485],{"xmlns":61},[63,486,487,491],{},[66,488,489],{},[69,490,153],{},[73,492,153],{"encoding":75},[49,494,496],{"className":495,"ariaHidden":80},[79],[49,497,499,502],{"className":498},[84],[49,500],{"className":501,"style":184},[88],[49,503,153],{"className":504,"style":188},[93,94]," 在 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2.2（临界点）","。若 ",[49,1409,1411,1438],{"className":1410,"translate":53},[52],[49,1412,1414],{"className":1413},[57],[59,1415,1416],{"xmlns":61},[63,1417,1418,1435],{},[66,1419,1420,1427,1429],{},[527,1421,1422,1424],{},[69,1423,520],{},[69,1425,1426],{},"c",[155,1428,285],{},[159,1430,1431,1433],{},[69,1432,164],{"mathvariant":163},[69,1434,71],{},[73,1436,1437],{"encoding":75},"x_c \\in 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满足",[49,1535,1537],{"className":1536,"translate":53},[752],[49,1538,1540,1572],{"className":1539,"translate":53},[52],[49,1541,1543],{"className":1542},[57],[59,1544,1545],{"xmlns":61,"display":761},[63,1546,1547,1569],{},[66,1548,1549,1551,1553,1555,1561,1563,1565,1567],{},[69,1550,769],{"mathvariant":768},[69,1552,153],{},[155,1554,298],{"stretchy":297},[527,1556,1557,1559],{},[69,1558,520],{},[69,1560,1426],{},[155,1562,307],{"stretchy":297},[155,1564,523],{},[292,1566,801],{},[155,1568,536],{"separator":80},[73,1570,1571],{"encoding":75},"\\nabla f(x_c) = 0,",[49,1573,1575,1642],{"className":1574,"ariaHidden":80},[79],[49,1576,1578,1581,1584,1587,1590,1630,1633,1636,1639],{"className":1577},[84],[49,1579],{"className":1580,"style":589},[88],[49,1582,769],{"className":1583},[93],[49,1585,153],{"className":1586,"style":188},[93,94],[49,1588,298],{"className":1589},[373],[49,1591,1593,1596],{"className":1592},[93],[49,1594,520],{"className":1595},[93,94],[49,1597,1599],{"className":1598},[218],[49,1600,1602,1622],{"className":1601},[222,605],[49,1603,1605,1619],{"className":1604},[226],[49,1606,1608],{"className":1607,"style":1466},[230],[49,1609,1610,1613],{"style":615},[49,1611],{"className":1612,"style":238},[237],[49,1614,1616],{"className":1615},[242,243,244,245],[49,1617,1426],{"className":1618},[93,94,245],[49,1620,629],{"className":1621},[628],[49,1623,1625],{"className":1624},[226],[49,1626,1628],{"className":1627,"style":636},[230],[49,1629],{},[49,1631,307],{"className":1632},[406],[49,1634],{"className":1635,"style":193},[192],[49,1637,523],{"className":1638},[197],[49,1640],{"className":1641,"style":193},[192],[49,1643,1645,1649,1652],{"className":1644},[84],[49,1646],{"className":1647,"style":1648},[88],"height:0.8389em;vertical-align:-0.1944em;",[49,1650,801],{"className":1651},[93],[49,1653,536],{"className":1654},[642],[11,1656,1657,1658,1686],{},"则称该点为 ",[49,1659,1661,1674],{"className":1660,"translate":53},[52],[49,1662,1664],{"className":1663},[57],[59,1665,1666],{"xmlns":61},[63,1667,1668,1672],{},[66,1669,1670],{},[69,1671,153],{},[73,1673,153],{"encoding":75},[49,1675,1677],{"className":1676,"ariaHidden":80},[79],[49,1678,1680,1683],{"className":1679},[84],[49,1681],{"className":1682,"style":184},[88],[49,1684,153],{"className":1685,"style":188},[93,94]," 的临界点。对开区域上的光滑函数，内部的局部极小值与局部极大值都必然是临界点。",[469,1688,1689,1861,2391,2476,2964],{},[11,1690,1691,407,1694,505,1722,1804,1805,1860],{},[28,1692,1693],{},"定义 2.3（Hessian）",[49,1695,1697,1710],{"className":1696,"translate":53},[52],[49,1698,1700],{"className":1699},[57],[59,1701,1702],{"xmlns":61},[63,1703,1704,1708],{},[66,1705,1706],{},[69,1707,153],{},[73,1709,153],{"encoding":75},[49,1711,1713],{"className":1712,"ariaHidden":80},[79],[49,1714,1716,1719],{"className":1715},[84],[49,1717],{"className":1718,"style":184},[88],[49,1720,153],{"className":1721,"style":188},[93,94],[49,1723,1725,1747],{"className":1724,"translate":53},[52],[49,1726,1728],{"className":1727},[57],[59,1729,1730],{"xmlns":61},[63,1731,1732,1744],{},[66,1733,1734,1736,1738],{},[69,1735,520],{},[155,1737,285],{},[159,1739,1740,1742],{},[69,1741,164],{"mathvariant":163},[69,1743,71],{},[73,1745,1746],{"encoding":75},"x\\in\\mathbb{R}^N",[49,1748,1750,1769],{"className":1749,"ariaHidden":80},[79],[49,1751,1753,1757,1760,1763,1766],{"className":1752},[84],[49,1754],{"className":1755,"style":1756},[88],"height:0.5782em;vertical-align:-0.0391em;",[49,1758,520],{"className":1759},[93,94],[49,1761],{"className":1762,"style":193},[192],[49,1764,285],{"className":1765},[197],[49,1767],{"className":1768,"style":193},[192],[49,1770,1772,1775],{"className":1771},[84],[49,1773],{"className":1774,"style":207},[88],[49,1776,1778,1781],{"className":1777},[93],[49,1779,164],{"className":1780},[93,214],[49,1782,1784],{"className":1783},[218],[49,1785,1787],{"className":1786},[222],[49,1788,1790],{"className":1789},[226],[49,1791,1793],{"className":1792,"style":207},[230],[49,1794,1795,1798],{"style":233},[49,1796],{"className":1797,"style":238},[237],[49,1799,1801],{"className":1800},[242,243,244,245],[49,1802,71],{"className":1803,"style":95},[93,94,245]," 处的 Hessian 是 ",[49,1806,1808,1827],{"className":1807,"translate":53},[52],[49,1809,1811],{"className":1810},[57],[59,1812,1813],{"xmlns":61},[63,1814,1815,1824],{},[66,1816,1817,1819,1822],{},[69,1818,71],{},[155,1820,1821],{},"×",[69,1823,71],{},[73,1825,1826],{"encoding":75},"N\\times N",[49,1828,1830,1851],{"className":1829,"ariaHidden":80},[79],[49,1831,1833,1837,1840,1844,1848],{"className":1832},[84],[49,1834],{"className":1835,"style":1836},[88],"height:0.7667em;vertical-align:-0.0833em;",[49,1838,71],{"className":1839,"style":95},[93,94],[49,1841],{"className":1842,"style":1843},[192],"margin-right:0.2222em;",[49,1845,1821],{"className":1846},[1847],"mbin",[49,1849],{"className":1850,"style":1843},[192],[49,1852,1854,1857],{"className":1853},[84],[49,1855],{"className":1856,"style":89},[88],[49,1858,71],{"className":1859,"style":95},[93,94]," 矩阵",[49,1862,1864],{"className":1863,"translate":53},[752],[49,1865,1867,1977],{"className":1866,"translate":53},[52],[49,1868,1870],{"className":1869},[57],[59,1871,1872],{"xmlns":61,"display":761},[63,1873,1874,1974],{},[66,1875,1876,1883,1885,1887,1889,1891,1897,1899,1901,1903,1905,1907,1909,1972],{},[527,1877,1878,1881],{},[69,1879,1880],{},"H",[69,1882,153],{},[155,1884,298],{"stretchy":297},[69,1886,520],{},[155,1888,307],{"stretchy":297},[155,1890,523],{},[159,1892,1893,1895],{},[69,1894,769],{"mathvariant":768},[292,1896,294],{},[69,1898,153],{},[155,1900,298],{"stretchy":297},[69,1902,520],{},[155,1904,307],{"stretchy":297},[155,1906,157],{},[155,1908,523],{},[1910,1911,1912,1958,1970],"msubsup",{},[66,1913,1914,1917,1949,1951,1953,1955],{},[155,1915,1916],{"fence":80},"[",[807,1918,1919,1929],{},[66,1920,1921,1927],{},[159,1922,1923,1925],{},[69,1924,813],{"mathvariant":768},[292,1926,294],{},[69,1928,153],{},[66,1930,1931,1933,1940,1942],{},[69,1932,813],{"mathvariant":768},[527,1934,1935,1937],{},[69,1936,520],{},[69,1938,1939],{},"i",[69,1941,813],{"mathvariant":768},[527,1943,1944,1946],{},[69,1945,520],{},[69,1947,1948],{},"j",[155,1950,298],{"stretchy":297},[69,1952,520],{},[155,1954,307],{"stretchy":297},[155,1956,1957],{"fence":80},"]",[66,1959,1960,1962,1964,1966,1968],{},[69,1961,1939],{},[155,1963,536],{"separator":80},[69,1965,1948],{},[155,1967,523],{},[292,1969,533],{},[69,1971,71],{},[69,1973,897],{"mathvariant":768},[73,1975,1976],{"encoding":75},"H_f(x) = \\nabla^2 f(x) := \\left[\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x)\\right]_{i,j=1}^N.",[49,1978,1980,2049,2107],{"className":1979,"ariaHidden":80},[79],[49,1981,1983,1987,2031,2034,2037,2040,2043,2046],{"className":1982},[84],[49,1984],{"className":1985,"style":1986},[88],"height:1.0361em;vertical-align:-0.2861em;",[49,1988,1990,1994],{"className":1989},[93],[49,1991,1880],{"className":1992,"style":1993},[93,94],"margin-right:0.0813em;",[49,1995,1997],{"className":1996},[218],[49,1998,2000,2022],{"className":1999},[222,605],[49,2001,2003,2019],{"className":2002},[226],[49,2004,2007],{"className":2005,"style":2006},[230],"height:0.3361em;",[49,2008,2010,2013],{"style":2009},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[49,2011],{"className":2012,"style":238},[237],[49,2014,2016],{"className":2015},[242,243,244,245],[49,2017,153],{"className":2018,"style":188},[93,94,245],[49,2020,629],{"className":2021},[628],[49,2023,2025],{"className":2024},[226],[49,2026,2029],{"className":2027,"style":2028},[230],"height:0.2861em;",[49,2030],{},[49,2032,298],{"className":2033},[373],[49,2035,520],{"className":2036},[93,94],[49,2038,307],{"className":2039},[406],[49,2041],{"className":2042,"style":193},[192],[49,2044,523],{"className":2045},[197],[49,2047],{"className":2048,"style":193},[192],[49,2050,2052,2056,2086,2089,2092,2095,2098,2101,2104],{"className":2051},[84],[49,2053],{"className":2054,"style":2055},[88],"height:1.1141em;vertical-align:-0.25em;",[49,2057,2059,2062],{"className":2058},[93],[49,2060,769],{"className":2061},[93],[49,2063,2065],{"className":2064},[218],[49,2066,2068],{"className":2067},[222],[49,2069,2071],{"className":2070},[226],[49,2072,2075],{"className":2073,"style":2074},[230],"height:0.8641em;",[49,2076,2077,2080],{"style":1387},[49,2078],{"className":2079,"style":238},[237],[49,2081,2083],{"className":2082},[242,243,244,245],[49,2084,294],{"className":2085},[93,245],[49,2087,153],{"className":2088,"style":188},[93,94],[49,2090,298],{"className":2091},[373],[49,2093,520],{"className":2094},[93,94],[49,2096,307],{"className":2097},[406],[49,2099],{"className":2100,"style":193},[192],[49,2102,931],{"className":2103},[197],[49,2105],{"className":2106,"style":193},[192],[49,2108,2110,2114,2385,2388],{"className":2109},[84],[49,2111],{"className":2112,"style":2113},[88],"height:2.8803em;vertical-align:-1.1579em;",[49,2115,2117,2321],{"className":2116},[650],[49,2118,2120,2128,2306,2309,2312,2315],{"className":2119},[650],[49,2121,2125],{"className":2122,"style":2124},[373,2123],"delimcenter","top:0em;",[49,2126,1916],{"className":2127},[951,244],[49,2129,2131,2134,2303],{"className":2130},[93],[49,2132],{"className":2133},[373,1037],[49,2135,2137],{"className":2136},[807],[49,2138,2140,2294],{"className":2139},[222,605],[49,2141,2143,2291],{"className":2142},[226],[49,2144,2147,2243,2251],{"className":2145,"style":2146},[230],"height:1.4911em;",[49,2148,2149,2152],{"style":1053},[49,2150],{"className":2151,"style":1057},[237],[49,2153,2155,2158,2199,2202],{"className":2154},[93],[49,2156,813],{"className":2157,"style":1064},[93],[49,2159,2161,2164],{"className":2160},[93],[49,2162,520],{"className":2163},[93,94],[49,2165,2167],{"className":2166},[218],[49,2168,2170,2191],{"className":2169},[222,605],[49,2171,2173,2188],{"className":2172},[226],[49,2174,2177],{"className":2175,"style":2176},[230],"height:0.3117em;",[49,2178,2179,2182],{"style":615},[49,2180],{"className":2181,"style":238},[237],[49,2183,2185],{"className":2184},[242,243,244,245],[49,2186,1939],{"className":2187},[93,94,245],[49,2189,629],{"className":2190},[628],[49,2192,2194],{"className":2193},[226],[49,2195,2197],{"className":2196,"style":636},[230],[49,2198],{},[49,2200,813],{"className":2201,"style":1064},[93],[49,2203,2205,2208],{"className":2204},[93],[49,2206,520],{"className":2207},[93,94],[49,2209,2211],{"className":2210},[218],[49,2212,2214,2235],{"className":2213},[222,605],[49,2215,2217,2232],{"className":2216},[226],[49,2218,2220],{"className":2219,"style":2176},[230],[49,2221,2222,2225],{"style":615},[49,2223],{"className":2224,"style":238},[237],[49,2226,2228],{"className":2227},[242,243,244,245],[49,2229,1948],{"className":2230,"style":2231},[93,94,245],"margin-right:0.0572em;",[49,2233,629],{"className":2234},[628],[49,2236,2238],{"className":2237},[226],[49,2239,2241],{"className":2240,"style":2028},[230],[49,2242],{},[49,2244,2245,2248],{"style":1107},[49,2246],{"className":2247,"style":1057},[237],[49,2249],{"className":2250,"style":1115},[1114],[49,2252,2253,2256],{"style":1118},[49,2254],{"className":2255,"style":1057},[237],[49,2257,2259,2288],{"className":2258},[93],[49,2260,2262,2265],{"className":2261},[93],[49,2263,813],{"className":2264,"style":1064},[93],[49,2266,2268],{"className":2267},[218],[49,2269,2271],{"className":2270},[222],[49,2272,2274],{"className":2273},[226],[49,2275,2277],{"className":2276,"style":358},[230],[49,2278,2279,2282],{"style":233},[49,2280],{"className":2281,"style":238},[237],[49,2283,2285],{"className":2284},[242,243,244,245],[49,2286,294],{"className":2287},[93,245],[49,2289,153],{"className":2290,"style":188},[93,94],[49,2292,629],{"className":2293},[628],[49,2295,2297],{"className":2296},[226],[49,2298,2301],{"className":2299,"style":2300},[230],"height:0.9721em;",[49,2302],{},[49,2304],{"className":2305},[406,1037],[49,2307,298],{"className":2308},[373],[49,2310,520],{"className":2311},[93,94],[49,2313,307],{"className":2314},[406],[49,2316,2318],{"className":2317,"style":2124},[406,2123],[49,2319,1957],{"className":2320},[951,244],[49,2322,2324],{"className":2323},[218],[49,2325,2327,2376],{"className":2326},[222,605],[49,2328,2330,2373],{"className":2329},[226],[49,2331,2334,2361],{"className":2332,"style":2333},[230],"height:1.7223em;",[49,2335,2337,2340],{"style":2336},"top:-1.6782em;margin-right:0.05em;",[49,2338],{"className":2339,"style":238},[237],[49,2341,2343],{"className":2342},[242,243,244,245],[49,2344,2346,2349,2352,2355,2358],{"className":2345},[93,245],[49,2347,1939],{"className":2348},[93,94,245],[49,2350,536],{"className":2351},[642,245],[49,2353,1948],{"className":2354,"style":2231},[93,94,245],[49,2356,523],{"className":2357},[197,245],[49,2359,533],{"className":2360},[93,245],[49,2362,2364,2367],{"style":2363},"top:-3.944em;margin-right:0.05em;",[49,2365],{"className":2366,"style":238},[237],[49,2368,2370],{"className":2369},[242,243,244,245],[49,2371,71],{"className":2372,"style":95},[93,94,245],[49,2374,629],{"className":2375},[628],[49,2377,2379],{"className":2378},[226],[49,2380,2383],{"className":2381,"style":2382},[230],"height:1.1579em;",[49,2384],{},[49,2386],{"className":2387,"style":646},[192],[49,2389,897],{"className":2390},[93],[11,2392,2393,2394,2475],{},"由于 ",[49,2395,2397,2419],{"className":2396,"translate":53},[52],[49,2398,2400],{"className":2399},[57],[59,2401,2402],{"xmlns":61},[63,2403,2404,2416],{},[66,2405,2406,2408,2410],{},[69,2407,153],{},[155,2409,285],{},[159,2411,2412,2414],{},[69,2413,290],{},[292,2415,294],{},[73,2417,2418],{"encoding":75},"f\\in C^2",[49,2420,2422,2440],{"className":2421,"ariaHidden":80},[79],[49,2423,2425,2428,2431,2434,2437],{"className":2424},[84],[49,2426],{"className":2427,"style":184},[88],[49,2429,153],{"className":2430,"style":188},[93,94],[49,2432],{"className":2433,"style":193},[192],[49,2435,285],{"className":2436},[197],[49,2438],{"className":2439,"style":193},[192],[49,2441,2443,2446],{"className":2442},[84],[49,2444],{"className":2445,"style":358},[88],[49,2447,2449,2452],{"className":2448},[93],[49,2450,290],{"className":2451,"style":345},[93,94],[49,2453,2455],{"className":2454},[218],[49,2456,2458],{"className":2457},[222],[49,2459,2461],{"className":2460},[226],[49,2462,2464],{"className":2463,"style":358},[230],[49,2465,2466,2469],{"style":233},[49,2467],{"className":2468,"style":238},[237],[49,2470,2472],{"className":2471},[242,243,244,245],[49,2473,294],{"className":2474},[93,245],"，由 Clairaut 定理可得",[49,2477,2479],{"className":2478,"translate":53},[752],[49,2480,2482,2570],{"className":2481,"translate":53},[52],[49,2483,2485],{"className":2484},[57],[59,2486,2487],{"xmlns":61,"display":761},[63,2488,2489,2567],{},[66,2490,2491,2521,2523,2525,2527,2529,2559,2561,2563,2565],{},[807,2492,2493,2503],{},[66,2494,2495,2501],{},[159,2496,2497,2499],{},[69,2498,813],{"mathvariant":768},[292,2500,294],{},[69,2502,153],{},[66,2504,2505,2507,2513,2515],{},[69,2506,813],{"mathvariant":768},[527,2508,2509,2511],{},[69,2510,520],{},[69,2512,1939],{},[69,2514,813],{"mathvariant":768},[527,2516,2517,2519],{},[69,2518,520],{},[69,2520,1948],{},[155,2522,298],{"stretchy":297},[69,2524,520],{},[155,2526,307],{"stretchy":297},[155,2528,523],{},[807,2530,2531,2541],{},[66,2532,2533,2539],{},[159,2534,2535,2537],{},[69,2536,813],{"mathvariant":768},[292,2538,294],{},[69,2540,153],{},[66,2542,2543,2545,2551,2553],{},[69,2544,813],{"mathvariant":768},[527,2546,2547,2549],{},[69,2548,520],{},[69,2550,1948],{},[69,2552,813],{"mathvariant":768},[527,2554,2555,2557],{},[69,2556,520],{},[69,2558,1939],{},[155,2560,298],{"stretchy":297},[69,2562,520],{},[155,2564,307],{"stretchy":297},[155,2566,536],{"separator":80},[73,2568,2569],{"encoding":75},"\\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x) = \\frac{\\partial^2 f}{\\partial x_j \\partial x_i}(x),",[49,2571,2573,2772],{"className":2572,"ariaHidden":80},[79],[49,2574,2576,2580,2754,2757,2760,2763,2766,2769],{"className":2575},[84],[49,2577],{"className":2578,"style":2579},[88],"height:2.4632em;vertical-align:-0.9721em;",[49,2581,2583,2586,2751],{"className":2582},[93],[49,2584],{"className":2585},[373,1037],[49,2587,2589],{"className":2588},[807],[49,2590,2592,2743],{"className":2591},[222,605],[49,2593,2595,2740],{"className":2594},[226],[49,2596,2598,2692,2700],{"className":2597,"style":2146},[230],[49,2599,2600,2603],{"style":1053},[49,2601],{"className":2602,"style":1057},[237],[49,2604,2606,2609,2649,2652],{"className":2605},[93],[49,2607,813],{"className":2608,"style":1064},[93],[49,2610,2612,2615],{"className":2611},[93],[49,2613,520],{"className":2614},[93,94],[49,2616,2618],{"className":2617},[218],[49,2619,2621,2641],{"className":2620},[222,605],[49,2622,2624,2638],{"className":2623},[226],[49,2625,2627],{"className":2626,"style":2176},[230],[49,2628,2629,2632],{"style":615},[49,2630],{"className":2631,"style":238},[237],[49,2633,2635],{"className":2634},[242,243,244,245],[49,2636,1939],{"className":2637},[93,94,245],[49,2639,629],{"className":2640},[628],[49,2642,2644],{"className":2643},[226],[49,2645,2647],{"className":2646,"style":636},[230],[49,2648],{},[49,2650,813],{"className":2651,"style":1064},[93],[49,2653,2655,2658],{"className":2654},[93],[49,2656,520],{"className":2657},[93,94],[49,2659,2661],{"className":2660},[218],[49,2662,2664,2684],{"className":2663},[222,605],[49,2665,2667,2681],{"className":2666},[226],[49,2668,2670],{"className":2669,"style":2176},[230],[49,2671,2672,2675],{"style":615},[49,2673],{"className":2674,"style":238},[237],[49,2676,2678],{"className":2677},[242,243,244,245],[49,2679,1948],{"className":2680,"style":2231},[93,94,245],[49,2682,629],{"className":2683},[628],[49,2685,2687],{"className":2686},[226],[49,2688,2690],{"className":2689,"style":2028},[230],[49,2691],{},[49,2693,2694,2697],{"style":1107},[49,2695],{"className":2696,"style":1057},[237],[49,2698],{"className":2699,"style":1115},[1114],[49,2701,2702,2705],{"style":1118},[49,2703],{"className":2704,"style":1057},[237],[49,2706,2708,2737],{"className":2707},[93],[49,2709,2711,2714],{"className":2710},[93],[49,2712,813],{"className":2713,"style":1064},[93],[49,2715,2717],{"className":2716},[218],[49,2718,2720],{"className":2719},[222],[49,2721,2723],{"className":2722},[226],[49,2724,2726],{"className":2725,"style":358},[230],[49,2727,2728,2731],{"style":233},[49,2729],{"className":2730,"style":238},[237],[49,2732,2734],{"className":2733},[242,243,244,245],[49,2735,294],{"className":2736},[93,245],[49,2738,153],{"className":2739,"style":188},[93,94],[49,2741,629],{"className":2742},[628],[49,2744,2746],{"className":2745},[226],[49,2747,2749],{"className":2748,"style":2300},[230],[49,2750],{},[49,2752],{"className":2753},[406,1037],[49,2755,298],{"className":2756},[373],[49,2758,520],{"className":2759},[93,94],[49,2761,307],{"className":2762},[406],[49,2764],{"className":2765,"style":193},[192],[49,2767,523],{"className":2768},[197],[49,2770],{"className":2771,"style":193},[192],[49,2773,2775,2778,2952,2955,2958,2961],{"className":2774},[84],[49,2776],{"className":2777,"style":2579},[88],[49,2779,2781,2784,2949],{"className":2780},[93],[49,2782],{"className":2783},[373,1037],[49,2785,2787],{"className":2786},[807],[49,2788,2790,2941],{"className":2789},[222,605],[49,2791,2793,2938],{"className":2792},[226],[49,2794,2796,2890,2898],{"className":2795,"style":2146},[230],[49,2797,2798,2801],{"style":1053},[49,2799],{"className":2800,"style":1057},[237],[49,2802,2804,2807,2847,2850],{"className":2803},[93],[49,2805,813],{"className":2806,"style":1064},[93],[49,2808,2810,2813],{"className":2809},[93],[49,2811,520],{"className":2812},[93,94],[49,2814,2816],{"className":2815},[218],[49,2817,2819,2839],{"className":2818},[222,605],[49,2820,2822,2836],{"className":2821},[226],[49,2823,2825],{"className":2824,"style":2176},[230],[49,2826,2827,2830],{"style":615},[49,2828],{"className":2829,"style":238},[237],[49,2831,2833],{"className":2832},[242,243,244,245],[49,2834,1948],{"className":2835,"style":2231},[93,94,245],[49,2837,629],{"className":2838},[628],[49,2840,2842],{"className":2841},[226],[49,2843,2845],{"className":2844,"style":2028},[230],[49,2846],{},[49,2848,813],{"className":2849,"style":1064},[93],[49,2851,2853,2856],{"className":2852},[93],[49,2854,520],{"className":2855},[93,94],[49,2857,2859],{"className":2858},[218],[49,2860,2862,2882],{"className":2861},[222,605],[49,2863,2865,2879],{"className":2864},[226],[49,2866,2868],{"className":2867,"style":2176},[230],[49,2869,2870,2873],{"style":615},[49,2871],{"className":2872,"style":238},[237],[49,2874,2876],{"className":2875},[242,243,244,245],[49,2877,1939],{"className":2878},[93,94,245],[49,2880,629],{"className":2881},[628],[49,2883,2885],{"className":2884},[226],[49,2886,2888],{"className":2887,"style":636},[230],[49,2889],{},[49,2891,2892,2895],{"style":1107},[49,2893],{"className":2894,"style":1057},[237],[49,2896],{"className":2897,"style":1115},[1114],[49,2899,2900,2903],{"style":1118},[49,2901],{"className":2902,"style":1057},[237],[49,2904,2906,2935],{"className":2905},[93],[49,2907,2909,2912],{"className":2908},[93],[49,2910,813],{"className":2911,"style":1064},[93],[49,2913,2915],{"className":2914},[218],[49,2916,2918],{"className":2917},[222],[49,2919,2921],{"className":2920},[226],[49,2922,2924],{"className":2923,"style":358},[230],[49,2925,2926,2929],{"style":233},[49,2927],{"className":2928,"style":238},[237],[49,2930,2932],{"className":2931},[242,243,244,245],[49,2933,294],{"className":2934},[93,245],[49,2936,153],{"className":2937,"style":188},[93,94],[49,2939,629],{"className":2940},[628],[49,2942,2944],{"className":2943},[226],[49,2945,2947],{"className":2946,"style":2300},[230],[49,2948],{},[49,2950],{"className":2951},[406,1037],[49,2953,298],{"className":2954},[373],[49,2956,520],{"className":2957},[93,94],[49,2959,307],{"className":2960},[406],[49,2962,536],{"className":2963},[642],[11,2965,2966,2967,3052],{},"因此 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x_0",[49,3088,3090,3109,3128],{"className":3089,"ariaHidden":80},[79],[49,3091,3093,3097,3100,3103,3106],{"className":3092},[84],[49,3094],{"className":3095,"style":3096},[88],"height:0.6944em;",[49,3098,3070],{"className":3099},[93,94],[49,3101],{"className":3102,"style":193},[192],[49,3104,523],{"className":3105},[197],[49,3107],{"className":3108,"style":193},[192],[49,3110,3112,3116,3119,3122,3125],{"className":3111},[84],[49,3113],{"className":3114,"style":3115},[88],"height:0.6667em;vertical-align:-0.0833em;",[49,3117,520],{"className":3118},[93,94],[49,3120],{"className":3121,"style":1843},[192],[49,3123,3077],{"className":3124},[1847],[49,3126],{"className":3127,"style":1843},[192],[49,3129,3131,3135],{"className":3130},[84],[49,3132],{"className":3133,"style":3134},[88],"height:0.5806em;vertical-align:-0.15em;",[49,3136,3138,3141],{"className":3137},[93],[49,3139,520],{"className":3140},[93,94],[49,3142,3144],{"className":3143},[218],[49,3145,3147,3167],{"className":3146},[222,605],[49,3148,3150,3164],{"className":3149},[226],[49,3151,3153],{"className":3152,"style":612},[230],[49,3154,3155,3158],{"style":615},[49,3156],{"className":3157,"style":238},[237],[49,3159,3161],{"className":3160},[242,243,244,245],[49,3162,801],{"className":3163},[93,245],[49,3165,629],{"className":3166},[628],[49,3168,3170],{"className":3169},[226],[49,3171,3173],{"className":3172,"style":636},[230],[49,3174],{},"，则",[49,3177,3179],{"className":3178,"translate":53},[752],[49,3180,3182,3320],{"className":3181,"translate":53},[52],[49,3183,3185],{"className":3184},[57],[59,3186,3187],{"xmlns":61,"display":761},[63,3188,3189,3317],{},[787,3190,3192],{"width":3191},"100%",[793,3193,3194,3197,3309,3311],{},[796,3195],{"width":3196},"50%",[796,3198,3199],{},[66,3200,3201,3203,3205,3211,3214,3216,3218,3220,3222,3224,3230,3232,3234,3236,3238,3240,3246,3253,3255,3257,3263,3269,3275,3277,3283,3285,3287,3289,3292,3294,3297,3299,3305,3307],{},[69,3202,153],{},[155,3204,298],{"stretchy":297},[527,3206,3207,3209],{},[69,3208,520],{},[292,3210,801],{},[155,3212,3213],{},"+",[69,3215,3070],{},[155,3217,307],{"stretchy":297},[155,3219,523],{},[69,3221,153],{},[155,3223,298],{"stretchy":297},[527,3225,3226,3228],{},[69,3227,520],{},[292,3229,801],{},[155,3231,307],{"stretchy":297},[155,3233,3213],{},[69,3235,769],{"mathvariant":768},[69,3237,153],{},[155,3239,298],{"stretchy":297},[527,3241,3242,3244],{},[69,3243,520],{},[292,3245,801],{},[159,3247,3248,3250],{},[155,3249,307],{"stretchy":297},[69,3251,3252],{},"T",[69,3254,3070],{},[155,3256,3213],{},[807,3258,3259,3261],{},[292,3260,533],{},[292,3262,294],{},[159,3264,3265,3267],{},[69,3266,3070],{},[69,3268,3252],{},[527,3270,3271,3273],{},[69,3272,1880],{},[69,3274,153],{},[155,3276,298],{"stretchy":297},[527,3278,3279,3281],{},[69,3280,520],{},[292,3282,801],{},[155,3284,307],{"stretchy":297},[69,3286,3070],{},[155,3288,3213],{},[69,3290,3291],{},"o",[155,3293,298],{"stretchy":297},[69,3295,3296],{"mathvariant":768},"∥",[69,3298,3070],{},[159,3300,3301,3303],{},[69,3302,3296],{"mathvariant":768},[292,3304,294],{},[155,3306,307],{"stretchy":297},[69,3308,897],{"mathvariant":768},[796,3310],{"width":3196},[796,3312,3313],{},[3314,3315,3316],"mtext",{},"(1)",[73,3318,3319],{"encoding":75},"f(x_0+h) = f(x_0) + \\nabla f(x_0)^T h + \\frac{1}{2}h^T H_f(x_0) h + o(\\|h\\|^2). 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",[49,3845,3847,3865],{"className":3846,"translate":53},[52],[49,3848,3850],{"className":3849},[57],[59,3851,3852],{"xmlns":61},[63,3853,3854,3862],{},[66,3855,3856],{},[527,3857,3858,3860],{},[69,3859,520],{},[69,3861,1426],{},[73,3863,3864],{"encoding":75},"x_c",[49,3866,3868],{"className":3867,"ariaHidden":80},[79],[49,3869,3871,3874],{"className":3870},[84],[49,3872],{"className":3873,"style":3134},[88],[49,3875,3877,3880],{"className":3876},[93],[49,3878,520],{"className":3879},[93,94],[49,3881,3883],{"className":3882},[218],[49,3884,3886,3906],{"className":3885},[222,605],[49,3887,3889,3903],{"className":3888},[226],[49,3890,3892],{"className":3891,"style":1466},[230],[49,3893,3894,3897],{"style":615},[49,3895],{"className":3896,"style":238},[237],[49,3898,3900],{"className":3899},[242,243,244,245],[49,3901,1426],{"className":3902},[93,94,245],[49,3904,629],{"className":3905},[628],[49,3907,3909],{"className":3908},[226],[49,3910,3912],{"className":3911,"style":636},[230],[49,3913],{}," 处，线性项消失，因此局部几何由下式决定到二阶：",[49,3916,3918],{"className":3917,"translate":53},[752],[49,3919,3921,4015],{"className":3920,"translate":53},[52],[49,3922,3924],{"className":3923},[57],[59,3925,3926],{"xmlns":61,"display":761},[63,3927,3928,4012],{},[66,3929,3930,3932,3934,3940,3942,3944,3946,3948,3950,3952,3958,3960,3962,3968,3974,3980,3982,3988,3990,3992,3994,3996,3998,4000,4002,4008,4010],{},[69,3931,153],{},[155,3933,298],{"stretchy":297},[527,3935,3936,3938],{},[69,3937,520],{},[69,3939,1426],{},[155,3941,3213],{},[69,3943,3070],{},[155,3945,307],{"stretchy":297},[155,3947,3077],{},[69,3949,153],{},[155,3951,298],{"stretchy":297},[527,3953,3954,3956],{},[69,3955,520],{},[69,3957,1426],{},[155,3959,307],{"stretchy":297},[155,3961,523],{},[807,3963,3964,3966],{},[292,3965,533],{},[292,3967,294],{},[159,3969,3970,3972],{},[69,3971,3070],{},[69,3973,3252],{},[527,3975,3976,3978],{},[69,3977,1880],{},[69,3979,153],{},[155,3981,298],{"stretchy":297},[527,3983,3984,3986],{},[69,3985,520],{},[69,3987,1426],{},[155,3989,307],{"stretchy":297},[69,3991,3070],{},[155,3993,3213],{},[69,3995,3291],{},[155,3997,298],{"stretchy":297},[69,3999,3296],{"mathvariant":768},[69,4001,3070],{},[159,4003,4004,4006],{},[69,4005,3296],{"mathvariant":768},[292,4007,294],{},[155,4009,307],{"stretchy":297},[69,4011,897],{"mathvariant":768},[73,4013,4014],{"encoding":75},"f(x_c+h) - f(x_c) = \\frac{1}{2}h^T H_f(x_c)h + o(\\|h\\|^2).",[49,4016,4018,4079,4100,4164,4359],{"className":4017,"ariaHidden":80},[79],[49,4019,4021,4024,4027,4030,4070,4073,4076],{"className":4020},[84],[49,4022],{"className":4023,"style":589},[88],[49,4025,153],{"className":4026,"style":188},[93,94],[49,4028,298],{"className":4029},[373],[49,4031,4033,4036],{"className":4032},[93],[49,4034,520],{"className":4035},[93,94],[49,4037,4039],{"className":4038},[218],[49,4040,4042,4062],{"className":4041},[222,605],[49,4043,4045,4059],{"className":4044},[226],[49,4046,4048],{"className":4047,"style":1466},[230],[49,4049,4050,4053],{"style":615},[49,4051],{"className":4052,"style":238},[237],[49,4054,4056],{"className":4055},[242,243,244,245],[49,4057,1426],{"className":4058},[93,94,245],[49,4060,629],{"className":4061},[628],[49,4063,4065],{"className":4064},[226],[49,4066,4068],{"className":4067,"style":636},[230],[49,4069],{},[49,4071],{"className":4072,"style":1843},[192],[49,4074,3213],{"className":4075},[1847],[49,4077],{"className":4078,"style":1843},[192],[49,4080,4082,4085,4088,4091,4094,4097],{"className":4081},[84],[49,4083],{"className":4084,"style":589},[88],[49,4086,3070],{"className":4087},[93,94],[49,4089,307],{"className":4090},[406],[49,4092],{"className":4093,"style":1843},[192],[49,4095,3077],{"className":4096},[1847],[49,4098],{"className":4099,"style":1843},[192],[49,4101,4103,4106,4109,4112,4152,4155,4158,4161],{"className":4102},[84],[49,4104],{"className":4105,"style":589},[88],[49,4107,153],{"className":4108,"style":188},[93,94],[49,4110,298],{"className":4111},[373],[49,4113,4115,4118],{"className":4114},[93],[49,4116,520],{"className":4117},[93,94],[49,4119,4121],{"className":4120},[218],[49,4122,4124,4144],{"className":4123},[222,605],[49,4125,4127,4141],{"className":4126},[226],[49,4128,4130],{"className":4129,"style":1466},[230],[49,4131,4132,4135],{"style":615},[49,4133],{"className":4134,"style":238},[237],[49,4136,4138],{"className":4137},[242,243,244,245],[49,4139,1426],{"className":4140},[93,94,245],[49,4142,629],{"className":4143},[628],[49,4145,4147],{"className":4146},[226],[49,4148,4150],{"className":4149,"style":636},[230],[49,4151],{},[49,4153,307],{"className":4154},[406],[49,4156],{"className":4157,"style":193},[192],[49,4159,523],{"className":4160},[197],[49,4162],{"className":4163,"style":193},[192],[49,4165,4167,4170,4232,4261,4301,4304,4344,4347,4350,4353,4356],{"className":4166},[84],[49,4168],{"className":4169,"style":3573},[88],[49,4171,4173,4176,4229],{"className":4172},[93],[49,4174],{"className":4175},[373,1037],[49,4177,4179],{"className":4178},[807],[49,4180,4182,4221],{"className":4181},[222,605],[49,4183,4185,4218],{"className":4184},[226],[49,4186,4188,4199,4207],{"className":4187,"style":3592},[230],[49,4189,4190,4193],{"style":1053},[49,4191],{"className":4192,"style":1057},[237],[49,4194,4196],{"className":4195},[93],[49,4197,294],{"className":4198},[93],[49,4200,4201,4204],{"style":1107},[49,4202],{"className":4203,"style":1057},[237],[49,4205],{"className":4206,"style":1115},[1114],[49,4208,4209,4212],{"style":1118},[49,4210],{"className":4211,"style":1057},[237],[49,4213,4215],{"className":4214},[93],[49,4216,533],{"className":4217},[93],[49,4219,629],{"className":4220},[628],[49,4222,4224],{"className":4223},[226],[49,4225,4227],{"className":4226,"style":3632},[230],[49,4228],{},[49,4230],{"className":4231},[406,1037],[49,4233,4235,4238],{"className":4234},[93],[49,4236,3070],{"className":4237},[93,94],[49,4239,4241],{"className":4240},[218],[49,4242,4244],{"className":4243},[222],[49,4245,4247],{"className":4246},[226],[49,4248,4250],{"className":4249,"style":1366},[230],[49,4251,4252,4255],{"style":1387},[49,4253],{"className":4254,"style":238},[237],[49,4256,4258],{"className":4257},[242,243,244,245],[49,4259,3252],{"className":4260,"style":3554},[93,94,245],[49,4262,4264,4267],{"className":4263},[93],[49,4265,1880],{"className":4266,"style":1993},[93,94],[49,4268,4270],{"className":4269},[218],[49,4271,4273,4293],{"className":4272},[222,605],[49,4274,4276,4290],{"className":4275},[226],[49,4277,4279],{"className":4278,"style":2006},[230],[49,4280,4281,4284],{"style":2009},[49,4282],{"className":4283,"style":238},[237],[49,4285,4287],{"className":4286},[242,243,244,245],[49,4288,153],{"className":4289,"style":188},[93,94,245],[49,4291,629],{"className":4292},[628],[49,4294,4296],{"className":4295},[226],[49,4297,4299],{"className":4298,"style":2028},[230],[49,4300],{},[49,4302,298],{"className":4303},[373],[49,4305,4307,4310],{"className":4306},[93],[49,4308,520],{"className":4309},[93,94],[49,4311,4313],{"className":4312},[218],[49,4314,4316,4336],{"className":4315},[222,605],[49,4317,4319,4333],{"className":4318},[226],[49,4320,4322],{"className":4321,"style":1466},[230],[49,4323,4324,4327],{"style":615},[49,4325],{"className":4326,"style":238},[237],[49,4328,4330],{"className":4329},[242,243,244,245],[49,4331,1426],{"className":4332},[93,94,245],[49,4334,629],{"className":4335},[628],[4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是实对称矩阵，谱定理给出由特征向量 ",[49,4542,4544,4575],{"className":4543,"translate":53},[52],[49,4545,4547],{"className":4546},[57],[59,4548,4549],{"xmlns":61},[63,4550,4551,4572],{},[66,4552,4553,4560,4562,4564,4566],{},[527,4554,4555,4558],{},[69,4556,4557],{},"v",[292,4559,533],{},[155,4561,536],{"separator":80},[155,4563,539],{},[155,4565,536],{"separator":80},[527,4567,4568,4570],{},[69,4569,4557],{},[69,4571,71],{},[73,4573,4574],{"encoding":75},"v_1,\\dots,v_N",[49,4576,4578],{"className":4577,"ariaHidden":80},[79],[49,4579,4581,4585,4627,4630,4633,4636,4639,4642,4645],{"className":4580},[84],[49,4582],{"className":4583,"style":4584},[88],"height:0.625em;vertical-align:-0.1944em;",[49,4586,4588,4592],{"className":4587},[93],[49,4589,4557],{"className":4590,"style":4591},[93,94],"margin-right:0.0359em;",[49,4593,4595],{"className":4594},[218],[49,4596,4598,4619],{"className":4597},[222,605],[49,4599,4601,4616],{"className":4600},[226],[49,4602,4604],{"className":4603,"style":612},[230],[49,4605,4607,4610],{"style":4606},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[49,4608],{"className":4609,"style":238},[237],[49,4611,4613],{"className":4612},[242,243,244,245],[49,4614,533],{"className":4615},[93,245],[49,4617,629],{"className":4618},[628],[49,4620,4622],{"className":4621},[226],[49,4623,4625],{"className":4624,"style":636},[230],[49,4626],{},[49,4628,536],{"className":4629},[642],[49,4631],{"className":4632,"style":646},[192],[49,4634,539],{"className":4635},[650],[49,4637],{"className":4638,"style":646},[192],[49,4640,536],{"className":4641},[642],[49,4643],{"className":4644,"style":646},[192],[49,4646,4648,4651],{"className":4647},[93],[49,4649,4557],{"className":4650,"style":4591},[93,94],[49,4652,4654],{"className":4653},[218],[49,4655,4657,4677],{"className":4656},[222,605],[49,4658,4660,4674],{"className":4659},[226],[49,4661,4663],{"className":4662,"style":678},[230],[49,4664,4665,4668],{"style":4606},[49,4666],{"className":4667,"style":238},[237],[49,4669,4671],{"className":4670},[242,243,244,245],[49,4672,71],{"className":4673,"style":95},[93,94,245],[49,4675,629],{"className":4676},[628],[49,4678,4680],{"className":4679},[226],[49,4681,4683],{"className":4682,"style":636},[230],[49,4684],{}," 组成的标准正交基，对应的实特征值为 ",[49,4687,4689,4720],{"className":4688,"translate":53},[52],[49,4690,4692],{"className":4691},[57],[59,4693,4694],{"xmlns":61},[63,4695,4696,4717],{},[66,4697,4698,4705,4707,4709,4711],{},[527,4699,4700,4703],{},[69,4701,4702],{},"λ",[292,4704,533],{},[155,4706,536],{"separator":80},[155,4708,539],{},[155,4710,536],{"separator":80},[527,4712,4713,4715],{},[69,4714,4702],{},[69,4716,71],{},[73,4718,4719],{"encoding":75},"\\lambda_1,\\dots,\\lambda_N",[49,4721,4723],{"className":4722,"ariaHidden":80},[79],[49,4724,4726,4729,4769,4772,4775,4778,4781,4784,4787],{"className":4725},[84],[49,4727],{"className":4728,"style":184},[88],[49,4730,4732,4735],{"className":4731},[93],[49,4733,4702],{"className":4734},[93,94],[49,4736,4738],{"className":4737},[218],[49,4739,4741,4761],{"className":4740},[222,605],[49,4742,4744,4758],{"className":4743},[226],[49,4745,4747],{"className":4746,"style":612},[230],[49,4748,4749,4752],{"style":615},[49,4750],{"className":4751,"style":238},[237],[49,4753,4755],{"className":4754},[242,243,244,245],[49,4756,533],{"className":4757},[93,245],[49,4759,629],{"className":4760},[628],[49,4762,4764],{"className":4763},[226],[49,4765,4767],{"className":4766,"style":636},[230],[49,4768],{},[49,4770,536],{"className":4771},[642],[49,4773],{"className":4774,"style":646},[192],[49,4776,539],{"className":4777},[650],[49,4779],{"className":4780,"style":646},[192],[49,4782,536],{"className":4783},[642],[49,4785],{"className":4786,"style":646},[192],[49,4788,4790,4793],{"className":4789},[93],[49,4791,4702],{"className":4792},[93,94],[49,4794,4796],{"className":4795},[218],[49,4797,4799,4819],{"className":4798},[222,605],[49,4800,4802,4816],{"className":4801},[226],[49,4803,4805],{"className":4804,"style":678},[230],[49,4806,4807,4810],{"style":615},[49,4808],{"className":4809,"style":238},[237],[49,4811,4813],{"className":4812},[242,243,244,245],[49,4814,71],{"className":4815,"style":95},[93,94,245],[49,4817,629],{"className":4818},[628],[49,4820,4822],{"className":4821},[226],[49,4823,4825],{"className":4824,"style":636},[230],[49,4826],{},"。若",[49,4829,4831],{"className":4830,"translate":53},[752],[49,4832,4834,4881],{"className":4833,"translate":53},[52],[49,4835,4837],{"className":4836},[57],[59,4838,4839],{"xmlns":61,"display":761},[63,4840,4841,4878],{},[66,4842,4843,4845,4847,4863,4870,4876],{},[69,4844,3070],{},[155,4846,523],{},[4848,4849,4850,4853,4861],"munderover",{},[155,4851,4852],{},"∑",[66,4854,4855,4857,4859],{},[69,4856,1939],{},[155,4858,523],{},[292,4860,533],{},[69,4862,71],{},[527,4864,4865,4868],{},[69,4866,4867],{},"a",[69,4869,1939],{},[527,4871,4872,4874],{},[69,4873,4557],{},[69,4875,1939],{},[155,4877,536],{"separator":80},[73,4879,4880],{"encoding":75},"h = \\sum_{i=1}^N a_i v_i,",[49,4882,4884,4902],{"className":4883,"ariaHidden":80},[79],[49,4885,4887,4890,4893,4896,4899],{"className":4886},[84],[49,4888],{"className":4889,"style":3096},[88],[49,4891,3070],{"className":4892},[93,94],[49,4894],{"className":4895,"style":193},[192],[49,4897,523],{"className":4898},[197],[49,4900],{"className":4901,"style":193},[192],[49,4903,4905,4909,4983,4986,5026,5066],{"className":4904},[84],[49,4906],{"className":4907,"style":4908},[88],"height:3.106em;vertical-align:-1.2777em;",[49,4910,4914],{"className":4911},[4912,4913],"mop","op-limits",[49,4915,4917,4974],{"className":4916},[222,605],[49,4918,4920,4971],{"className":4919},[226],[49,4921,4924,4946,4959],{"className":4922,"style":4923},[230],"height:1.8283em;",[49,4925,4927,4931],{"style":4926},"top:-1.8723em;margin-left:0em;",[49,4928],{"className":4929,"style":4930},[237],"height:3.05em;",[49,4932,4934],{"className":4933},[242,243,244,245],[49,4935,4937,4940,4943],{"className":4936},[93,245],[49,4938,1939],{"className":4939},[93,94,245],[49,4941,523],{"className":4942},[197,245],[49,4944,533],{"className":4945},[93,245],[49,4947,4949,4952],{"style":4948},"top:-3.05em;",[49,4950],{"className":4951,"style":4930},[237],[49,4953,4954],{},[49,4955,4852],{"className":4956},[4912,4957,4958],"op-symbol","large-op",[49,4960,4962,4965],{"style":4961},"top:-4.3em;margin-left:0em;",[49,4963],{"className":4964,"style":4930},[237],[49,4966,4968],{"className":4967},[242,243,244,245],[49,4969,71],{"className":4970,"style":95},[93,94,245],[49,4972,629],{"className":4973},[628],[49,4975,4977],{"className":4976},[226],[49,4978,4981],{"className":4979,"style":4980},[230],"height:1.2777em;",[49,4982],{},[49,4984],{"className":4985,"style":646},[192],[49,4987,4989,4992],{"className":4988},[93],[49,4990,4867],{"className":4991},[93,94],[49,4993,4995],{"className":4994},[218],[49,4996,4998,5018],{"className":4997},[222,605],[49,4999,5001,5015],{"className":5000},[226],[49,5002,5004],{"className":5003,"style":2176},[230],[49,5005,5006,5009],{"style":615},[49,5007],{"className":5008,"style":238},[237],[49,5010,5012],{"className":5011},[242,243,244,245],[49,5013,1939],{"className":5014},[93,94,245],[49,5016,629],{"className":5017},[628],[49,5019,5021],{"className":5020},[226],[49,5022,5024],{"className":5023,"style":636},[230],[49,5025],{},[49,5027,5029,5032],{"className":5028},[93],[49,5030,4557],{"className":5031,"style":4591},[93,94],[49,5033,5035],{"className":5034},[218],[49,5036,5038,5058],{"className":5037},[222,605],[49,5039,5041,5055],{"className":5040},[226],[49,5042,5044],{"className":5043,"style":2176},[230],[49,5045,5046,5049],{"style":4606},[49,5047],{"className":5048,"style":238},[237],[49,5050,5052],{"className":5051},[242,243,244,245],[49,5053,1939],{"className":5054},[93,94,245],[49,5056,629],{"className":5057},[628],[49,5059,5061],{"className":5060},[226],[49,5062,5064],{"className":5063,"style":636},[230],[49,5065],{},[49,5067,536],{"className":5068},[642],[11,5070,5071],{},"则",[49,5073,5075],{"className":5074,"translate":53},[752],[49,5076,5078,5161],{"className":5077,"translate":53},[52],[49,5079,5081],{"className":5080},[57],[59,5082,5083],{"xmlns":61,"display":761},[63,5084,5085,5158],{},[787,5086,5087],{"width":3191},[793,5088,5089,5091,5151,5153],{},[796,5090],{"width":3196},[796,5092,5093],{},[66,5094,5095,5101,5107,5109,5115,5117,5119,5121,5135,5141,5149],{},[159,5096,5097,5099],{},[69,5098,3070],{},[69,5100,3252],{},[527,5102,5103,5105],{},[69,5104,1880],{},[69,5106,153],{},[155,5108,298],{"stretchy":297},[527,5110,5111,5113],{},[69,5112,520],{},[69,5114,1426],{},[155,5116,307],{"stretchy":297},[69,5118,3070],{},[155,5120,523],{},[4848,5122,5123,5125,5133],{},[155,5124,4852],{},[66,5126,5127,5129,5131],{},[69,5128,1939],{},[155,5130,523],{},[292,5132,533],{},[69,5134,71],{},[527,5136,5137,5139],{},[69,5138,4702],{},[69,5140,1939],{},[1910,5142,5143,5145,5147],{},[69,5144,4867],{},[69,5146,1939],{},[292,5148,294],{},[69,5150,897],{"mathvariant":768},[796,5152],{"width":3196},[796,5154,5155],{},[3314,5156,5157],{},"(2)",[73,5159,5160],{"encoding":75},"h^T H_f(x_c) h = \\sum_{i=1}^N \\lambda_i a_i^2. \\tag{2}",[49,5162,5164,5298,5467],{"className":5163,"ariaHidden":80},[79],[49,5165,5167,5171,5200,5240,5243,5283,5286,5289,5292,5295],{"className":5166},[84],[49,5168],{"className":5169,"style":5170},[88],"height:1.1774em;vertical-align:-0.2861em;",[49,5172,5174,5177],{"className":5173},[93],[49,5175,3070],{"className":5176},[93,94],[49,5178,5180],{"className":5179},[218],[49,5181,5183],{"className":5182},[222],[49,5184,5186],{"className":5185},[226],[49,5187,5189],{"className":5188,"style":1366},[230],[49,5190,5191,5194],{"style":1387},[49,5192],{"className":5193,"style":238},[237],[49,5195,5197],{"className":5196},[242,243,244,245],[49,5198,3252],{"className":5199,"style":3554},[93,94,245],[49,5201,5203,5206],{"className":5202},[93],[49,5204,1880],{"className":5205,"style":1993},[93,94],[49,5207,5209],{"className":5208},[218],[49,5210,5212,5232],{"className":5211},[222,605],[49,5213,5215,5229],{"className":5214},[226],[49,5216,5218],{"className":5217,"style":2006},[230],[49,5219,5220,5223],{"style":2009},[49,5221],{"className":5222,"style":238},[237],[49,5224,5226],{"className":5225},[242,243,244,245],[49,5227,153],{"className":5228,"style":188},[93,94,245],[49,5230,629],{"className":5231},[628],[49,5233,5235],{"className":5234},[226],[49,5236,5238],{"className":5237,"style":2028},[230],[49,5239],{},[49,5241,298],{"className":5242},[373],[49,5244,5246,5249],{"className":5245},[93],[49,5247,520],{"className":5248},[93,94],[49,5250,5252],{"className":5251},[218],[49,5253,5255,5275],{"className":5254},[222,605],[49,5256,5258,5272],{"className":5257},[226],[49,5259,5261],{"className":5260,"style":1466},[230],[49,5262,5263,5266],{"style":615},[49,5264],{"className":5265,"style":238},[237],[49,5267,5269],{"className":5268},[242,243,244,245],[49,5270,1426],{"className":5271},[93,94,245],[49,5273,629],{"className":5274},[628],[49,5276,5278],{"className":5277},[226],[49,5279,5281],{"className":5280,"style":636},[230],[49,5282],{},[49,5284,307],{"className":5285},[406],[49,5287,3070],{"className":5288},[93,94],[49,5290],{"className":5291,"style":193},[192],[49,5293,523],{"className":5294},[197],[49,5296],{"className":5297,"style":193},[192],[49,5299,5301,5304,5368,5371,5411,5464],{"className":5300},[84],[49,5302],{"className":5303,"style":4908},[88],[49,5305,5307],{"className":5306},[4912,4913],[49,5308,5310,5360],{"className":5309},[222,605],[49,5311,5313,5357],{"className":5312},[226],[49,5314,5316,5336,5346],{"className":5315,"style":4923},[230],[49,5317,5318,5321],{"style":4926},[49,5319],{"className":5320,"style":4930},[237],[49,5322,5324],{"className":5323},[242,243,244,245],[49,5325,5327,5330,5333],{"className":5326},[93,245],[49,5328,1939],{"className":5329},[93,94,245],[49,5331,523],{"className":5332},[197,245],[49,5334,533],{"className":5335},[93,245],[49,5337,5338,5341],{"style":4948},[49,5339],{"className":5340,"style":4930},[237],[49,5342,5343],{},[49,5344,4852],{"className":5345},[4912,4957,4958],[49,5347,5348,5351],{"style":4961},[49,5349],{"className":5350,"style":4930},[237],[49,5352,5354],{"className":5353},[242,243,244,245],[49,5355,71],{"className":5356,"style":95},[93,94,245],[49,5358,629],{"className":5359},[628],[49,5361,5363],{"className":5362},[226],[49,5364,5366],{"className":5365,"style":4980},[230],[49,5367],{},[49,5369],{"className":5370,"style":646},[192],[49,5372,5374,5377],{"className":5373},[93],[49,5375,4702],{"className":5376},[93,94],[49,5378,5380],{"className":5379},[218],[49,5381,5383,5403],{"className":5382},[222,605],[49,5384,5386,5400],{"className":5385},[226],[49,5387,5389],{"className":5388,"style":2176},[230],[49,5390,5391,5394],{"style":615},[49,5392],{"className":5393,"style":238},[237],[49,5395,5397],{"className":5396},[242,243,244,245],[49,5398,1939],{"className":5399},[93,94,245],[49,5401,629],{"className":5402},[628],[49,5404,5406],{"className":5405},[226],[49,5407,5409],{"className":5408,"style":636},[230],[49,5410],{},[49,5412,5414,5417],{"className":5413},[93],[49,5415,4867],{"className":5416},[93,94],[49,5418,5420],{"className":5419},[218],[49,5421,5423,5455],{"className":5422},[222,605],[49,5424,5426,5452],{"className":5425},[226],[49,5427,5429,5441],{"className":5428,"style":2074},[230],[49,5430,5432,5435],{"style":5431},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[49,5433],{"className":5434,"style":238},[237],[49,5436,5438],{"className":5437},[242,243,244,245],[49,5439,1939],{"className":5440},[93,94,245],[49,5442,5443,5446],{"style":1387},[49,5444],{"className":5445,"style":238},[237],[49,5447,5449],{"className":5448},[242,243,244,245],[49,5450,294],{"className":5451},[93,245],[49,5453,629],{"className":5454},[628],[49,5456,5458],{"className":5457},[226],[49,5459,5462],{"className":5460,"style":5461},[230],"height:0.247em;",[49,5463],{},[49,5465,897],{"className":5466},[93],[49,5468,5470,5473],{"className":5469},[3821],[49,5471],{"className":5472,"style":4908},[88],[49,5474,5476,5479,5485],{"className":5475},[93,3828],[49,5477,298],{"className":5478},[93],[49,5480,5482],{"className":5481},[93],[49,5483,294],{"className":5484},[93],[49,5486,307],{"className":5487},[93],[11,5489,5490,5491,5523],{},"因此，",[28,5492,5493,5494,5522],{},"Hessian 的特征值就是 ",[49,5495,5497,5510],{"className":5496,"translate":53},[52],[49,5498,5500],{"className":5499},[57],[59,5501,5502],{"xmlns":61},[63,5503,5504,5508],{},[66,5505,5506],{},[69,5507,153],{},[73,5509,153],{"encoding":75},[49,5511,5513],{"className":5512,"ariaHidden":80},[79],[49,5514,5516,5519],{"className":5515},[84],[49,5517],{"className":5518,"style":184},[88],[49,5520,153],{"className":5521,"style":188},[93,94]," 在临界点处的主二阶曲率","——每个特征值的符号，直接告诉我们函数沿对应方向「向上弯」还是「向下弯」。",[469,5525,5526,5728,6326,6528,6758],{},[11,5527,5528,5531,5532,5601,5602,5727],{},[28,5529,5530],{},"定义 2.4（非退化极小值、极大值与鞍点）","。设 ",[49,5533,5535,5552],{"className":5534,"translate":53},[52],[49,5536,5538],{"className":5537},[57],[59,5539,5540],{"xmlns":61},[63,5541,5542,5550],{},[66,5543,5544],{},[527,5545,5546,5548],{},[69,5547,520],{},[69,5549,1426],{},[73,5551,3864],{"encoding":75},[49,5553,5555],{"className":5554,"ariaHidden":80},[79],[49,5556,5558,5561],{"className":5557},[84],[49,5559],{"className":5560,"style":3134},[88],[49,5562,5564,5567],{"className":5563},[93],[49,5565,520],{"className":5566},[93,94],[49,5568,5570],{"className":5569},[218],[49,5571,5573,5593],{"className":5572},[222,605],[49,5574,5576,5590],{"className":5575},[226],[49,5577,5579],{"className":5578,"style":1466},[230],[49,5580,5581,5584],{"style":615},[49,5582],{"className":5583,"style":238},[237],[49,5585,5587],{"className":5586},[242,243,244,245],[49,5588,1426],{"className":5589},[93,94,245],[49,5591,629],{"className":5592},[628],[49,5594,5596],{"className":5595},[226],[49,5597,5599],{"className":5598,"style":636},[230],[49,5600],{}," 是非退化临界点，即 ",[49,5603,5605,5632],{"className":5604,"translate":53},[52],[49,5606,5608],{"className":5607},[57],[59,5609,5610],{"xmlns":61},[63,5611,5612,5630],{},[66,5613,5614,5620,5622,5628],{},[527,5615,5616,5618],{},[69,5617,1880],{},[69,5619,153],{},[155,5621,298],{"stretchy":297},[527,5623,5624,5626],{},[69,5625,520],{},[69,5627,1426],{},[155,5629,307],{"stretchy":297},[73,5631,4444],{"encoding":75},[49,5633,5635],{"className":5634,"ariaHidden":80},[79],[49,5636,5638,5641,5681,5684,5724],{"className":5637},[84],[49,5639],{"className":5640,"style":1986},[88],[49,5642,5644,5647],{"className":5643},[93],[49,5645,1880],{"className":5646,"style":1993},[93,94],[49,5648,5650],{"className":5649},[218],[49,5651,5653,5673],{"className":5652},[222,605],[49,5654,5656,5670],{"className":5655},[226],[49,5657,5659],{"className":5658,"style":2006},[230],[49,5660,5661,5664],{"style":2009},[49,5662],{"className":5663,"style":238},[237],[49,5665,5667],{"className":5666},[242,243,244,245],[49,5668,153],{"className":5669,"style":188},[93,94,245],[49,5671,629],{"className":5672},[628],[49,5674,5676],{"className":5675},[226],[49,5677,5679],{"className":5678,"style":2028},[230],[49,5680],{},[49,5682,298],{"className":5683},[373],[49,5685,5687,5690],{"className":5686},[93],[49,5688,520],{"className":5689},[93,94],[49,5691,5693],{"className":5692},[218],[49,5694,5696,5716],{"className":5695},[222,605],[49,5697,5699,5713],{"className":5698},[226],[49,5700,5702],{"className":5701,"style":1466},[230],[49,5703,5704,5707],{"style":615},[49,5705],{"className":5706,"style":238},[237],[49,5708,5710],{"className":5709},[242,243,244,245],[49,5711,1426],{"className":5712},[93,94,245],[49,5714,629],{"className":5715},[628],[49,5717,5719],{"className":5718},[226],[49,5720,5722],{"className":5721,"style":636},[230],[49,5723],{},[49,5725,307],{"className":5726},[406]," 没有零特征值。",[5729,5730,5731,5930,6128],"ul",{},[21,5732,5733,5734,5859,5860,5929],{},"若 ",[49,5735,5737,5764],{"className":5736,"translate":53},[52],[49,5738,5740],{"className":5739},[57],[59,5741,5742],{"xmlns":61},[63,5743,5744,5762],{},[66,5745,5746,5752,5754,5760],{},[527,5747,5748,5750],{},[69,5749,1880],{},[69,5751,153],{},[155,5753,298],{"stretchy":297},[527,5755,5756,5758],{},[69,5757,520],{},[69,5759,1426],{},[155,5761,307],{"stretchy":297},[73,5763,4444],{"encoding":75},[49,5765,5767],{"className":5766,"ariaHidden":80},[79],[49,5768,5770,5773,5813,5816,5856],{"className":5769},[84],[49,5771],{"className":5772,"style":1986},[88],[49,5774,5776,5779],{"className":5775},[93],[49,5777,1880],{"className":5778,"style":1993},[93,94],[49,5780,5782],{"className":5781},[218],[49,5783,5785,5805],{"className":5784},[222,605],[49,5786,5788,5802],{"className":5787},[226],[49,5789,5791],{"className":5790,"style":2006},[230],[49,5792,5793,5796],{"style":2009},[49,5794],{"className":5795,"style":238},[237],[49,5797,5799],{"className":5798},[242,243,244,245],[49,5800,153],{"className":5801,"style":188},[93,94,245],[49,5803,629],{"className":5804},[628],[49,5806,5808],{"className":5807},[226],[49,5809,5811],{"className":5810,"style":2028},[230],[49,5812],{},[49,5814,298],{"className":5815},[373],[49,5817,5819,5822],{"className":5818},[93],[49,5820,520],{"className":5821},[93,94],[49,5823,5825],{"className":5824},[218],[49,5826,5828,5848],{"className":5827},[222,605],[49,5829,5831,5845],{"className":5830},[226],[49,5832,5834],{"className":5833,"style":1466},[230],[49,5835,5836,5839],{"style":615},[49,5837],{"className":5838,"style":238},[237],[49,5840,5842],{"className":5841},[242,243,244,245],[49,5843,1426],{"className":5844},[93,94,245],[49,5846,629],{"className":5847},[628],[49,5849,5851],{"className":5850},[226],[49,5852,5854],{"className":5853,"style":636},[230],[49,5855],{},[49,5857,307],{"className":5858},[406]," 的所有特征值都为正，则 ",[49,5861,5863,5880],{"className":5862,"translate":53},[52],[49,5864,5866],{"className":5865},[57],[59,5867,5868],{"xmlns":61},[63,5869,5870,5878],{},[66,5871,5872],{},[527,5873,5874,5876],{},[69,5875,520],{},[69,5877,1426],{},[73,5879,3864],{"encoding":75},[49,5881,5883],{"className":5882,"ariaHidden":80},[79],[49,5884,5886,5889],{"className":5885},[84],[49,5887],{"className":5888,"style":3134},[88],[49,5890,5892,5895],{"className":5891},[93],[49,5893,520],{"className":5894},[93,94],[49,5896,5898],{"className":5897},[218],[49,5899,5901,5921],{"className":5900},[222,605],[49,5902,5904,5918],{"className":5903},[226],[49,5905,5907],{"className":5906,"style":1466},[230],[49,5908,5909,5912],{"style":615},[49,5910],{"className":5911,"style":238},[237],[49,5913,5915],{"className":5914},[242,243,244,245],[49,5916,1426],{"className":5917},[93,94,245],[49,5919,629],{"className":5920},[628],[49,5922,5924],{"className":5923},[226],[49,5925,5927],{"className":5926,"style":636},[230],[49,5928],{}," 是非退化局部极小值。",[21,5931,5733,5932,6057,6058,6127],{},[49,5933,5935,5962],{"className":5934,"translate":53},[52],[49,5936,5938],{"className":5937},[57],[59,5939,5940],{"xmlns":61},[63,5941,5942,5960],{},[66,5943,5944,5950,5952,5958],{},[527,5945,5946,5948],{},[69,5947,1880],{},[69,5949,153],{},[155,5951,298],{"stretchy":297},[527,5953,5954,5956],{},[69,5955,520],{},[69,5957,1426],{},[155,5959,307],{"stretchy":297},[73,5961,4444],{"encoding":75},[49,5963,5965],{"className":5964,"ariaHidden":80},[79],[49,5966,5968,5971,6011,6014,6054],{"className":5967},[84],[49,5969],{"className":5970,"style":1986},[88],[49,5972,5974,5977],{"className":5973},[93],[49,5975,1880],{"className":5976,"style":1993},[93,94],[49,5978,5980],{"className":5979},[218],[49,5981,5983,6003],{"className":5982},[222,605],[49,5984,5986,6000],{"className":5985},[226],[49,5987,5989],{"className":5988,"style":2006},[230],[49,5990,5991,5994],{"style":2009},[49,5992],{"className":5993,"style":238},[237],[49,5995,5997],{"className":5996},[242,243,244,245],[49,5998,153],{"className":5999,"style":188},[93,94,245],[49,6001,629],{"className":6002},[628],[49,6004,6006],{"className":6005},[226],[49,6007,6009],{"className":6008,"style":2028},[230],[49,6010],{},[49,6012,298],{"className":6013},[373],[49,6015,6017,6020],{"className":6016},[93],[49,6018,520],{"className":6019},[93,94],[49,6021,6023],{"className":6022},[218],[49,6024,6026,6046],{"className":6025},[222,605],[49,6027,6029,6043],{"className":6028},[226],[49,6030,6032],{"className":6031,"style":1466},[230],[49,6033,6034,6037],{"style":615},[49,6035],{"className":6036,"style":238},[237],[49,6038,6040],{"className":6039},[242,243,244,245],[49,6041,1426],{"className":6042},[93,94,245],[49,6044,629],{"className":6045},[628],[49,6047,6049],{"className":6048},[226],[49,6050,6052],{"className":6051,"style":636},[230],[49,6053],{},[49,6055,307],{"className":6056},[406]," 的所有特征值都为负，则 ",[49,6059,6061,6078],{"className":6060,"translate":53},[52],[49,6062,6064],{"className":6063},[57],[59,6065,6066],{"xmlns":61},[63,6067,6068,6076],{},[66,6069,6070],{},[527,6071,6072,6074],{},[69,6073,520],{},[69,6075,1426],{},[73,6077,3864],{"encoding":75},[49,6079,6081],{"className":6080,"ariaHidden":80},[79],[49,6082,6084,6087],{"className":6083},[84],[49,6085],{"className":6086,"style":3134},[88],[49,6088,6090,6093],{"className":6089},[93],[49,6091,520],{"className":6092},[93,94],[49,6094,6096],{"className":6095},[218],[49,6097,6099,6119],{"className":6098},[222,605],[49,6100,6102,6116],{"className":6101},[226],[49,6103,6105],{"className":6104,"style":1466},[230],[49,6106,6107,6110],{"style":615},[49,6108],{"className":6109,"style":238},[237],[49,6111,6113],{"className":6112},[242,243,244,245],[49,6114,1426],{"className":6115},[93,94,245],[49,6117,629],{"className":6118},[628],[49,6120,6122],{"className":6121},[226],[49,6123,6125],{"className":6124,"style":636},[230],[49,6126],{}," 是非退化局部极大值。",[21,6129,5733,6130,6255,6256,6325],{},[49,6131,6133,6160],{"className":6132,"translate":53},[52],[49,6134,6136],{"className":6135},[57],[59,6137,6138],{"xmlns":61},[63,6139,6140,6158],{},[66,6141,6142,6148,6150,6156],{},[527,6143,6144,6146],{},[69,6145,1880],{},[69,6147,153],{},[155,6149,298],{"stretchy":297},[527,6151,6152,6154],{},[69,6153,520],{},[69,6155,1426],{},[155,6157,307],{"stretchy":297},[73,6159,4444],{"encoding":75},[49,6161,6163],{"className":6162,"ariaHidden":80},[79],[49,6164,6166,6169,6209,6212,6252],{"className":6165},[84],[49,6167],{"className":6168,"style":1986},[88],[49,6170,6172,6175],{"className":6171},[93],[49,6173,1880],{"className":6174,"style":1993},[93,94],[49,6176,6178],{"className":6177},[218],[49,6179,6181,6201],{"className":6180},[222,605],[49,6182,6184,6198],{"className":6183},[226],[49,6185,6187],{"className":6186,"style":2006},[230],[49,6188,6189,6192],{"style":2009},[49,6190],{"className":6191,"style":238},[237],[49,6193,6195],{"className":6194},[242,243,244,245],[49,6196,153],{"className":6197,"style":188},[93,94,245],[49,6199,629],{"className":6200},[628],[49,6202,6204],{"className":6203},[226],[49,6205,6207],{"className":6206,"style":2028},[230],[49,6208],{},[49,6210,298],{"className":6211},[373],[49,6213,6215,6218],{"className":6214},[93],[49,6216,520],{"className":6217},[93,94],[49,6219,6221],{"className":6220},[218],[49,6222,6224,6244],{"className":6223},[222,605],[49,6225,6227,6241],{"className":6226},[226],[49,6228,6230],{"className":6229,"style":1466},[230],[49,6231,6232,6235],{"style":615},[49,6233],{"className":6234,"style":238},[237],[49,6236,6238],{"className":6237},[242,243,244,245],[49,6239,1426],{"className":6240},[93,94,245],[49,6242,629],{"className":6243},[628],[49,6245,6247],{"className":6246},[226],[49,6248,6250],{"className":6249,"style":636},[230],[49,6251],{},[49,6253,307],{"className":6254},[406]," 至少有一个正特征值和至少一个负特征值，则 ",[49,6257,6259,6276],{"className":6258,"translate":53},[52],[49,6260,6262],{"className":6261},[57],[59,6263,6264],{"xmlns":61},[63,6265,6266,6274],{},[66,6267,6268],{},[527,6269,6270,6272],{},[69,6271,520],{},[69,6273,1426],{},[73,6275,3864],{"encoding":75},[49,6277,6279],{"className":6278,"ariaHidden":80},[79],[49,6280,6282,6285],{"className":6281},[84],[49,6283],{"className":6284,"style":3134},[88],[49,6286,6288,6291],{"className":6287},[93],[49,6289,520],{"className":6290},[93,94],[49,6292,6294],{"className":6293},[218],[49,6295,6297,6317],{"className":6296},[222,605],[49,6298,6300,6314],{"className":6299},[226],[49,6301,6303],{"className":6302,"style":1466},[230],[49,6304,6305,6308],{"style":615},[49,6306],{"className":6307,"style":238},[237],[49,6309,6311],{"className":6310},[242,243,244,245],[49,6312,1426],{"className":6313},[93,94,245],[49,6315,629],{"className":6316},[628],[49,6318,6320],{"className":6319},[226],[49,6321,6323],{"className":6322,"style":636},[230],[49,6324],{}," 是鞍点。",[11,6327,6328,6331,6332,6401,6402,6527],{},[28,6329,6330],{},"定义 2.5（Morse 指数）","。非退化临界点 ",[49,6333,6335,6352],{"className":6334,"translate":53},[52],[49,6336,6338],{"className":6337},[57],[59,6339,6340],{"xmlns":61},[63,6341,6342,6350],{},[66,6343,6344],{},[527,6345,6346,6348],{},[69,6347,520],{},[69,6349,1426],{},[73,6351,3864],{"encoding":75},[49,6353,6355],{"className":6354,"ariaHidden":80},[79],[49,6356,6358,6361],{"className":6357},[84],[49,6359],{"className":6360,"style":3134},[88],[49,6362,6364,6367],{"className":6363},[93],[49,6365,520],{"className":6366},[93,94],[49,6368,6370],{"className":6369},[218],[49,6371,6373,6393],{"className":6372},[222,605],[49,6374,6376,6390],{"className":6375},[226],[49,6377,6379],{"className":6378,"style":1466},[230],[49,6380,6381,6384],{"style":615},[49,6382],{"className":6383,"style":238},[237],[49,6385,6387],{"className":6386},[242,243,244,245],[49,6388,1426],{"className":6389},[93,94,245],[49,6391,629],{"className":6392},[628],[49,6394,6396],{"className":6395},[226],[49,6397,6399],{"className":6398,"style":636},[230],[49,6400],{}," 的 Morse 指数是 ",[49,6403,6405,6432],{"className":6404,"translate":53},[52],[49,6406,6408],{"className":6407},[57],[59,6409,6410],{"xmlns":61},[63,6411,6412,6430],{},[66,6413,6414,6420,6422,6428],{},[527,6415,6416,6418],{},[69,6417,1880],{},[69,6419,153],{},[155,6421,298],{"stretchy":297},[527,6423,6424,6426],{},[69,6425,520],{},[69,6427,1426],{},[155,6429,307],{"stretchy":297},[73,6431,4444],{"encoding":75},[49,6433,6435],{"className":6434,"ariaHidden":80},[79],[49,6436,6438,6441,6481,6484,6524],{"className":6437},[84],[49,6439],{"className":6440,"style":1986},[88],[49,6442,6444,6447],{"className":6443},[93],[49,6445,1880],{"className":6446,"style":1993},[93,94],[49,6448,6450],{"className":6449},[218],[49,6451,6453,6473],{"className":6452},[222,605],[49,6454,6456,6470],{"className":6455},[226],[49,6457,6459],{"className":6458,"style":2006},[230],[49,6460,6461,6464],{"style":2009},[49,6462],{"className":6463,"style":238},[237],[49,6465,6467],{"className":6466},[242,243,244,245],[49,6468,153],{"className":6469,"style":188},[93,94,245],[49,6471,629],{"className":6472},[628],[49,6474,6476],{"className":6475},[226],[49,6477,6479],{"className":6478,"style":2028},[230],[49,6480],{},[49,6482,298],{"className":6483},[373],[49,6485,6487,6490],{"className":6486},[93],[49,6488,520],{"className":6489},[93,94],[49,6491,6493],{"className":6492},[218],[49,6494,6496,6516],{"className":6495},[222,605],[49,6497,6499,6513],{"className":6498},[226],[49,6500,6502],{"className":6501,"style":1466},[230],[49,6503,6504,6507],{"style":615},[49,6505],{"className":6506,"style":238},[237],[49,6508,6510],{"className":6509},[242,243,244,245],[49,6511,1426],{"className":6512},[93,94,245],[49,6514,629],{"className":6515},[628],[49,6517,6519],{"className":6518},[226],[49,6520,6522],{"className":6521,"style":636},[230],[49,6523],{},[49,6525,307],{"className":6526},[406]," 的负特征值个数",[49,6529,6531],{"className":6530,"translate":53},[752],[49,6532,6534,6592],{"className":6533,"translate":53},[52],[49,6535,6537],{"className":6536},[57],[59,6538,6539],{"xmlns":61,"display":761},[63,6540,6541,6589],{},[66,6542,6543,6546,6549,6551,6557,6559,6561,6563,6566,6569,6571,6573,6579,6582,6584,6587],{},[69,6544,6545],{"mathvariant":768},"index",[155,6547,6548],{},"⁡",[155,6550,298],{"stretchy":297},[527,6552,6553,6555],{},[69,6554,520],{},[69,6556,1426],{},[155,6558,307],{"stretchy":297},[155,6560,157],{},[155,6562,523],{},[69,6564,6565],{"mathvariant":768},"#",[155,6567,6568],{"stretchy":297},"{",[69,6570,1939],{},[155,6572,157],{},[527,6574,6575,6577],{},[69,6576,4702],{},[69,6578,1939],{},[155,6580,6581],{},"\u003C",[292,6583,801],{},[155,6585,6586],{"stretchy":297},"}",[69,6588,897],{"mathvariant":768},[73,6590,6591],{"encoding":75},"\\operatorname{index}(x_c) := \\#\\{i : \\lambda_i \u003C 0\\}.",[49,6593,6595,6663,6687,6743],{"className":6594,"ariaHidden":80},[79],[49,6596,6598,6601,6608,6611,6651,6654,6657,6660],{"className":6597},[84],[49,6599],{"className":6600,"style":589},[88],[49,6602,6604],{"className":6603},[4912],[49,6605,6545],{"className":6606},[93,6607],"mathrm",[49,6609,298],{"className":6610},[373],[49,6612,6614,6617],{"className":6613},[93],[49,6615,520],{"className":6616},[93,94],[49,6618,6620],{"className":6619},[218],[49,6621,6623,6643],{"className":6622},[222,605],[49,6624,6626,6640],{"className":6625},[226],[49,6627,6629],{"className":6628,"style":1466},[230],[49,6630,6631,6634],{"style":615},[49,6632],{"className":6633,"style":238},[237],[49,6635,6637],{"className":6636},[242,243,244,245],[49,6638,1426],{"className":6639},[93,94,245],[49,6641,629],{"className":6642},[628],[49,6644,6646],{"className":6645},[226],[49,6647,6649],{"className":6648,"style":636},[230],[49,6650],{},[49,6652,307],{"className":6653},[406],[49,6655],{"className":6656,"style":193},[192],[49,6658,931],{"className":6659},[197],[49,6661],{"className":6662,"style":193},[192],[49,6664,6666,6669,6672,6675,6678,6681,6684],{"className":6665},[84],[49,6667],{"className":6668,"style":589},[88],[49,6670,6565],{"className":6671},[93],[49,6673,6568],{"className":6674},[373],[49,6676,1939],{"className":6677},[93,94],[49,6679],{"className":6680,"style":193},[192],[49,6682,157],{"className":6683},[197],[49,6685],{"className":6686,"style":193},[192],[49,6688,6690,6694,6734,6737,6740],{"className":6689},[84],[49,6691],{"className":6692,"style":6693},[88],"height:0.8444em;vertical-align:-0.15em;",[49,6695,6697,6700],{"className":6696},[93],[49,6698,4702],{"className":6699},[93,94],[49,6701,6703],{"className":6702},[218],[49,6704,6706,6726],{"className":6705},[222,605],[49,6707,6709,6723],{"className":6708},[226],[49,6710,6712],{"className":6711,"style":2176},[230],[49,6713,6714,6717],{"style":615},[49,6715],{"className":6716,"style":238},[237],[49,6718,6720],{"className":6719},[242,243,244,245],[49,6721,1939],{"className":6722},[93,94,245],[49,6724,629],{"className":6725},[628],[49,6727,6729],{"className":6728},[226],[49,6730,6732],{"className":6731,"style":636},[230],[49,6733],{},[49,6735],{"className":6736,"style":193},[192],[49,6738,6581],{"className":6739},[197],[49,6741],{"className":6742,"style":193},[192],[49,6744,6746,6749,6752,6755],{"className":6745},[84],[49,6747],{"className":6748,"style":589},[88],[49,6750,801],{"className":6751},[93],[49,6753,6586],{"className":6754},[406],[49,6756,897],{"className":6757},[93],[11,6759,6760,6761,6789,6790,6842],{},"非退化局部极小值的指数为 0，非退化局部极大值的指数为 ",[49,6762,6764,6777],{"className":6763,"translate":53},[52],[49,6765,6767],{"className":6766},[57],[59,6768,6769],{"xmlns":61},[63,6770,6771,6775],{},[66,6772,6773],{},[69,6774,71],{},[73,6776,71],{"encoding":75},[49,6778,6780],{"className":6779,"ariaHidden":80},[79],[49,6781,6783,6786],{"className":6782},[84],[49,6784],{"className":6785,"style":89},[88],[49,6787,71],{"className":6788,"style":95},[93,94],"，鞍点的指数在 1 到 ",[49,6791,6793,6811],{"className":6792,"translate":53},[52],[49,6794,6796],{"className":6795},[57],[59,6797,6798],{"xmlns":61},[63,6799,6800,6808],{},[66,6801,6802,6804,6806],{},[69,6803,71],{},[155,6805,3077],{},[292,6807,533],{},[73,6809,6810],{"encoding":75},"N-1",[49,6812,6814,6832],{"className":6813,"ariaHidden":80},[79],[49,6815,6817,6820,6823,6826,6829],{"className":6816},[84],[49,6818],{"className":6819,"style":1836},[88],[49,6821,71],{"className":6822,"style":95},[93,94],[49,6824],{"className":6825,"style":1843},[192],[49,6827,3077],{"className":6828},[1847],[49,6830],{"className":6831,"style":1843},[192],[49,6833,6835,6839],{"className":6834},[84],[49,6836],{"className":6837,"style":6838},[88],"height:0.6444em;",[49,6840,533],{"className":6841},[93]," 之间。",[11,6844,6845,6846,6874,6875,6903],{},"高维的困难此刻清晰可见。在 1 维中，非退化极小值只要求一个正的二阶导数。在 ",[49,6847,6849,6862],{"className":6848,"translate":53},[52],[49,6850,6852],{"className":6851},[57],[59,6853,6854],{"xmlns":61},[63,6855,6856,6860],{},[66,6857,6858],{},[69,6859,71],{},[73,6861,71],{"encoding":75},[49,6863,6865],{"className":6864,"ariaHidden":80},[79],[49,6866,6868,6871],{"className":6867},[84],[49,6869],{"className":6870,"style":89},[88],[49,6872,71],{"className":6873,"style":95},[93,94]," 维中，非退化局部极小值要求 ",[49,6876,6878,6891],{"className":6877,"translate":53},[52],[49,6879,6881],{"className":6880},[57],[59,6882,6883],{"xmlns":61},[63,6884,6885,6889],{},[66,6886,6887],{},[69,6888,71],{},[73,6890,71],{"encoding":75},[49,6892,6894],{"className":6893,"ariaHidden":80},[79],[49,6895,6897,6900],{"className":6896},[84],[49,6898],{"className":6899,"style":89},[88],[49,6901,71],{"className":6902,"style":95},[93,94]," 个同时成立的正曲率条件：",[49,6905,6907],{"className":6906,"translate":53},[752],[49,6908,6910,6969],{"className":6909,"translate":53},[52],[49,6911,6913],{"className":6912},[57],[59,6914,6915],{"xmlns":61,"display":761},[63,6916,6917,6966],{},[66,6918,6919,6925,6928,6930,6932,6935,6941,6943,6945,6947,6949,6951,6953,6955,6961,6963],{},[527,6920,6921,6923],{},[69,6922,4702],{},[292,6924,533],{},[155,6926,6927],{},">",[292,6929,801],{},[155,6931,536],{"separator":80},[3314,6933,6934],{}," ",[527,6936,6937,6939],{},[69,6938,4702],{},[292,6940,294],{},[155,6942,6927],{},[292,6944,801],{},[155,6946,536],{"separator":80},[3314,6948,6934],{},[155,6950,539],{},[155,6952,536],{"separator":80},[3314,6954,6934],{},[527,6956,6957,6959],{},[69,6958,4702],{},[69,6960,71],{},[155,6962,6927],{},[292,6964,6965],{},"0.",[73,6967,6968],{"encoding":75},"\\lambda_1 > 0,\\ \\lambda_2 > 0,\\ \\dots,\\ \\lambda_N > 0.",[49,6970,6972,7027,7094,7176],{"className":6971,"ariaHidden":80},[79],[49,6973,6975,6978,7018,7021,7024],{"className":6974},[84],[49,6976],{"className":6977,"style":6693},[88],[49,6979,6981,6984],{"className":6980},[93],[49,6982,4702],{"className":6983},[93,94],[49,6985,6987],{"className":6986},[218],[49,6988,6990,7010],{"className":6989},[222,605],[49,6991,6993,7007],{"className":6992},[226],[49,6994,6996],{"className":6995,"style":612},[230],[49,6997,6998,7001],{"style":615},[49,6999],{"className":7000,"style":238},[237],[49,7002,7004],{"className":7003},[242,243,244,245],[49,7005,533],{"className":7006},[93,245],[49,7008,629],{"className":7009},[628],[49,7011,7013],{"className":7012},[226],[49,7014,7016],{"className":7015,"style":636},[230],[49,7017],{},[49,7019],{"className":7020,"style":193},[192],[49,7022,6927],{"className":7023},[197],[49,7025],{"className":7026,"style":193},[192],[49,7028,7030,7033,7036,7039,7042,7045,7085,7088,7091],{"className":7029},[84],[49,7031],{"className":7032,"style":184},[88],[49,7034,801],{"className":7035},[93],[49,7037,536],{"className":7038},[642],[49,7040,6934],{"className":7041},[192],[49,7043],{"className":7044,"style":646},[192],[49,7046,7048,7051],{"className":7047},[93],[49,7049,4702],{"className":7050},[93,94],[49,7052,7054],{"className":7053},[218],[49,7055,7057,7077],{"className":7056},[222,605],[49,7058,7060,7074],{"className":7059},[226],[49,7061,7063],{"className":7062,"style":612},[230],[49,7064,7065,7068],{"style":615},[49,7066],{"className":7067,"style":238},[237],[49,7069,7071],{"className":7070},[242,243,244,245],[49,7072,294],{"className":7073},[93,245],[49,7075,629],{"className":7076},[628],[49,7078,7080],{"className":7079},[226],[49,7081,7083],{"className":7082,"style":636},[230],[49,7084],{},[49,7086],{"className":7087,"style":193},[192],[49,7089,6927],{"className":7090},[197],[49,7092],{"className":7093,"style":193},[192],[49,7095,7097,7100,7103,7106,7109,7112,7115,7118,7121,7124,7127,7167,7170,7173],{"className":7096},[84],[49,7098],{"className":7099,"style":184},[88],[49,7101,801],{"className":7102},[93],[49,7104,536],{"className":7105},[642],[49,7107,6934],{"className":7108},[192],[49,7110],{"className":7111,"style":646},[192],[49,7113,539],{"className":7114},[650],[49,7116],{"className":7117,"style":646},[192],[49,7119,536],{"className":7120},[642],[49,7122,6934],{"className":7123},[192],[49,7125],{"className":7126,"style":646},[192],[49,7128,7130,7133],{"className":7129},[93],[49,7131,4702],{"className":7132},[93,94],[49,7134,7136],{"className":7135},[218],[49,7137,7139,7159],{"className":7138},[222,605],[49,7140,7142,7156],{"className":7141},[226],[49,7143,7145],{"className":7144,"style":678},[230],[49,7146,7147,7150],{"style":615},[49,7148],{"className":7149,"style":238},[237],[49,7151,7153],{"className":7152},[242,243,244,245],[49,7154,71],{"className":7155,"style":95},[93,94,245],[49,7157,629],{"className":7158},[628],[49,7160,7162],{"className":7161},[226],[49,7163,7165],{"className":7164,"style":636},[230],[49,7166],{},[49,7168],{"className":7169,"style":193},[192],[49,7171,6927],{"className":7172},[197],[49,7174],{"className":7175,"style":193},[192],[49,7177,7179,7182],{"className":7178},[84],[49,7180],{"className":7181,"style":6838},[88],[49,7183,6965],{"className":7184},[93],[11,7186,7187],{},"相比之下，鞍点只要求至少一个正特征值和至少一个负特征值。当 Hessian 是随机的、且对正曲率没有强偏好时，后者显然容易满足得多。",[39,7189,7190],{"id":7190},"临界点本身就稀少",[11,7192,7193],{},"在讨论「临界点是不是极小值」之前，先确认一个更基础、但常被忽视的事实：临界点本身在空间中就是「测度为零」的稀疏集合。这个结论不难证明，但它还不是本文要讲的高维现象——它对一维函数同样成立。",[469,7195,7196,7434,7642,7674],{},[11,7197,7198,1407,7201,7259,7260,7363,7364,7433],{},[28,7199,7200],{},"定义 3.1（Morse 函数）",[49,7202,7204,7221],{"className":7203,"translate":53},[52],[49,7205,7207],{"className":7206},[57],[59,7208,7209],{"xmlns":61},[63,7210,7211,7219],{},[66,7212,7213],{},[159,7214,7215,7217],{},[69,7216,290],{},[292,7218,294],{},[73,7220,428],{"encoding":75},[49,7222,7224],{"className":7223,"ariaHidden":80},[79],[49,7225,7227,7230],{"className":7226},[84],[49,7228],{"className":7229,"style":358},[88],[49,7231,7233,7236],{"className":7232},[93],[49,7234,290],{"className":7235,"style":345},[93,94],[49,7237,7239],{"className":7238},[218],[49,7240,7242],{"className":7241},[222],[49,7243,7245],{"className":7244},[226],[49,7246,7248],{"className":7247,"style":358},[230],[49,7249,7250,7253],{"style":233},[49,7251],{"className":7252,"style":238},[237],[49,7254,7256],{"className":7255},[242,243,244,245],[49,7257,294],{"className":7258},[93,245]," 函数 ",[49,7261,7263,7289],{"className":7262,"translate":53},[52],[49,7264,7266],{"className":7265},[57],[59,7267,7268],{"xmlns":61},[63,7269,7270,7286],{},[66,7271,7272,7274,7276,7282,7284],{},[69,7273,153],{},[155,7275,157],{},[159,7277,7278,7280],{},[69,7279,164],{"mathvariant":163},[69,7281,71],{},[155,7283,169],{},[69,7285,164],{"mathvariant":163},[73,7287,7288],{"encoding":75},"f:\\mathbb{R}^N\\to\\mathbb{R}",[49,7290,7292,7310,7354],{"className":7291,"ariaHidden":80},[79],[49,7293,7295,7298,7301,7304,7307],{"className":7294},[84],[49,7296],{"className":7297,"style":184},[88],[49,7299,153],{"className":7300,"style":188},[93,94],[49,7302],{"className":7303,"style":193},[192],[49,7305,157],{"className":7306},[197],[49,7308],{"className":7309,"style":193},[192],[49,7311,7313,7316,7345,7348,7351],{"className":7312},[84],[49,7314],{"className":7315,"style":207},[88],[49,7317,7319,7322],{"className":7318},[93],[49,7320,164],{"className":7321},[93,214],[49,7323,7325],{"className":7324},[218],[49,7326,7328],{"className":7327},[222],[49,7329,7331],{"className":7330},[226],[49,7332,7334],{"className":7333,"style":207},[230],[49,7335,7336,7339],{"style":233},[49,7337],{"className":7338,"style":238},[237],[49,7340,7342],{"className":7341},[242,243,244,245],[49,7343,71],{"className":7344,"style":95},[93,94,245],[49,7346],{"className":7347,"style":193},[192],[49,7349,169],{"className":7350},[197],[49,7352],{"className":7353,"style":193},[192],[49,7355,7357,7360],{"className":7356},[84],[49,7358],{"className":7359,"style":264},[88],[49,7361,164],{"className":7362},[93,214]," 的每个临界点都是非退化的，即对每个临界点 ",[49,7365,7367,7384],{"className":7366,"translate":53},[52],[49,7368,7370],{"className":7369},[57],[59,7371,7372],{"xmlns":61},[63,7373,7374,7382],{},[66,7375,7376],{},[527,7377,7378,7380],{},[69,7379,520],{},[69,7381,1426],{},[73,7383,3864],{"encoding":75},[49,7385,7387],{"className":7386,"ariaHidden":80},[79],[49,7388,7390,7393],{"className":7389},[84],[49,7391],{"className":7392,"style":3134},[88],[49,7394,7396,7399],{"className":7395},[93],[49,7397,520],{"className":7398},[93,94],[49,7400,7402],{"className":7401},[218],[49,7403,7405,7425],{"className":7404},[222,605],[49,7406,7408,7422],{"className":7407},[226],[49,7409,7411],{"className":7410,"style":1466},[230],[49,7412,7413,7416],{"style":615},[49,7414],{"className":7415,"style":238},[237],[49,7417,7419],{"className":7418},[242,243,244,245],[49,7420,1426],{"className":7421},[93,94,245],[49,7423,629],{"className":7424},[628],[49,7426,7428],{"className":7427},[226],[49,7429,7431],{"className":7430,"style":636},[230],[49,7432],{}," 有",[49,7435,7437],{"className":7436,"translate":53},[752],[49,7438,7440,7480],{"className":7439,"translate":53},[52],[49,7441,7443],{"className":7442},[57],[59,7444,7445],{"xmlns":61,"display":761},[63,7446,7447,7477],{},[66,7448,7449,7452,7454,7460,7462,7468,7470,7473,7475],{},[69,7450,7451],{},"det",[155,7453,6548],{},[527,7455,7456,7458],{},[69,7457,1880],{},[69,7459,153],{},[155,7461,298],{"stretchy":297},[527,7463,7464,7466],{},[69,7465,520],{},[69,7467,1426],{},[155,7469,307],{"stretchy":297},[155,7471,7472],{"mathvariant":768},"≠",[292,7474,801],{},[155,7476,536],{"separator":80},[73,7478,7479],{"encoding":75},"\\det H_f(x_c) \\neq 0,",[49,7481,7483,7630],{"className":7482,"ariaHidden":80},[79],[49,7484,7486,7489,7492,7495,7535,7538,7578,7581,7584,7627],{"className":7485},[84],[49,7487],{"className":7488,"style":1986},[88],[49,7490,7451],{"className":7491},[4912],[49,7493],{"className":7494,"style":646},[192],[49,7496,7498,7501],{"className":7497},[93],[49,7499,1880],{"className":7500,"style":1993},[93,94],[49,7502,7504],{"className":7503},[218],[49,7505,7507,7527],{"className":7506},[222,605],[49,7508,7510,7524],{"className":7509},[226],[49,7511,7513],{"className":7512,"style":2006},[230],[49,7514,7515,7518],{"style":2009},[49,7516],{"className":7517,"style":238},[237],[49,7519,7521],{"className":7520},[242,243,244,245],[49,7522,153],{"className":7523,"style":188},[93,94,245],[49,7525,629],{"className":7526},[628],[49,7528,7530],{"className":7529},[226],[49,7531,7533],{"className":7532,"style":2028},[230],[49,7534],{},[49,7536,298],{"className":7537},[373],[49,7539,7541,7544],{"className":7540},[93],[49,7542,520],{"className":7543},[93,94],[49,7545,7547],{"className":7546},[218],[49,7548,7550,7570],{"className":7549},[222,605],[49,7551,7553,7567],{"className":7552},[226],[49,7554,7556],{"className":7555,"style":1466},[230],[49,7557,7558,7561],{"style":615},[49,7559],{"className":7560,"style":238},[237],[49,7562,7564],{"className":7563},[242,243,244,245],[49,7565,1426],{"className":7566},[93,94,245],[49,7568,629],{"className":7569},[628],[49,7571,7573],{"className":7572},[226],[49,7574,7576],{"className":7575,"style":636},[230],[49,7577],{},[49,7579,307],{"className":7580},[406],[49,7582],{"className":7583,"style":193},[192],[49,7585,7587,7620,7624],{"className":7586},[197],[49,7588,7590],{"className":7589},[197],[49,7591,7594],{"className":7592},[93,7593],"vbox",[49,7595,7598],{"className":7596},[7597],"thinbox",[49,7599,7602,7605,7616],{"className":7600},[7601],"rlap",[49,7603],{"className":7604,"style":184},[88],[49,7606,7609],{"className":7607},[7608],"inner",[49,7610,7612],{"className":7611},[93],[49,7613,7615],{"className":7614},[197],"",[49,7617],{"className":7618},[7619],"fix",[49,7621],{"className":7622},[192,7623],"nobreak",[49,7625,523],{"className":7626},[197],[49,7628],{"className":7629,"style":193},[192],[49,7631,7633,7636,7639],{"className":7632},[84],[49,7634],{"className":7635,"style":1648},[88],[49,7637,801],{"className":7638},[93],[49,7640,536],{"className":7641},[642],[11,7643,7644,7645,7673],{},"则称 ",[49,7646,7648,7661],{"className":7647,"translate":53},[52],[49,7649,7651],{"className":7650},[57],[59,7652,7653],{"xmlns":61},[63,7654,7655,7659],{},[66,7656,7657],{},[69,7658,153],{},[73,7660,153],{"encoding":75},[49,7662,7664],{"className":7663,"ariaHidden":80},[79],[49,7665,7667,7670],{"className":7666},[84],[49,7668],{"className":7669,"style":184},[88],[49,7671,153],{"className":7672,"style":188},[93,94]," 为 Morse 函数。",[11,7675,7676,1407,7679,7707,7708,7767],{},[28,7677,7678],{},"命题 3.2",[49,7680,7682,7695],{"className":7681,"translate":53},[52],[49,7683,7685],{"className":7684},[57],[59,7686,7687],{"xmlns":61},[63,7688,7689,7693],{},[66,7690,7691],{},[69,7692,153],{},[73,7694,153],{"encoding":75},[49,7696,7698],{"className":7697,"ariaHidden":80},[79],[49,7699,7701,7704],{"className":7700},[84],[49,7702],{"className":7703,"style":184},[88],[49,7705,153],{"className":7706,"style":188},[93,94]," 是 Morse 函数，则其临界点是孤立的。特别地，在 ",[49,7709,7711,7729],{"className":7710,"translate":53},[52],[49,7712,7714],{"className":7713},[57],[59,7715,7716],{"xmlns":61},[63,7717,7718,7726],{},[66,7719,7720],{},[159,7721,7722,7724],{},[69,7723,164],{"mathvariant":163},[69,7725,71],{},[73,7727,7728],{"encoding":75},"\\mathbb{R}^N",[49,7730,7732],{"className":7731,"ariaHidden":80},[79],[49,7733,7735,7738],{"className":7734},[84],[49,7736],{"className":7737,"style":207},[88],[49,7739,7741,7744],{"className":7740},[93],[49,7742,164],{"className":7743},[93,214],[49,7745,7747],{"className":7746},[218],[49,7748,7750],{"className":7749},[222],[49,7751,7753],{"className":7752},[226],[49,7754,7756],{"className":7755,"style":207},[230],[49,7757,7758,7761],{"style":233},[49,7759],{"className":7760,"style":238},[237],[49,7762,7764],{"className":7763},[242,243,244,245],[49,7765,71],{"className":7766,"style":95},[93,94,245]," 的每个紧子集中，只有有限多个临界点，因此临界点集具有 Lebesgue 测度零。",[11,7769,7770,7774,7775,7913,7914,7983,7984,1407,8109,8178,8179,8304,8305,505,8339,8408,8409,8478,8479,8512,8513],{},[7771,7772,7773],"em",{},"证明","：临界点是梯度映射 ",[49,7776,7778,7810],{"className":7777,"translate":53},[52],[49,7779,7781],{"className":7780},[57],[59,7782,7783],{"xmlns":61},[63,7784,7785,7807],{},[66,7786,7787,7789,7791,7793,7799,7801],{},[69,7788,769],{"mathvariant":768},[69,7790,153],{},[155,7792,157],{},[159,7794,7795,7797],{},[69,7796,164],{"mathvariant":163},[69,7798,71],{},[155,7800,169],{},[159,7802,7803,7805],{},[69,7804,164],{"mathvariant":163},[69,7806,71],{},[73,7808,7809],{"encoding":75},"\\nabla f: \\mathbb{R}^N \\to \\mathbb{R}^N",[49,7811,7813,7834,7878],{"className":7812,"ariaHidden":80},[79],[49,7814,7816,7819,7822,7825,7828,7831],{"className":7815},[84],[49,7817],{"className":7818,"style":184},[88],[49,7820,769],{"className":7821},[93],[49,7823,153],{"className":7824,"style":188},[93,94],[49,7826],{"className":7827,"style":193},[192],[49,7829,157],{"className":7830},[197],[49,7832],{"className":7833,"style":193},[192],[49,7835,7837,7840,7869,7872,7875],{"className":7836},[84],[49,7838],{"className":7839,"style":207},[88],[49,7841,7843,7846],{"className":7842},[93],[49,7844,164],{"className":7845},[93,214],[49,7847,7849],{"className":7848},[218],[49,7850,7852],{"className":7851},[222],[49,7853,7855],{"className":7854},[226],[49,7856,7858],{"className":7857,"style":207},[230],[49,7859,7860,7863],{"style":233},[49,7861],{"className":7862,"style":238},[237],[49,7864,7866],{"className":7865},[242,243,244,245],[49,7867,71],{"className":7868,"style":95},[93,94,245],[49,7870],{"className":7871,"style":193},[192],[49,7873,169],{"className":7874},[197],[49,7876],{"className":7877,"style":193},[192],[49,7879,7881,7884],{"className":7880},[84],[49,7882],{"className":7883,"style":207},[88],[49,7885,7887,7890],{"className":7886},[93],[49,7888,164],{"className":7889},[93,214],[49,7891,7893],{"className":7892},[218],[49,7894,7896],{"className":7895},[222],[49,7897,7899],{"className":7898},[226],[49,7900,7902],{"className":7901,"style":207},[230],[49,7903,7904,7907],{"style":233},[49,7905],{"className":7906,"style":238},[237],[49,7908,7910],{"className":7909},[242,243,244,245],[49,7911,71],{"className":7912,"style":95},[93,94,245]," 的零点。该映射在 ",[49,7915,7917,7934],{"className":7916,"translate":53},[52],[49,7918,7920],{"className":7919},[57],[59,7921,7922],{"xmlns":61},[63,7923,7924,7932],{},[66,7925,7926],{},[527,7927,7928,7930],{},[69,7929,520],{},[69,7931,1426],{},[73,7933,3864],{"encoding":75},[49,7935,7937],{"className":7936,"ariaHidden":80},[79],[49,7938,7940,7943],{"className":7939},[84],[49,7941],{"className":7942,"style":3134},[88],[49,7944,7946,7949],{"className":7945},[93],[49,7947,520],{"className":7948},[93,94],[49,7950,7952],{"className":7951},[218],[49,7953,7955,7975],{"className":7954},[222,605],[49,7956,7958,7972],{"className":7957},[226],[49,7959,7961],{"className":7960,"style":1466},[230],[49,7962,7963,7966],{"style":615},[49,7964],{"className":7965,"style":238},[237],[49,7967,7969],{"className":7968},[242,243,244,245],[49,7970,1426],{"className":7971},[93,94,245],[49,7973,629],{"className":7974},[628],[49,7976,7978],{"className":7977},[226],[49,7979,7981],{"className":7980,"style":636},[230],[49,7982],{}," 处的导数就是 Hessian ",[49,7985,7987,8014],{"className":7986,"translate":53},[52],[49,7988,7990],{"className":7989},[57],[59,7991,7992],{"xmlns":61},[63,7993,7994,8012],{},[66,7995,7996,8002,8004,8010],{},[527,7997,7998,8000],{},[69,7999,1880],{},[69,8001,153],{},[155,8003,298],{"stretchy":297},[527,8005,8006,8008],{},[69,8007,520],{},[69,8009,1426],{},[155,8011,307],{"stretchy":297},[73,8013,4444],{"encoding":75},[49,8015,8017],{"className":8016,"ariaHidden":80},[79],[49,8018,8020,8023,8063,8066,8106],{"className":8019},[84],[49,8021],{"className":8022,"style":1986},[88],[49,8024,8026,8029],{"className":8025},[93],[49,8027,1880],{"className":8028,"style":1993},[93,94],[49,8030,8032],{"className":8031},[218],[49,8033,8035,8055],{"className":8034},[222,605],[49,8036,8038,8052],{"className":8037},[226],[49,8039,8041],{"className":8040,"style":2006},[230],[49,8042,8043,8046],{"style":2009},[49,8044],{"className":8045,"style":238},[237],[49,8047,8049],{"className":8048},[242,243,244,245],[49,8050,153],{"className":8051,"style":188},[93,94,245],[49,8053,629],{"className":8054},[628],[49,8056,8058],{"className":8057},[226],[49,8059,8061],{"className":8060,"style":2028},[230],[49,8062],{},[49,8064,298],{"className":8065},[373],[49,8067,8069,8072],{"className":8068},[93],[49,8070,520],{"className":8071},[93,94],[49,8073,8075],{"className":8074},[218],[49,8076,8078,8098],{"className":8077},[222,605],[49,8079,8081,8095],{"className":8080},[226],[49,8082,8084],{"className":8083,"style":1466},[230],[49,8085,8086,8089],{"style":615},[49,8087],{"className":8088,"style":238},[237],[49,8090,8092],{"className":8091},[242,243,244,245],[49,8093,1426],{"className":8094},[93,94,245],[49,8096,629],{"className":8097},[628],[49,8099,8101],{"className":8100},[226],[49,8102,8104],{"className":8103,"style":636},[230],[49,8105],{},[49,8107,307],{"className":8108},[406],[49,8110,8112,8129],{"className":8111,"translate":53},[52],[49,8113,8115],{"className":8114},[57],[59,8116,8117],{"xmlns":61},[63,8118,8119,8127],{},[66,8120,8121],{},[527,8122,8123,8125],{},[69,8124,520],{},[69,8126,1426],{},[73,8128,3864],{"encoding":75},[49,8130,8132],{"className":8131,"ariaHidden":80},[79],[49,8133,8135,8138],{"className":8134},[84],[49,8136],{"className":8137,"style":3134},[88],[49,8139,8141,8144],{"className":8140},[93],[49,8142,520],{"className":8143},[93,94],[49,8145,8147],{"className":8146},[218],[49,8148,8150,8170],{"className":8149},[222,605],[49,8151,8153,8167],{"className":8152},[226],[49,8154,8156],{"className":8155,"style":1466},[230],[49,8157,8158,8161],{"style":615},[49,8159],{"className":8160,"style":238},[237],[49,8162,8164],{"className":8163},[242,243,244,245],[49,8165,1426],{"className":8166},[93,94,245],[49,8168,629],{"className":8169},[628],[49,8171,8173],{"className":8172},[226],[49,8174,8176],{"className":8175,"style":636},[230],[49,8177],{}," 非退化，则 ",[49,8180,8182,8209],{"className":8181,"translate":53},[52],[49,8183,8185],{"className":8184},[57],[59,8186,8187],{"xmlns":61},[63,8188,8189,8207],{},[66,8190,8191,8197,8199,8205],{},[527,8192,8193,8195],{},[69,8194,1880],{},[69,8196,153],{},[155,8198,298],{"stretchy":297},[527,8200,8201,8203],{},[69,8202,520],{},[69,8204,1426],{},[155,8206,307],{"stretchy":297},[73,8208,4444],{"encoding":75},[49,8210,8212],{"className":8211,"ariaHidden":80},[79],[49,8213,8215,8218,8258,8261,8301],{"className":8214},[84],[49,8216],{"className":8217,"style":1986},[88],[49,8219,8221,8224],{"className":8220},[93],[49,8222,1880],{"className":8223,"style":1993},[93,94],[49,8225,8227],{"className":8226},[218],[49,8228,8230,8250],{"className":8229},[222,605],[49,8231,8233,8247],{"className":8232},[226],[49,8234,8236],{"className":8235,"style":2006},[230],[49,8237,8238,8241],{"style":2009},[49,8239],{"className":8240,"style":238},[237],[49,8242,8244],{"className":8243},[242,243,244,245],[49,8245,153],{"className":8246,"style":188},[93,94,245],[49,8248,629],{"className":8249},[628],[49,8251,8253],{"className":8252},[226],[49,8254,8256],{"className":8255,"style":2028},[230],[49,8257],{},[49,8259,298],{"className":8260},[373],[49,8262,8264,8267],{"className":8263},[93],[49,8265,520],{"className":8266},[93,94],[49,8268,8270],{"className":8269},[218],[49,8271,8273,8293],{"className":8272},[222,605],[49,8274,8276,8290],{"className":8275},[226],[49,8277,8279],{"className":8278,"style":1466},[230],[49,8280,8281,8284],{"style":615},[49,8282],{"className":8283,"style":238},[237],[49,8285,8287],{"className":8286},[242,243,244,245],[49,8288,1426],{"className":8289},[93,94,245],[49,8291,629],{"className":8292},[628],[49,8294,8296],{"className":8295},[226],[49,8297,8299],{"className":8298,"style":636},[230],[49,8300],{},[49,8302,307],{"className":8303},[406]," 可逆。由反函数定理，",[49,8306,8308,8324],{"className":8307,"translate":53},[52],[49,8309,8311],{"className":8310},[57],[59,8312,8313],{"xmlns":61},[63,8314,8315,8321],{},[66,8316,8317,8319],{},[69,8318,769],{"mathvariant":768},[69,8320,153],{},[73,8322,8323],{"encoding":75},"\\nabla f",[49,8325,8327],{"className":8326,"ariaHidden":80},[79],[49,8328,8330,8333,8336],{"className":8329},[84],[49,8331],{"className":8332,"style":184},[88],[49,8334,769],{"className":8335},[93],[49,8337,153],{"className":8338,"style":188},[93,94],[49,8340,8342,8359],{"className":8341,"translate":53},[52],[49,8343,8345],{"className":8344},[57],[59,8346,8347],{"xmlns":61},[63,8348,8349,8357],{},[66,8350,8351],{},[527,8352,8353,8355],{},[69,8354,520],{},[69,8356,1426],{},[73,8358,3864],{"encoding":75},[49,8360,8362],{"className":8361,"ariaHidden":80},[79],[49,8363,8365,8368],{"className":8364},[84],[49,8366],{"className":8367,"style":3134},[88],[49,8369,8371,8374],{"className":8370},[93],[49,8372,520],{"className":8373},[93,94],[49,8375,8377],{"className":8376},[218],[49,8378,8380,8400],{"className":8379},[222,605],[49,8381,8383,8397],{"className":8382},[226],[49,8384,8386],{"className":8385,"style":1466},[230],[49,8387,8388,8391],{"style":615},[49,8389],{"className":8390,"style":238},[237],[49,8392,8394],{"className":8393},[242,243,244,245],[49,8395,1426],{"className":8396},[93,94,245],[49,8398,629],{"className":8399},[628],[49,8401,8403],{"className":8402},[226],[49,8404,8406],{"className":8405,"style":636},[230],[49,8407],{}," 附近局部可逆，因此 ",[49,8410,8412,8429],{"className":8411,"translate":53},[52],[49,8413,8415],{"className":8414},[57],[59,8416,8417],{"xmlns":61},[63,8418,8419,8427],{},[66,8420,8421],{},[527,8422,8423,8425],{},[69,8424,520],{},[69,8426,1426],{},[73,8428,3864],{"encoding":75},[49,8430,8432],{"className":8431,"ariaHidden":80},[79],[49,8433,8435,8438],{"className":8434},[84],[49,8436],{"className":8437,"style":3134},[88],[49,8439,8441,8444],{"className":8440},[93],[49,8442,520],{"className":8443},[93,94],[49,8445,8447],{"className":8446},[218],[49,8448,8450,8470],{"className":8449},[222,605],[49,8451,8453,8467],{"className":8452},[226],[49,8454,8456],{"className":8455,"style":1466},[230],[49,8457,8458,8461],{"style":615},[49,8459],{"className":8460,"style":238},[237],[49,8462,8464],{"className":8463},[242,243,244,245],[49,8465,1426],{"className":8466},[93,94,245],[49,8468,629],{"className":8469},[628],[49,8471,8473],{"className":8472},[226],[49,8474,8476],{"className":8475,"style":636},[230],[49,8477],{}," 是 ",[49,8480,8482,8497],{"className":8481,"translate":53},[52],[49,8483,8485],{"className":8484},[57],[59,8486,8487],{"xmlns":61},[63,8488,8489,8495],{},[66,8490,8491,8493],{},[69,8492,769],{"mathvariant":768},[69,8494,153],{},[73,8496,8323],{"encoding":75},[49,8498,8500],{"className":8499,"ariaHidden":80},[79],[49,8501,8503,8506,8509],{"className":8502},[84],[49,8504],{"className":8505,"style":184},[88],[49,8507,769],{"className":8508},[93],[49,8510,153],{"className":8511,"style":188},[93,94]," 在某个邻域内的唯一零点。于是每个临界点都是孤立的。由于临界点集是闭集，紧区域内若有无限多个临界点，就会有一个同样为临界的聚点，这与孤立性矛盾。因此任何紧区域中只有有限多个临界点。有限集具有 Lebesgue 测度零。",[49,8514,8516,8531],{"className":8515,"translate":53},[52],[49,8517,8519],{"className":8518},[57],[59,8520,8521],{"xmlns":61},[63,8522,8523,8528],{},[66,8524,8525],{},[69,8526,8527],{"mathvariant":768},"■",[73,8529,8530],{"encoding":75},"\\blacksquare",[49,8532,8534],{"className":8533,"ariaHidden":80},[79],[49,8535,8537,8541],{"className":8536},[84],[49,8538],{"className":8539,"style":8540},[88],"height:0.675em;",[49,8542,8527],{"className":8543},[93,8544],"amsrm",[11,8546,8547,8548,8606,8607,8610],{},"这说明：如果从 ",[49,8549,8551,8568],{"className":8550,"translate":53},[52],[49,8552,8554],{"className":8553},[57],[59,8555,8556],{"xmlns":61},[63,8557,8558,8566],{},[66,8559,8560],{},[159,8561,8562,8564],{},[69,8563,164],{"mathvariant":163},[69,8565,71],{},[73,8567,7728],{"encoding":75},[49,8569,8571],{"className":8570,"ariaHidden":80},[79],[49,8572,8574,8577],{"className":8573},[84],[49,8575],{"className":8576,"style":207},[88],[49,8578,8580,8583],{"className":8579},[93],[49,8581,164],{"className":8582},[93,214],[49,8584,8586],{"className":8585},[218],[49,8587,8589],{"className":8588},[222],[49,8590,8592],{"className":8591},[226],[49,8593,8595],{"className":8594,"style":207},[230],[49,8596,8597,8600],{"style":233},[49,8598],{"className":8599,"style":238},[237],[49,8601,8603],{"className":8602},[242,243,244,245],[49,8604,71],{"className":8605,"style":95},[93,94,245]," 中的一个盒子里均匀采样一点，恰好落在临界点上的概率为零。但真正有趣的问题是",[28,8608,8609],{},"条件性的","：",[469,8612,8613],{},[11,8614,8615],{},"既然我们已经身处某个高维随机函数的临界点，那么这个临界点是非退化局部极小值的概率是多少？",[11,8617,8618],{},"这个问题的答案，由 Hessian 特征值的符号来回答。",[39,8620,8621],{"id":8621},"马鞍面",[11,8623,8624],{},"在进入随机矩阵论证之前，先看一个最简单的鞍点例子，建立手感。",[11,8626,8627,8628,505,8713,8765],{},"函数 ",[49,8629,8631,8662],{"className":8630,"translate":53},[52],[49,8632,8634],{"className":8633},[57],[59,8635,8636],{"xmlns":61},[63,8637,8638,8659],{},[66,8639,8640,8642,8644,8646,8648,8651,8653,8655,8657],{},[69,8641,153],{},[155,8643,298],{"stretchy":297},[69,8645,520],{},[155,8647,536],{"separator":80},[69,8649,8650],{},"y",[155,8652,307],{"stretchy":297},[155,8654,523],{},[69,8656,520],{},[69,8658,8650],{},[73,8660,8661],{"encoding":75},"f(x,y) = xy",[49,8663,8665,8701],{"className":8664,"ariaHidden":80},[79],[49,8666,8668,8671,8674,8677,8680,8683,8686,8689,8692,8695,8698],{"className":8667},[84],[49,8669],{"className":8670,"style":589},[88],[49,8672,153],{"className":8673,"style":188},[93,94],[49,8675,298],{"className":8676},[373],[49,8678,520],{"className":8679},[93,94],[49,8681,536],{"className":8682},[642],[49,8684],{"className":8685,"style":646},[192],[49,8687,8650],{"className":8688,"style":4591},[93,94],[49,8690,307],{"className":8691},[406],[49,8693],{"className":8694,"style":193},[192],[49,8696,523],{"className":8697},[197],[49,8699],{"className":8700,"style":193},[192],[49,8702,8704,8707,8710],{"className":8703},[84],[49,8705],{"className":8706,"style":4584},[88],[49,8708,520],{"className":8709},[93,94],[49,8711,8650],{"className":8712,"style":4591},[93,94],[49,8714,8716,8738],{"className":8715,"translate":53},[52],[49,8717,8719],{"className":8718},[57],[59,8720,8721],{"xmlns":61},[63,8722,8723,8735],{},[66,8724,8725,8727,8729,8731,8733],{},[155,8726,298],{"stretchy":297},[292,8728,801],{},[155,8730,536],{"separator":80},[292,8732,801],{},[155,8734,307],{"stretchy":297},[73,8736,8737],{"encoding":75},"(0,0)",[49,8739,8741],{"className":8740,"ariaHidden":80},[79],[49,8742,8744,8747,8750,8753,8756,8759,8762],{"className":8743},[84],[49,8745],{"className":8746,"style":589},[88],[49,8748,298],{"className":8749},[373],[49,8751,801],{"className":8752},[93],[49,8754,536],{"className":8755},[642],[49,8757],{"className":8758,"style":646},[192],[49,8760,801],{"className":8761},[93],[49,8763,307],{"className":8764},[406]," 处有一个临界点，因为",[49,8767,8769],{"className":8768,"translate":53},[752],[49,8770,8772,8815],{"className":8771,"translate":53},[52],[49,8773,8775],{"className":8774},[57],[59,8776,8777],{"xmlns":61,"display":761},[63,8778,8779,8812],{},[66,8780,8781,8783,8785,8787,8789,8791,8793,8795,8797,8810],{},[69,8782,769],{"mathvariant":768},[69,8784,153],{},[155,8786,298],{"stretchy":297},[69,8788,520],{},[155,8790,536],{"separator":80},[69,8792,8650],{},[155,8794,307],{"stretchy":297},[155,8796,523],{},[66,8798,8799,8801,8808],{},[155,8800,298],{"fence":80},[807,8802,8804,8806],{"linethickness":8803},"0px",[69,8805,8650],{},[69,8807,520],{},[155,8809,307],{"fence":80},[155,8811,536],{"separator":80},[73,8813,8814],{"encoding":75},"\\nabla f(x,y) = \\binom{y}{x},",[49,8816,8818,8857],{"className":8817,"ariaHidden":80},[79],[49,8819,8821,8824,8827,8830,8833,8836,8839,8842,8845,8848,8851,8854],{"className":8820},[84],[49,8822],{"className":8823,"style":589},[88],[49,8825,769],{"className":8826},[93],[49,8828,153],{"className":8829,"style":188},[93,94],[49,8831,298],{"className":8832},[373],[49,8834,520],{"className":8835},[93,94],[49,8837,536],{"className":8838},[642],[49,8840],{"className":8841,"style":646},[192],[49,8843,8650],{"className":8844,"style":4591},[93,94],[49,8846,307],{"className":8847},[406],[49,8849],{"className":8850,"style":193},[192],[49,8852,523],{"className":8853},[197],[49,8855],{"className":8856,"style":193},[192],[49,8858,8860,8864,8925],{"className":8859},[84],[49,8861],{"className":8862,"style":8863},[88],"height:2.4em;vertical-align:-0.95em;",[49,8865,8867,8873,8919],{"className":8866},[93],[49,8868,8870],{"className":8869,"style":2124},[373,2123],[49,8871,298],{"className":8872},[951,244],[49,8874,8876],{"className":8875},[807],[49,8877,8879,8911],{"className":8878},[222,605],[49,8880,8882,8908],{"className":8881},[226],[49,8883,8886,8897],{"className":8884,"style":8885},[230],"height:1.1076em;",[49,8887,8888,8891],{"style":1053},[49,8889],{"className":8890,"style":1057},[237],[49,8892,8894],{"className":8893},[93],[49,8895,520],{"className":8896},[93,94],[49,8898,8899,8902],{"style":1118},[49,8900],{"className":8901,"style":1057},[237],[49,8903,8905],{"className":8904},[93],[49,8906,8650],{"className":8907,"style":4591},[93,94],[49,8909,629],{"className":8910},[628],[49,8912,8914],{"className":8913},[226],[49,8915,8917],{"className":8916,"style":3632},[230],[49,8918],{},[49,8920,8922],{"className":8921,"style":2124},[406,2123],[49,8923,307],{"className":8924},[951,244],[49,8926,536],{"className":8927},[642],[11,8929,8930,8931,9015],{},"所以 ",[49,8932,8934,8964],{"className":8933,"translate":53},[52],[49,8935,8937],{"className":8936},[57],[59,8938,8939],{"xmlns":61},[63,8940,8941,8961],{},[66,8942,8943,8945,8947,8949,8951,8953,8955,8957,8959],{},[69,8944,769],{"mathvariant":768},[69,8946,153],{},[155,8948,298],{"stretchy":297},[292,8950,801],{},[155,8952,536],{"separator":80},[292,8954,801],{},[155,8956,307],{"stretchy":297},[155,8958,523],{},[292,8960,801],{},[73,8962,8963],{"encoding":75},"\\nabla f(0,0) = 0",[49,8965,8967,9006],{"className":8966,"ariaHidden":80},[79],[49,8968,8970,8973,8976,8979,8982,8985,8988,8991,8994,8997,9000,9003],{"className":8969},[84],[49,8971],{"className":8972,"style":589},[88],[49,8974,769],{"className":8975},[93],[49,8977,153],{"className":8978,"style":188},[93,94],[49,8980,298],{"className":8981},[373],[49,8983,801],{"className":8984},[93],[49,8986,536],{"className":8987},[642],[49,8989],{"className":8990,"style":646},[192],[49,8992,801],{"className":8993},[93],[49,8995,307],{"className":8996},[406],[49,8998],{"className":8999,"style":193},[192],[49,9001,523],{"className":9002},[197],[49,9004],{"className":9005,"style":193},[192],[49,9007,9009,9012],{"className":9008},[84],[49,9010],{"className":9011,"style":6838},[88],[49,9013,801],{"className":9014},[93],"。它的 Hessian 是",[49,9017,9019],{"className":9018,"translate":53},[752],[49,9020,9022,9091],{"className":9021,"translate":53},[52],[49,9023,9025],{"className":9024},[57],[59,9026,9027],{"xmlns":61,"display":761},[63,9028,9029,9088],{},[66,9030,9031,9037,9039,9041,9043,9045,9047,9049,9086],{},[527,9032,9033,9035],{},[69,9034,1880],{},[69,9036,153],{},[155,9038,298],{"stretchy":297},[292,9040,801],{},[155,9042,536],{"separator":80},[292,9044,801],{},[155,9046,307],{"stretchy":297},[155,9048,523],{},[66,9050,9051,9053,9084],{},[155,9052,298],{"fence":80},[787,9054,9056,9070],{"rowspacing":789,"columnalign":9055,"columnspacing":791},"center 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",[49,9304,9306,9319],{"className":9305,"translate":53},[52],[49,9307,9309],{"className":9308},[57],[59,9310,9311],{"xmlns":61},[63,9312,9313,9317],{},[66,9314,9315],{},[292,9316,533],{},[73,9318,533],{"encoding":75},[49,9320,9322],{"className":9321,"ariaHidden":80},[79],[49,9323,9325,9328],{"className":9324},[84],[49,9326],{"className":9327,"style":6838},[88],[49,9329,533],{"className":9330},[93]," 和 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",[49,9372,9374,9396],{"className":9373,"translate":53},[52],[49,9375,9377],{"className":9376},[57],[59,9378,9379],{"xmlns":61},[63,9380,9381,9393],{},[66,9382,9383,9385,9387,9389,9391],{},[155,9384,298],{"stretchy":297},[69,9386,520],{},[155,9388,536],{"separator":80},[69,9390,8650],{},[155,9392,307],{"stretchy":297},[73,9394,9395],{"encoding":75},"(x,y)",[49,9397,9399],{"className":9398,"ariaHidden":80},[79],[49,9400,9402,9405,9408,9411,9414,9417,9420],{"className":9401},[84],[49,9403],{"className":9404,"style":589},[88],[49,9406,298],{"className":9407},[373],[49,9409,520],{"className":9410},[93,94],[49,9412,536],{"className":9413},[642],[49,9415],{"className":9416,"style":646},[192],[49,9418,8650],{"className":9419,"style":4591},[93,94],[49,9421,307],{"className":9422},[406]," 坐标中，二阶导数 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与 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都为零，而混合导数 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非零——主曲率方向并不是坐标轴。定义旋转后的坐标",[49,9668,9670],{"className":9669,"translate":53},[752],[49,9671,9673,9730],{"className":9672,"translate":53},[52],[49,9674,9676],{"className":9675},[57],[59,9677,9678],{"xmlns":61,"display":761},[63,9679,9680,9727],{},[66,9681,9682,9685,9687,9702,9704,9707,9709,9711,9725],{},[69,9683,9684],{},"u",[155,9686,523],{},[807,9688,9689,9697],{},[66,9690,9691,9693,9695],{},[69,9692,520],{},[155,9694,3213],{},[69,9696,8650],{},[9698,9699,9700],"msqrt",{},[292,9701,294],{},[155,9703,536],{"separator":80},[192,9705],{"width":9706},"2em",[69,9708,4557],{},[155,9710,523],{},[807,9712,9713,9721],{},[66,9714,9715,9717,9719],{},[69,9716,520],{},[155,9718,3077],{},[69,9720,8650],{},[9698,9722,9723],{},[292,9724,294],{},[155,9726,536],{"separator":80},[73,9728,9729],{"encoding":75},"u = \\frac{x+y}{\\sqrt2}, \\qquad v = 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= \\frac{u+v}{\\sqrt2}\\cdot\\frac{u-v}{\\sqrt2} = 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是对角的：",[49,10891,10893],{"className":10892,"translate":53},[752],[49,10894,10896,10954],{"className":10895,"translate":53},[52],[49,10897,10899],{"className":10898},[57],[59,10900,10901],{"xmlns":61,"display":761},[63,10902,10903,10951],{},[66,10904,10905,10907,10909,10949],{},[69,10906,1880],{},[155,10908,523],{},[66,10910,10911,10913,10947],{},[155,10912,298],{"fence":80},[787,10914,10915,10929],{"rowspacing":789,"columnalign":9055,"columnspacing":791},[793,10916,10917,10923],{},[796,10918,10919],{},[799,10920,10921],{"scriptlevel":801,"displaystyle":297},[292,10922,533],{},[796,10924,10925],{},[799,10926,10927],{"scriptlevel":801,"displaystyle":297},[292,10928,801],{},[793,10930,10931,10937],{},[796,10932,10933],{},[799,10934,10935],{"scriptlevel":801,"displaystyle":297},[292,10936,801],{},[796,10938,10939],{},[799,10940,10941],{"scriptlevel":801,"displaystyle":297},[66,10942,10943,10945],{},[155,10944,3077],{},[292,10946,533],{},[155,10948,307],{"fence":80},[69,10950,897],{"mathvariant":768},[73,10952,10953],{"encoding":75},"H = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}.",[49,10955,10957,10975],{"className":10956,"ariaHidden":80},[79],[49,10958,10960,10963,10966,10969,10972],{"className":10959},[84],[49,10961],{"className":10962,"style":89},[88],[49,10964,1880],{"className":10965,"style":1993},[93,94],[49,10967],{"className":10968,"style":193},[192],[49,10970,523],{"className":10971},[197],[49,10973],{"className":10974,"style":193},[192],[49,10976,10978,10981,11101,11104],{"className":10977},[84],[49,10979],{"className":10980,"style":8863},[88],[49,10982,10984,10990,11095],{"className":10983},[650],[49,10985,10987],{"className":10986,"style":2124},[373,2123],[49,10988,298],{"className":10989},[951,244],[49,10991,10993],{"className":10992},[93],[49,10994,10996,11041,11044,11047],{"className":10995},[787],[49,10997,10999],{"className":10998},[1010],[49,11000,11002,11033],{"className":11001},[222,605],[49,11003,11005,11030],{"className":11004},[226],[49,11006,11008,11019],{"className":11007,"style":9200},[230],[49,11009,11010,11013],{"style":9203},[49,11011],{"className":11012,"style":1057},[237],[49,11014,11016],{"className":11015},[93],[49,11017,533],{"className":11018},[93],[49,11020,11021,11024],{"style":9215},[49,11022],{"className":11023,"style":1057},[237],[49,11025,11027],{"className":11026},[93],[49,11028,801],{"className":11029},[93],[49,11031,629],{"className":11032},[628],[49,11034,11036],{"className":11035},[226],[49,11037,11039],{"className":11038,"style":9234},[230],[49,11040],{},[49,11042],{"className":11043,"style":1006},[1005],[49,11045],{"className":11046,"style":1006},[1005],[49,11048,11050],{"className":11049},[1010],[49,11051,11053,11087],{"className":11052},[222,605],[49,11054,11056,11084],{"className":11055},[226],[49,11057,11059,11070],{"className":11058,"style":9200},[230],[49,11060,11061,11064],{"style":9203},[49,11062],{"className":11063,"style":1057},[237],[49,11065,11067],{"className":11066},[93],[49,11068,801],{"className":11069},[93],[49,11071,11072,11075],{"style":9215},[49,11073],{"className":11074,"style":1057},[237],[49,11076,11078,11081],{"className":11077},[93],[49,11079,3077],{"className":11080},[93],[49,11082,533],{"className":11083},[93],[49,11085,629],{"className":11086},[628],[49,11088,11090],{"className":11089},[226],[49,11091,11093],{"className":11092,"style":9234},[230],[49,11094],{},[49,11096,11098],{"className":11097,"style":2124},[406,2123],[49,11099,307],{"className":11100},[951,244],[49,11102],{"className":11103,"style":646},[192],[49,11105,897],{"className":11106},[93],[11,11108,11109,11110,11138,11139,11167],{},"此刻鞍点结构一目了然：函数在 ",[49,11111,11113,11126],{"className":11112,"translate":53},[52],[49,11114,11116],{"className":11115},[57],[59,11117,11118],{"xmlns":61},[63,11119,11120,11124],{},[66,11121,11122],{},[69,11123,9684],{},[73,11125,9684],{"encoding":75},[49,11127,11129],{"className":11128,"ariaHidden":80},[79],[49,11130,11132,11135],{"className":11131},[84],[49,11133],{"className":11134,"style":570},[88],[49,11136,9684],{"className":11137},[93,94]," 方向向上弯曲，在 ",[49,11140,11142,11155],{"className":11141,"translate":53},[52],[49,11143,11145],{"className":11144},[57],[59,11146,11147],{"xmlns":61},[63,11148,11149,11153],{},[66,11150,11151],{},[69,11152,4557],{},[73,11154,4557],{"encoding":75},[49,11156,11158],{"className":11157,"ariaHidden":80},[79],[49,11159,11161,11164],{"className":11160},[84],[49,11162],{"className":11163,"style":570},[88],[49,11165,4557],{"className":11166,"style":4591},[93,94]," 方向向下弯曲。",[11,11169,11170,11171,11174,11175,11178],{},"不变的事实并不是 Hessian 在某个选定坐标系中的具体矩阵元素，而是它的",[28,11172,11173],{},"符号分布","：一个正特征值、一个负特征值。这是",[28,11176,11177],{},"西尔维斯特惯性定律（Sylvester's law of inertia）"," 的一个特例——该定律说，实二次型的正、负、零方向的数目，在可逆线性坐标变换下保持不变。",[39,11180,11181],{"id":11181},"随机矩阵的视角",[11,11183,11184],{},"现在回到主线问题：假设我们随机站在高维景观的某个临界点上，它是局部极小值的概率有多大？",[11,11186,11187],{},"第一个启发式是把典型临界点处的 Hessian 建模为一个随机实对称矩阵。这个模型对每个随机函数并非字面上都正确——Hessian 可能与函数值、梯度或问题中其它结构相关——但它抓住了核心机制：如果 Hessian 没有强的正偏向，它的特征值谱就应当同时含有正负号。",[11189,11190,11191],"h3",{"id":11191},"简化启发",[11,11193,11194,11195,11229],{},"仅仅出于直觉，假设每个 Hessian 特征值取正的概率为 ",[49,11196,11198,11217],{"className":11197,"translate":53},[52],[49,11199,11201],{"className":11200},[57],[59,11202,11203],{"xmlns":61},[63,11204,11205,11214],{},[66,11206,11207,11209,11212],{},[292,11208,533],{},[69,11210,11211],{"mathvariant":768},"\u002F",[292,11213,294],{},[73,11215,11216],{"encoding":75},"1\u002F2",[49,11218,11220],{"className":11219,"ariaHidden":80},[79],[49,11221,11223,11226],{"className":11222},[84],[49,11224],{"className":11225,"style":589},[88],[49,11227,11216],{"className":11228},[93],"，且这些符号相互独立。那么",[49,11231,11233],{"className":11232,"translate":53},[752],[49,11234,11236,11306],{"className":11235,"translate":53},[52],[49,11237,11239],{"className":11238},[57],[59,11240,11241],{"xmlns":61,"display":761},[63,11242,11243,11303],{},[66,11244,11245,11248,11250,11253,11255,11257,11259,11261,11267,11269,11271,11273,11275,11277,11283,11285,11287,11289,11291,11301],{},[69,11246,11247],{"mathvariant":163},"P",[155,11249,298],{"stretchy":297},[3314,11251,11252],{},"非退化局部极小值",[155,11254,307],{"stretchy":297},[155,11256,523],{},[69,11258,11247],{"mathvariant":163},[155,11260,298],{"stretchy":297},[527,11262,11263,11265],{},[69,11264,4702],{},[292,11266,533],{},[155,11268,6927],{},[292,11270,801],{},[155,11272,536],{"separator":80},[155,11274,539],{},[155,11276,536],{"separator":80},[527,11278,11279,11281],{},[69,11280,4702],{},[69,11282,71],{},[155,11284,6927],{},[292,11286,801],{},[155,11288,307],{"stretchy":297},[155,11290,523],{},[159,11292,11293,11295],{},[292,11294,294],{},[66,11296,11297,11299],{},[155,11298,3077],{},[69,11300,71],{},[69,11302,897],{"mathvariant":768},[73,11304,11305],{"encoding":75},"\\mathbb{P}(\\text{非退化局部极小值}) = \\mathbb{P}(\\lambda_1>0,\\dots,\\lambda_N>0) = 2^{-N}.",[49,11307,11309,11340,11401,11477,11498],{"className":11308,"ariaHidden":80},[79],[49,11310,11312,11315,11318,11321,11328,11331,11334,11337],{"className":11311},[84],[49,11313],{"className":11314,"style":589},[88],[49,11316,11247],{"className":11317},[93,214],[49,11319,298],{"className":11320},[373],[49,11322,11324],{"className":11323},[93,3828],[49,11325,11252],{"className":11326},[93,11327],"cjk_fallback",[49,11329,307],{"className":11330},[406],[49,11332],{"className":11333,"style":193},[192],[49,11335,523],{"className":11336},[197],[49,11338],{"className":11339,"style":193},[192],[49,11341,11343,11346,11349,11352,11392,11395,11398],{"className":11342},[84],[49,11344],{"className":11345,"style":589},[88],[49,11347,11247],{"className":11348},[93,214],[49,11350,298],{"className":11351},[373],[49,11353,11355,11358],{"className":11354},[93],[49,11356,4702],{"className":11357},[93,94],[49,11359,11361],{"className":11360},[218],[49,11362,11364,11384],{"className":11363},[222,605],[49,11365,11367,11381],{"className":11366},[226],[49,11368,11370],{"className":11369,"style":612},[230],[49,11371,11372,11375],{"style":615},[49,11373],{"className":11374,"style":238},[237],[49,11376,11378],{"className":11377},[242,243,244,245],[49,11379,533],{"className":11380},[93,245],[49,11382,629],{"className":11383},[628],[49,11385,11387],{"className":11386},[226],[49,11388,11390],{"className":11389,"style":636},[230],[49,11391],{},[49,11393],{"className":11394,"style":193},[192],[49,11396,6927],{"className":11397},[197],[49,11399],{"className":11400,"style":193},[192],[49,11402,11404,11407,11410,11413,11416,11419,11422,11425,11428,11468,11471,11474],{"className":11403},[84],[49,11405],{"className":11406,"style":184},[88],[49,11408,801],{"className":11409},[93],[49,11411,536],{"className":11412},[642],[49,11414],{"className":11415,"style":646},[192],[49,11417,539],{"className":11418},[650],[49,11420],{"className":11421,"style":646},[192],[49,11423,536],{"className":11424},[642],[49,11426],{"className":11427,"style":646},[192],[49,11429,11431,11434],{"className":11430},[93],[49,11432,4702],{"className":11433},[93,94],[49,11435,11437],{"className":11436},[218],[49,11438,11440,11460],{"className":11439},[222,605],[49,11441,11443,11457],{"className":11442},[226],[49,11444,11446],{"className":11445,"style":678},[230],[49,11447,11448,11451],{"style":615},[49,11449],{"className":11450,"style":238},[237],[49,11452,11454],{"className":11453},[242,243,244,245],[49,11455,71],{"className":11456,"style":95},[93,94,245],[49,11458,629],{"className":11459},[628],[49,11461,11463],{"className":11462},[226],[49,11464,11466],{"className":11465,"style":636},[230],[49,11467],{},[49,11469],{"className":11470,"style":193},[192],[49,11472,6927],{"className":11473},[197],[49,11475],{"className":11476,"style":193},[192],[49,11478,11480,11483,11486,11489,11492,11495],{"className":11479},[84],[49,11481],{"className":11482,"style":589},[88],[49,11484,801],{"className":11485},[93],[49,11487,307],{"className":11488},[406],[49,11490],{"className":11491,"style":193},[192],[49,11493,523],{"className":11494},[197],[49,11496],{"className":11497,"style":193},[192],[49,11499,11501,11504,11539],{"className":11500},[84],[49,11502],{"className":11503,"style":1366},[88],[49,11505,11507,11510],{"className":11506},[93],[49,11508,294],{"className":11509},[93],[49,11511,11513],{"className":11512},[218],[49,11514,11516],{"className":11515},[222],[49,11517,11519],{"className":11518},[226],[49,11520,11522],{"className":11521,"style":1366},[230],[49,11523,11524,11527],{"style":1387},[49,11525],{"className":11526,"style":238},[237],[49,11528,11530],{"className":11529},[242,243,244,245],[49,11531,11533,11536],{"className":11532},[93,245],[49,11534,3077],{"className":11535},[93,245],[49,11537,71],{"className":11538,"style":95},[93,94,245],[49,11540,897],{"className":11541},[93],[11,11543,11544,11545,11573],{},"这已经在 ",[49,11546,11548,11561],{"className":11547,"translate":53},[52],[49,11549,11551],{"className":11550},[57],[59,11552,11553],{"xmlns":61},[63,11554,11555,11559],{},[66,11556,11557],{},[69,11558,71],{},[73,11560,71],{"encoding":75},[49,11562,11564],{"className":11563,"ariaHidden":80},[79],[49,11565,11567,11570],{"className":11566},[84],[49,11568],{"className":11569,"style":89},[88],[49,11571,71],{"className":11572,"style":95},[93,94]," 上指数衰减。独立性假设对对称随机矩阵的特征值并不成立——特征值彼此强烈排斥——但「正定很罕见」这一结论仍然正确，而且真正的随机矩阵估计还要更尖锐。",[11189,11575,11577],{"id":11576},"goe-hessian-与半圆律","GOE Hessian 与半圆律",[11,11579,11580,11583,11584,11634,11635,11687,11688,11734],{},[28,11581,11582],{},"高斯正交系综（GOE）"," 是中心化随机实对称矩阵的标准模型。非正式地说，",[49,11585,11587,11604],{"className":11586,"translate":53},[52],[49,11588,11590],{"className":11589},[57],[59,11591,11592],{"xmlns":61},[63,11593,11594,11602],{},[66,11595,11596,11598,11600],{},[69,11597,71],{},[155,11599,1821],{},[69,11601,71],{},[73,11603,1826],{"encoding":75},[49,11605,11607,11625],{"className":11606,"ariaHidden":80},[79],[49,11608,11610,11613,11616,11619,11622],{"className":11609},[84],[49,11611],{"className":11612,"style":1836},[88],[49,11614,71],{"className":11615,"style":95},[93,94],[49,11617],{"className":11618,"style":1843},[192],[49,11620,1821],{"className":11621},[1847],[49,11623],{"className":11624,"style":1843},[192],[49,11626,11628,11631],{"className":11627},[84],[49,11629],{"className":11630,"style":89},[88],[49,11632,71],{"className":11633,"style":95},[93,94]," 的 GOE 矩阵是其独立元素为高斯的对称矩阵，归一化方式使得特征值在 ",[49,11636,11638,11657],{"className":11637,"translate":53},[52],[49,11639,11641],{"className":11640},[57],[59,11642,11643],{"xmlns":61},[63,11644,11645,11654],{},[66,11646,11647,11649,11651],{},[69,11648,71],{},[155,11650,169],{},[69,11652,11653],{"mathvariant":768},"∞",[73,11655,11656],{"encoding":75},"N\\to\\infty",[49,11658,11660,11678],{"className":11659,"ariaHidden":80},[79],[49,11661,11663,11666,11669,11672,11675],{"className":11662},[84],[49,11664],{"className":11665,"style":89},[88],[49,11667,71],{"className":11668,"style":95},[93,94],[49,11670],{"className":11671,"style":193},[192],[49,11673,169],{"className":11674},[197],[49,11676],{"className":11677,"style":193},[192],[49,11679,11681,11684],{"className":11680},[84],[49,11682],{"className":11683,"style":570},[88],[49,11685,11653],{"className":11686},[93]," 时保持在 ",[49,11689,11691,11712],{"className":11690,"translate":53},[52],[49,11692,11694],{"className":11693},[57],[59,11695,11696],{"xmlns":61},[63,11697,11698,11709],{},[66,11699,11700,11703,11705,11707],{},[69,11701,11702],{},"O",[155,11704,298],{"stretchy":297},[292,11706,533],{},[155,11708,307],{"stretchy":297},[73,11710,11711],{"encoding":75},"O(1)",[49,11713,11715],{"className":11714,"ariaHidden":80},[79],[49,11716,11718,11721,11725,11728,11731],{"className":11717},[84],[49,11719],{"className":11720,"style":589},[88],[49,11722,11702],{"className":11723,"style":11724},[93,94],"margin-right:0.0278em;",[49,11726,298],{"className":11727},[373],[49,11729,533],{"className":11730},[93],[49,11732,307],{"className":11733},[406]," 量级。",[11,11736,11737,11740,11741,11791,11792,11829,11830,407],{},[28,11738,11739],{},"维格纳半圆律（Wigner's semicircle law）"," 表明，中心化 GOE 矩阵的经验特征值分布在 ",[49,11742,11744,11761],{"className":11743,"translate":53},[52],[49,11745,11747],{"className":11746},[57],[59,11748,11749],{"xmlns":61},[63,11750,11751,11759],{},[66,11752,11753,11755,11757],{},[69,11754,71],{},[155,11756,169],{},[69,11758,11653],{"mathvariant":768},[73,11760,11656],{"encoding":75},[49,11762,11764,11782],{"className":11763,"ariaHidden":80},[79],[49,11765,11767,11770,11773,11776,11779],{"className":11766},[84],[49,11768],{"className":11769,"style":89},[88],[49,11771,71],{"className":11772,"style":95},[93,94],[49,11774],{"className":11775,"style":193},[192],[49,11777,169],{"className":11778},[197],[49,11780],{"className":11781,"style":193},[192],[49,11783,11785,11788],{"className":11784},[84],[49,11786],{"className":11787,"style":570},[88],[49,11789,11653],{"className":11790},[93]," 时收敛到关于零点对称的确定性半圆分布。因此，典型的大 GOE 矩阵有大量正特征值和大量负特征值，它的指数通常接近 ",[49,11793,11795,11813],{"className":11794,"translate":53},[52],[49,11796,11798],{"className":11797},[57],[59,11799,11800],{"xmlns":61},[63,11801,11802,11810],{},[66,11803,11804,11806,11808],{},[69,11805,71],{},[69,11807,11211],{"mathvariant":768},[292,11809,294],{},[73,11811,11812],{"encoding":75},"N\u002F2",[49,11814,11816],{"className":11815,"ariaHidden":80},[79],[49,11817,11819,11822,11825],{"className":11818},[84],[49,11820],{"className":11821,"style":589},[88],[49,11823,71],{"className":11824,"style":95},[93,94],[49,11826,11828],{"className":11827},[93],"\u002F2","，而不是接近 0 或 ",[49,11831,11833,11846],{"className":11832,"translate":53},[52],[49,11834,11836],{"className":11835},[57],[59,11837,11838],{"xmlns":61},[63,11839,11840,11844],{},[66,11841,11842],{},[69,11843,71],{},[73,11845,71],{"encoding":75},[49,11847,11849],{"className":11848,"ariaHidden":80},[79],[49,11850,11852,11855],{"className":11851},[84],[49,11853],{"className":11854,"style":89},[88],[49,11856,71],{"className":11857,"style":95},[93,94],[11,11859,11860],{},"然而，一个临界点要成为非退化局部极小值，Hessian 必须正定。在 GOE 模型中，这就是事件",[49,11862,11864],{"className":11863,"translate":53},[752],[49,11865,11867,11898],{"className":11866,"translate":53},[52],[49,11868,11870],{"className":11869},[57],[59,11871,11872],{"xmlns":61,"display":761},[63,11873,11874,11895],{},[66,11875,11876,11885,11887,11889,11891,11893],{},[527,11877,11878,11880,11883],{},[69,11879,4702],{},[69,11881,11882],{},"min",[155,11884,6548],{},[155,11886,298],{"stretchy":297},[69,11888,1880],{},[155,11890,307],{"stretchy":297},[155,11892,6927],{},[292,11894,6965],{},[73,11896,11897],{"encoding":75},"\\lambda_{\\min}(H) > 0.",[49,11899,11901,11980],{"className":11900,"ariaHidden":80},[79],[49,11902,11904,11907,11962,11965,11968,11971,11974,11977],{"className":11903},[84],[49,11905],{"className":11906,"style":589},[88],[49,11908,11910,11913],{"className":11909},[93],[49,11911,4702],{"className":11912},[93,94],[49,11914,11916],{"className":11915},[218],[49,11917,11919,11954],{"className":11918},[222,605],[49,11920,11922,11951],{"className":11921},[226],[49,11923,11926],{"className":11924,"style":11925},[230],"height:0.3175em;",[49,11927,11928,11931],{"style":615},[49,11929],{"className":11930,"style":238},[237],[49,11932,11934],{"className":11933},[242,243,244,245],[49,11935,11937],{"className":11936},[93,245],[49,11938,11940,11944,11947],{"className":11939},[4912,245],[49,11941,11943],{"className":11942},[245],"m",[49,11945,1939],{"className":11946},[245],[49,11948,11950],{"className":11949},[245],"n",[49,11952,629],{"className":11953},[628],[49,11955,11957],{"className":11956},[226],[49,11958,11960],{"className":11959,"style":636},[230],[49,11961],{},[49,11963,298],{"className":11964},[373],[49,11966,1880],{"className":11967,"style":1993},[93,94],[49,11969,307],{"className":11970},[406],[49,11972],{"className":11973,"style":193},[192],[49,11975,6927],{"className":11976},[197],[49,11978],{"className":11979,"style":193},[192],[49,11981,11983,11986],{"className":11982},[84],[49,11984],{"className":11985,"style":6838},[88],[49,11987,6965],{"className":11988},[93],[11,11990,11991],{},"这个事件迫使整个特征值谱——其自然的极限形状以零为中心——平移到正的一侧。这是一个大偏差事件。",[469,11993,11994,12224,12640],{},[11,11995,11996,5531,11999,8478,12070,12120,12121,12172,12173,12223],{},[28,11997,11998],{},"定理 4.1（GOE 正定呈指数级罕见）",[49,12000,12002,12020],{"className":12001,"translate":53},[52],[49,12003,12005],{"className":12004},[57],[59,12006,12007],{"xmlns":61},[63,12008,12009,12017],{},[66,12010,12011],{},[527,12012,12013,12015],{},[69,12014,1880],{},[69,12016,71],{},[73,12018,12019],{"encoding":75},"H_N",[49,12021,12023],{"className":12022,"ariaHidden":80},[79],[49,12024,12026,12030],{"className":12025},[84],[49,12027],{"className":12028,"style":12029},[88],"height:0.8333em;vertical-align:-0.15em;",[49,12031,12033,12036],{"className":12032},[93],[49,12034,1880],{"className":12035,"style":1993},[93,94],[49,12037,12039],{"className":12038},[218],[49,12040,12042,12062],{"className":12041},[222,605],[49,12043,12045,12059],{"className":12044},[226],[49,12046,12048],{"className":12047,"style":678},[230],[49,12049,12050,12053],{"style":2009},[49,12051],{"className":12052,"style":238},[237],[49,12054,12056],{"className":12055},[242,243,244,245],[49,12057,71],{"className":12058,"style":95},[93,94,245],[49,12060,629],{"className":12061},[628],[49,12063,12065],{"className":12064},[226],[49,12066,12068],{"className":12067,"style":636},[230],[49,12069],{},[49,12071,12073,12090],{"className":12072,"translate":53},[52],[49,12074,12076],{"className":12075},[57],[59,12077,12078],{"xmlns":61},[63,12079,12080,12088],{},[66,12081,12082,12084,12086],{},[69,12083,71],{},[155,12085,1821],{},[69,12087,71],{},[73,12089,1826],{"encoding":75},[49,12091,12093,12111],{"className":12092,"ariaHidden":80},[79],[49,12094,12096,12099,12102,12105,12108],{"className":12095},[84],[49,12097],{"className":12098,"style":1836},[88],[49,12100,71],{"className":12101,"style":95},[93,94],[49,12103],{"className":12104,"style":1843},[192],[49,12106,1821],{"className":12107},[1847],[49,12109],{"className":12110,"style":1843},[192],[49,12112,12114,12117],{"className":12113},[84],[49,12115],{"className":12116,"style":89},[88],[49,12118,71],{"className":12119,"style":95},[93,94]," 中心化 GOE 矩阵，采用通常的随机矩阵标度。则存在常数 ",[49,12122,12124,12142],{"className":12123,"translate":53},[52],[49,12125,12127],{"className":12126},[57],[59,12128,12129],{"xmlns":61},[63,12130,12131,12139],{},[66,12132,12133,12135,12137],{},[69,12134,1426],{},[155,12136,6927],{},[292,12138,801],{},[73,12140,12141],{"encoding":75},"c>0",[49,12143,12145,12163],{"className":12144,"ariaHidden":80},[79],[49,12146,12148,12151,12154,12157,12160],{"className":12147},[84],[49,12149],{"className":12150,"style":1756},[88],[49,12152,1426],{"className":12153},[93,94],[49,12155],{"className":12156,"style":193},[192],[49,12158,6927],{"className":12159},[197],[49,12161],{"className":12162,"style":193},[192],[49,12164,12166,12169],{"className":12165},[84],[49,12167],{"className":12168,"style":6838},[88],[49,12170,801],{"className":12171},[93],"，使得在 ",[49,12174,12176,12193],{"className":12175,"translate":53},[52],[49,12177,12179],{"className":12178},[57],[59,12180,12181],{"xmlns":61},[63,12182,12183,12191],{},[66,12184,12185,12187,12189],{},[69,12186,71],{},[155,12188,169],{},[69,12190,11653],{"mathvariant":768},[73,12192,11656],{"encoding":75},[49,12194,12196,12214],{"className":12195,"ariaHidden":80},[79],[49,12197,12199,12202,12205,12208,12211],{"className":12198},[84],[49,12200],{"className":12201,"style":89},[88],[49,12203,71],{"className":12204,"style":95},[93,94],[49,12206],{"className":12207,"style":193},[192],[49,12209,169],{"className":12210},[197],[49,12212],{"className":12213,"style":193},[192],[49,12215,12217,12220],{"className":12216},[84],[49,12218],{"className":12219,"style":570},[88],[49,12221,11653],{"className":12222},[93]," 时",[49,12225,12227],{"className":12226,"translate":53},[752],[49,12228,12230,12323],{"className":12229,"translate":53},[52],[49,12231,12233],{"className":12232},[57],[59,12234,12235],{"xmlns":61,"display":761},[63,12236,12237,12320],{},[66,12238,12239,12241,12243,12249,12251,12253,12255,12257,12259,12267,12269,12275,12277,12279,12281,12283,12285,12288,12290,12292,12294,12296,12302,12304,12306,12308,12314,12316,12318],{},[69,12240,11247],{"mathvariant":163},[155,12242,298],{"stretchy":297},[527,12244,12245,12247],{},[69,12246,1880],{},[69,12248,71],{},[3314,12250,30],{},[155,12252,307],{"stretchy":297},[155,12254,523],{},[69,12256,11247],{"mathvariant":163},[155,12258,298],{"stretchy":297},[527,12260,12261,12263,12265],{},[69,12262,4702],{},[69,12264,11882],{},[155,12266,6548],{},[155,12268,298],{"stretchy":297},[527,12270,12271,12273],{},[69,12272,1880],{},[69,12274,71],{},[155,12276,307],{"stretchy":297},[155,12278,6927],{},[292,12280,801],{},[155,12282,307],{"stretchy":297},[155,12284,523],{},[69,12286,12287],{},"exp",[155,12289,6548],{},[155,12291,6568],{"stretchy":297},[155,12293,3077],{},[69,12295,1426],{},[159,12297,12298,12300],{},[69,12299,71],{},[292,12301,294],{},[155,12303,3213],{},[69,12305,3291],{},[155,12307,298],{"stretchy":297},[159,12309,12310,12312],{},[69,12311,71],{},[292,12313,294],{},[155,12315,307],{"stretchy":297},[155,12317,6586],{"stretchy":297},[69,12319,897],{"mathvariant":768},[73,12321,12322],{"encoding":75},"\\mathbb{P}(H_N \\text{正定}) = \\mathbb{P}(\\lambda_{\\min}(H_N)>0) = \\exp\\{-cN^2 + o(N^2)\\}.",[49,12324,12326,12396,12515,12536,12592],{"className":12325,"ariaHidden":80},[79],[49,12327,12329,12332,12335,12338,12378,12384,12387,12390,12393],{"className":12328},[84],[49,12330],{"className":12331,"style":589},[88],[49,12333,11247],{"className":12334},[93,214],[49,12336,298],{"className":12337},[373],[49,12339,12341,12344],{"className":12340},[93],[49,12342,1880],{"className":12343,"style":1993},[93,94],[49,12345,12347],{"className":12346},[218],[49,12348,12350,12370],{"className":12349},[222,605],[49,12351,12353,12367],{"className":12352},[226],[49,12354,12356],{"className":12355,"style":678},[230],[49,12357,12358,12361],{"style":2009},[49,12359],{"className":12360,"style":238},[237],[49,12362,12364],{"className":12363},[242,243,244,245],[49,12365,71],{"className":12366,"style":95},[93,94,245],[49,12368,629],{"className":12369},[628],[49,12371,12373],{"className":12372},[226],[49,12374,12376],{"className":12375,"style":636},[230],[49,12377],{},[49,12379,12381],{"className":12380},[93,3828],[49,12382,30],{"className":12383},[93,11327],[49,12385,307],{"className":12386},[406],[49,12388],{"className":12389,"style":193},[192],[49,12391,523],{"className":12392},[197],[49,12394],{"className":12395,"style":193},[192],[49,12397,12399,12402,12405,12408,12460,12463,12503,12506,12509,12512],{"className":12398},[84],[49,12400],{"className":12401,"style":589},[88],[49,12403,11247],{"className":12404},[93,214],[49,12406,298],{"className":12407},[373],[49,12409,12411,12414],{"className":12410},[93],[49,12412,4702],{"className":12413},[93,94],[49,12415,12417],{"className":12416},[218],[49,12418,12420,12452],{"className":12419},[222,605],[49,12421,12423,12449],{"className":12422},[226],[49,12424,12426],{"className":12425,"style":11925},[230],[49,12427,12428,12431],{"style":615},[49,12429],{"className":12430,"style":238},[237],[49,12432,12434],{"className":12433},[242,243,244,245],[49,12435,12437],{"className":12436},[93,245],[49,12438,12440,12443,12446],{"className":12439},[4912,245],[49,12441,11943],{"className":12442},[245],[49,12444,1939],{"className":12445},[245],[49,12447,11950],{"className":12448},[245],[49,12450,629],{"className":12451},[628],[49,12453,12455],{"className":12454},[226],[49,12456,12458],{"className":12457,"style":636},[230],[49,12459],{},[49,12461,298],{"className":12462},[373],[49,12464,12466,12469],{"className":12465},[93],[49,12467,1880],{"className":12468,"style":1993},[93,94],[49,12470,12472],{"className":12471},[218],[49,12473,12475,12495],{"className":12474},[222,605],[49,12476,12478,12492],{"className":12477},[226],[49,12479,12481],{"className":12480,"style":678},[230],[49,12482,12483,12486],{"style":2009},[49,12484],{"className":12485,"style":238},[237],[49,12487,12489],{"className":12488},[242,243,244,245],[49,12490,71],{"className":12491,"style":95},[93,94,245],[49,12493,629],{"className":12494},[628],[49,12496,12498],{"className":12497},[226],[49,12499,12501],{"className":12500,"style":636},[230],[49,12502],{},[49,12504,307],{"className":12505},[406],[49,12507],{"className":12508,"style":193},[192],[49,12510,6927],{"className":12511},[197],[49,12513],{"className":12514,"style":193},[192],[49,12516,12518,12521,12524,12527,12530,12533],{"className":12517},[84],[49,12519],{"className":12520,"style":589},[88],[49,12522,801],{"className":12523},[93],[49,12525,307],{"className":12526},[406],[49,12528],{"className":12529,"style":193},[192],[49,12531,523],{"className":12532},[197],[49,12534],{"className":12535,"style":193},[192],[49,12537,12539,12542,12545,12548,12551,12554,12583,12586,12589],{"className":12538},[84],[49,12540],{"className":12541,"style":2055},[88],[49,12543,12287],{"className":12544},[4912],[49,12546,6568],{"className":12547},[373],[49,12549,3077],{"className":12550},[93],[49,12552,1426],{"className":12553},[93,94],[49,12555,12557,12560],{"className":12556},[93],[49,12558,71],{"className":12559,"style":95},[93,94],[49,12561,12563],{"className":12562},[218],[49,12564,12566],{"className":12565},[222],[49,12567,12569],{"className":12568},[226],[49,12570,12572],{"className":12571,"style":2074},[230],[49,12573,12574,12577],{"style":1387},[49,12575],{"className":12576,"style":238},[237],[49,12578,12580],{"className":12579},[242,243,244,245],[49,12581,294],{"className":12582},[93,245],[49,12584],{"className":12585,"style":1843},[192],[49,12587,3213],{"className":12588},[1847],[49,12590],{"className":12591,"style":1843},[192],[49,12593,12595,12598,12601,12604,12633,12637],{"className":12594},[84],[49,12596],{"className":12597,"style":2055},[88],[49,12599,3291],{"className":12600},[93,94],[49,12602,298],{"className":12603},[373],[49,12605,12607,12610],{"className":12606},[93],[49,12608,71],{"className":12609,"style":95},[93,94],[49,12611,12613],{"className":12612},[218],[49,12614,12616],{"className":12615},[222],[49,12617,12619],{"className":12618},[226],[49,12620,12622],{"className":12621,"style":2074},[230],[49,12623,12624,12627],{"style":1387},[49,12625],{"className":12626,"style":238},[237],[49,12628,12630],{"className":12629},[242,243,244,245],[49,12631,294],{"className":12632},[93,245],[49,12634,12636],{"className":12635},[406],")}",[49,12638,897],{"className":12639},[93],[11,12641,12642,12643,12674,12675,12772,12773,12875],{},"更精确地说，对高斯 ",[49,12644,12646,12661],{"className":12645,"translate":53},[52],[49,12647,12649],{"className":12648},[57],[59,12650,12651],{"xmlns":61},[63,12652,12653,12658],{},[66,12654,12655],{},[69,12656,12657],{},"β",[73,12659,12660],{"encoding":75},"\\beta",[49,12662,12664],{"className":12663,"ariaHidden":80},[79],[49,12665,12667,12670],{"className":12666},[84],[49,12668],{"className":12669,"style":184},[88],[49,12671,12657],{"className":12672,"style":12673},[93,94],"margin-right:0.0528em;"," 系综，指数具有 ",[49,12676,12678,12712],{"className":12677,"translate":53},[52],[49,12679,12681],{"className":12680},[57],[59,12682,12683],{"xmlns":61},[63,12684,12685,12709],{},[66,12686,12687,12689,12692,12695,12697,12699,12701,12703],{},[69,12688,12657],{},[3314,12690,12691],{}," ",[69,12693,12694],{},"θ",[155,12696,298],{"stretchy":297},[292,12698,801],{},[155,12700,307],{"stretchy":297},[3314,12702,12691],{},[159,12704,12705,12707],{},[69,12706,71],{},[292,12708,294],{},[73,12710,12711],{"encoding":75},"\\beta\\,\\theta(0)\\,N^2",[49,12713,12715],{"className":12714,"ariaHidden":80},[79],[49,12716,12718,12722,12725,12728,12731,12734,12737,12740,12743],{"className":12717},[84],[49,12719],{"className":12720,"style":12721},[88],"height:1.0641em;vertical-align:-0.25em;",[49,12723,12657],{"className":12724,"style":12673},[93,94],[49,12726],{"className":12727,"style":646},[192],[49,12729,12694],{"className":12730,"style":11724},[93,94],[49,12732,298],{"className":12733},[373],[49,12735,801],{"className":12736},[93],[49,12738,307],{"className":12739},[406],[49,12741],{"className":12742,"style":646},[192],[49,12744,12746,12749],{"className":12745},[93],[49,12747,71],{"className":12748,"style":95},[93,94],[49,12750,12752],{"className":12751},[218],[49,12753,12755],{"className":12754},[222],[49,12756,12758],{"className":12757},[226],[49,12759,12761],{"className":12760,"style":358},[230],[49,12762,12763,12766],{"style":233},[49,12764],{"className":12765,"style":238},[237],[49,12767,12769],{"className":12768},[242,243,244,245],[49,12770,294],{"className":12771},[93,245]," 的形式，其中 ",[49,12774,12776,12815],{"className":12775,"translate":53},[52],[49,12777,12779],{"className":12778},[57],[59,12780,12781],{"xmlns":61},[63,12782,12783,12812],{},[66,12784,12785,12787,12789,12791,12793,12795,12797,12800,12802,12805,12807,12809],{},[69,12786,12694],{},[155,12788,298],{"stretchy":297},[292,12790,801],{},[155,12792,307],{"stretchy":297},[155,12794,523],{},[155,12796,298],{"stretchy":297},[69,12798,12799],{},"log",[155,12801,6548],{},[292,12803,12804],{},"3",[155,12806,307],{"stretchy":297},[69,12808,11211],{"mathvariant":768},[292,12810,12811],{},"4",[73,12813,12814],{"encoding":75},"\\theta(0) = (\\log 3)\u002F4",[49,12816,12818,12845],{"className":12817,"ariaHidden":80},[79],[49,12819,12821,12824,12827,12830,12833,12836,12839,12842],{"className":12820},[84],[49,12822],{"className":12823,"style":589},[88],[49,12825,12694],{"className":12826,"style":11724},[93,94],[49,12828,298],{"className":12829},[373],[49,12831,801],{"className":12832},[93],[49,12834,307],{"className":12835},[406],[49,12837],{"className":12838,"style":193},[192],[49,12840,523],{"className":12841},[197],[49,12843],{"className":12844,"style":193},[192],[49,12846,12848,12851,12854,12862,12865,12868,12871],{"className":12847},[84],[49,12849],{"className":12850,"style":589},[88],[49,12852,298],{"className":12853},[373],[49,12855,12857,12858],{"className":12856},[4912],"lo",[49,12859,12861],{"style":12860},"margin-right:0.0139em;","g",[49,12863],{"className":12864,"style":646},[192],[49,12866,12804],{"className":12867},[93],[49,12869,307],{"className":12870},[406],[49,12872,12874],{"className":12873},[93],"\u002F4","（Dean & Majumdar 的结果与约定）。",[11,12877,12878,12879,12938,12939,12942,12943,13012],{},"精确常数不如标度重要。要点在于：正定不仅是「不太可能」，对中心化 GOE 矩阵而言，它在 ",[49,12880,12882,12900],{"className":12881,"translate":53},[52],[49,12883,12885],{"className":12884},[57],[59,12886,12887],{"xmlns":61},[63,12888,12889,12897],{},[66,12890,12891],{},[159,12892,12893,12895],{},[69,12894,71],{},[292,12896,294],{},[73,12898,12899],{"encoding":75},"N^2",[49,12901,12903],{"className":12902,"ariaHidden":80},[79],[49,12904,12906,12909],{"className":12905},[84],[49,12907],{"className":12908,"style":358},[88],[49,12910,12912,12915],{"className":12911},[93],[49,12913,71],{"className":12914,"style":95},[93,94],[49,12916,12918],{"className":12917},[218],[49,12919,12921],{"className":12920},[222],[49,12922,12924],{"className":12923},[226],[49,12925,12927],{"className":12926,"style":358},[230],[49,12928,12929,12932],{"style":233},[49,12930],{"className":12931,"style":238},[237],[49,12933,12935],{"className":12934},[242,243,244,245],[49,12936,294],{"className":12937},[93,245]," 上呈",[28,12940,12941],{},"指数级","不可能——比前面粗糙估计里的 ",[49,12944,12946,12968],{"className":12945,"translate":53},[52],[49,12947,12949],{"className":12948},[57],[59,12950,12951],{"xmlns":61},[63,12952,12953,12965],{},[66,12954,12955],{},[159,12956,12957,12959],{},[292,12958,294],{},[66,12960,12961,12963],{},[155,12962,3077],{},[69,12964,71],{},[73,12966,12967],{"encoding":75},"2^{-N}",[49,12969,12971],{"className":12970,"ariaHidden":80},[79],[49,12972,12974,12977],{"className":12973},[84],[49,12975],{"className":12976,"style":207},[88],[49,12978,12980,12983],{"className":12979},[93],[49,12981,294],{"className":12982},[93],[49,12984,12986],{"className":12985},[218],[49,12987,12989],{"className":12988},[222],[49,12990,12992],{"className":12991},[226],[49,12993,12995],{"className":12994,"style":207},[230],[49,12996,12997,13000],{"style":233},[49,12998],{"className":12999,"style":238},[237],[49,13001,13003],{"className":13002},[242,243,244,245],[49,13004,13006,13009],{"className":13005},[93,245],[49,13007,3077],{"className":13008},[93,245],[49,13010,71],{"className":13011,"style":95},[93,94,245]," 还要猛烈。于是，在随机 Hessian 模型中，",[49,13014,13016],{"className":13015,"translate":53},[752],[49,13017,13019,13062],{"className":13018,"translate":53},[52],[49,13020,13022],{"className":13021},[57],[59,13023,13024],{"xmlns":61,"display":761},[63,13025,13026,13059],{},[66,13027,13028,13030,13032,13035,13037,13040,13042,13044,13046,13048,13050,13053,13055,13057],{},[69,13029,11247],{"mathvariant":163},[155,13031,298],{"stretchy":297},[3314,13033,13034],{},"临界点是鞍点",[155,13036,307],{"stretchy":297},[155,13038,13039],{},"⟶",[292,13041,533],{},[155,13043,536],{"separator":80},[192,13045],{"width":9706},[69,13047,11247],{"mathvariant":163},[155,13049,298],{"stretchy":297},[3314,13051,13052],{},"临界点是非退化局部极小值",[155,13054,307],{"stretchy":297},[155,13056,13039],{},[292,13058,6965],{},[73,13060,13061],{"encoding":75},"\\mathbb{P}(\\text{临界点是鞍点}) \\longrightarrow 1, \\qquad \\mathbb{P}(\\text{临界点是非退化局部极小值}) \\longrightarrow 0.",[49,13063,13065,13095,13137],{"className":13064,"ariaHidden":80},[79],[49,13066,13068,13071,13074,13077,13083,13086,13089,13092],{"className":13067},[84],[49,13069],{"className":13070,"style":589},[88],[49,13072,11247],{"className":13073},[93,214],[49,13075,298],{"className":13076},[373],[49,13078,13080],{"className":13079},[93,3828],[49,13081,13034],{"className":13082},[93,11327],[49,13084,307],{"className":13085},[406],[49,13087],{"className":13088,"style":193},[192],[49,13090,13039],{"className":13091},[197],[49,13093],{"className":13094,"style":193},[192],[49,13096,13098,13101,13104,13107,13110,13113,13116,13119,13125,13128,13131,13134],{"className":13097},[84],[49,13099],{"className":13100,"style":589},[88],[49,13102,533],{"className":13103},[93],[49,13105,536],{"className":13106},[642],[49,13108],{"className":13109,"style":9896},[192],[49,13111],{"className":13112,"style":646},[192],[49,13114,11247],{"className":13115},[93,214],[49,13117,298],{"className":13118},[373],[49,13120,13122],{"className":13121},[93,3828],[49,13123,13052],{"className":13124},[93,11327],[49,13126,307],{"className":13127},[406],[49,13129],{"className":13130,"style":193},[192],[49,13132,13039],{"className":13133},[197],[49,13135],{"className":13136,"style":193},[192],[49,13138,13140,13143],{"className":13139},[84],[49,13141],{"className":13142,"style":6838},[88],[49,13144,6965],{"className":13145},[93],[11189,13147,13148],{"id":13148},"更明确的几何诠释",[11,13150,3843,13151,13220],{},[49,13152,13154,13171],{"className":13153,"translate":53},[52],[49,13155,13157],{"className":13156},[57],[59,13158,13159],{"xmlns":61},[63,13160,13161,13169],{},[66,13162,13163],{},[527,13164,13165,13167],{},[69,13166,520],{},[69,13168,1426],{},[73,13170,3864],{"encoding":75},[49,13172,13174],{"className":13173,"ariaHidden":80},[79],[49,13175,13177,13180],{"className":13176},[84],[49,13178],{"className":13179,"style":3134},[88],[49,13181,13183,13186],{"className":13182},[93],[49,13184,520],{"className":13185},[93,94],[49,13187,13189],{"className":13188},[218],[49,13190,13192,13212],{"className":13191},[222,605],[49,13193,13195,13209],{"className":13194},[226],[49,13196,13198],{"className":13197,"style":1466},[230],[49,13199,13200,13203],{"style":615},[49,13201],{"className":13202,"style":238},[237],[49,13204,13206],{"className":13205},[242,243,244,245],[49,13207,1426],{"className":13208},[93,94,245],[49,13210,629],{"className":13211},[628],[49,13213,13215],{"className":13214},[226],[49,13216,13218],{"className":13217,"style":636},[230],[49,13219],{}," 处，二阶近似为",[49,13222,13224],{"className":13223,"translate":53},[752],[49,13225,13227,13304],{"className":13226,"translate":53},[52],[49,13228,13230],{"className":13229},[57],[59,13231,13232],{"xmlns":61,"display":761},[63,13233,13234,13301],{},[66,13235,13236,13238,13240,13246,13248,13250,13252,13254,13256,13258,13264,13266,13269,13275,13281,13287,13289,13295,13297,13299],{},[69,13237,153],{},[155,13239,298],{"stretchy":297},[527,13241,13242,13244],{},[69,13243,520],{},[69,13245,1426],{},[155,13247,3213],{},[69,13249,3070],{},[155,13251,307],{"stretchy":297},[155,13253,3077],{},[69,13255,153],{},[155,13257,298],{"stretchy":297},[527,13259,13260,13262],{},[69,13261,520],{},[69,13263,1426],{},[155,13265,307],{"stretchy":297},[155,13267,13268],{},"≈",[807,13270,13271,13273],{},[292,13272,533],{},[292,13274,294],{},[159,13276,13277,13279],{},[69,13278,3070],{},[69,13280,3252],{},[527,13282,13283,13285],{},[69,13284,1880],{},[69,13286,153],{},[155,13288,298],{"stretchy":297},[527,13290,13291,13293],{},[69,13292,520],{},[69,13294,1426],{},[155,13296,307],{"stretchy":297},[69,13298,3070],{},[69,13300,897],{"mathvariant":768},[73,13302,13303],{"encoding":75},"f(x_c+h) - f(x_c) \\approx \\frac{1}{2}h^T H_f(x_c) h.",[49,13305,13307,13368,13389,13453],{"className":13306,"ariaHidden":80},[79],[49,13308,13310,13313,13316,13319,13359,13362,13365],{"className":13309},[84],[49,13311],{"className":13312,"style":589},[88],[49,13314,153],{"className":13315,"style":188},[93,94],[49,13317,298],{"className":13318},[373],[49,13320,13322,13325],{"className":13321},[93],[49,13323,520],{"className":13324},[93,94],[49,13326,13328],{"className":13327},[218],[49,13329,13331,13351],{"className":13330},[222,605],[49,13332,13334,13348],{"className":13333},[226],[49,13335,13337],{"className":13336,"style":1466},[230],[49,13338,13339,13342],{"style":615},[49,13340],{"className":13341,"style":238},[237],[49,13343,13345],{"className":13344},[242,243,244,245],[49,13346,1426],{"className":13347},[93,94,245],[49,13349,629],{"className":13350},[628],[49,13352,13354],{"className":13353},[226],[49,13355,13357],{"className":13356,"style":636},[230],[49,13358],{},[49,13360],{"className":13361,"style":1843},[192],[49,13363,3213],{"className":13364},[1847],[49,13366],{"className":13367,"style":1843},[192],[49,13369,13371,13374,13377,13380,13383,13386],{"className":13370},[84],[49,13372],{"className":13373,"style":589},[88],[49,13375,3070],{"className":13376},[93,94],[49,13378,307],{"className":13379},[406],[49,13381],{"className":13382,"style":1843},[192],[49,13384,3077],{"className":13385},[1847],[49,13387],{"className":13388,"style":1843},[192],[49,13390,13392,13395,13398,13401,13441,13444,13447,13450],{"className":13391},[84],[49,13393],{"className":13394,"style":589},[88],[49,13396,153],{"className":13397,"style":188},[93,94],[49,13399,298],{"className":13400},[373],[49,13402,13404,13407],{"className":13403},[93],[49,13405,520],{"className":13406},[93,94],[49,13408,13410],{"className":13409},[218],[49,13411,13413,13433],{"className":13412},[222,605],[49,13414,13416,13430],{"className":13415},[226],[49,13417,13419],{"className":13418,"style":1466},[230],[49,13420,13421,13424],{"style":615},[49,13422],{"className":13423,"style":238},[237],[49,13425,13427],{"className":13426},[242,243,244,245],[49,13428,1426],{"className":13429},[93,94,245],[49,13431,629],{"className":13432},[628],[49,13434,13436],{"className":13435},[226],[49,13437,13439],{"className":13438,"style":636},[230],[49,13440],{},[49,13442,307],{"className":13443},[406],[49,13445],{"className":13446,"style":193},[192],[49,13448,13268],{"className":13449},[197],[49,13451],{"className":13452,"style":193},[192],[49,13454,13456,13459,13521,13550,13590,13593,13633,13636,13639],{"className":13455},[84],[49,13457],{"className":13458,"style":3573},[88],[49,13460,13462,13465,13518],{"className":13461},[93],[49,13463],{"className":13464},[373,1037],[49,13466,13468],{"className":13467},[807],[49,13469,13471,13510],{"className":13470},[222,605],[49,13472,13474,13507],{"className":13473},[226],[49,13475,13477,13488,13496],{"className":13476,"style":3592},[230],[49,13478,13479,13482],{"style":1053},[49,13480],{"className":13481,"style":1057},[237],[49,13483,13485],{"className":13484},[93],[49,13486,294],{"className":13487},[93],[49,13489,13490,13493],{"style":1107},[49,13491],{"className":13492,"style":1057},[237],[49,13494],{"className":13495,"style":1115},[1114],[49,13497,13498,13501],{"style":1118},[49,13499],{"className":13500,"style":1057},[237],[49,13502,13504],{"className":13503},[93],[49,13505,533],{"className":13506},[93],[49,13508,629],{"className":13509},[628],[49,13511,13513],{"className":13512},[226],[49,13514,13516],{"className":13515,"style":3632},[230],[49,13517],{},[49,13519],{"className":13520},[406,1037],[49,13522,13524,13527],{"className":13523},[93],[49,13525,3070],{"className":13526},[93,94],[49,13528,13530],{"className":13529},[218],[49,13531,13533],{"className":13532},[222],[49,13534,13536],{"className":13535},[226],[49,13537,13539],{"className":13538,"style":1366},[230],[49,13540,13541,13544],{"style":1387},[49,13542],{"className":13543,"style":238},[237],[49,13545,13547],{"className":13546},[242,243,244,245],[49,13548,3252],{"className":13549,"style":3554},[93,94,245],[49,13551,13553,13556],{"className":13552},[93],[49,13554,1880],{"className":13555,"style":1993},[93,94],[49,13557,13559],{"className":13558},[218],[49,13560,13562,13582],{"className":13561},[222,605],[49,13563,13565,13579],{"className":13564},[226],[49,13566,13568],{"className":13567,"style":2006},[230],[49,13569,13570,13573],{"style":2009},[49,13571],{"className":13572,"style":238},[237],[49,13574,13576],{"className":13575},[242,243,244,245],[49,13577,153],{"className":13578,"style":188},[93,94,245],[49,13580,629],{"className":13581},[628],[49,13583,13585],{"className":13584},[226],[49,13586,13588],{"className":13587,"style":2028},[230],[49,13589],{},[49,13591,298],{"className":13592},[373],[49,13594,13596,13599],{"className":13595},[93],[49,13597,520],{"className":13598},[93,94],[49,13600,13602],{"className":13601},[218],[49,13603,13605,13625],{"className":13604},[222,605],[49,13606,13608,13622],{"className":13607},[226],[49,13609,13611],{"className":13610,"style":1466},[230],[49,13612,13613,13616],{"style":615},[49,13614],{"className":13615,"style":238},[237],[49,13617,13619],{"className":13618},[242,243,244,245],[49,13620,1426],{"className":13621},[93,94,245],[49,13623,629],{"className":13624},[628],[49,13626,13628],{"className":13627},[226],[49,13629,13631],{"className":13630,"style":636},[230],[49,13632],{},[49,13634,307],{"className":13635},[406],[49,13637,3070],{"className":13638},[93,94],[49,13640,897],{"className":13641},[93],[11,13643,5733,13644,13769,13770,13799,13800,13828],{},[49,13645,13647,13674],{"className":13646,"translate":53},[52],[49,13648,13650],{"className":13649},[57],[59,13651,13652],{"xmlns":61},[63,13653,13654,13672],{},[66,13655,13656,13662,13664,13670],{},[527,13657,13658,13660],{},[69,13659,1880],{},[69,13661,153],{},[155,13663,298],{"stretchy":297},[527,13665,13666,13668],{},[69,13667,520],{},[69,13669,1426],{},[155,13671,307],{"stretchy":297},[73,13673,4444],{"encoding":75},[49,13675,13677],{"className":13676,"ariaHidden":80},[79],[49,13678,13680,13683,13723,13726,13766],{"className":13679},[84],[49,13681],{"className":13682,"style":1986},[88],[49,13684,13686,13689],{"className":13685},[93],[49,13687,1880],{"className":13688,"style":1993},[93,94],[49,13690,13692],{"className":13691},[218],[49,13693,13695,13715],{"className":13694},[222,605],[49,13696,13698,13712],{"className":13697},[226],[49,13699,13701],{"className":13700,"style":2006},[230],[49,13702,13703,13706],{"style":2009},[49,13704],{"className":13705,"style":238},[237],[49,13707,13709],{"className":13708},[242,243,244,245],[49,13710,153],{"className":13711,"style":188},[93,94,245],[49,13713,629],{"className":13714},[628],[49,13716,13718],{"className":13717},[226],[49,13719,13721],{"className":13720,"style":2028},[230],[49,13722],{},[49,13724,298],{"className":13725},[373],[49,13727,13729,13732],{"className":13728},[93],[49,13730,520],{"className":13731},[93,94],[49,13733,13735],{"className":13734},[218],[49,13736,13738,13758],{"className":13737},[222,605],[49,13739,13741,13755],{"className":13740},[226],[49,13742,13744],{"className":13743,"style":1466},[230],[49,13745,13746,13749],{"style":615},[49,13747],{"className":13748,"style":238},[237],[49,13750,13752],{"className":13751},[242,243,244,245],[49,13753,1426],{"className":13754},[93,94,245],[49,13756,629],{"className":13757},[628],[49,13759,13761],{"className":13760},[226],[49,13762,13764],{"className":13763,"style":636},[230],[49,13765],{},[49,13767,307],{"className":13768},[406]," 只要有一个负特征值 ",[49,13771,13773,13787],{"className":13772,"translate":53},[52],[49,13774,13776],{"className":13775},[57],[59,13777,13778],{"xmlns":61},[63,13779,13780,13784],{},[66,13781,13782],{},[69,13783,4702],{},[73,13785,13786],{"encoding":75},"\\lambda",[49,13788,13790],{"className":13789,"ariaHidden":80},[79],[49,13791,13793,13796],{"className":13792},[84],[49,13794],{"className":13795,"style":3096},[88],[49,13797,4702],{"className":13798},[93,94],"，对应特征向量 ",[49,13801,13803,13816],{"className":13802,"translate":53},[52],[49,13804,13806],{"className":13805},[57],[59,13807,13808],{"xmlns":61},[63,13809,13810,13814],{},[66,13811,13812],{},[69,13813,4557],{},[73,13815,4557],{"encoding":75},[49,13817,13819],{"className":13818,"ariaHidden":80},[79],[49,13820,13822,13825],{"className":13821},[84],[49,13823],{"className":13824,"style":570},[88],[49,13826,4557],{"className":13827,"style":4591},[93,94],"，那么",[49,13830,13832],{"className":13831,"translate":53},[752],[49,13833,13835,13877],{"className":13834,"translate":53},[52],[49,13836,13838],{"className":13837},[57],[59,13839,13840],{"xmlns":61,"display":761},[63,13841,13842,13874],{},[66,13843,13844,13850,13856,13858,13864,13866,13868,13870,13872],{},[159,13845,13846,13848],{},[69,13847,4557],{},[69,13849,3252],{},[527,13851,13852,13854],{},[69,13853,1880],{},[69,13855,153],{},[155,13857,298],{"stretchy":297},[527,13859,13860,13862],{},[69,13861,520],{},[69,13863,1426],{},[155,13865,307],{"stretchy":297},[69,13867,4557],{},[155,13869,6581],{},[292,13871,801],{},[155,13873,536],{"separator":80},[73,13875,13876],{"encoding":75},"v^T H_f(x_c) v \u003C 0,",[49,13878,13880,14013],{"className":13879,"ariaHidden":80},[79],[49,13881,13883,13886,13915,13955,13958,13998,14001,14004,14007,14010],{"className":13882},[84],[49,13884],{"className":13885,"style":5170},[88],[49,13887,13889,13892],{"className":13888},[93],[49,13890,4557],{"className":13891,"style":4591},[93,94],[49,13893,13895],{"className":13894},[218],[49,13896,13898],{"className":13897},[222],[49,13899,13901],{"className":13900},[226],[49,13902,13904],{"className":13903,"style":1366},[230],[49,13905,13906,13909],{"style":1387},[49,13907],{"className":13908,"style":238},[237],[49,13910,13912],{"className":13911},[242,243,244,245],[49,13913,3252],{"className":13914,"style":3554},[93,94,245],[49,13916,13918,13921],{"className":13917},[93],[49,13919,1880],{"className":13920,"style":1993},[93,94],[49,13922,13924],{"className":13923},[218],[49,13925,13927,13947],{"className":13926},[222,605],[49,13928,13930,13944],{"className":13929},[226],[49,13931,13933],{"className":13932,"style":2006},[230],[49,13934,13935,13938],{"style":2009},[49,13936],{"className":13937,"style":238},[237],[49,13939,13941],{"className":13940},[242,243,244,245],[49,13942,153],{"className":13943,"style":188},[93,94,245],[49,13945,629],{"className":13946},[628],[49,13948,13950],{"className":13949},[226],[49,13951,13953],{"className":13952,"style":2028},[230],[49,13954],{},[49,13956,298],{"className":13957},[373],[49,13959,13961,13964],{"className":13960},[93],[49,13962,520],{"className":13963},[93,94],[49,13965,13967],{"className":13966},[218],[49,13968,13970,13990],{"className":13969},[222,605],[49,13971,13973,13987],{"className":13972},[226],[49,13974,13976],{"className":13975,"style":1466},[230],[49,13977,13978,13981],{"style":615},[49,13979],{"className":13980,"style":238},[237],[49,13982,13984],{"className":13983},[242,243,244,245],[49,13985,1426],{"className":13986},[93,94,245],[49,13988,629],{"className":13989},[628],[49,13991,13993],{"className":13992},[226],[49,13994,13996],{"className":13995,"style":636},[230],[49,13997],{},[49,13999,307],{"className":14000},[406],[49,14002,4557],{"className":14003,"style":4591},[93,94],[49,14005],{"className":14006,"style":193},[192],[49,14008,6581],{"className":14009},[197],[49,14011],{"className":14012,"style":193},[192],[49,14014,14016,14019,14022],{"className":14015},[84],[49,14017],{"className":14018,"style":1648},[88],[49,14020,801],{"className":14021},[93],[49,14023,536],{"className":14024},[642],[11,14026,14027,14028,14056,14057,14085],{},"沿 ",[49,14029,14031,14044],{"className":14030,"translate":53},[52],[49,14032,14034],{"className":14033},[57],[59,14035,14036],{"xmlns":61},[63,14037,14038,14042],{},[66,14039,14040],{},[69,14041,4557],{},[73,14043,4557],{"encoding":75},[49,14045,14047],{"className":14046,"ariaHidden":80},[79],[49,14048,14050,14053],{"className":14049},[84],[49,14051],{"className":14052,"style":570},[88],[49,14054,4557],{"className":14055,"style":4591},[93,94]," 移动一小段距离会使 ",[49,14058,14060,14073],{"className":14059,"translate":53},[52],[49,14061,14063],{"className":14062},[57],[59,14064,14065],{"xmlns":61},[63,14066,14067,14071],{},[66,14068,14069],{},[69,14070,153],{},[73,14072,153],{"encoding":75},[49,14074,14076],{"className":14075,"ariaHidden":80},[79],[49,14077,14079,14082],{"className":14078},[84],[49,14080],{"className":14081,"style":184},[88],[49,14083,153],{"className":14084,"style":188},[93,94]," 在二阶意义上减小：",[49,14087,14089],{"className":14088,"translate":53},[752],[49,14090,14092,14181],{"className":14091,"translate":53},[52],[49,14093,14095],{"className":14094},[57],[59,14096,14097],{"xmlns":61,"display":761},[63,14098,14099,14178],{},[66,14100,14101,14103,14105,14111,14113,14116,14118,14120,14122,14124,14126,14132,14134,14136,14142,14148,14150,14156,14162,14164,14170,14172,14174,14176],{},[69,14102,153],{},[155,14104,298],{"stretchy":297},[527,14106,14107,14109],{},[69,14108,520],{},[69,14110,1426],{},[155,14112,3213],{},[69,14114,14115],{},"t",[69,14117,4557],{},[155,14119,307],{"stretchy":297},[155,14121,3077],{},[69,14123,153],{},[155,14125,298],{"stretchy":297},[527,14127,14128,14130],{},[69,14129,520],{},[69,14131,1426],{},[155,14133,307],{"stretchy":297},[155,14135,13268],{},[807,14137,14138,14140],{},[292,14139,533],{},[292,14141,294],{},[159,14143,14144,14146],{},[69,14145,14115],{},[292,14147,294],{},[3314,14149,12691],{},[159,14151,14152,14154],{},[69,14153,4557],{},[69,14155,3252],{},[527,14157,14158,14160],{},[69,14159,1880],{},[69,14161,153],{},[155,14163,298],{"stretchy":297},[527,14165,14166,14168],{},[69,14167,520],{},[69,14169,1426],{},[155,14171,307],{"stretchy":297},[69,14173,4557],{},[155,14175,6581],{},[292,14177,6965],{},[73,14179,14180],{"encoding":75},"f(x_c+tv) - f(x_c) \\approx \\frac{1}{2}t^2\\, v^T H_f(x_c) v \u003C 0.",[49,14182,14184,14245,14269,14333,14560],{"className":14183,"ariaHidden":80},[79],[49,14185,14187,14190,14193,14196,14236,14239,14242],{"className":14186},[84],[49,14188],{"className":14189,"style":589},[88],[49,14191,153],{"className":14192,"style":188},[93,94],[49,14194,298],{"className":14195},[373],[49,14197,14199,14202],{"className":14198},[93],[49,14200,520],{"className":14201},[93,94],[49,14203,14205],{"className":14204},[218],[49,14206,14208,14228],{"className":14207},[222,605],[49,14209,14211,14225],{"className":14210},[226],[49,14212,14214],{"className":14213,"style":1466},[230],[49,14215,14216,14219],{"style":615},[49,14217],{"className":14218,"style":238},[237],[49,14220,14222],{"className":14221},[242,243,244,245],[49,14223,1426],{"className":14224},[93,94,245],[49,14226,629],{"className":14227},[628],[49,14229,14231],{"className":14230},[226],[49,14232,14234],{"className":14233,"style":636},[230],[49,14235],{},[49,14237],{"className":14238,"style":1843},[192],[49,14240,3213],{"className":14241},[1847],[49,14243],{"className":14244,"style":1843},[192],[49,14246,14248,14251,14254,14257,14260,14263,14266],{"className":14247},[84],[49,14249],{"className":14250,"style":589},[88],[49,14252,14115],{"className":14253},[93,94],[49,14255,4557],{"className":14256,"style":4591},[93,94],[49,14258,307],{"className":14259},[406],[49,14261],{"className":14262,"style":1843},[192],[49,14264,3077],{"className":14265},[1847],[49,14267],{"className":14268,"style":1843},[192],[49,14270,14272,14275,14278,14281,14321,14324,14327,14330],{"className":14271},[84],[49,14273],{"className":14274,"style":589},[88],[49,14276,153],{"className":14277,"style":188},[93,94],[49,14279,298],{"className":14280},[373],[49,14282,14284,14287],{"className":14283},[93],[49,14285,520],{"className":14286},[93,94],[49,14288,14290],{"className":14289},[218],[49,14291,14293,14313],{"className":14292},[222,605],[49,14294,14296,14310],{"className":14295},[226],[49,14297,14299],{"className":14298,"style":1466},[230],[49,14300,14301,14304],{"style":615},[49,14302],{"className":14303,"style":238},[237],[49,14305,14307],{"className":14306},[242,243,244,245],[49,14308,1426],{"className":14309},[93,94,245],[49,14311,629],{"className":14312},[628],[49,14314,14316],{"className":14315},[226],[49,14317,14319],{"className":14318,"style":636},[230],[49,14320],{},[49,14322,307],{"className":14323},[406],[49,14325],{"className":14326,"style":193},[192],[49,14328,13268],{"className":14329},[197],[49,14331],{"className":14332,"style":193},[192],[49,14334,14336,14339,14401,14430,14433,14462,14502,14505,14545,14548,14551,14554,14557],{"className":14335},[84],[49,14337],{"className":14338,"style":3573},[88],[49,14340,14342,14345,14398],{"className":14341},[93],[49,14343],{"className":14344},[373,1037],[49,14346,14348],{"className":14347},[807],[49,14349,14351,14390],{"className":14350},[222,605],[49,14352,14354,14387],{"className":14353},[226],[49,14355,14357,14368,14376],{"className":14356,"style":3592},[230],[49,14358,14359,14362],{"style":1053},[49,14360],{"className":14361,"style":1057},[237],[49,14363,14365],{"className":14364},[93],[49,14366,294],{"className":14367},[93],[49,14369,14370,14373],{"style":1107},[49,14371],{"className":14372,"style":1057},[237],[49,14374],{"className":14375,"style":1115},[1114],[49,14377,14378,14381],{"style":1118},[49,14379],{"className":14380,"style":1057},[237],[49,14382,14384],{"className":14383},[93],[49,14385,533],{"className":14386},[93],[49,14388,629],{"className":14389},[628],[49,14391,14393],{"className":14392},[226],[49,14394,14396],{"className":14395,"style":3632},[230],[49,14397],{},[49,14399],{"className":14400},[406,1037],[49,14402,14404,14407],{"className":14403},[93],[49,14405,14115],{"className":14406},[93,94],[49,14408,14410],{"className":14409},[218],[49,14411,14413],{"className":14412},[222],[49,14414,14416],{"className":14415},[226],[49,14417,14419],{"className":14418,"style":2074},[230],[49,14420,14421,14424],{"style":1387},[49,14422],{"className":14423,"style":238},[237],[49,14425,14427],{"className":14426},[242,243,244,245],[49,14428,294],{"className":14429},[93,245],[49,14431],{"className":14432,"style":646},[192],[49,14434,14436,14439],{"className":14435},[93],[49,14437,4557],{"className":14438,"style":4591},[93,94],[49,14440,14442],{"className":14441},[218],[49,14443,14445],{"className":14444},[222],[49,14446,14448],{"className":14447},[226],[49,14449,14451],{"className":14450,"style":1366},[230],[49,14452,14453,14456],{"style":1387},[49,14454],{"className":14455,"style":238},[237],[49,14457,14459],{"className":14458},[242,243,244,245],[49,14460,3252],{"className":14461,"style":3554},[93,94,245],[49,14463,14465,14468],{"className":14464},[93],[49,14466,1880],{"className":14467,"style":1993},[93,94],[49,14469,14471],{"className":14470},[218],[49,14472,14474,14494],{"className":14473},[222,605],[49,14475,14477,14491],{"className":14476},[226],[49,14478,14480],{"className":14479,"style":2006},[230],[49,14481,14482,14485],{"style":2009},[49,14483],{"className":14484,"style":238},[237],[49,14486,14488],{"className":14487},[242,243,244,245],[49,14489,153],{"className":14490,"style":188},[93,94,245],[49,14492,629],{"className":14493},[628],[49,14495,14497],{"className":14496},[226],[49,14498,14500],{"className":14499,"style":2028},[230],[49,14501],{},[49,14503,298],{"className":14504},[373],[49,14506,14508,14511],{"className":14507},[93],[49,14509,520],{"className":14510},[93,94],[49,14512,14514],{"className":14513},[218],[49,14515,14517,14537],{"className":14516},[222,605],[49,14518,14520,14534],{"className":14519},[226],[49,14521,14523],{"className":14522,"style":1466},[230],[49,14524,14525,14528],{"style":615},[49,14526],{"className":14527,"style":238},[237],[49,14529,14531],{"className":14530},[242,243,244,245],[49,14532,1426],{"className":14533},[93,94,245],[49,14535,629],{"className":14536},[628],[49,14538,14540],{"className":14539},[226],[49,14541,14543],{"className":14542,"style":636},[230],[49,14544],{},[49,14546,307],{"className":14547},[406],[49,14549,4557],{"className":14550,"style":4591},[93,94],[49,14552],{"className":14553,"style":193},[192],[49,14555,6581],{"className":14556},[197],[49,14558],{"className":14559,"style":193},[192],[49,14561,14563,14566],{"className":14562},[84],[49,14564],{"className":14565,"style":6838},[88],[49,14567,6965],{"className":14568},[93],[11,14570,5490,14571,14574],{},[28,14572,14573],{},"单个负特征值就摧毁了非退化局部极小值的可能性","。在高维中，随机 Hessian 有很多机会获得这样的负方向。只有当这些方向都不向下弯曲时，非退化局部极小值才能存活。",[39,14576,14578],{"id":14577},"kacrice-公式","Kac–Rice 公式",[11,14580,14581,14582,407],{},"随机矩阵论证解释了为什么随机 Hessian 不太可能正定。要把这一论证与实际随机函数联系起来，需要一种计数临界点并按指数分类的方法。标准工具是 ",[28,14583,14578],{},[11,14585,138,14586,14614,14615,14699,14700,14778],{},[49,14587,14589,14602],{"className":14588,"translate":53},[52],[49,14590,14592],{"className":14591},[57],[59,14593,14594],{"xmlns":61},[63,14595,14596,14600],{},[66,14597,14598],{},[69,14599,153],{},[73,14601,153],{"encoding":75},[49,14603,14605],{"className":14604,"ariaHidden":80},[79],[49,14606,14608,14611],{"className":14607},[84],[49,14609],{"className":14610,"style":184},[88],[49,14612,153],{"className":14613,"style":188},[93,94]," 是区域 ",[49,14616,14618,14642],{"className":14617,"translate":53},[52],[49,14619,14621],{"className":14620},[57],[59,14622,14623],{"xmlns":61},[63,14624,14625,14639],{},[66,14626,14627,14630,14633],{},[69,14628,14629],{},"D",[155,14631,14632],{},"⊂",[159,14634,14635,14637],{},[69,14636,164],{"mathvariant":163},[69,14638,71],{},[73,14640,14641],{"encoding":75},"D\\subset\\mathbb{R}^N",[49,14643,14645,14664],{"className":14644,"ariaHidden":80},[79],[49,14646,14648,14652,14655,14658,14661],{"className":14647},[84],[49,14649],{"className":14650,"style":14651},[88],"height:0.7224em;vertical-align:-0.0391em;",[49,14653,14629],{"className":14654,"style":11724},[93,94],[49,14656],{"className":14657,"style":193},[192],[49,14659,14632],{"className":14660},[197],[49,14662],{"className":14663,"style":193},[192],[49,14665,14667,14670],{"className":14666},[84],[49,14668],{"className":14669,"style":207},[88],[49,14671,14673,14676],{"className":14672},[93],[49,14674,164],{"className":14675},[93,214],[49,14677,14679],{"className":14678},[218],[49,14680,14682],{"className":14681},[222],[49,14683,14685],{"className":14684},[226],[49,14686,14688],{"className":14687,"style":207},[230],[49,14689,14690,14693],{"style":233},[49,14691],{"className":14692,"style":238},[237],[49,14694,14696],{"className":14695},[242,243,244,245],[49,14697,71],{"className":14698,"style":95},[93,94,245]," 上足够正则的随机场。对 ",[49,14701,14703,14727],{"className":14702,"translate":53},[52],[49,14704,14706],{"className":14705},[57],[59,14707,14708],{"xmlns":61},[63,14709,14710,14724],{},[66,14711,14712,14714,14717,14720,14722],{},[292,14713,801],{},[155,14715,14716],{},"≤",[69,14718,14719],{},"k",[155,14721,14716],{},[69,14723,71],{},[73,14725,14726],{"encoding":75},"0\\le k\\le N",[49,14728,14730,14749,14769],{"className":14729,"ariaHidden":80},[79],[49,14731,14733,14737,14740,14743,14746],{"className":14732},[84],[49,14734],{"className":14735,"style":14736},[88],"height:0.7804em;vertical-align:-0.136em;",[49,14738,801],{"className":14739},[93],[49,14741],{"className":14742,"style":193},[192],[49,14744,14716],{"className":14745},[197],[49,14747],{"className":14748,"style":193},[192],[49,14750,14752,14756,14760,14763,14766],{"className":14751},[84],[49,14753],{"className":14754,"style":14755},[88],"height:0.8304em;vertical-align:-0.136em;",[49,14757,14719],{"className":14758,"style":14759},[93,94],"margin-right:0.0315em;",[49,14761],{"className":14762,"style":193},[192],[49,14764,14716],{"className":14765},[197],[49,14767],{"className":14768,"style":193},[192],[49,14770,14772,14775],{"className":14771},[84],[49,14773],{"className":14774,"style":89},[88],[49,14776,71],{"className":14777,"style":95},[93,94],"，定义",[49,14780,14782],{"className":14781,"translate":53},[752],[49,14783,14785,14871],{"className":14784,"translate":53},[52],[49,14786,14788],{"className":14787},[57],[59,14789,14790],{"xmlns":61,"display":761},[63,14791,14792,14868],{},[66,14793,14794,14800,14802,14804,14806,14808,14810,14812,14814,14816,14818,14820,14822,14824,14826,14828,14830,14832,14834,14836,14838,14840,14842,14844,14846,14852,14854,14856,14858,14860,14862,14864,14866],{},[527,14795,14796,14798],{},[69,14797,290],{},[69,14799,14719],{},[155,14801,298],{"stretchy":297},[69,14803,14629],{},[155,14805,307],{"stretchy":297},[155,14807,157],{},[155,14809,523],{},[69,14811,6565],{"mathvariant":768},[155,14813,6568],{"stretchy":297},[69,14815,520],{},[155,14817,285],{},[69,14819,14629],{},[155,14821,157],{},[69,14823,769],{"mathvariant":768},[69,14825,153],{},[155,14827,298],{"stretchy":297},[69,14829,520],{},[155,14831,307],{"stretchy":297},[155,14833,523],{},[292,14835,801],{},[155,14837,536],{"separator":80},[3314,14839,6934],{},[69,14841,6545],{"mathvariant":768},[155,14843,6548],{},[155,14845,298],{"stretchy":297},[527,14847,14848,14850],{},[69,14849,1880],{},[69,14851,153],{},[155,14853,298],{"stretchy":297},[69,14855,520],{},[155,14857,307],{"stretchy":297},[155,14859,307],{"stretchy":297},[155,14861,523],{},[69,14863,14719],{},[155,14865,6586],{"stretchy":297},[155,14867,536],{"separator":80},[73,14869,14870],{"encoding":75},"C_k(D) := \\#\\{x\\in D : \\nabla f(x)=0,\\ \\operatorname{index}(H_f(x))=k\\},",[49,14872,14874,14939,14963,14981,15011,15097],{"className":14873,"ariaHidden":80},[79],[49,14875,14877,14880,14921,14924,14927,14930,14933,14936],{"className":14876},[84],[49,14878],{"className":14879,"style":589},[88],[49,14881,14883,14886],{"className":14882},[93],[49,14884,290],{"className":14885,"style":345},[93,94],[49,14887,14889],{"className":14888},[218],[49,14890,14892,14913],{"className":14891},[222,605],[49,14893,14895,14910],{"className":14894},[226],[49,14896,14898],{"className":14897,"style":2006},[230],[49,14899,14901,14904],{"style":14900},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[49,14902],{"className":14903,"style":238},[237],[49,14905,14907],{"className":14906},[242,243,244,245],[49,14908,14719],{"className":14909,"style":14759},[93,94,245],[49,14911,629],{"className":14912},[628],[49,14914,14916],{"className":14915},[226],[49,14917,14919],{"className":14918,"style":636},[230],[49,14920],{},[49,14922,298],{"className":14923},[373],[49,14925,14629],{"className":14926,"style":11724},[93,94],[49,14928,307],{"className":14929},[406],[49,14931],{"className":14932,"style":193},[192],[49,14934,931],{"className":14935},[197],[49,14937],{"className":14938,"style":193},[192],[49,14940,14942,14945,14948,14951,14954,14957,14960],{"className":14941},[84],[49,14943],{"className":14944,"style":589},[88],[49,14946,6565],{"className":14947},[93],[49,14949,6568],{"className":14950},[373],[49,14952,520],{"className":14953},[93,94],[49,14955],{"className":14956,"style":193},[192],[49,14958,285],{"className":14959},[197],[49,14961],{"className":14962,"style":193},[192],[49,14964,14966,14969,14972,14975,14978],{"className":14965},[84],[49,14967],{"className":14968,"style":89},[88],[49,14970,14629],{"className":14971,"style":11724},[93,94],[49,14973],{"className":14974,"style":193},[192],[49,14976,157],{"className":14977},[197],[49,14979],{"className":14980,"style":193},[192],[49,14982,14984,14987,14990,14993,14996,14999,15002,15005,15008],{"className":14983},[84],[49,14985],{"className":14986,"style":589},[88],[49,14988,769],{"className":14989},[93],[49,14991,153],{"className":14992,"style":188},[93,94],[49,14994,298],{"className":14995},[373],[49,14997,520],{"className":14998},[93,94],[49,15000,307],{"className":15001},[406],[49,15003],{"className":15004,"style":193},[192],[49,15006,523],{"className":15007},[197],[49,15009],{"className":15010,"style":193},[192],[49,15012,15014,15017,15020,15023,15026,15029,15035,15038,15078,15081,15084,15088,15091,15094],{"className":15013},[84],[49,15015],{"className":15016,"style":1986},[88],[49,15018,801],{"className":15019},[93],[49,15021,536],{"className":15022},[642],[49,15024,6934],{"className":15025},[192],[49,15027],{"className":15028,"style":646},[192],[49,15030,15032],{"className":15031},[4912],[49,15033,6545],{"className":15034},[93,6607],[49,15036,298],{"className":15037},[373],[49,15039,15041,15044],{"className":15040},[93],[49,15042,1880],{"className":15043,"style":1993},[93,94],[49,15045,15047],{"className":15046},[218],[49,15048,15050,15070],{"className":15049},[222,605],[49,15051,15053,15067],{"className":15052},[226],[49,15054,15056],{"className":15055,"style":2006},[230],[49,15057,15058,15061],{"style":2009},[49,15059],{"className":15060,"style":238},[237],[49,15062,15064],{"className":15063},[242,243,244,245],[49,15065,153],{"className":15066,"style":188},[93,94,245],[49,15068,629],{"className":15069},[628],[49,15071,15073],{"className":15072},[226],[49,15074,15076],{"className":15075,"style":2028},[230],[49,15077],{},[49,15079,298],{"className":15080},[373],[49,15082,520],{"className":15083},[93,94],[49,15085,15087],{"className":15086},[406],"))",[49,15089],{"className":15090,"style":193},[192],[49,15092,523],{"className":15093},[197],[49,15095],{"className":15096,"style":193},[192],[49,15098,15100,15103,15106,15109],{"className":15099},[84],[49,15101],{"className":15102,"style":589},[88],[49,15104,14719],{"className":15105,"style":14759},[93,94],[49,15107,6586],{"className":15108},[406],[49,15110,536],{"className":15111},[642],[11,15113,15114,15115,15143,15144,15172],{},"即 ",[49,15116,15118,15131],{"className":15117,"translate":53},[52],[49,15119,15121],{"className":15120},[57],[59,15122,15123],{"xmlns":61},[63,15124,15125,15129],{},[66,15126,15127],{},[69,15128,14629],{},[73,15130,14629],{"encoding":75},[49,15132,15134],{"className":15133,"ariaHidden":80},[79],[49,15135,15137,15140],{"className":15136},[84],[49,15138],{"className":15139,"style":89},[88],[49,15141,14629],{"className":15142,"style":11724},[93,94]," 中指数为 ",[49,15145,15147,15160],{"className":15146,"translate":53},[52],[49,15148,15150],{"className":15149},[57],[59,15151,15152],{"xmlns":61},[63,15153,15154,15158],{},[66,15155,15156],{},[69,15157,14719],{},[73,15159,14719],{"encoding":75},[49,15161,15163],{"className":15162,"ariaHidden":80},[79],[49,15164,15166,15169],{"className":15165},[84],[49,15167],{"className":15168,"style":3096},[88],[49,15170,14719],{"className":15171,"style":14759},[93,94]," 的临界点个数。在正则性与非退化性假设下，Kac–Rice 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f(x)}(0)",[49,15874,15876],{"className":15875,"ariaHidden":80},[79],[49,15877,15879,15883,15938,15941,15944],{"className":15878},[84],[49,15880],{"className":15881,"style":15882},[88],"height:1.1052em;vertical-align:-0.3552em;",[49,15884,15886,15889],{"className":15885},[93],[49,15887,11],{"className":15888},[93,94],[49,15890,15892],{"className":15891},[218],[49,15893,15895,15930],{"className":15894},[222,605],[49,15896,15898,15927],{"className":15897},[226],[49,15899,15901],{"className":15900,"style":15751},[230],[49,15902,15903,15906],{"style":15754},[49,15904],{"className":15905,"style":238},[237],[49,15907,15909],{"className":15908},[242,243,244,245],[49,15910,15912,15915,15918,15921,15924],{"className":15911},[93,245],[49,15913,769],{"className":15914},[93,245],[49,15916,153],{"className":15917,"style":188},[93,94,245],[49,15919,298],{"className":15920},[373,245],[49,15922,520],{"className":15923},[93,94,245],[49,15925,307],{"className":15926},[406,245],[49,15928,629],{"className":15929},[628],[49,15931,15933],{"className":15932},[226],[49,15934,15936],{"className":15935,"style":15788},[230],[49,15937],{},[49,15939,298],{"className":15940},[373],[49,15942,801],{"className":15943},[93],[49,15945,307],{"className":15946},[406]," 是梯度在零处的密度。行列式因子的出现，是因为梯度映射的零点要按局部雅可比体积变化来计数。指示函数则选出所需的指数。",[11,15949,15950,15951,8610],{},"方程 (3) 使问题变得精确。计数非退化局部极小值就是取 ",[49,15952,15954,15972],{"className":15953,"translate":53},[52],[49,15955,15957],{"className":15956},[57],[59,15958,15959],{"xmlns":61},[63,15960,15961,15969],{},[66,15962,15963,15965,15967],{},[69,15964,14719],{},[155,15966,523],{},[292,15968,801],{},[73,15970,15971],{"encoding":75},"k=0",[49,15973,15975,15993],{"className":15974,"ariaHidden":80},[79],[49,15976,15978,15981,15984,15987,15990],{"className":15977},[84],[49,15979],{"className":15980,"style":3096},[88],[49,15982,14719],{"className":15983,"style":14759},[93,94],[49,15985],{"className":15986,"style":193},[192],[49,15988,523],{"className":15989},[197],[49,15991],{"className":15992,"style":193},[192],[49,15994,15996,15999],{"className":15995},[84],[49,15997],{"className":15998,"style":6838},[88],[49,16000,801],{"className":16001},[93],[49,16003,16005],{"className":16004,"translate":53},[752],[49,16006,16008,16085],{"className":16007,"translate":53},[52],[49,16009,16011],{"className":16010},[57],[59,16012,16013],{"xmlns":61,"display":761},[63,16014,16015,16082],{},[66,16016,16017,16023,16025,16027,16029,16031,16033,16035,16037,16039,16041,16043,16045,16047,16049,16051,16053,16055,16057,16059,16061,16067,16069,16071,16073,16076,16078,16080],{},[527,16018,16019,16021],{},[69,16020,290],{},[292,16022,801],{},[155,16024,298],{"stretchy":297},[69,16026,14629],{},[155,16028,307],{"stretchy":297},[155,16030,523],{},[69,16032,6565],{"mathvariant":768},[155,16034,6568],{"stretchy":297},[69,16036,520],{},[155,16038,285],{},[69,16040,14629],{},[155,16042,157],{},[69,16044,769],{"mathvariant":768},[69,16046,153],{},[155,16048,298],{"stretchy":297},[69,16050,520],{},[155,16052,307],{"stretchy":297},[155,16054,523],{},[292,16056,801],{},[155,16058,536],{"separator":80},[3314,16060,6934],{},[527,16062,16063,16065],{},[69,16064,1880],{},[69,16066,153],{},[155,16068,298],{"stretchy":297},[69,16070,520],{},[155,16072,307],{"stretchy":297},[155,16074,16075],{},"≻",[292,16077,801],{},[155,16079,6586],{"stretchy":297},[69,16081,897],{"mathvariant":768},[73,16083,16084],{"encoding":75},"C_0(D) = \\#\\{x\\in D : \\nabla f(x)=0,\\ H_f(x)\\succ 0\\}.",[49,16086,16088,16152,16176,16194,16224,16300],{"className":16087,"ariaHidden":80},[79],[49,16089,16091,16094,16134,16137,16140,16143,16146,16149],{"className":16090},[84],[49,16092],{"className":16093,"style":589},[88],[49,16095,16097,16100],{"className":16096},[93],[49,16098,290],{"className":16099,"style":345},[93,94],[49,16101,16103],{"className":16102},[218],[49,16104,16106,16126],{"className":16105},[222,605],[49,16107,16109,16123],{"className":16108},[226],[49,16110,16112],{"className":16111,"style":612},[230],[49,16113,16114,16117],{"style":14900},[49,16115],{"className":16116,"style":238},[237],[49,16118,16120],{"className":16119},[242,243,244,245],[49,16121,801],{"className":16122},[93,245],[49,16124,629],{"className":16125},[628],[49,16127,16129],{"className":16128},[226],[49,16130,16132],{"className":16131,"style":636},[230],[49,16133],{},[49,16135,298],{"className":16136},[373],[49,16138,14629],{"className":16139,"style":11724},[93,94],[49,16141,307],{"className":16142},[406],[49,16144],{"className":16145,"style":193},[192],[49,16147,523],{"className":16148},[197],[49,16150],{"className":16151,"style":193},[192],[49,16153,16155,16158,16161,16164,16167,16170,16173],{"className":16154},[84],[49,16156],{"className":16157,"style":589},[88],[49,16159,6565],{"className":16160},[93],[49,16162,6568],{"className":16163},[373],[49,16165,520],{"className":16166},[93,94],[49,16168],{"className":16169,"style":193},[192],[49,16171,285],{"className":16172},[197],[49,16174],{"className":16175,"style":193},[192],[49,16177,16179,16182,16185,16188,16191],{"className":16178},[84],[49,16180],{"className":16181,"style":89},[88],[49,16183,14629],{"className":16184,"style":11724},[93,94],[49,16186],{"className":16187,"style":193},[192],[49,16189,157],{"className":16190},[197],[49,16192],{"className":16193,"style":193},[192],[49,16195,16197,16200,16203,16206,16209,16212,16215,16218,16221],{"className":16196},[84],[49,16198],{"className":16199,"style":589},[88],[49,16201,769],{"className":16202},[93],[49,16204,153],{"className":16205,"style":188},[93,94],[49,16207,298],{"className":16208},[373],[49,16210,520],{"className":16211},[93,94],[49,16213,307],{"className":16214},[406],[49,16216],{"className":16217,"style":193},[192],[49,16219,523],{"className":16220},[197],[49,16222],{"className":16223,"style":193},[192],[49,16225,16227,16230,16233,16236,16239,16242,16282,16285,16288,16291,16294,16297],{"className":16226},[84],[49,16228],{"className":16229,"style":1986},[88],[49,16231,801],{"className":16232},[93],[49,16234,536],{"className":16235},[642],[49,16237,6934],{"className":16238},[192],[49,16240],{"className":16241,"style":646},[192],[49,16243,16245,16248],{"className":16244},[93],[49,16246,1880],{"className":16247,"style":1993},[93,94],[49,16249,16251],{"className":16250},[218],[49,16252,16254,16274],{"className":16253},[222,605],[49,16255,16257,16271],{"className":16256},[226],[49,16258,16260],{"className":16259,"style":2006},[230],[49,16261,16262,16265],{"style":2009},[49,16263],{"className":16264,"style":238},[237],[49,16266,16268],{"className":16267},[242,243,244,245],[49,16269,153],{"className":16270,"style":188},[93,94,245],[49,16272,629],{"className":16273},[628],[49,16275,16277],{"className":16276},[226],[49,16278,16280],{"className":16279,"style":2028},[230],[49,16281],{},[49,16283,298],{"className":16284},[373],[49,16286,520],{"className":16287},[93,94],[49,16289,307],{"className":16290},[406],[49,16292],{"className":16293,"style":193},[192],[49,16295,16075],{"className":16296},[197],[49,16298],{"className":16299,"style":193},[192],[49,16301,16303,16306,16309,16312],{"className":16302},[84],[49,16304],{"className":16305,"style":589},[88],[49,16307,801],{"className":16308},[93],[49,16310,6586],{"className":16311},[406],[49,16313,897],{"className":16314},[93],[11,16316,16317,16318,16413],{},"计数鞍点则是对 ",[49,16319,16321,16347],{"className":16320,"translate":53},[52],[49,16322,16324],{"className":16323},[57],[59,16325,16326],{"xmlns":61},[63,16327,16328,16344],{},[66,16329,16330,16332,16334,16336,16338,16340,16342],{},[292,16331,533],{},[155,16333,14716],{},[69,16335,14719],{},[155,16337,14716],{},[69,16339,71],{},[155,16341,3077],{},[292,16343,533],{},[73,16345,16346],{"encoding":75},"1\\le k\\le N-1",[49,16348,16350,16368,16386,16404],{"className":16349,"ariaHidden":80},[79],[49,16351,16353,16356,16359,16362,16365],{"className":16352},[84],[49,16354],{"className":16355,"style":14736},[88],[49,16357,533],{"className":16358},[93],[49,16360],{"className":16361,"style":193},[192],[49,16363,14716],{"className":16364},[197],[49,16366],{"className":16367,"style":193},[192],[49,16369,16371,16374,16377,16380,16383],{"className":16370},[84],[49,16372],{"className":16373,"style":14755},[88],[49,16375,14719],{"className":16376,"style":14759},[93,94],[49,16378],{"className":16379,"style":193},[192],[49,16381,14716],{"className":16382},[197],[49,16384],{"className":16385,"style":193},[192],[49,16387,16389,16392,16395,16398,16401],{"className":16388},[84],[49,16390],{"className":16391,"style":1836},[88],[49,16393,71],{"className":16394,"style":95},[93,94],[49,16396],{"className":16397,"style":1843},[192],[49,16399,3077],{"className":16400},[1847],[49,16402],{"className":16403,"style":1843},[192],[49,16405,16407,16410],{"className":16406},[84],[49,16408],{"className":16409,"style":6838},[88],[49,16411,533],{"className":16412},[93]," 求和。",[11,16415,16416,16417,16488],{},"对许多高斯随机场，在 ",[49,16418,16420,16446],{"className":16419,"translate":53},[52],[49,16421,16423],{"className":16422},[57],[59,16424,16425],{"xmlns":61},[63,16426,16427,16443],{},[66,16428,16429,16431,16433,16435,16437,16439,16441],{},[69,16430,769],{"mathvariant":768},[69,16432,153],{},[155,16434,298],{"stretchy":297},[69,16436,520],{},[155,16438,307],{"stretchy":297},[155,16440,523],{},[292,16442,801],{},[73,16444,16445],{"encoding":75},"\\nabla f(x)=0",[49,16447,16449,16479],{"className":16448,"ariaHidden":80},[79],[49,16450,16452,16455,16458,16461,16464,16467,16470,16473,16476],{"className":16451},[84],[49,16453],{"className":16454,"style":589},[88],[49,16456,769],{"className":16457},[93],[49,16459,153],{"className":16460,"style":188},[93,94],[49,16462,298],{"className":16463},[373],[49,16465,520],{"className":16466},[93,94],[49,16468,307],{"className":16469},[406],[49,16471],{"className":16472,"style":193},[192],[49,16474,523],{"className":16475},[197],[49,16477],{"className":16478,"style":193},[192],[49,16480,16482,16485],{"className":16481},[84],[49,16483],{"className":16484,"style":6838},[88],[49,16486,801],{"className":16487},[93]," 条件下，Hessian 的条件分布是一个正交不变的高斯矩阵，通常是 GOE 型矩阵加上一个依赖于场值的标量平移——这正是随机矩阵理论严格介入之处，Hessian 的指数分布控制着每一类临界点的期望个数。",[11,16490,16491,16492,16520],{},"在自旋玻璃模型中，这一方案已被详细执行。Bray 与 Dean 计算了大维空间上高斯场的平均临界点个数，把它作为能量与指数的函数。Aufinger、Ben Arous 与 Černý 利用 Kac–Rice 与 GOE 特征值渐近，对球面 ",[49,16493,16495,16508],{"className":16494,"translate":53},[52],[49,16496,16498],{"className":16497},[57],[59,16499,16500],{"xmlns":61},[63,16501,16502,16506],{},[66,16503,16504],{},[69,16505,11],{},[73,16507,11],{"encoding":75},[49,16509,16511],{"className":16510,"ariaHidden":80},[79],[49,16512,16514,16517],{"className":16513},[84],[49,16515],{"className":16516,"style":4584},[88],[49,16518,11],{"className":16519},[93,94],"-自旋自旋玻璃哈密顿量的复杂度给出了严格的渐近分析。这些工作展示了一种典型模式：临界点可能是指数级众多的，但主导指数通常不是 0——相反，在景观的大部分区域，占主导的临界点具有正比例的负特征值，即它们是鞍点。",[11,16522,16523],{},"因此，严格的概率陈述不是「每个高维函数都只有很少的局部极小值」，而是更接近：",[469,16525,16526],{},[11,16527,16528],{},"在广泛的高维随机景观模型中，期望的临界点计数以鞍点为主，而指数为零的临界点只构成一个趋于消失或指数级更小的比例，除非处于特殊的低能区域。",[39,16530,16532],{"id":16531},"morse-理论","Morse 理论",[11,16534,16535],{},"Morse 理论为指数提供了拓扑诠释。它本身并不证明极小值是罕见的，而是解释为什么指数是应当追踪的正确不变量。",[11,16537,138,16538,16612,16613,16641,16642,16670],{},[49,16539,16541,16564],{"className":16540,"translate":53},[52],[49,16542,16544],{"className":16543},[57],[59,16545,16546],{"xmlns":61},[63,16547,16548,16561],{},[66,16549,16550,16552,16554,16557,16559],{},[69,16551,153],{},[155,16553,157],{},[69,16555,16556],{},"M",[155,16558,169],{},[69,16560,164],{"mathvariant":163},[73,16562,16563],{"encoding":75},"f: M\\to\\mathbb{R}",[49,16565,16567,16585,16603],{"className":16566,"ariaHidden":80},[79],[49,16568,16570,16573,16576,16579,16582],{"className":16569},[84],[49,16571],{"className":16572,"style":184},[88],[49,16574,153],{"className":16575,"style":188},[93,94],[49,16577],{"className":16578,"style":193},[192],[49,16580,157],{"className":16581},[197],[49,16583],{"className":16584,"style":193},[192],[49,16586,16588,16591,16594,16597,16600],{"className":16587},[84],[49,16589],{"className":16590,"style":89},[88],[49,16592,16556],{"className":16593,"style":95},[93,94],[49,16595],{"className":16596,"style":193},[192],[49,16598,169],{"className":16599},[197],[49,16601],{"className":16602,"style":193},[192],[49,16604,16606,16609],{"className":16605},[84],[49,16607],{"className":16608,"style":264},[88],[49,16610,164],{"className":16611},[93,214]," 是紧光滑流形 ",[49,16614,16616,16629],{"className":16615,"translate":53},[52],[49,16617,16619],{"className":16618},[57],[59,16620,16621],{"xmlns":61},[63,16622,16623,16627],{},[66,16624,16625],{},[69,16626,16556],{},[73,16628,16556],{"encoding":75},[49,16630,16632],{"className":16631,"ariaHidden":80},[79],[49,16633,16635,16638],{"className":16634},[84],[49,16636],{"className":16637,"style":89},[88],[49,16639,16556],{"className":16640,"style":95},[93,94]," 上的 Morse 函数。当抬高阈值 ",[49,16643,16645,16658],{"className":16644,"translate":53},[52],[49,16646,16648],{"className":16647},[57],[59,16649,16650],{"xmlns":61},[63,16651,16652,16656],{},[66,16653,16654],{},[69,16655,4867],{},[73,16657,4867],{"encoding":75},[49,16659,16661],{"className":16660,"ariaHidden":80},[79],[49,16662,16664,16667],{"className":16663},[84],[49,16665],{"className":16666,"style":570},[88],[49,16668,4867],{"className":16669},[93,94]," 并考察下水平集",[49,16672,16674],{"className":16673,"translate":53},[752],[49,16675,16677,16723],{"className":16676,"translate":53},[52],[49,16678,16680],{"className":16679},[57],[59,16681,16682],{"xmlns":61,"display":761},[63,16683,16684,16720],{},[66,16685,16686,16692,16694,16696,16698,16700,16702,16704,16706,16708,16710,16712,16714,16716,16718],{},[527,16687,16688,16690],{},[69,16689,16556],{},[69,16691,4867],{},[155,16693,157],{},[155,16695,523],{},[155,16697,6568],{"stretchy":297},[69,16699,520],{},[155,16701,285],{},[69,16703,16556],{},[155,16705,157],{},[69,16707,153],{},[155,16709,298],{"stretchy":297},[69,16711,520],{},[155,16713,307],{"stretchy":297},[155,16715,14716],{},[69,16717,4867],{},[155,16719,6586],{"stretchy":297},[73,16721,16722],{"encoding":75},"M_a := \\{x\\in M : f(x)\\le a\\}",[49,16724,16726,16782,16803,16821,16848],{"className":16725,"ariaHidden":80},[79],[49,16727,16729,16732,16773,16776,16779],{"className":16728},[84],[49,16730],{"className":16731,"style":12029},[88],[49,16733,16735,16738],{"className":16734},[93],[49,16736,16556],{"className":16737,"style":95},[93,94],[49,16739,16741],{"className":16740},[218],[49,16742,16744,16765],{"className":16743},[222,605],[49,16745,16747,16762],{"className":16746},[226],[49,16748,16750],{"className":16749,"style":1466},[230],[49,16751,16753,16756],{"style":16752},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[49,16754],{"className":16755,"style":238},[237],[49,16757,16759],{"className":16758},[242,243,244,245],[49,16760,4867],{"className":16761},[93,94,245],[49,16763,629],{"className":16764},[628],[49,16766,16768],{"className":16767},[226],[49,16769,16771],{"className":16770,"style":636},[230],[49,16772],{},[49,16774],{"className":16775,"style":193},[192],[49,16777,931],{"className":16778},[197],[49,16780],{"className":16781,"style":193},[192],[49,16783,16785,16788,16791,16794,16797,16800],{"className":16784},[84],[49,16786],{"className":16787,"style":589},[88],[49,16789,6568],{"className":16790},[373],[49,16792,520],{"className":16793},[93,94],[49,16795],{"className":16796,"style":193},[192],[49,16798,285],{"className":16799},[197],[49,16801],{"className":16802,"style":193},[192],[49,16804,16806,16809,16812,16815,16818],{"className":16805},[84],[49,16807],{"className":16808,"style":89},[88],[49,16810,16556],{"className":16811,"style":95},[93,94],[49,16813],{"className":16814,"style":193},[192],[49,16816,157],{"className":16817},[197],[49,16819],{"className":16820,"style":193},[192],[49,16822,16824,16827,16830,16833,16836,16839,16842,16845],{"className":16823},[84],[49,16825],{"className":16826,"style":589},[88],[49,16828,153],{"className":16829,"style":188},[93,94],[49,16831,298],{"className":16832},[373],[49,16834,520],{"className":16835},[93,94],[49,16837,307],{"className":16838},[406],[49,16840],{"className":16841,"style":193},[192],[49,16843,14716],{"className":16844},[197],[49,16846],{"className":16847,"style":193},[192],[49,16849,16851,16854,16857],{"className":16850},[84],[49,16852],{"className":16853,"style":589},[88],[49,16855,4867],{"className":16856},[93,94],[49,16858,6586],{"className":16859},[406],[11,16861,16862,16863,16933,16934,16962,16963,16991,16992,17020],{},"时，",[49,16864,16866,16884],{"className":16865,"translate":53},[52],[49,16867,16869],{"className":16868},[57],[59,16870,16871],{"xmlns":61},[63,16872,16873,16881],{},[66,16874,16875],{},[527,16876,16877,16879],{},[69,16878,16556],{},[69,16880,4867],{},[73,16882,16883],{"encoding":75},"M_a",[49,16885,16887],{"className":16886,"ariaHidden":80},[79],[49,16888,16890,16893],{"className":16889},[84],[49,16891],{"className":16892,"style":12029},[88],[49,16894,16896,16899],{"className":16895},[93],[49,16897,16556],{"className":16898,"style":95},[93,94],[49,16900,16902],{"className":16901},[218],[49,16903,16905,16925],{"className":16904},[222,605],[49,16906,16908,16922],{"className":16907},[226],[49,16909,16911],{"className":16910,"style":1466},[230],[49,16912,16913,16916],{"style":16752},[49,16914],{"className":16915,"style":238},[237],[49,16917,16919],{"className":16918},[242,243,244,245],[49,16920,4867],{"className":16921},[93,94,245],[49,16923,629],{"className":16924},[628],[49,16926,16928],{"className":16927},[226],[49,16929,16931],{"className":16930,"style":636},[230],[49,16932],{}," 的拓扑只在 ",[49,16935,16937,16950],{"className":16936,"translate":53},[52],[49,16938,16940],{"className":16939},[57],[59,16941,16942],{"xmlns":61},[63,16943,16944,16948],{},[66,16945,16946],{},[69,16947,4867],{},[73,16949,4867],{"encoding":75},[49,16951,16953],{"className":16952,"ariaHidden":80},[79],[49,16954,16956,16959],{"className":16955},[84],[49,16957],{"className":16958,"style":570},[88],[49,16960,4867],{"className":16961},[93,94]," 经过临界值时改变。若临界点的指数为 ",[49,16964,16966,16979],{"className":16965,"translate":53},[52],[49,16967,16969],{"className":16968},[57],[59,16970,16971],{"xmlns":61},[63,16972,16973,16977],{},[66,16974,16975],{},[69,16976,14719],{},[73,16978,14719],{"encoding":75},[49,16980,16982],{"className":16981,"ariaHidden":80},[79],[49,16983,16985,16988],{"className":16984},[84],[49,16986],{"className":16987,"style":3096},[88],[49,16989,14719],{"className":16990,"style":14759},[93,94],"，那么粗略地说，穿过该水平会连接一个 ",[49,16993,16995,17008],{"className":16994,"translate":53},[52],[49,16996,16998],{"className":16997},[57],[59,16999,17000],{"xmlns":61},[63,17001,17002,17006],{},[66,17003,17004],{},[69,17005,14719],{},[73,17007,14719],{"encoding":75},[49,17009,17011],{"className":17010,"ariaHidden":80},[79],[49,17012,17014,17017],{"className":17013},[84],[49,17015],{"className":17016,"style":3096},[88],[49,17018,14719],{"className":17019,"style":14759},[93,94]," 维手柄：局部极小值的指数为 0，因此它会为下水平集创造出新的连通分支。指数为 1 的鞍点可以把分支合并。更高指数的鞍点则创造出更高维的隧道与空洞。",[11,17022,17023,17024,8610],{},"一个推论是欧拉示性数的 ",[28,17025,17026],{},"Morse 关系",[49,17028,17030],{"className":17029,"translate":53},[752],[49,17031,17033,17105],{"className":17032,"translate":53},[52],[49,17034,17036],{"className":17035},[57],[59,17037,17038],{"xmlns":61,"display":761},[63,17039,17040,17102],{},[787,17041,17042],{"width":3191},[793,17043,17044,17046,17095,17097],{},[796,17045],{"width":3196},[796,17047,17048],{},[66,17049,17050,17064,17066,17068,17070,17076,17082,17084,17087,17089,17091,17093],{},[4848,17051,17052,17054,17062],{},[155,17053,4852],{},[66,17055,17056,17058,17060],{},[69,17057,14719],{},[155,17059,523],{},[292,17061,801],{},[69,17063,71],{},[155,17065,298],{"stretchy":297},[155,17067,3077],{},[292,17069,533],{},[159,17071,17072,17074],{},[155,17073,307],{"stretchy":297},[69,17075,14719],{},[527,17077,17078,17080],{},[69,17079,290],{},[69,17081,14719],{},[155,17083,523],{},[69,17085,17086],{},"χ",[155,17088,298],{"stretchy":297},[69,17090,16556],{},[155,17092,307],{"stretchy":297},[155,17094,536],{"separator":80},[796,17096],{"width":3196},[796,17098,17099],{},[3314,17100,17101],{},"(4)",[73,17103,17104],{"encoding":75},"\\sum_{k=0}^{N} (-1)^k C_k = \\chi(M), \\tag{4}",[49,17106,17108,17272,17293],{"className":17107,"ariaHidden":80},[79],[49,17109,17111,17115,17184,17187,17190,17193,17223,17263,17266,17269],{"className":17110},[84],[49,17112],{"className":17113,"style":17114},[88],"height:3.1304em;vertical-align:-1.3021em;",[49,17116,17118],{"className":17117},[4912,4913],[49,17119,17121,17175],{"className":17120},[222,605],[49,17122,17124,17172],{"className":17123},[226],[49,17125,17127,17148,17158],{"className":17126,"style":4923},[230],[49,17128,17130,17133],{"style":17129},"top:-1.8479em;margin-left:0em;",[49,17131],{"className":17132,"style":4930},[237],[49,17134,17136],{"className":17135},[242,243,244,245],[49,17137,17139,17142,17145],{"className":17138},[93,245],[49,17140,14719],{"className":17141,"style":14759},[93,94,245],[49,17143,523],{"className":17144},[197,245],[49,17146,801],{"className":17147},[93,245],[49,17149,17150,17153],{"style":4948},[49,17151],{"className":17152,"style":4930},[237],[49,17154,17155],{},[49,17156,4852],{"className":17157},[4912,4957,4958],[49,17159,17160,17163],{"style":4961},[49,17161],{"className":17162,"style":4930},[237],[49,17164,17166],{"className":17165},[242,243,244,245],[49,17167,17169],{"className":17168},[93,245],[49,17170,71],{"className":17171,"style":95},[93,94,245],[49,17173,629],{"className":17174},[628],[49,17176,17178],{"className":17177},[226],[49,17179,17182],{"className":17180,"style":17181},[230],"height:1.3021em;",[49,17183],{},[49,17185,298],{"className":17186},[373],[49,17188,3077],{"className":17189},[93],[49,17191,533],{"className":17192},[93],[49,17194,17196,17199],{"className":17195},[406],[49,17197,307],{"className":17198},[406],[49,17200,17202],{"className":17201},[218],[49,17203,17205],{"className":17204},[222],[49,17206,17208],{"className":17207},[226],[49,17209,17212],{"className":17210,"style":17211},[230],"height:0.8991em;",[49,17213,17214,17217],{"style":1387},[49,17215],{"className":17216,"style":238},[237],[49,17218,17220],{"className":17219},[242,243,244,245],[49,17221,14719],{"className":17222,"style":14759},[93,94,245],[49,17224,17226,17229],{"className":17225},[93],[49,17227,290],{"className":17228,"style":345},[93,94],[49,17230,17232],{"className":17231},[218],[49,17233,17235,17255],{"className":17234},[222,605],[49,17236,17238,17252],{"className":17237},[226],[49,17239,17241],{"className":17240,"style":2006},[230],[49,17242,17243,17246],{"style":14900},[49,17244],{"className":17245,"style":238},[237],[49,17247,17249],{"className":17248},[242,243,244,245],[49,17250,14719],{"className":17251,"style":14759},[93,94,245],[49,17253,629],{"className":17254},[628],[49,17256,17258],{"className":17257},[226],[49,17259,17261],{"className":17260,"style":636},[230],[49,17262],{},[49,17264],{"className":17265,"style":193},[192],[49,17267,523],{"className":17268},[197],[49,17270],{"className":17271,"style":193},[192],[49,17273,17275,17278,17281,17284,17287,17290],{"className":17274},[84],[49,17276],{"className":17277,"style":589},[88],[49,17279,17086],{"className":17280},[93,94],[49,17282,298],{"className":17283},[373],[49,17285,16556],{"className":17286,"style":95},[93,94],[49,17288,307],{"className":17289},[406],[49,17291,536],{"className":17292},[642],[49,17294,17296,17299],{"className":17295},[3821],[49,17297],{"className":17298,"style":17114},[88],[49,17300,17302,17305,17311],{"className":17301},[93,3828],[49,17303,298],{"className":17304},[93],[49,17306,17308],{"className":17307},[93],[49,17309,12811],{"className":17310},[93],[49,17312,307],{"className":17313},[93],[11,17315,17316,17317,17387,17388,17416,17417,8478,17461,17489,17490,17533],{},"其中 ",[49,17318,17320,17338],{"className":17319,"translate":53},[52],[49,17321,17323],{"className":17322},[57],[59,17324,17325],{"xmlns":61},[63,17326,17327,17335],{},[66,17328,17329],{},[527,17330,17331,17333],{},[69,17332,290],{},[69,17334,14719],{},[73,17336,17337],{"encoding":75},"C_k",[49,17339,17341],{"className":17340,"ariaHidden":80},[79],[49,17342,17344,17347],{"className":17343},[84],[49,17345],{"className":17346,"style":12029},[88],[49,17348,17350,17353],{"className":17349},[93],[49,17351,290],{"className":17352,"style":345},[93,94],[49,17354,17356],{"className":17355},[218],[49,17357,17359,17379],{"className":17358},[222,605],[49,17360,17362,17376],{"className":17361},[226],[49,17363,17365],{"className":17364,"style":2006},[230],[49,17366,17367,17370],{"style":14900},[49,17368],{"className":17369,"style":238},[237],[49,17371,17373],{"className":17372},[242,243,244,245],[49,17374,14719],{"className":17375,"style":14759},[93,94,245],[49,17377,629],{"className":17378},[628],[49,17380,17382],{"className":17381},[226],[49,17383,17385],{"className":17384,"style":636},[230],[49,17386],{}," 是指数为 ",[49,17389,17391,17404],{"className":17390,"translate":53},[52],[49,17392,17394],{"className":17393},[57],[59,17395,17396],{"xmlns":61},[63,17397,17398,17402],{},[66,17399,17400],{},[69,17401,14719],{},[73,17403,14719],{"encoding":75},[49,17405,17407],{"className":17406,"ariaHidden":80},[79],[49,17408,17410,17413],{"className":17409},[84],[49,17411],{"className":17412,"style":3096},[88],[49,17414,14719],{"className":17415,"style":14759},[93,94]," 的临界点个数，",[49,17418,17420,17440],{"className":17419,"translate":53},[52],[49,17421,17423],{"className":17422},[57],[59,17424,17425],{"xmlns":61},[63,17426,17427,17437],{},[66,17428,17429,17431,17433,17435],{},[69,17430,17086],{},[155,17432,298],{"stretchy":297},[69,17434,16556],{},[155,17436,307],{"stretchy":297},[73,17438,17439],{"encoding":75},"\\chi(M)",[49,17441,17443],{"className":17442,"ariaHidden":80},[79],[49,17444,17446,17449,17452,17455,17458],{"className":17445},[84],[49,17447],{"className":17448,"style":589},[88],[49,17450,17086],{"className":17451},[93,94],[49,17453,298],{"className":17454},[373],[49,17456,16556],{"className":17457,"style":95},[93,94],[49,17459,307],{"className":17460},[406],[49,17462,17464,17477],{"className":17463,"translate":53},[52],[49,17465,17467],{"className":17466},[57],[59,17468,17469],{"xmlns":61},[63,17470,17471,17475],{},[66,17472,17473],{},[69,17474,16556],{},[73,17476,16556],{"encoding":75},[49,17478,17480],{"className":17479,"ariaHidden":80},[79],[49,17481,17483,17486],{"className":17482},[84],[49,17484],{"className":17485,"style":89},[88],[49,17487,16556],{"className":17488,"style":95},[93,94]," 的欧拉示性数。由于在复杂的随机景观中，",[49,17491,17493,17512],{"className":17492,"translate":53},[52],[49,17494,17496],{"className":17495},[57],[59,17497,17498],{"xmlns":61},[63,17499,17500,17510],{},[66,17501,17502,17504,17506,17508],{},[69,17503,17086],{},[155,17505,298],{"stretchy":297},[69,17507,16556],{},[155,17509,307],{"stretchy":297},[73,17511,17439],{"encoding":75},[49,17513,17515],{"className":17514,"ariaHidden":80},[79],[49,17516,17518,17521,17524,17527,17530],{"className":17517},[84],[49,17519],{"className":17520,"style":589},[88],[49,17522,17086],{"className":17523},[93,94],[49,17525,298],{"className":17526},[373],[49,17528,16556],{"className":17529,"style":95},[93,94],[49,17531,307],{"className":17532},[406]," 通常相对于临界点总数很小，许多临界点必须在这个交错求和中相互抵消——相邻指数的鞍点天然适合这种抵消。",[11,17535,17536],{},"这一拓扑图景与概率图景一致：随机高维景观常常含有大量散布在各个指数上的鞍点。但重要的是不要夸大拓扑论证：球面上的函数可以只有一个极小值和一个极大值。Morse 理论约束着可能的临界点计数。随机矩阵理论与 Kac–Rice 则解释了为什么在随机高维模型中，计数的绝大部分通常是鞍点式的。",[11189,17538,17540],{"id":17539},"坐标不变性morse-引理","坐标不变性：Morse 引理",[11,17542,17543,17544,17547],{},"在非退化临界点附近，经过光滑坐标变换后，每个光滑函数都看起来像它的 Hessian 标准形——这正是 ",[28,17545,17546],{},"Morse 引理"," 的内容。",[469,17549,17550,17889],{},[11,17551,17552,5531,17555,17627,17628,17387,17680,17708,17709,17737,17738,17888],{},[28,17553,17554],{},"定理 B.1（Morse 引理）",[49,17556,17558,17579],{"className":17557,"translate":53},[52],[49,17559,17561],{"className":17560},[57],[59,17562,17563],{"xmlns":61},[63,17564,17565,17577],{},[66,17566,17567,17569,17571,17573,17575],{},[69,17568,153],{},[155,17570,157],{},[69,17572,16556],{},[155,17574,169],{},[69,17576,164],{"mathvariant":163},[73,17578,16563],{"encoding":75},[49,17580,17582,17600,17618],{"className":17581,"ariaHidden":80},[79],[49,17583,17585,17588,17591,17594,17597],{"className":17584},[84],[49,17586],{"className":17587,"style":184},[88],[49,17589,153],{"className":17590,"style":188},[93,94],[49,17592],{"className":17593,"style":193},[192],[49,17595,157],{"className":17596},[197],[49,17598],{"className":17599,"style":193},[192],[49,17601,17603,17606,17609,17612,17615],{"className":17602},[84],[49,17604],{"className":17605,"style":89},[88],[49,17607,16556],{"className":17608,"style":95},[93,94],[49,17610],{"className":17611,"style":193},[192],[49,17613,169],{"className":17614},[197],[49,17616],{"className":17617,"style":193},[192],[49,17619,17621,17624],{"className":17620},[84],[49,17622],{"className":17623,"style":264},[88],[49,17625,164],{"className":17626},[93,214]," 光滑，",[49,17629,17631,17649],{"className":17630,"translate":53},[52],[49,17632,17634],{"className":17633},[57],[59,17635,17636],{"xmlns":61},[63,17637,17638,17646],{},[66,17639,17640,17642,17644],{},[69,17641,11],{},[155,17643,285],{},[69,17645,16556],{},[73,17647,17648],{"encoding":75},"p\\in M",[49,17650,17652,17671],{"className":17651,"ariaHidden":80},[79],[49,17653,17655,17659,17662,17665,17668],{"className":17654},[84],[49,17656],{"className":17657,"style":17658},[88],"height:0.7335em;vertical-align:-0.1944em;",[49,17660,11],{"className":17661},[93,94],[49,17663],{"className":17664,"style":193},[192],[49,17666,285],{"className":17667},[197],[49,17669],{"className":17670,"style":193},[192],[49,17672,17674,17677],{"className":17673},[84],[49,17675],{"className":17676,"style":89},[88],[49,17678,16556],{"className":17679,"style":95},[93,94],[49,17681,17683,17696],{"className":17682,"translate":53},[52],[49,17684,17686],{"className":17685},[57],[59,17687,17688],{"xmlns":61},[63,17689,17690,17694],{},[66,17691,17692],{},[69,17693,14719],{},[73,17695,14719],{"encoding":75},[49,17697,17699],{"className":17698,"ariaHidden":80},[79],[49,17700,17702,17705],{"className":17701},[84],[49,17703],{"className":17704,"style":3096},[88],[49,17706,14719],{"className":17707,"style":14759},[93,94]," 的非退化临界点。则以 ",[49,17710,17712,17725],{"className":17711,"translate":53},[52],[49,17713,17715],{"className":17714},[57],[59,17716,17717],{"xmlns":61},[63,17718,17719,17723],{},[66,17720,17721],{},[69,17722,11],{},[73,17724,11],{"encoding":75},[49,17726,17728],{"className":17727,"ariaHidden":80},[79],[49,17729,17731,17734],{"className":17730},[84],[49,17732],{"className":17733,"style":4584},[88],[49,17735,11],{"className":17736},[93,94]," 为中心存在局部坐标 ",[49,17739,17741,17775],{"className":17740,"translate":53},[52],[49,17742,17744],{"className":17743},[57],[59,17745,17746],{"xmlns":61},[63,17747,17748,17772],{},[66,17749,17750,17752,17758,17760,17762,17764,17770],{},[155,17751,298],{"stretchy":297},[527,17753,17754,17756],{},[69,17755,9684],{},[292,17757,533],{},[155,17759,536],{"separator":80},[155,17761,539],{},[155,17763,536],{"separator":80},[527,17765,17766,17768],{},[69,17767,9684],{},[69,17769,71],{},[155,17771,307],{"stretchy":297},[73,17773,17774],{"encoding":75},"(u_1,\\dots,u_N)",[49,17776,17778],{"className":17777,"ariaHidden":80},[79],[49,17779,17781,17784,17787,17827,17830,17833,17836,17839,17842,17845,17885],{"className":17780},[84],[49,17782],{"className":17783,"style":589},[88],[49,17785,298],{"className":17786},[373],[49,17788,17790,17793],{"className":17789},[93],[49,17791,9684],{"className":17792},[93,94],[49,17794,17796],{"className":17795},[218],[49,17797,17799,17819],{"className":17798},[222,605],[49,17800,17802,17816],{"className":17801},[226],[49,17803,17805],{"className":17804,"style":612},[230],[49,17806,17807,17810],{"style":615},[49,17808],{"className":17809,"style":238},[237],[49,17811,17813],{"className":17812},[242,243,244,245],[49,17814,533],{"className":17815},[93,245],[49,17817,629],{"className":17818},[628],[49,17820,17822],{"className":17821},[226],[49,17823,17825],{"className":17824,"style":636},[230],[49,17826],{},[49,17828,536],{"className":17829},[642],[49,17831],{"className":17832,"style":646},[192],[49,17834,539],{"className":17835},[650],[49,17837],{"className":17838,"style":646},[192],[49,17840,536],{"className":17841},[642],[49,17843],{"className":17844,"style":646},[192],[49,17846,17848,17851],{"className":17847},[93],[49,17849,9684],{"className":17850},[93,94],[49,17852,17854],{"className":17853},[218],[49,17855,17857,17877],{"className":17856},[222,605],[49,17858,17860,17874],{"className":17859},[226],[49,17861,17863],{"className":17862,"style":678},[230],[49,17864,17865,17868],{"style":615},[49,17866],{"className":17867,"style":238},[237],[49,17869,17871],{"className":17870},[242,243,244,245],[49,17872,71],{"className":17873,"style":95},[93,94,245],[49,17875,629],{"className":17876},[628],[49,17878,17880],{"className":17879},[226],[49,17881,17883],{"className":17882,"style":636},[230],[49,17884],{},[49,17886,307],{"className":17887},[406],"，使得",[49,17890,17892],{"className":17891,"translate":53},[752],[49,17893,17895,17982],{"className":17894,"translate":53},[52],[49,17896,17898],{"className":17897},[57],[59,17899,17900],{"xmlns":61,"display":761},[63,17901,17902,17979],{},[66,17903,17904,17906,17908,17910,17912,17914,17916,17918,17920,17922,17924,17932,17934,17937,17939,17947,17949,17963,17965,17967,17969,17977],{},[69,17905,153],{},[155,17907,298],{"stretchy":297},[69,17909,9684],{},[155,17911,307],{"stretchy":297},[155,17913,523],{},[69,17915,153],{},[155,17917,298],{"stretchy":297},[69,17919,11],{},[155,17921,307],{"stretchy":297},[155,17923,3077],{},[1910,17925,17926,17928,17930],{},[69,17927,9684],{},[292,17929,533],{},[292,17931,294],{},[155,17933,3077],{},[155,17935,17936],{},"⋯",[155,17938,3077],{},[1910,17940,17941,17943,17945],{},[69,17942,9684],{},[69,17944,14719],{},[292,17946,294],{},[155,17948,3213],{},[1910,17950,17951,17953,17961],{},[69,17952,9684],{},[66,17954,17955,17957,17959],{},[69,17956,14719],{},[155,17958,3213],{},[292,17960,533],{},[292,17962,294],{},[155,17964,3213],{},[155,17966,17936],{},[155,17968,3213],{},[1910,17970,17971,17973,17975],{},[69,17972,9684],{},[69,17974,71],{},[292,17976,294],{},[69,17978,897],{"mathvariant":768},[73,17980,17981],{"encoding":75},"f(u) = f(p) - u_1^2 - \\dots - u_k^2 + u_{k+1}^2 + \\dots + u_N^2.",[49,17983,17985,18012,18039,18106,18124,18190,18267,18285],{"className":17984,"ariaHidden":80},[79],[49,17986,17988,17991,17994,17997,18000,18003,18006,18009],{"className":17987},[84],[49,17989],{"className":17990,"style":589},[88],[49,17992,153],{"className":17993,"style":188},[93,94],[49,17995,298],{"className":17996},[373],[49,17998,9684],{"className":17999},[93,94],[49,18001,307],{"className":18002},[406],[49,18004],{"className":18005,"style":193},[192],[49,18007,523],{"className":18008},[197],[49,18010],{"className":18011,"style":193},[192],[49,18013,18015,18018,18021,18024,18027,18030,18033,18036],{"className":18014},[84],[49,18016],{"className":18017,"style":589},[88],[49,18019,153],{"className":18020,"style":188},[93,94],[49,18022,298],{"className":18023},[373],[49,18025,11],{"className":18026},[93,94],[49,18028,307],{"className":18029},[406],[49,18031],{"className":18032,"style":1843},[192],[49,18034,3077],{"className":18035},[1847],[49,18037],{"className":18038,"style":1843},[192],[49,18040,18042,18046,18097,18100,18103],{"className":18041},[84],[49,18043],{"className":18044,"style":18045},[88],"height:1.1111em;vertical-align:-0.247em;",[49,18047,18049,18052],{"className":18048},[93],[49,18050,9684],{"className":18051},[93,94],[49,18053,18055],{"className":18054},[218],[49,18056,18058,18089],{"className":18057},[222,605],[49,18059,18061,18086],{"className":18060},[226],[49,18062,18064,18075],{"className":18063,"style":2074},[230],[49,18065,18066,18069],{"style":5431},[49,18067],{"className":18068,"style":238},[237],[49,18070,18072],{"className":18071},[242,243,244,245],[49,18073,533],{"className":18074},[93,245],[49,18076,18077,18080],{"style":1387},[49,18078],{"className":18079,"style":238},[237],[49,18081,18083],{"className":18082},[242,243,244,245],[49,18084,294],{"className":18085},[93,245],[49,18087,629],{"className":18088},[628],[49,18090,18092],{"className":18091},[226],[49,18093,18095],{"className":18094,"style":5461},[230],[49,18096],{},[49,18098],{"className":18099,"style":1843},[192],[49,18101,3077],{"className":18102},[1847],[49,18104],{"className":18105,"style":1843},[192],[49,18107,18109,18112,18115,18118,18121],{"className":18108},[84],[49,18110],{"className":18111,"style":3115},[88],[49,18113,17936],{"className":18114},[650],[49,18116],{"className":18117,"style":1843},[192],[49,18119,3077],{"className":18120},[1847],[49,18122],{"className":18123,"style":1843},[192],[49,18125,18127,18130,18181,18184,18187],{"className":18126},[84],[49,18128],{"className":18129,"style":18045},[88],[49,18131,18133,18136],{"className":18132},[93],[49,18134,9684],{"className":18135},[93,94],[49,18137,18139],{"className":18138},[218],[49,18140,18142,18173],{"className":18141},[222,605],[49,18143,18145,18170],{"className":18144},[226],[49,18146,18148,18159],{"className":18147,"style":2074},[230],[49,18149,18150,18153],{"style":5431},[49,18151],{"className":18152,"style":238},[237],[49,18154,18156],{"className":18155},[242,243,244,245],[49,18157,14719],{"className":18158,"style":14759},[93,94,245],[49,18160,18161,18164],{"style":1387},[49,18162],{"className":18163,"style":238},[237],[49,18165,18167],{"className":18166},[242,243,244,245],[49,18168,294],{"className":18169},[93,245],[49,18171,629],{"className":18172},[628],[49,18174,18176],{"className":18175},[226],[49,18177,18179],{"className":18178,"style":5461},[230],[49,18180],{},[49,18182],{"className":18183,"style":1843},[192],[49,18185,3213],{"className":18186},[1847],[49,18188],{"className":18189,"style":1843},[192],[49,18191,18193,18197,18258,18261,18264],{"className":18192},[84],[49,18194],{"className":18195,"style":18196},[88],"height:1.1694em;vertical-align:-0.3053em;",[49,18198,18200,18203],{"className":18199},[93],[49,18201,9684],{"className":18202},[93,94],[49,18204,18206],{"className":18205},[218],[49,18207,18209,18249],{"className":18208},[222,605],[49,18210,18212,18246],{"className":18211},[226],[49,18213,18215,18235],{"className":18214,"style":2074},[230],[49,18216,18217,18220],{"style":5431},[49,18218],{"className":18219,"style":238},[237],[49,18221,18223],{"className":18222},[242,243,244,245],[49,18224,18226,18229,18232],{"className":18225},[93,245],[49,18227,14719],{"className":18228,"style":14759},[93,94,245],[49,18230,3213],{"className":18231},[1847,245],[49,18233,533],{"className":18234},[93,245],[49,18236,18237,18240],{"style":1387},[49,18238],{"className":18239,"style":238},[237],[49,18241,18243],{"className":18242},[242,243,244,245],[49,18244,294],{"className":18245},[93,245],[49,18247,629],{"className":18248},[628],[49,18250,18252],{"className":18251},[226],[49,18253,18256],{"className":18254,"style":18255},[230],"height:0.3053em;",[49,18257],{},[49,18259],{"className":18260,"style":1843},[192],[49,18262,3213],{"className":18263},[1847],[49,18265],{"className":18266,"style":1843},[192],[49,18268,18270,18273,18276,18279,18282],{"className":18269},[84],[49,18271],{"className":18272,"style":3115},[88],[49,18274,17936],{"className":18275},[650],[49,18277],{"className":18278,"style":1843},[192],[49,18280,3213],{"className":18281},[1847],[49,18283],{"className":18284,"style":1843},[192],[49,18286,18288,18291,18342],{"className":18287},[84],[49,18289],{"className":18290,"style":18045},[88],[49,18292,18294,18297],{"className":18293},[93],[49,18295,9684],{"className":18296},[93,94],[49,18298,18300],{"className":18299},[218],[49,18301,18303,18334],{"className":18302},[222,605],[49,18304,18306,18331],{"className":18305},[226],[49,18307,18309,18320],{"className":18308,"style":2074},[230],[49,18310,18311,18314],{"style":5431},[49,18312],{"className":18313,"style":238},[237],[49,18315,18317],{"className":18316},[242,243,244,245],[49,18318,71],{"className":18319,"style":95},[93,94,245],[49,18321,18322,18325],{"style":1387},[49,18323],{"className":18324,"style":238},[237],[49,18326,18328],{"className":18327},[242,243,244,245],[49,18329,294],{"className":18330},[93,245],[49,18332,629],{"className":18333},[628],[49,18335,18337],{"className":18336},[226],[49,18338,18340],{"className":18339,"style":5461},[230],[49,18341],{},[49,18343,897],{"className":18344},[93],[11,18346,18347],{},"因此，在光滑局部坐标变换意义下，非退化临界点完全由其指数分类。非退化局部极小值的标准形为",[49,18349,18351],{"className":18350,"translate":53},[752],[49,18352,18354,18400],{"className":18353,"translate":53},[52],[49,18355,18357],{"className":18356},[57],[59,18358,18359],{"xmlns":61,"display":761},[63,18360,18361,18397],{},[66,18362,18363,18365,18367,18369,18371,18373,18381,18383,18385,18387,18395],{},[69,18364,153],{},[155,18366,298],{"stretchy":297},[69,18368,11],{},[155,18370,307],{"stretchy":297},[155,18372,3213],{},[1910,18374,18375,18377,18379],{},[69,18376,9684],{},[292,18378,533],{},[292,18380,294],{},[155,18382,3213],{},[155,18384,17936],{},[155,18386,3213],{},[1910,18388,18389,18391,18393],{},[69,18390,9684],{},[69,18392,71],{},[292,18394,294],{},[155,18396,536],{"separator":80},[73,18398,18399],{"encoding":75},"f(p) + u_1^2 + \\dots + u_N^2,",[49,18401,18403,18430,18496,18514],{"className":18402,"ariaHidden":80},[79],[49,18404,18406,18409,18412,18415,18418,18421,18424,18427],{"className":18405},[84],[49,18407],{"className":18408,"style":589},[88],[49,18410,153],{"className":18411,"style":188},[93,94],[49,18413,298],{"className":18414},[373],[49,18416,11],{"className":18417},[93,94],[49,18419,307],{"className":18420},[406],[49,18422],{"className":18423,"style":1843},[192],[49,18425,3213],{"className":18426},[1847],[49,18428],{"className":18429,"style":1843},[192],[49,18431,18433,18436,18487,18490,18493],{"className":18432},[84],[49,18434],{"className":18435,"style":18045},[88],[49,18437,18439,18442],{"className":18438},[93],[49,18440,9684],{"className":18441},[93,94],[49,18443,18445],{"className":18444},[218],[49,18446,18448,18479],{"className":18447},[222,605],[49,18449,18451,18476],{"className":18450},[226],[49,18452,18454,18465],{"className":18453,"style":2074},[230],[49,18455,18456,18459],{"style":5431},[49,18457],{"className":18458,"style":238},[237],[49,18460,18462],{"className":18461},[242,243,244,245],[49,18463,533],{"className":18464},[93,245],[49,18466,18467,18470],{"style":1387},[49,18468],{"className":18469,"style":238},[237],[49,18471,18473],{"className":18472},[242,243,244,245],[49,18474,294],{"className":18475},[93,245],[49,18477,629],{"className":18478},[628],[49,18480,18482],{"className":18481},[226],[49,18483,18485],{"className":18484,"style":5461},[230],[49,18486],{},[49,18488],{"className":18489,"style":1843},[192],[49,18491,3213],{"className":18492},[1847],[49,18494],{"className":18495,"style":1843},[192],[49,18497,18499,18502,18505,18508,18511],{"className":18498},[84],[49,18500],{"className":18501,"style":3115},[88],[49,18503,17936],{"className":18504},[650],[49,18506],{"className":18507,"style":1843},[192],[49,18509,3213],{"className":18510},[1847],[49,18512],{"className":18513,"style":1843},[192],[49,18515,18517,18520,18571],{"className":18516},[84],[49,18518],{"className":18519,"style":18045},[88],[49,18521,18523,18526],{"className":18522},[93],[49,18524,9684],{"className":18525},[93,94],[49,18527,18529],{"className":18528},[218],[49,18530,18532,18563],{"className":18531},[222,605],[49,18533,18535,18560],{"className":18534},[226],[49,18536,18538,18549],{"className":18537,"style":2074},[230],[49,18539,18540,18543],{"style":5431},[49,18541],{"className":18542,"style":238},[237],[49,18544,18546],{"className":18545},[242,243,244,245],[49,18547,71],{"className":18548,"style":95},[93,94,245],[49,18550,18551,18554],{"style":1387},[49,18552],{"className":18553,"style":238},[237],[49,18555,18557],{"className":18556},[242,243,244,245],[49,18558,294],{"className":18559},[93,245],[49,18561,629],{"className":18562},[628],[49,18564,18566],{"className":18565},[226],[49,18567,18569],{"className":18568,"style":5461},[230],[49,18570],{},[49,18572,536],{"className":18573},[642],[11,18575,18576],{},"非退化局部极大值的标准形为",[49,18578,18580],{"className":18579,"translate":53},[752],[49,18581,18583,18629],{"className":18582,"translate":53},[52],[49,18584,18586],{"className":18585},[57],[59,18587,18588],{"xmlns":61,"display":761},[63,18589,18590,18626],{},[66,18591,18592,18594,18596,18598,18600,18602,18610,18612,18614,18616,18624],{},[69,18593,153],{},[155,18595,298],{"stretchy":297},[69,18597,11],{},[155,18599,307],{"stretchy":297},[155,18601,3077],{},[1910,18603,18604,18606,18608],{},[69,18605,9684],{},[292,18607,533],{},[292,18609,294],{},[155,18611,3077],{},[155,18613,17936],{},[155,18615,3077],{},[1910,18617,18618,18620,18622],{},[69,18619,9684],{},[69,18621,71],{},[292,18623,294],{},[155,18625,536],{"separator":80},[73,18627,18628],{"encoding":75},"f(p) - u_1^2 - \\dots - u_N^2,",[49,18630,18632,18659,18725,18743],{"className":18631,"ariaHidden":80},[79],[49,18633,18635,18638,18641,18644,18647,18650,18653,18656],{"className":18634},[84],[49,18636],{"className":18637,"style":589},[88],[49,18639,153],{"className":18640,"style":188},[93,94],[49,18642,298],{"className":18643},[373],[49,18645,11],{"className":18646},[93,94],[49,18648,307],{"className":18649},[406],[49,18651],{"className":18652,"style":1843},[192],[49,18654,3077],{"className":18655},[1847],[49,18657],{"className":18658,"style":1843},[192],[49,18660,18662,18665,18716,18719,18722],{"className":18661},[84],[49,18663],{"className":18664,"style":18045},[88],[49,18666,18668,18671],{"className":18667},[93],[49,18669,9684],{"className":18670},[93,94],[49,18672,18674],{"className":18673},[218],[49,18675,18677,18708],{"className":18676},[222,605],[49,18678,18680,18705],{"className":18679},[226],[49,18681,18683,18694],{"className":18682,"style":2074},[230],[49,18684,18685,18688],{"style":5431},[49,18686],{"className":18687,"style":238},[237],[49,18689,18691],{"className":18690},[242,243,244,245],[49,18692,533],{"className":18693},[93,245],[49,18695,18696,18699],{"style":1387},[49,18697],{"className":18698,"style":238},[237],[49,18700,18702],{"className":18701},[242,243,244,245],[49,18703,294],{"className":18704},[93,245],[49,18706,629],{"className":18707},[628],[49,18709,18711],{"className":18710},[226],[49,18712,18714],{"className":18713,"style":5461},[230],[49,18715],{},[49,18717],{"className":18718,"style":1843},[192],[49,18720,3077],{"className":18721},[1847],[49,18723],{"className":18724,"style":1843},[192],[49,18726,18728,18731,18734,18737,18740],{"className":18727},[84],[49,18729],{"className":18730,"style":3115},[88],[49,18732,17936],{"className":18733},[650],[49,18735],{"className":18736,"style":1843},[192],[49,18738,3077],{"className":18739},[1847],[49,18741],{"className":18742,"style":1843},[192],[49,18744,18746,18749,18800],{"className":18745},[84],[49,18747],{"className":18748,"style":18045},[88],[49,18750,18752,18755],{"className":18751},[93],[49,18753,9684],{"className":18754},[93,94],[49,18756,18758],{"className":18757},[218],[49,18759,18761,18792],{"className":18760},[222,605],[49,18762,18764,18789],{"className":18763},[226],[49,18765,18767,18778],{"className":18766,"style":2074},[230],[49,18768,18769,18772],{"style":5431},[49,18770],{"className":18771,"style":238},[237],[49,18773,18775],{"className":18774},[242,243,244,245],[49,18776,71],{"className":18777,"style":95},[93,94,245],[49,18779,18780,18783],{"style":1387},[49,18781],{"className":18782,"style":238},[237],[49,18784,18786],{"className":18785},[242,243,244,245],[49,18787,294],{"className":18788},[93,245],[49,18790,629],{"className":18791},[628],[49,18793,18795],{"className":18794},[226],[49,18796,18798],{"className":18797,"style":5461},[230],[49,18799],{},[49,18801,536],{"className":18802},[642],[11,18804,18805,18806,18834,18835,18863,18864,18915],{},"而指数为 ",[49,18807,18809,18822],{"className":18808,"translate":53},[52],[49,18810,18812],{"className":18811},[57],[59,18813,18814],{"xmlns":61},[63,18815,18816,18820],{},[66,18817,18818],{},[69,18819,14719],{},[73,18821,14719],{"encoding":75},[49,18823,18825],{"className":18824,"ariaHidden":80},[79],[49,18826,18828,18831],{"className":18827},[84],[49,18829],{"className":18830,"style":3096},[88],[49,18832,14719],{"className":18833,"style":14759},[93,94]," 的鞍点有 ",[49,18836,18838,18851],{"className":18837,"translate":53},[52],[49,18839,18841],{"className":18840},[57],[59,18842,18843],{"xmlns":61},[63,18844,18845,18849],{},[66,18846,18847],{},[69,18848,14719],{},[73,18850,14719],{"encoding":75},[49,18852,18854],{"className":18853,"ariaHidden":80},[79],[49,18855,18857,18860],{"className":18856},[84],[49,18858],{"className":18859,"style":3096},[88],[49,18861,14719],{"className":18862,"style":14759},[93,94]," 个向下的方向和 ",[49,18865,18867,18885],{"className":18866,"translate":53},[52],[49,18868,18870],{"className":18869},[57],[59,18871,18872],{"xmlns":61},[63,18873,18874,18882],{},[66,18875,18876,18878,18880],{},[69,18877,71],{},[155,18879,3077],{},[69,18881,14719],{},[73,18883,18884],{"encoding":75},"N-k",[49,18886,18888,18906],{"className":18887,"ariaHidden":80},[79],[49,18889,18891,18894,18897,18900,18903],{"className":18890},[84],[49,18892],{"className":18893,"style":1836},[88],[49,18895,71],{"className":18896,"style":95},[93,94],[49,18898],{"className":18899,"style":1843},[192],[49,18901,3077],{"className":18902},[1847],[49,18904],{"className":18905,"style":1843},[192],[49,18907,18909,18912],{"className":18908},[84],[49,18910],{"className":18911,"style":3096},[88],[49,18913,14719],{"className":18914,"style":14759},[93,94]," 个向上的方向。这正是「指数」成为高维临界点自然语言的原因。",[39,18917,18918],{"id":18918},"局限性",[11,18920,18921],{},"任何听起来很「美」的结论都需要打上补丁，这个也不例外。",[18,18923,18924,19096,19250,19256],{},[21,18925,18926,18929,18930,18988,18989,19095],{},[28,18927,18928],{},"不是所有函数都适用。"," ",[49,18931,18933,18950],{"className":18932,"translate":53},[52],[49,18934,18936],{"className":18935},[57],[59,18937,18938],{"xmlns":61},[63,18939,18940,18948],{},[66,18941,18942],{},[159,18943,18944,18946],{},[69,18945,164],{"mathvariant":163},[69,18947,71],{},[73,18949,7728],{"encoding":75},[49,18951,18953],{"className":18952,"ariaHidden":80},[79],[49,18954,18956,18959],{"className":18955},[84],[49,18957],{"className":18958,"style":207},[88],[49,18960,18962,18965],{"className":18961},[93],[49,18963,164],{"className":18964},[93,214],[49,18966,18968],{"className":18967},[218],[49,18969,18971],{"className":18970},[222],[49,18972,18974],{"className":18973},[226],[49,18975,18977],{"className":18976,"style":207},[230],[49,18978,18979,18982],{"style":233},[49,18980],{"className":18981,"style":238},[237],[49,18983,18985],{"className":18984},[242,243,244,245],[49,18986,71],{"className":18987,"style":95},[93,94,245]," 上的强凸函数，例如 ",[49,18990,18992,19024],{"className":18991,"translate":53},[52],[49,18993,18995],{"className":18994},[57],[59,18996,18997],{"xmlns":61},[63,18998,18999,19021],{},[66,19000,19001,19003,19005,19007,19009,19011,19013,19015],{},[69,19002,153],{},[155,19004,298],{"stretchy":297},[69,19006,520],{},[155,19008,307],{"stretchy":297},[155,19010,523],{},[69,19012,3296],{"mathvariant":768},[69,19014,520],{},[159,19016,19017,19019],{},[69,19018,3296],{"mathvariant":768},[292,19020,294],{},[73,19022,19023],{"encoding":75},"f(x)=\\|x\\|^2",[49,19025,19027,19054],{"className":19026,"ariaHidden":80},[79],[49,19028,19030,19033,19036,19039,19042,19045,19048,19051],{"className":19029},[84],[49,19031],{"className":19032,"style":589},[88],[49,19034,153],{"className":19035,"style":188},[93,94],[49,19037,298],{"className":19038},[373],[49,19040,520],{"className":19041},[93,94],[49,19043,307],{"className":19044},[406],[49,19046],{"className":19047,"style":193},[192],[49,19049,523],{"className":19050},[197],[49,19052],{"className":19053,"style":193},[192],[49,19055,19057,19060,19063,19066],{"className":19056},[84],[49,19058],{"className":19059,"style":12721},[88],[49,19061,3296],{"className":19062},[93],[49,19064,520],{"className":19065},[93,94],[49,19067,19069,19072],{"className":19068},[93],[49,19070,3296],{"className":19071},[93],[49,19073,19075],{"className":19074},[218],[49,19076,19078],{"className":19077},[222],[49,19079,19081],{"className":19080},[226],[49,19082,19084],{"className":19083,"style":358},[230],[49,19085,19086,19089],{"style":233},[49,19087],{"className":19088,"style":238},[237],[49,19090,19092],{"className":19091},[242,243,244,245],[49,19093,294],{"className":19094},[93,245],"，只有一个临界点，而且它是严格的全局极小值，没有鞍点。因此，仅仅高维并不意味着极小值稀缺——起决定作用的是 Hessian 有没有系统性地偏向正定。",[21,19097,19098,19101,19102,505,19198,19249],{},[28,19099,19100],{},"正定是严格极小值的充分条件，而非必要条件。"," 退化的严格极小值是存在的：一维函数 ",[49,19103,19105,19133],{"className":19104,"translate":53},[52],[49,19106,19108],{"className":19107},[57],[59,19109,19110],{"xmlns":61},[63,19111,19112,19130],{},[66,19113,19114,19116,19118,19120,19122,19124],{},[69,19115,153],{},[155,19117,298],{"stretchy":297},[69,19119,520],{},[155,19121,307],{"stretchy":297},[155,19123,523],{},[159,19125,19126,19128],{},[69,19127,520],{},[292,19129,12811],{},[73,19131,19132],{"encoding":75},"f(x)=x^4",[49,19134,19136,19163],{"className":19135,"ariaHidden":80},[79],[49,19137,19139,19142,19145,19148,19151,19154,19157,19160],{"className":19138},[84],[49,19140],{"className":19141,"style":589},[88],[49,19143,153],{"className":19144,"style":188},[93,94],[49,19146,298],{"className":19147},[373],[49,19149,520],{"className":19150},[93,94],[49,19152,307],{"className":19153},[406],[49,19155],{"className":19156,"style":193},[192],[49,19158,523],{"className":19159},[197],[49,19161],{"className":19162,"style":193},[192],[49,19164,19166,19169],{"className":19165},[84],[49,19167],{"className":19168,"style":358},[88],[49,19170,19172,19175],{"className":19171},[93],[49,19173,520],{"className":19174},[93,94],[49,19176,19178],{"className":19177},[218],[49,19179,19181],{"className":19180},[222],[49,19182,19184],{"className":19183},[226],[49,19185,19187],{"className":19186,"style":358},[230],[49,19188,19189,19192],{"style":233},[49,19190],{"className":19191,"style":238},[237],[49,19193,19195],{"className":19194},[242,243,244,245],[49,19196,12811],{"className":19197},[93,245],[49,19199,19201,19219],{"className":19200,"translate":53},[52],[49,19202,19204],{"className":19203},[57],[59,19205,19206],{"xmlns":61},[63,19207,19208,19216],{},[66,19209,19210,19212,19214],{},[69,19211,520],{},[155,19213,523],{},[292,19215,801],{},[73,19217,19218],{"encoding":75},"x=0",[49,19220,19222,19240],{"className":19221,"ariaHidden":80},[79],[49,19223,19225,19228,19231,19234,19237],{"className":19224},[84],[49,19226],{"className":19227,"style":570},[88],[49,19229,520],{"className":19230},[93,94],[49,19232],{"className":19233,"style":193},[192],[49,19235,523],{"className":19236},[197],[49,19238],{"className":19239,"style":193},[192],[49,19241,19243,19246],{"className":19242},[84],[49,19244],{"className":19245,"style":6838},[88],[49,19247,801],{"className":19248},[93]," 处有严格局部极小值，但其 Hessian 在那里为零。因此，Hessian 符号论证针对的是非退化的、即 Morse 的临界点。",[21,19251,19252,19255],{},[28,19253,19254],{},"结论依赖于随机景观系综。"," 中心化、类 GOE 的 Hessian 导致鞍点主导。Hessian 强烈向正方向平移的模型可以有许多极小值。在高斯场模型中，以极低函数值为条件，可以把 Hessian 谱向上平移，使极小值在景观底部附近更可能出现。",[21,19257,19258,19261,19262,19290],{},[28,19259,19260],{},"罕见并不意味着不存在。"," 即使极小值在所有临界点中的占比趋于零，在某些模型中极小值的绝对数目仍可能随 ",[49,19263,19265,19278],{"className":19264,"translate":53},[52],[49,19266,19268],{"className":19267},[57],[59,19269,19270],{"xmlns":61},[63,19271,19272,19276],{},[66,19273,19274],{},[69,19275,71],{},[73,19277,71],{"encoding":75},[49,19279,19281],{"className":19280,"ariaHidden":80},[79],[49,19282,19284,19287],{"className":19283},[84],[49,19285],{"className":19286,"style":89},[88],[49,19288,71],{"className":19289,"style":95},[93,94]," 增长。变得罕见的只是「随机选出的临界点是极小值」这一概率。",[11,19292,19293,19294,19297],{},"有了这些限定，可以更准确地表述为：",[28,19295,19296],{},"在许多 Hessian 谱近似无偏的高维随机景观中，一个典型的非退化临界点压倒性地倾向于成为鞍点。"," Morse 临界点中的局部极小值要求所有 Hessian 特征值为正，而这一正定事件随维度增长呈指数级罕见。",[39,19299,19300],{"id":19300},"优化实践的启发",[11,19302,19303],{},"这有助于解释为什么高维优化常被描述为「鞍点逃逸问题」，而非单纯的「坏局部极小值问题」。这一陈述不应被过度解读：它并不证明梯度下降总是有效，也不是说所有实际目标都表现得像 GOE 随机矩阵——真实的损失函数具有架构、数据、对称性、退化与约束，远比理想化模型复杂。",[11,19305,19306],{},"尽管如此，几何上的教训是有用的：",[18,19308,19309,19312,19315],{},[21,19310,19311],{},"高维中，绝大多数临界点是鞍点，其负特征值方向提供了逃离的「出口」（负曲率方向）。",[21,19313,19314],{},"因此，优化卡住的主因不是「掉进局部极小值的陷阱」，而是在高原或鞍点附近「停滞」。",[21,19316,19317],{},"算法如 SGD（随机梯度下降）自带的噪声，恰好能高效地把优化过程推离这些鞍点。而二阶算法（如牛顿法）反而需要谨慎处理负曲率方向，否则容易被带偏。",[11,19319,19320],{},"最后提醒一点：上述结论适用于随机、无结构化的景观。在深度学习中，由于对称性（如神经元置换）和平坦极小值的存在，景观结构更加复杂，但「鞍点主导」这一高维几何直觉依然成立。",[39,19322,19323],{"id":19323},"结论",[11,19325,19326,19354,19355,19383],{},[49,19327,19329,19342],{"className":19328,"translate":53},[52],[49,19330,19332],{"className":19331},[57],[59,19333,19334],{"xmlns":61},[63,19335,19336,19340],{},[66,19337,19338],{},[69,19339,71],{},[73,19341,71],{"encoding":75},[49,19343,19345],{"className":19344,"ariaHidden":80},[79],[49,19346,19348,19351],{"className":19347},[84],[49,19349],{"className":19350,"style":89},[88],[49,19352,71],{"className":19353,"style":95},[93,94]," 维中的非退化局部极小值必须通过 ",[49,19356,19358,19371],{"className":19357,"translate":53},[52],[49,19359,19361],{"className":19360},[57],[59,19362,19363],{"xmlns":61},[63,19364,19365,19369],{},[66,19366,19367],{},[69,19368,71],{},[73,19370,71],{"encoding":75},[49,19372,19374],{"className":19373,"ariaHidden":80},[79],[49,19375,19377,19380],{"className":19376},[84],[49,19378],{"className":19379,"style":89},[88],[49,19381,71],{"className":19382,"style":95},[93,94]," 个同时成立的曲率检验——在临界点处，每个 Hessian 特征值都必须为正。鞍点只需要正负号的混合。随机矩阵理论表明，大型中心化对称矩阵天然具有这样的混合：它们的谱散布在零附近，有大量正、负特征值。「所有特征值都落在正侧」这一事件是一个大偏差。",[11,19385,19386,19387,19415,19416,19444],{},"这解释了高维的直觉。临界点本身是光滑景观中体积为零的特征，但在临界点之中，决定性的问题是 Hessian 的符号。在随机模型中，典型的符号其指数靠近中间，而不是极端：非退化局部极小值指数为 0，非退化局部极大值指数为 ",[49,19388,19390,19403],{"className":19389,"translate":53},[52],[49,19391,19393],{"className":19392},[57],[59,19394,19395],{"xmlns":61},[63,19396,19397,19401],{},[66,19398,19399],{},[69,19400,71],{},[73,19402,71],{"encoding":75},[49,19404,19406],{"className":19405,"ariaHidden":80},[79],[49,19407,19409,19412],{"className":19408},[84],[49,19410],{"className":19411,"style":89},[88],[49,19413,71],{"className":19414,"style":95},[93,94],"，鞍点占据两者之间的所有指数。随着 ",[49,19417,19419,19432],{"className":19418,"translate":53},[52],[49,19420,19422],{"className":19421},[57],[59,19423,19424],{"xmlns":61},[63,19425,19426,19430],{},[66,19427,19428],{},[69,19429,71],{},[73,19431,71],{"encoding":75},[49,19433,19435],{"className":19434,"ariaHidden":80},[79],[49,19436,19438,19441],{"className":19437},[84],[49,19439],{"className":19440,"style":89},[88],[49,19442,71],{"className":19443,"style":95},[93,94]," 增长，中间压倒了极端。",[11,19446,19447],{},"因此，最好把这一观点理解为一条稳健的概率原则，而非关于每个高维函数的定理：",[469,19449,19450],{},[11,19451,19452],{},"在高维随机景观中，指数为零的 Morse 临界点是例外，而鞍点才是默认。",[11,19454,19455],{},"这正是深度学习等现代优化能够成功的理论基石之一：绝大多数临界点是鞍点，其负曲率方向提供了逃离的出口。带噪声的一阶方法能高效地把优化过程推开这些鞍点。",[19457,19458,19459],"h4",{"id":19459},"参考",[11,19461,19462,19463,19469],{},"参考来源：",[4867,19464,19468],{"href":19465,"rel":19466},"https:\u002F\u002Fwww.grantstenger.com\u002Flocal-minima",[19467],"nofollow","《Why Local Minima Are Rare in High-Dimensional Landscapes》","，Grant Stenger.",{"title":19471,"searchDepth":19472,"depth":19472,"links":19473},"",2,[19474,19475,19476,19477,19478,19484,19485,19488,19489,19490],{"id":41,"depth":19472,"text":41},{"id":134,"depth":19472,"text":135},{"id":7190,"depth":19472,"text":7190},{"id":8621,"depth":19472,"text":8621},{"id":11181,"depth":19472,"text":11181,"children":19479},[19480,19482,19483],{"id":11191,"depth":19481,"text":11191},3,{"id":11576,"depth":19481,"text":11577},{"id":13148,"depth":19481,"text":13148},{"id":14577,"depth":19472,"text":14578},{"id":16531,"depth":19472,"text":16532,"children":19486},[19487],{"id":17539,"depth":19481,"text":17540},{"id":18918,"depth":19472,"text":18918},{"id":19300,"depth":19472,"text":19300},{"id":19323,"depth":19472,"text":19323},false,true,"md",null,{},"2026-08-22","\u002Fblog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions",{"title":6,"description":13},"blog\u002F2026\u002F2026-08-22-are-local-minima-rare-in-high-dimensions","高维空间中的局部极小值极为罕见——因为一个真正的局部极小值要求所有方向都向上弯曲，也就是 Hessian 矩阵的所有特征值都必须为正。而高维空间里，随机的临界点几乎总有某个方向可以向下走，也就是鞍点。这正是为什么神经网络训练更像是「逃离鞍点」的游戏，而不是「跳出局部最优」的游戏。",[19502],"machine-learning","th-V2v3CHy_yZh0Uqf-q8gOGr1caUKcukdZ3CZuUCbw",{"id":19505,"title":19506,"body":19507,"description":19511,"draft":19491,"enableComment":19492,"extension":19493,"image":19471,"important":19491,"location":19494,"meta":24602,"navigation":19492,"ogImage":19494,"onday":24603,"path":24604,"seo":24605,"stem":24606,"summary":24607,"tags":24608,"__hash__":24609},"blog\u002Fblog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao.md","研究关系的三种视角",{"type":8,"value":19508,"toc":24594},[19509,19512,19515,19518,19522,19525,19733,20387,20578,20582,20585,20971,20975,20978,21446,21717,22525,22528,22531,22591,22936,23031,23305,23308,23311,23314,23423,23808,23811,23814,23887,23890,24588,24591],[11,19510,19511],{},"哲学里有一种命题：「存在即关系」，在数理研究中同样也有很深的体现。听起来很抽象，但它其实是对世界本质是什么的一种深刻回答。理解事物不是先存在、再去建立关系，而是事物在建立关系的过程中，才得以成为事物。它最著名的提出者是 20 世纪法国哲学家怀特海（过程哲学），但其思想渊源可以追溯到德国哲学家莱布尼茨（单子论）和古希腊的赫拉克利特（万物流动）。",[11,19513,19514],{},"所以，研究「存在」也是研究「关系」。而关系通常有三种视角：几何、拓扑和因果。",[11,19516,19517],{},"拓扑提供稳健的骨架，几何赋予精细的尺度，因果指明干预的方向。三者共同构成智能系统理解世界的先验地基。更准确的说法或许是：几何和拓扑是关于「关系的静态结构」（分别是度量化的和非度量化的），而因果是关于「关系的动态\u002F生成机制」，它回答的不是「这两者如何关联」，而是「如果我改变一个，另一个会怎样」。",[39,19519,19521],{"id":19520},"几何度量结构","几何（度量结构）",[11,19523,19524],{},"几何关注的是「测量」。基本问题是多接近、多相似，或者一个具体的度量数值。——它需要一个度量结构（距离、内积、相似度函数），通常要求满足对称性、三角不等式等公理。典型例子：嵌入空间里两个向量的余弦相似度、社会心理学里的「亲密度量表」。研究的是关系的强度。几何关系是最重的——信息量最大，但也最脆弱，度量方式一变，结论可能就变。",[11,19526,19527,19528,19582,19583,19732],{},"一个度量空间是二元组 ",[49,19529,19531,19554],{"className":19530,"translate":53},[52],[49,19532,19534],{"className":19533},[57],[59,19535,19536],{"xmlns":61},[63,19537,19538,19551],{},[66,19539,19540,19542,19545,19547,19549],{},[155,19541,298],{"stretchy":297},[69,19543,19544],{},"X",[155,19546,536],{"separator":80},[69,19548,15334],{},[155,19550,307],{"stretchy":297},[73,19552,19553],{"encoding":75},"(X, d)",[49,19555,19557],{"className":19556,"ariaHidden":80},[79],[49,19558,19560,19563,19566,19570,19573,19576,19579],{"className":19559},[84],[49,19561],{"className":19562,"style":589},[88],[49,19564,298],{"className":19565},[373],[49,19567,19544],{"className":19568,"style":19569},[93,94],"margin-right:0.0785em;",[49,19571,536],{"className":19572},[642],[49,19574],{"className":19575,"style":646},[192],[49,19577,15334],{"className":19578},[93,94],[49,19580,307],{"className":19581},[406],"，其中 ",[49,19584,19586,19621],{"className":19585,"translate":53},[52],[49,19587,19589],{"className":19588},[57],[59,19590,19591],{"xmlns":61},[63,19592,19593,19618],{},[66,19594,19595,19597,19599,19601,19603,19605,19607],{},[69,19596,15334],{},[155,19598,157],{},[69,19600,19544],{},[155,19602,1821],{},[69,19604,19544],{},[155,19606,169],{},[527,19608,19609,19611],{},[69,19610,164],{"mathvariant":163},[66,19612,19613,19616],{},[155,19614,19615],{},"≥",[292,19617,801],{},[73,19619,19620],{"encoding":75},"d: 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满足：",[49,19734,19736],{"className":19735,"translate":53},[752],[49,19737,19739,19916],{"className":19738,"translate":53},[52],[49,19740,19742],{"className":19741},[57],[59,19743,19744],{"xmlns":61,"display":761},[63,19745,19746,19913],{},[66,19747,19748,19750],{},[155,19749,6568],{"fence":80},[787,19751,19754],{"rowspacing":19752,"columnalign":19753,"columnspacing":791},"0.36em","left left",[793,19755,19756],{},[796,19757,19758],{},[799,19759,19760],{"scriptlevel":801,"displaystyle":297},[787,19761,19764,19812,19856],{"rowspacing":19762,"columnalign":19763,"columnspacing":846},"0.25em","right 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ML 里，我们常用的是更强的结构：内积空间 ",[49,20391,20393,20425],{"className":20392,"translate":53},[52],[49,20394,20396],{"className":20395},[57],[59,20397,20398],{"xmlns":61},[63,20399,20400,20422],{},[66,20401,20402,20404,20406,20408,20411,20413,20415,20417,20420],{},[155,20403,298],{"stretchy":297},[69,20405,19544],{},[155,20407,536],{"separator":80},[155,20409,20410],{"stretchy":297},"⟨",[155,20412,10425],{},[155,20414,536],{"separator":80},[155,20416,10425],{},[155,20418,20419],{"stretchy":297},"⟩",[155,20421,307],{"stretchy":297},[73,20423,20424],{"encoding":75},"(X, \\langle \\cdot,\\cdot\\rangle)",[49,20426,20428],{"className":20427,"ariaHidden":80},[79],[49,20429,20431,20434,20437,20440,20443,20446,20449,20452,20455,20458,20461],{"className":20430},[84],[49,20432],{"className":20433,"style":589},[88],[49,20435,298],{"className":20436},[373],[49,20438,19544],{"className":20439,"style":19569},[93,94],[49,20441,536],{"className":20442},[642],[49,20444],{"className":20445,"style":646},[192],[49,20447,20410],{"className":20448},[373],[49,20450,10425],{"className":20451},[93],[49,20453,536],{"className":20454},[642],[49,20456],{"className":20457,"style":646},[192],[49,20459,10425],{"className":20460},[93],[49,20462,20464],{"className":20463},[406],"⟩)","，度量由内积诱导 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\\|x-y\\|",[49,20506,20508,20544,20565],{"className":20507,"ariaHidden":80},[79],[49,20509,20511,20514,20517,20520,20523,20526,20529,20532,20535,20538,20541],{"className":20510},[84],[49,20512],{"className":20513,"style":589},[88],[49,20515,15334],{"className":20516},[93,94],[49,20518,298],{"className":20519},[373],[49,20521,520],{"className":20522},[93,94],[49,20524,536],{"className":20525},[642],[49,20527],{"className":20528,"style":646},[192],[49,20530,8650],{"className":20531,"style":4591},[93,94],[49,20533,307],{"className":20534},[406],[49,20536],{"className":20537,"style":193},[192],[49,20539,523],{"className":20540},[197],[49,20542],{"className":20543,"style":193},[192],[49,20545,20547,20550,20553,20556,20559,20562],{"className":20546},[84],[49,20548],{"className":20549,"style":589},[88],[49,20551,3296],{"className":20552},[93],[49,20554,520],{"className":20555},[93,94],[49,20557],{"className":20558,"style":1843},[192],[49,20560,3077],{"className":20561},[1847],[49,20563],{"className":20564,"style":1843},[192],[49,20566,20568,20571,20574],{"className":20567},[84],[49,20569],{"className":20570,"style":589},[88],[49,20572,8650],{"className":20573,"style":4591},[93,94],[49,20575,3296],{"className":20576},[93],"，这才有了余弦相似度、点积注意力这些操作。度量结构的信息量最大，因为它不仅告诉你「是否有关系」，还告诉你「关系有多强」——但代价是它对表示方式极度敏感：换一个嵌入模型、换一个归一化方式，同一批数据算出来的相似度可以完全不同。",[39,20579,20581],{"id":20580},"拓扑连通结构","拓扑（连通结构）",[11,20583,20584],{},"拓扑关注的是「连通」。把度量忘掉，只问是否有路径可达、结构是否连通、是否存在环\u002F洞。在同胚变换（连续拉伸、不撕裂）下不变的东西才算拓扑性质。典型例子：社交网络里的连通分量、图的最短路径存在性（而非具体长度）。拓扑关系比几何轻，研究的是关系的骨架，但换来的是更强的稳健性——不在乎具体尺度，只在乎结构骨架。",[11,20586,20587,20588,20642,20643,8478,20672,20700,20701,20767,20768,20771,20772,20843,20844,20872,20873,20970],{},"一个拓扑空间是二元组 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的一族「开集」，满足对任意并、有限交封闭，且 ",[49,20702,20704,20727],{"className":20703,"translate":53},[52],[49,20705,20707],{"className":20706},[57],[59,20708,20709],{"xmlns":61},[63,20710,20711,20724],{},[66,20712,20713,20716,20718,20720,20722],{},[69,20714,20715],{"mathvariant":768},"∅",[155,20717,536],{"separator":80},[69,20719,19544],{},[155,20721,285],{},[69,20723,20608],{},[73,20725,20726],{"encoding":75},"\\emptyset, X \\in \\tau",[49,20728,20730,20758],{"className":20729,"ariaHidden":80},[79],[49,20731,20733,20737,20740,20743,20746,20749,20752,20755],{"className":20732},[84],[49,20734],{"className":20735,"style":20736},[88],"height:0.9444em;vertical-align:-0.1944em;",[49,20738,20715],{"className":20739},[93],[49,20741,536],{"className":20742},[642],[49,20744],{"className":20745,"style":646},[192],[49,20747,19544],{"className":20748,"style":19569},[93,94],[49,20750],{"className":20751,"style":193},[192],[49,20753,285],{"className":20754},[197],[49,20756],{"className":20757,"style":193},[192],[49,20759,20761,20764],{"className":20760},[84],[49,20762],{"className":20763,"style":570},[88],[49,20765,20608],{"className":20766,"style":20638},[93,94],"。拓扑结构不需要度量：任何度量空间都能诱导出一个拓扑（用开球生成开集），但反过来不成立——拓扑空间不一定可度量化。关键的不变量是在",[28,20769,20770],{},"同胚","（连续双射且逆映射也连续）下保持不变的性质，比如连通性、紧致性、亏格（洞的个数）。用代数拓扑的语言，贝蒂数 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数的是 ",[49,20845,20847,20860],{"className":20846,"translate":53},[52],[49,20848,20850],{"className":20849},[57],[59,20851,20852],{"xmlns":61},[63,20853,20854,20858],{},[66,20855,20856],{},[69,20857,14719],{},[73,20859,14719],{"encoding":75},[49,20861,20863],{"className":20862,"ariaHidden":80},[79],[49,20864,20866,20869],{"className":20865},[84],[49,20867],{"className":20868,"style":3096},[88],[49,20870,14719],{"className":20871,"style":14759},[93,94]," 维「洞」的个数，这是持久同调（persistent homology）在数据分析里常用的工具——它告诉你数据云的「形状骨架」，而完全不关心具体的尺度参数。这解释了为什么拓扑比几何「轻」：它是几何结构经过遗忘函子 ",[49,20874,20876,20912],{"className":20875,"translate":53},[52],[49,20877,20879],{"className":20878},[57],[59,20880,20881],{"xmlns":61},[63,20882,20883,20909],{},[66,20884,20885,20888,20890,20899,20901],{},[69,20886,20887],{},"U",[155,20889,157],{},[66,20891,20892,20894,20897],{},[69,20893,16556],{"mathvariant":15255},[69,20895,20896],{"mathvariant":15255},"e",[69,20898,14115],{"mathvariant":15255},[155,20900,169],{},[66,20902,20903,20905,20907],{},[69,20904,3252],{"mathvariant":15255},[69,20906,3291],{"mathvariant":15255},[69,20908,11],{"mathvariant":15255},[73,20910,20911],{"encoding":75},"U: \\mathbf{Met} \\to \\mathbf{Top}",[49,20913,20915,20933,20956],{"className":20914,"ariaHidden":80},[79],[49,20916,20918,20921,20924,20927,20930],{"className":20917},[84],[49,20919],{"className":20920,"style":89},[88],[49,20922,20887],{"className":20923,"style":95},[93,94],[49,20925],{"className":20926,"style":193},[192],[49,20928,157],{"className":20929},[197],[49,20931],{"className":20932,"style":193},[192],[49,20934,20936,20940,20947,20950,20953],{"className":20935},[84],[49,20937],{"className":20938,"style":20939},[88],"height:0.6861em;",[49,20941,20943],{"className":20942},[93],[49,20944,20946],{"className":20945},[93,15556],"Met",[49,20948],{"className":20949,"style":193},[192],[49,20951,169],{"className":20952},[197],[49,20954],{"className":20955,"style":193},[192],[49,20957,20959,20963],{"className":20958},[84],[49,20960],{"className":20961,"style":20962},[88],"height:0.8805em;vertical-align:-0.1944em;",[49,20964,20966],{"className":20965},[93],[49,20967,20969],{"className":20968},[93,15556],"Top"," 投影之后剩下的东西。",[39,20972,20974],{"id":20973},"因果约束结构","因果（约束结构）",[11,20976,20977],{},"因果关注的是「约束」。这一支和前两支有本质区别：几何和拓扑通常是对称的（A 到 B 的距离=B 到 A 的距离；A 连着 B 等价于 B 连着 A），而因果关系本质上是非对称的——「X 决定\u002F限制 Y 的取值范围」不等于反过来成立。因果结构需要额外的信息才能识别（干预、反事实、时间先后），单纯的相关性数据（哪怕是几何或拓扑意义上完整的）不足以确定因果方向。这也是 Pearl 那套 do-calculus 要解决的问题。在 ML 中，几何\u002F拓扑模型在独立同分布（i.i.d.）下表现很好，但只要测试分布变了（协变量偏移），它们就会崩。而因果结构（比如有向无环图 DAG）告诉你的是「机制」——只要干预机制不变，哪怕输入分布变了，模型依然能泛化。这也是为什么说因果是稳健性的终极来源。",[11,20979,20980,20981,19582,21070,21098,21099,21272,21273,21445],{},"结构因果模型（SCM）是一个三元组 ",[49,20982,20984,21016],{"className":20983,"translate":53},[52],[49,20985,20987],{"className":20986},[57],[59,20988,20989],{"xmlns":61},[63,20990,20991,21013],{},[66,20992,20993,20995,20997,20999,21001,21003,21006,21008,21011],{},[69,20994,16556],{},[155,20996,523],{},[155,20998,20410],{"stretchy":297},[69,21000,20887],{},[155,21002,536],{"separator":80},[69,21004,21005],{},"V",[155,21007,536],{"separator":80},[69,21009,21010],{},"F",[155,21012,20419],{"stretchy":297},[73,21014,21015],{"encoding":75},"M = \\langle U, V, F \\rangle",[49,21017,21019,21037],{"className":21018,"ariaHidden":80},[79],[49,21020,21022,21025,21028,21031,21034],{"className":21021},[84],[49,21023],{"className":21024,"style":89},[88],[49,21026,16556],{"className":21027,"style":95},[93,94],[49,21029],{"className":21030,"style":193},[192],[49,21032,523],{"className":21033},[197],[49,21035],{"className":21036,"style":193},[192],[49,21038,21040,21043,21046,21049,21052,21055,21058,21061,21064,21067],{"className":21039},[84],[49,21041],{"className":21042,"style":589},[88],[49,21044,20410],{"className":21045},[373],[49,21047,20887],{"className":21048,"style":95},[93,94],[49,21050,536],{"className":21051},[642],[49,21053],{"className":21054,"style":646},[192],[49,21056,21005],{"className":21057,"style":1843},[93,94],[49,21059,536],{"className":21060},[642],[49,21062],{"className":21063,"style":646},[192],[49,21065,21010],{"className":21066,"style":3554},[93,94],[49,21068,20419],{"className":21069},[406],[49,21071,21073,21086],{"className":21072,"translate":53},[52],[49,21074,21076],{"className":21075},[57],[59,21077,21078],{"xmlns":61},[63,21079,21080,21084],{},[66,21081,21082],{},[69,21083,20887],{},[73,21085,20887],{"encoding":75},[49,21087,21089],{"className":21088,"ariaHidden":80},[79],[49,21090,21092,21095],{"className":21091},[84],[49,21093],{"className":21094,"style":89},[88],[49,21096,20887],{"className":21097,"style":95},[93,94]," 是外生变量，",[49,21100,21102,21140],{"className":21101,"translate":53},[52],[49,21103,21105],{"className":21104},[57],[59,21106,21107],{"xmlns":61},[63,21108,21109,21137],{},[66,21110,21111,21113,21115,21117,21123,21125,21127,21129,21135],{},[69,21112,21005],{},[155,21114,523],{},[155,21116,6568],{"stretchy":297},[527,21118,21119,21121],{},[69,21120,21005],{},[292,21122,533],{},[155,21124,536],{"separator":80},[155,21126,539],{},[155,21128,536],{"separator":80},[527,21130,21131,21133],{},[69,21132,21005],{},[69,21134,11950],{},[155,21136,6586],{"stretchy":297},[73,21138,21139],{"encoding":75},"V = \\{V_1, \\dots, V_n\\}",[49,21141,21143,21161],{"className":21142,"ariaHidden":80},[79],[49,21144,21146,21149,21152,21155,21158],{"className":21145},[84],[49,21147],{"className":21148,"style":89},[88],[49,21150,21005],{"className":21151,"style":1843},[93,94],[49,21153],{"className":21154,"style":193},[192],[49,21156,523],{"className":21157},[197],[49,21159],{"className":21160,"style":193},[192],[49,21162,21164,21167,21170,21211,21214,21217,21220,21223,21226,21229,21269],{"className":21163},[84],[49,21165],{"className":21166,"style":589},[88],[49,21168,6568],{"className":21169},[373],[49,21171,21173,21176],{"className":21172},[93],[49,21174,21005],{"className":21175,"style":1843},[93,94],[49,21177,21179],{"className":21178},[218],[49,21180,21182,21203],{"className":21181},[222,605],[49,21183,21185,21200],{"className":21184},[226],[49,21186,21188],{"className":21187,"style":612},[230],[49,21189,21191,21194],{"style":21190},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[49,21192],{"className":21193,"style":238},[237],[49,21195,21197],{"className":21196},[242,243,244,245],[49,21198,533],{"className":21199},[93,245],[49,21201,629],{"className":21202},[628],[49,21204,21206],{"className":21205},[226],[49,21207,21209],{"className":21208,"style":636},[230],[49,21210],{},[49,21212,536],{"className":21213},[642],[49,21215],{"className":21216,"style":646},[192],[49,21218,539],{"className":21219},[650],[49,21221],{"className":21222,"style":646},[192],[49,21224,536],{"className":21225},[642],[49,21227],{"className":21228,"style":646},[192],[49,21230,21232,21235],{"className":21231},[93],[49,21233,21005],{"className":21234,"style":1843},[93,94],[49,21236,21238],{"className":21237},[218],[49,21239,21241,21261],{"className":21240},[222,605],[49,21242,21244,21258],{"className":21243},[226],[49,21245,21247],{"className":21246,"style":1466},[230],[49,21248,21249,21252],{"style":21190},[49,21250],{"className":21251,"style":238},[237],[49,21253,21255],{"className":21254},[242,243,244,245],[49,21256,11950],{"className":21257},[93,94,245],[49,21259,629],{"className":21260},[628],[49,21262,21264],{"className":21263},[226],[49,21265,21267],{"className":21266,"style":636},[230],[49,21268],{},[49,21270,6586],{"className":21271},[406]," 是内生变量，",[49,21274,21276,21314],{"className":21275,"translate":53},[52],[49,21277,21279],{"className":21278},[57],[59,21280,21281],{"xmlns":61},[63,21282,21283,21311],{},[66,21284,21285,21287,21289,21291,21297,21299,21301,21303,21309],{},[69,21286,21010],{},[155,21288,523],{},[155,21290,6568],{"stretchy":297},[527,21292,21293,21295],{},[69,21294,153],{},[292,21296,533],{},[155,21298,536],{"separator":80},[155,21300,539],{},[155,21302,536],{"separator":80},[527,21304,21305,21307],{},[69,21306,153],{},[69,21308,11950],{},[155,21310,6586],{"stretchy":297},[73,21312,21313],{"encoding":75},"F = \\{f_1, \\dots, f_n\\}",[49,21315,21317,21335],{"className":21316,"ariaHidden":80},[79],[49,21318,21320,21323,21326,21329,21332],{"className":21319},[84],[49,21321],{"className":21322,"style":89},[88],[49,21324,21010],{"className":21325,"style":3554},[93,94],[49,21327],{"className":21328,"style":193},[192],[49,21330,523],{"className":21331},[197],[49,21333],{"className":21334,"style":193},[192],[49,21336,21338,21341,21344,21384,21387,21390,21393,21396,21399,21402,21442],{"className":21337},[84],[49,21339],{"className":21340,"style":589},[88],[49,21342,6568],{"className":21343},[373],[49,21345,21347,21350],{"className":21346},[93],[49,21348,153],{"className":21349,"style":188},[93,94],[49,21351,21353],{"className":21352},[218],[49,21354,21356,21376],{"className":21355},[222,605],[49,21357,21359,21373],{"className":21358},[226],[49,21360,21362],{"className":21361,"style":612},[230],[49,21363,21364,21367],{"style":9478},[49,21365],{"className":21366,"style":238},[237],[49,21368,21370],{"className":21369},[242,243,244,245],[49,21371,533],{"className":21372},[93,245],[49,21374,629],{"className":21375},[628],[49,21377,21379],{"className":21378},[226],[49,21380,21382],{"className":21381,"style":636},[230],[49,21383],{},[49,21385,536],{"className":21386},[642],[49,21388],{"className":21389,"style":646},[192],[49,21391,539],{"className":21392},[650],[49,21394],{"className":21395,"style":646},[192],[49,21397,536],{"className":21398},[642],[49,21400],{"className":21401,"style":646},[192],[49,21403,21405,21408],{"className":21404},[93],[49,21406,153],{"className":21407,"style":188},[93,94],[49,21409,21411],{"className":21410},[218],[49,21412,21414,21434],{"className":21413},[222,605],[49,21415,21417,21431],{"className":21416},[226],[49,21418,21420],{"className":21419,"style":1466},[230],[49,21421,21422,21425],{"style":9478},[49,21423],{"className":21424,"style":238},[237],[49,21426,21428],{"className":21427},[242,243,244,245],[49,21429,11950],{"className":21430},[93,94,245],[49,21432,629],{"className":21433},[628],[49,21435,21437],{"className":21436},[226],[49,21438,21440],{"className":21439,"style":636},[230],[49,21441],{},[49,21443,6586],{"className":21444},[406]," 是一组函数，满足",[49,21447,21449],{"className":21448,"translate":53},[752],[49,21450,21452,21508],{"className":21451,"translate":53},[52],[49,21453,21455],{"className":21454},[57],[59,21456,21457],{"xmlns":61,"display":761},[63,21458,21459,21505],{},[66,21460,21461,21467,21469,21471,21477,21479,21485,21487,21493,21495,21497,21503],{},[527,21462,21463,21465],{},[69,21464,21005],{},[69,21466,1939],{},[155,21468,157],{},[155,21470,523],{},[527,21472,21473,21475],{},[69,21474,153],{},[69,21476,1939],{},[155,21478,298],{"stretchy":297},[66,21480,21481,21483],{},[69,21482,11247],{"mathvariant":768},[69,21484,4867],{"mathvariant":768},[155,21486,298],{"stretchy":297},[527,21488,21489,21491],{},[69,21490,21005],{},[69,21492,1939],{},[155,21494,307],{"stretchy":297},[155,21496,536],{"separator":80},[527,21498,21499,21501],{},[69,21500,20887],{},[69,21502,1939],{},[155,21504,307],{"stretchy":297},[73,21506,21507],{"encoding":75},"V_i := f_i(\\mathrm{Pa}(V_i), U_i)",[49,21509,21511,21566],{"className":21510,"ariaHidden":80},[79],[49,21512,21514,21517,21557,21560,21563],{"className":21513},[84],[49,21515],{"className":21516,"style":12029},[88],[49,21518,21520,21523],{"className":21519},[93],[49,21521,21005],{"className":21522,"style":1843},[93,94],[49,21524,21526],{"className":21525},[218],[49,21527,21529,21549],{"className":21528},[222,605],[49,21530,21532,21546],{"className":21531},[226],[49,21533,21535],{"className":21534,"style":2176},[230],[49,21536,21537,21540],{"style":21190},[49,21538],{"className":21539,"style":238},[237],[49,21541,21543],{"className":21542},[242,243,244,245],[49,21544,1939],{"className":21545},[93,94,245],[49,21547,629],{"className":21548},[628],[49,21550,21552],{"className":21551},[226],[49,21553,21555],{"className":21554,"style":636},[230],[49,21556],{},[49,21558],{"className":21559,"style":193},[192],[49,21561,931],{"className":21562},[197],[49,21564],{"className":21565,"style":193},[192],[49,21567,21569,21572,21612,21615,21622,21625,21665,21668,21671,21674,21714],{"className":21568},[84],[49,21570],{"className":21571,"style":589},[88],[49,21573,21575,21578],{"className":21574},[93],[49,21576,153],{"className":21577,"style":188},[93,94],[49,21579,21581],{"className":21580},[218],[49,21582,21584,21604],{"className":21583},[222,605],[49,21585,21587,21601],{"className":21586},[226],[49,21588,21590],{"className":21589,"style":2176},[230],[49,21591,21592,21595],{"style":9478},[49,21593],{"className":21594,"style":238},[237],[49,21596,21598],{"className":21597},[242,243,244,245],[49,21599,1939],{"className":21600},[93,94,245],[49,21602,629],{"className":21603},[628],[49,21605,21607],{"className":21606},[226],[49,21608,21610],{"className":21609,"style":636},[230],[49,21611],{},[49,21613,298],{"className":21614},[373],[49,21616,21618],{"className":21617},[93],[49,21619,21621],{"className":21620},[93,6607],"Pa",[49,21623,298],{"className":21624},[373],[49,21626,21628,21631],{"className":21627},[93],[49,21629,21005],{"className":21630,"style":1843},[93,94],[49,21632,21634],{"className":21633},[218],[49,21635,21637,21657],{"className":21636},[222,605],[49,21638,21640,21654],{"className":21639},[226],[49,21641,21643],{"className":21642,"style":2176},[230],[49,21644,21645,21648],{"style":21190},[49,21646],{"className":21647,"style":238},[237],[49,21649,21651],{"className":21650},[242,243,244,245],[49,21652,1939],{"className":21653},[93,94,245],[49,21655,629],{"className":21656},[628],[49,21658,21660],{"className":21659},[226],[49,21661,21663],{"className":21662,"style":636},[230],[49,21664],{},[49,21666,307],{"className":21667},[406],[49,21669,536],{"className":21670},[642],[49,21672],{"className":21673,"style":646},[192],[49,21675,21677,21680],{"className":21676},[93],[49,21678,20887],{"className":21679,"style":95},[93,94],[49,21681,21683],{"className":21682},[218],[49,21684,21686,21706],{"className":21685},[222,605],[49,21687,21689,21703],{"className":21688},[226],[49,21690,21692],{"className":21691,"style":2176},[230],[49,21693,21694,21697],{"style":16752},[49,21695],{"className":21696,"style":238},[237],[49,21698,21700],{"className":21699},[242,243,244,245],[49,21701,1939],{"className":21702},[93,94,245],[49,21704,629],{"className":21705},[628],[49,21707,21709],{"className":21708},[226],[49,21710,21712],{"className":21711,"style":636},[230],[49,21713],{},[49,21715,307],{"className":21716},[406],[11,21718,21719,21720,21750,21751,21780,21781,21784,21785,21814,21815,21949,21950,22105,22106,22158,22159,22267,22268,22296,22297,22325,22326,22354,22355,22443,22444,22495,22496,22524],{},"赋值符号 ",[49,21721,21723,21738],{"className":21722,"translate":53},[52],[49,21724,21726],{"className":21725},[57],[59,21727,21728],{"xmlns":61},[63,21729,21730,21736],{},[66,21731,21732,21734],{},[155,21733,157],{},[155,21735,523],{},[73,21737,931],{"encoding":75},[49,21739,21741],{"className":21740,"ariaHidden":80},[79],[49,21742,21744,21747],{"className":21743},[84],[49,21745],{"className":21746,"style":570},[88],[49,21748,931],{"className":21749},[197],"（而非 ",[49,21752,21754,21767],{"className":21753,"translate":53},[52],[49,21755,21757],{"className":21756},[57],[59,21758,21759],{"xmlns":61},[63,21760,21761,21765],{},[66,21762,21763],{},[155,21764,523],{},[73,21766,523],{"encoding":75},[49,21768,21770],{"className":21769,"ariaHidden":80},[79],[49,21771,21773,21777],{"className":21772},[84],[49,21774],{"className":21775,"style":21776},[88],"height:0.3669em;",[49,21778,523],{"className":21779},[197],"）是关键——它表示这是一个",[28,21782,21783],{},"机制","，不是一个可逆的代数等式。这组机制天然诱导一张有向无环图（DAG）",[49,21786,21788,21802],{"className":21787,"translate":53},[52],[49,21789,21791],{"className":21790},[57],[59,21792,21793],{"xmlns":61},[63,21794,21795,21800],{},[66,21796,21797],{},[69,21798,21799],{},"G",[73,21801,21799],{"encoding":75},[49,21803,21805],{"className":21804,"ariaHidden":80},[79],[49,21806,21808,21811],{"className":21807},[84],[49,21809],{"className":21810,"style":89},[88],[49,21812,21799],{"className":21813},[93,94],"，边 ",[49,21816,21818,21844],{"className":21817,"translate":53},[52],[49,21819,21821],{"className":21820},[57],[59,21822,21823],{"xmlns":61},[63,21824,21825,21841],{},[66,21826,21827,21833,21835],{},[527,21828,21829,21831],{},[69,21830,21005],{},[69,21832,1948],{},[155,21834,169],{},[527,21836,21837,21839],{},[69,21838,21005],{},[69,21840,1939],{},[73,21842,21843],{"encoding":75},"V_j \\to V_i",[49,21845,21847,21903],{"className":21846,"ariaHidden":80},[79],[49,21848,21850,21854,21894,21897,21900],{"className":21849},[84],[49,21851],{"className":21852,"style":21853},[88],"height:0.9694em;vertical-align:-0.2861em;",[49,21855,21857,21860],{"className":21856},[93],[49,21858,21005],{"className":21859,"style":1843},[93,94],[49,21861,21863],{"className":21862},[218],[49,21864,21866,21886],{"className":21865},[222,605],[49,21867,21869,21883],{"className":21868},[226],[49,21870,21872],{"className":21871,"style":2176},[230],[49,21873,21874,21877],{"style":21190},[49,21875],{"className":21876,"style":238},[237],[49,21878,21880],{"className":21879},[242,243,244,245],[49,21881,1948],{"className":21882,"style":2231},[93,94,245],[49,21884,629],{"className":21885},[628],[49,21887,21889],{"className":21888},[226],[49,21890,21892],{"className":21891,"style":2028},[230],[49,21893],{},[49,21895],{"className":21896,"style":193},[192],[49,21898,169],{"className":21899},[197],[49,21901],{"className":21902,"style":193},[192],[49,21904,21906,21909],{"className":21905},[84],[49,21907],{"className":21908,"style":12029},[88],[49,21910,21912,21915],{"className":21911},[93],[49,21913,21005],{"className":21914,"style":1843},[93,94],[49,21916,21918],{"className":21917},[218],[49,21919,21921,21941],{"className":21920},[222,605],[49,21922,21924,21938],{"className":21923},[226],[49,21925,21927],{"className":21926,"style":2176},[230],[49,21928,21929,21932],{"style":21190},[49,21930],{"className":21931,"style":238},[237],[49,21933,21935],{"className":21934},[242,243,244,245],[49,21936,1939],{"className":21937},[93,94,245],[49,21939,629],{"className":21940},[628],[49,21942,21944],{"className":21943},[226],[49,21945,21947],{"className":21946,"style":636},[230],[49,21948],{}," 存在当且仅当 ",[49,21951,21953,21989],{"className":21952,"translate":53},[52],[49,21954,21956],{"className":21955},[57],[59,21957,21958],{"xmlns":61},[63,21959,21960,21986],{},[66,21961,21962,21968,21970,21976,21978,21984],{},[527,21963,21964,21966],{},[69,21965,21005],{},[69,21967,1948],{},[155,21969,285],{},[66,21971,21972,21974],{},[69,21973,11247],{"mathvariant":768},[69,21975,4867],{"mathvariant":768},[155,21977,298],{"stretchy":297},[527,21979,21980,21982],{},[69,21981,21005],{},[69,21983,1939],{},[155,21985,307],{"stretchy":297},[73,21987,21988],{"encoding":75},"V_j \\in \\mathrm{Pa}(V_i)",[49,21990,21992,22047],{"className":21991,"ariaHidden":80},[79],[49,21993,21995,21998,22038,22041,22044],{"className":21994},[84],[49,21996],{"className":21997,"style":21853},[88],[49,21999,22001,22004],{"className":22000},[93],[49,22002,21005],{"className":22003,"style":1843},[93,94],[49,22005,22007],{"className":22006},[218],[49,22008,22010,22030],{"className":22009},[222,605],[49,22011,22013,22027],{"className":22012},[226],[49,22014,22016],{"className":22015,"style":2176},[230],[49,22017,22018,22021],{"style":21190},[49,22019],{"className":22020,"style":238},[237],[49,22022,22024],{"className":22023},[242,243,244,245],[49,22025,1948],{"className":22026,"style":2231},[93,94,245],[49,22028,629],{"className":22029},[628],[49,22031,22033],{"className":22032},[226],[49,22034,22036],{"className":22035,"style":2028},[230],[49,22037],{},[49,22039],{"className":22040,"style":193},[192],[49,22042,285],{"className":22043},[197],[49,22045],{"className":22046,"style":193},[192],[49,22048,22050,22053,22059,22062,22102],{"className":22049},[84],[49,22051],{"className":22052,"style":589},[88],[49,22054,22056],{"className":22055},[93],[49,22057,21621],{"className":22058},[93,6607],[49,22060,298],{"className":22061},[373],[49,22063,22065,22068],{"className":22064},[93],[49,22066,21005],{"className":22067,"style":1843},[93,94],[49,22069,22071],{"className":22070},[218],[49,22072,22074,22094],{"className":22073},[222,605],[49,22075,22077,22091],{"className":22076},[226],[49,22078,22080],{"className":22079,"style":2176},[230],[49,22081,22082,22085],{"style":21190},[49,22083],{"className":22084,"style":238},[237],[49,22086,22088],{"className":22087},[242,243,244,245],[49,22089,1939],{"className":22090},[93,94,245],[49,22092,629],{"className":22093},[628],[49,22095,22097],{"className":22096},[226],[49,22098,22100],{"className":22099,"style":636},[230],[49,22101],{},[49,22103,307],{"className":22104},[406],"。因果推断的核心操作是 ",[49,22107,22109,22133],{"className":22108,"translate":53},[52],[49,22110,22112],{"className":22111},[57],[59,22113,22114],{"xmlns":61},[63,22115,22116,22130],{},[66,22117,22118,22124,22126,22128],{},[66,22119,22120,22122],{},[69,22121,15334],{"mathvariant":768},[69,22123,3291],{"mathvariant":768},[155,22125,298],{"stretchy":297},[155,22127,10425],{},[155,22129,307],{"stretchy":297},[73,22131,22132],{"encoding":75},"\\mathrm{do}(\\cdot)",[49,22134,22136],{"className":22135,"ariaHidden":80},[79],[49,22137,22139,22142,22149,22152,22155],{"className":22138},[84],[49,22140],{"className":22141,"style":589},[88],[49,22143,22145],{"className":22144},[93],[49,22146,22148],{"className":22147},[93,6607],"do",[49,22150,298],{"className":22151},[373],[49,22153,10425],{"className":22154},[93],[49,22156,307],{"className":22157},[406]," 算子：",[49,22160,22162,22201],{"className":22161,"translate":53},[52],[49,22163,22165],{"className":22164},[57],[59,22166,22167],{"xmlns":61},[63,22168,22169,22198],{},[66,22170,22171,22173,22175,22178,22180,22186,22188,22190,22192,22194,22196],{},[69,22172,11247],{},[155,22174,298],{"stretchy":297},[69,22176,22177],{},"Y",[155,22179,15232],{},[66,22181,22182,22184],{},[69,22183,15334],{"mathvariant":768},[69,22185,3291],{"mathvariant":768},[155,22187,298],{"stretchy":297},[69,22189,19544],{},[155,22191,523],{},[69,22193,520],{},[155,22195,307],{"stretchy":297},[155,22197,307],{"stretchy":297},[73,22199,22200],{"encoding":75},"P(Y \\mid \\mathrm{do}(X=x))",[49,22202,22204,22228,22255],{"className":22203,"ariaHidden":80},[79],[49,22205,22207,22210,22213,22216,22219,22222,22225],{"className":22206},[84],[49,22208],{"className":22209,"style":589},[88],[49,22211,11247],{"className":22212,"style":3554},[93,94],[49,22214,298],{"className":22215},[373],[49,22217,22177],{"className":22218,"style":1843},[93,94],[49,22220],{"className":22221,"style":193},[192],[49,22223,15232],{"className":22224},[197],[49,22226],{"className":22227,"style":193},[192],[49,22229,22231,22234,22240,22243,22246,22249,22252],{"className":22230},[84],[49,22232],{"className":22233,"style":589},[88],[49,22235,22237],{"className":22236},[93],[49,22238,22148],{"className":22239},[93,6607],[49,22241,298],{"className":22242},[373],[49,22244,19544],{"className":22245,"style":19569},[93,94],[49,22247],{"className":22248,"style":193},[192],[49,22250,523],{"className":22251},[197],[49,22253],{"className":22254,"style":193},[192],[49,22256,22258,22261,22264],{"className":22257},[84],[49,22259],{"className":22260,"style":589},[88],[49,22262,520],{"className":22263},[93,94],[49,22265,15087],{"className":22266},[406]," 表示「把 ",[49,22269,22271,22284],{"className":22270,"translate":53},[52],[49,22272,22274],{"className":22273},[57],[59,22275,22276],{"xmlns":61},[63,22277,22278,22282],{},[66,22279,22280],{},[69,22281,19544],{},[73,22283,19544],{"encoding":75},[49,22285,22287],{"className":22286,"ariaHidden":80},[79],[49,22288,22290,22293],{"className":22289},[84],[49,22291],{"className":22292,"style":89},[88],[49,22294,19544],{"className":22295,"style":19569},[93,94]," 的赋值机制替换为常数 ",[49,22298,22300,22313],{"className":22299,"translate":53},[52],[49,22301,22303],{"className":22302},[57],[59,22304,22305],{"xmlns":61},[63,22306,22307,22311],{},[66,22308,22309],{},[69,22310,520],{},[73,22312,520],{"encoding":75},[49,22314,22316],{"className":22315,"ariaHidden":80},[79],[49,22317,22319,22322],{"className":22318},[84],[49,22320],{"className":22321,"style":570},[88],[49,22323,520],{"className":22324},[93,94],"，切断它与所有父节点的连接」之后 ",[49,22327,22329,22342],{"className":22328,"translate":53},[52],[49,22330,22332],{"className":22331},[57],[59,22333,22334],{"xmlns":61},[63,22335,22336,22340],{},[66,22337,22338],{},[69,22339,22177],{},[73,22341,22177],{"encoding":75},[49,22343,22345],{"className":22344,"ariaHidden":80},[79],[49,22346,22348,22351],{"className":22347},[84],[49,22349],{"className":22350,"style":89},[88],[49,22352,22177],{"className":22353,"style":1843},[93,94]," 的分布，这与条件概率 ",[49,22356,22358,22386],{"className":22357,"translate":53},[52],[49,22359,22361],{"className":22360},[57],[59,22362,22363],{"xmlns":61},[63,22364,22365,22383],{},[66,22366,22367,22369,22371,22373,22375,22377,22379,22381],{},[69,22368,11247],{},[155,22370,298],{"stretchy":297},[69,22372,22177],{},[155,22374,15232],{},[69,22376,19544],{},[155,22378,523],{},[69,22380,520],{},[155,22382,307],{"stretchy":297},[73,22384,22385],{"encoding":75},"P(Y \\mid X=x)",[49,22387,22389,22413,22431],{"className":22388,"ariaHidden":80},[79],[49,22390,22392,22395,22398,22401,22404,22407,22410],{"className":22391},[84],[49,22393],{"className":22394,"style":589},[88],[49,22396,11247],{"className":22397,"style":3554},[93,94],[49,22399,298],{"className":22400},[373],[49,22402,22177],{"className":22403,"style":1843},[93,94],[49,22405],{"className":22406,"style":193},[192],[49,22408,15232],{"className":22409},[197],[49,22411],{"className":22412,"style":193},[192],[49,22414,22416,22419,22422,22425,22428],{"className":22415},[84],[49,22417],{"className":22418,"style":89},[88],[49,22420,19544],{"className":22421,"style":19569},[93,94],[49,22423],{"className":22424,"style":193},[192],[49,22426,523],{"className":22427},[197],[49,22429],{"className":22430,"style":193},[192],[49,22432,22434,22437,22440],{"className":22433},[84],[49,22435],{"className":22436,"style":589},[88],[49,22438,520],{"className":22439},[93,94],[49,22441,307],{"className":22442},[406],"——「观察到 ",[49,22445,22447,22465],{"className":22446,"translate":53},[52],[49,22448,22450],{"className":22449},[57],[59,22451,22452],{"xmlns":61},[63,22453,22454,22462],{},[66,22455,22456,22458,22460],{},[69,22457,19544],{},[155,22459,523],{},[69,22461,520],{},[73,22463,22464],{"encoding":75},"X=x",[49,22466,22468,22486],{"className":22467,"ariaHidden":80},[79],[49,22469,22471,22474,22477,22480,22483],{"className":22470},[84],[49,22472],{"className":22473,"style":89},[88],[49,22475,19544],{"className":22476,"style":19569},[93,94],[49,22478],{"className":22479,"style":193},[192],[49,22481,523],{"className":22482},[197],[49,22484],{"className":22485,"style":193},[192],[49,22487,22489,22492],{"className":22488},[84],[49,22490],{"className":22491,"style":570},[88],[49,22493,520],{"className":22494},[93,94]," 之后 ",[49,22497,22499,22512],{"className":22498,"translate":53},[52],[49,22500,22502],{"className":22501},[57],[59,22503,22504],{"xmlns":61},[63,22505,22506,22510],{},[66,22507,22508],{},[69,22509,22177],{},[73,22511,22177],{"encoding":75},[49,22513,22515],{"className":22514,"ariaHidden":80},[79],[49,22516,22518,22521],{"className":22517},[84],[49,22519],{"className":22520,"style":89},[88],[49,22522,22177],{"className":22523,"style":1843},[93,94]," 的分布」——一般是不相等的。两者的差异，正是相关不等于因果的形式化版本。",[39,22526,22527],{"id":22527},"因果与拓扑",[11,22529,22530],{},"一个自然的问题是：几何遗忘度量得到拓扑，那因果是不是遗忘更多东西之后剩下的更轻的骨架？答案是否定的。",[11,22532,22533,22534,22537,22538,22541,22542,22545,22546,22590],{},"几何到拓扑的「变轻」是",[28,22535,22536],{},"同一个信息维度上的压缩","——都是对称关系，只是分辨率从「连续的数值」降到了「是否连通」的二值\u002F离散判断。但因果结构引入了一个几何和拓扑都没有的新维度：",[28,22539,22540],{},"方向性\u002F非对称性","，并且这个方向性原则上",[28,22543,22544],{},"不能","从纯观测的联合分布 ",[49,22547,22549,22569],{"className":22548,"translate":53},[52],[49,22550,22552],{"className":22551},[57],[59,22553,22554],{"xmlns":61},[63,22555,22556,22566],{},[66,22557,22558,22560,22562,22564],{},[69,22559,11247],{},[155,22561,298],{"stretchy":297},[69,22563,21005],{},[155,22565,307],{"stretchy":297},[73,22567,22568],{"encoding":75},"P(V)",[49,22570,22572],{"className":22571,"ariaHidden":80},[79],[49,22573,22575,22578,22581,22584,22587],{"className":22574},[84],[49,22576],{"className":22577,"style":589},[88],[49,22579,11247],{"className":22580,"style":3554},[93,94],[49,22582,298],{"className":22583},[373],[49,22585,21005],{"className":22586,"style":1843},[93,94],[49,22588,307],{"className":22589},[406]," 中单独用几何或拓扑手段识别出来。这就是可识别性问题（identifiability）：",[469,22592,22593],{},[11,22594,22595,22598,22599,22642,22643,22700,22701,22752,22753,22752,22804,22879,22880,407],{},[28,22596,22597],{},"命题（因果不可从观测唯一识别）。"," 对于两个变量 ",[49,22600,22602,22620],{"className":22601,"translate":53},[52],[49,22603,22605],{"className":22604},[57],[59,22606,22607],{"xmlns":61},[63,22608,22609,22617],{},[66,22610,22611,22613,22615],{},[69,22612,19544],{},[155,22614,536],{"separator":80},[69,22616,22177],{},[73,22618,22619],{"encoding":75},"X, Y",[49,22621,22623],{"className":22622,"ariaHidden":80},[79],[49,22624,22626,22630,22633,22636,22639],{"className":22625},[84],[49,22627],{"className":22628,"style":22629},[88],"height:0.8778em;vertical-align:-0.1944em;",[49,22631,19544],{"className":22632,"style":19569},[93,94],[49,22634,536],{"className":22635},[642],[49,22637],{"className":22638,"style":646},[192],[49,22640,22177],{"className":22641,"style":1843},[93,94],"，仅凭观测联合分布 ",[49,22644,22646,22670],{"className":22645,"translate":53},[52],[49,22647,22649],{"className":22648},[57],[59,22650,22651],{"xmlns":61},[63,22652,22653,22667],{},[66,22654,22655,22657,22659,22661,22663,22665],{},[69,22656,11247],{},[155,22658,298],{"stretchy":297},[69,22660,19544],{},[155,22662,536],{"separator":80},[69,22664,22177],{},[155,22666,307],{"stretchy":297},[73,22668,22669],{"encoding":75},"P(X,Y)",[49,22671,22673],{"className":22672,"ariaHidden":80},[79],[49,22674,22676,22679,22682,22685,22688,22691,22694,22697],{"className":22675},[84],[49,22677],{"className":22678,"style":589},[88],[49,22680,11247],{"className":22681,"style":3554},[93,94],[49,22683,298],{"className":22684},[373],[49,22686,19544],{"className":22687,"style":19569},[93,94],[49,22689,536],{"className":22690},[642],[49,22692],{"className":22693,"style":646},[192],[49,22695,22177],{"className":22696,"style":1843},[93,94],[49,22698,307],{"className":22699},[406],"，一般无法在 ",[49,22702,22704,22722],{"className":22703,"translate":53},[52],[49,22705,22707],{"className":22706},[57],[59,22708,22709],{"xmlns":61},[63,22710,22711,22719],{},[66,22712,22713,22715,22717],{},[69,22714,19544],{},[155,22716,169],{},[69,22718,22177],{},[73,22720,22721],{"encoding":75},"X \\to Y",[49,22723,22725,22743],{"className":22724,"ariaHidden":80},[79],[49,22726,22728,22731,22734,22737,22740],{"className":22727},[84],[49,22729],{"className":22730,"style":89},[88],[49,22732,19544],{"className":22733,"style":19569},[93,94],[49,22735],{"className":22736,"style":193},[192],[49,22738,169],{"className":22739},[197],[49,22741],{"className":22742,"style":193},[192],[49,22744,22746,22749],{"className":22745},[84],[49,22747],{"className":22748,"style":89},[88],[49,22750,22177],{"className":22751,"style":1843},[93,94],"、",[49,22754,22756,22774],{"className":22755,"translate":53},[52],[49,22757,22759],{"className":22758},[57],[59,22760,22761],{"xmlns":61},[63,22762,22763,22771],{},[66,22764,22765,22767,22769],{},[69,22766,22177],{},[155,22768,169],{},[69,22770,19544],{},[73,22772,22773],{"encoding":75},"Y \\to X",[49,22775,22777,22795],{"className":22776,"ariaHidden":80},[79],[49,22778,22780,22783,22786,22789,22792],{"className":22779},[84],[49,22781],{"className":22782,"style":89},[88],[49,22784,22177],{"className":22785,"style":1843},[93,94],[49,22787],{"className":22788,"style":193},[192],[49,22790,169],{"className":22791},[197],[49,22793],{"className":22794,"style":193},[192],[49,22796,22798,22801],{"className":22797},[84],[49,22799],{"className":22800,"style":89},[88],[49,22802,19544],{"className":22803,"style":19569},[93,94],[49,22805,22807,22831],{"className":22806,"translate":53},[52],[49,22808,22810],{"className":22809},[57],[59,22811,22812],{"xmlns":61},[63,22813,22814,22828],{},[66,22815,22816,22818,22821,22824,22826],{},[69,22817,19544],{},[155,22819,22820],{},"←",[69,22822,22823],{},"Z",[155,22825,169],{},[69,22827,22177],{},[73,22829,22830],{"encoding":75},"X \\leftarrow Z \\to Y",[49,22832,22834,22852,22870],{"className":22833,"ariaHidden":80},[79],[49,22835,22837,22840,22843,22846,22849],{"className":22836},[84],[49,22838],{"className":22839,"style":89},[88],[49,22841,19544],{"className":22842,"style":19569},[93,94],[49,22844],{"className":22845,"style":193},[192],[49,22847,22820],{"className":22848},[197],[49,22850],{"className":22851,"style":193},[192],[49,22853,22855,22858,22861,22864,22867],{"className":22854},[84],[49,22856],{"className":22857,"style":89},[88],[49,22859,22823],{"className":22860,"style":345},[93,94],[49,22862],{"className":22863,"style":193},[192],[49,22865,169],{"className":22866},[197],[49,22868],{"className":22869,"style":193},[192],[49,22871,22873,22876],{"className":22872},[84],[49,22874],{"className":22875,"style":89},[88],[49,22877,22177],{"className":22878,"style":1843},[93,94],"（混杂）等因果结构之间做出唯一判定——它们可以诱导出完全相同的 ",[49,22881,22883,22906],{"className":22882,"translate":53},[52],[49,22884,22886],{"className":22885},[57],[59,22887,22888],{"xmlns":61},[63,22889,22890,22904],{},[66,22891,22892,22894,22896,22898,22900,22902],{},[69,22893,11247],{},[155,22895,298],{"stretchy":297},[69,22897,19544],{},[155,22899,536],{"separator":80},[69,22901,22177],{},[155,22903,307],{"stretchy":297},[73,22905,22669],{"encoding":75},[49,22907,22909],{"className":22908,"ariaHidden":80},[79],[49,22910,22912,22915,22918,22921,22924,22927,22930,22933],{"className":22911},[84],[49,22913],{"className":22914,"style":589},[88],[49,22916,11247],{"className":22917,"style":3554},[93,94],[49,22919,298],{"className":22920},[373],[49,22922,19544],{"className":22923,"style":19569},[93,94],[49,22925,536],{"className":22926},[642],[49,22928],{"className":22929,"style":646},[192],[49,22931,22177],{"className":22932,"style":1843},[93,94],[49,22934,307],{"className":22935},[406],[11,22937,22938,22939,22973,22974,20642,23002,23030],{},"要打破这种不可识别性，需要额外的信息源：随机对照实验（真正的 ",[49,22940,22942,22958],{"className":22941,"translate":53},[52],[49,22943,22945],{"className":22944},[57],[59,22946,22947],{"xmlns":61},[63,22948,22949,22955],{},[66,22950,22951,22953],{},[69,22952,15334],{"mathvariant":768},[69,22954,3291],{"mathvariant":768},[73,22956,22957],{"encoding":75},"\\mathrm{do}",[49,22959,22961],{"className":22960,"ariaHidden":80},[79],[49,22962,22964,22967],{"className":22963},[84],[49,22965],{"className":22966,"style":3096},[88],[49,22968,22970],{"className":22969},[93],[49,22971,22148],{"className":22972},[93,6607],"）、时间先后顺序、结构假设（如可加噪声模型、非高斯性，如 LiNGAM 方法利用非高斯噪声打破对称性）。这说明因果结构不是几何\u002F拓扑那条压缩链上的下一站，而是一个正交的轴——即便你拥有关于系统的完整几何信息（所有点对的精确距离）和完整拓扑信息（所有连通关系），你依然可能无法回答「如果我强行改变 ",[49,22975,22977,22990],{"className":22976,"translate":53},[52],[49,22978,22980],{"className":22979},[57],[59,22981,22982],{"xmlns":61},[63,22983,22984,22988],{},[66,22985,22986],{},[69,22987,19544],{},[73,22989,19544],{"encoding":75},[49,22991,22993],{"className":22992,"ariaHidden":80},[79],[49,22994,22996,22999],{"className":22995},[84],[49,22997],{"className":22998,"style":89},[88],[49,23000,19544],{"className":23001,"style":19569},[93,94],[49,23003,23005,23018],{"className":23004,"translate":53},[52],[49,23006,23008],{"className":23007},[57],[59,23009,23010],{"xmlns":61},[63,23011,23012,23016],{},[66,23013,23014],{},[69,23015,22177],{},[73,23017,22177],{"encoding":75},[49,23019,23021],{"className":23020,"ariaHidden":80},[79],[49,23022,23024,23027],{"className":23023},[84],[49,23025],{"className":23026,"style":89},[88],[49,23028,22177],{"className":23029,"style":1843},[93,94]," 会怎样」这个问题。这正是 Pearl 提出因果之梯（Ladder of Causation）的意义所在：",[23032,23033,23034,23053],"table",{},[23035,23036,23037],"thead",{},[23038,23039,23040,23044,23047,23050],"tr",{},[23041,23042,23043],"th",{},"层级",[23041,23045,23046],{},"问题类型",[23041,23048,23049],{},"例子",[23041,23051,23052],{},"所需信息",[23054,23055,23056,23136,23234],"tbody",{},[23038,23057,23058,23062,23065,23133],{},[23059,23060,23061],"td",{},"1. 关联",[23059,23063,23064],{},"看见（seeing）",[23059,23066,23067],{},[49,23068,23070,23094],{"className":23069,"translate":53},[52],[49,23071,23073],{"className":23072},[57],[59,23074,23075],{"xmlns":61},[63,23076,23077,23091],{},[66,23078,23079,23081,23083,23085,23087,23089],{},[69,23080,11247],{},[155,23082,298],{"stretchy":297},[69,23084,22177],{},[155,23086,15232],{},[69,23088,19544],{},[155,23090,307],{"stretchy":297},[73,23092,23093],{"encoding":75},"P(Y\\mid X)",[49,23095,23097,23121],{"className":23096,"ariaHidden":80},[79],[49,23098,23100,23103,23106,23109,23112,23115,23118],{"className":23099},[84],[49,23101],{"className":23102,"style":589},[88],[49,23104,11247],{"className":23105,"style":3554},[93,94],[49,23107,298],{"className":23108},[373],[49,23110,22177],{"className":23111,"style":1843},[93,94],[49,23113],{"className":23114,"style":193},[192],[49,23116,15232],{"className":23117},[197],[49,23119],{"className":23120,"style":193},[192],[49,23122,23124,23127,23130],{"className":23123},[84],[49,23125],{"className":23126,"style":589},[88],[49,23128,19544],{"className":23129,"style":19569},[93,94],[49,23131,307],{"className":23132},[406],[23059,23134,23135],{},"观测数据即可，几何\u002F拓扑视角覆盖此层",[23038,23137,23138,23141,23144,23231],{},[23059,23139,23140],{},"2. 干预",[23059,23142,23143],{},"行动（doing）",[23059,23145,23146],{},[49,23147,23149,23183],{"className":23148,"translate":53},[52],[49,23150,23152],{"className":23151},[57],[59,23153,23154],{"xmlns":61},[63,23155,23156,23180],{},[66,23157,23158,23160,23162,23164,23166,23172,23174,23176,23178],{},[69,23159,11247],{},[155,23161,298],{"stretchy":297},[69,23163,22177],{},[155,23165,15232],{},[66,23167,23168,23170],{},[69,23169,15334],{"mathvariant":768},[69,23171,3291],{"mathvariant":768},[155,23173,298],{"stretchy":297},[69,23175,19544],{},[155,23177,307],{"stretchy":297},[155,23179,307],{"stretchy":297},[73,23181,23182],{"encoding":75},"P(Y\\mid \\mathrm{do}(X))",[49,23184,23186,23210],{"className":23185,"ariaHidden":80},[79],[49,23187,23189,23192,23195,23198,23201,23204,23207],{"className":23188},[84],[49,23190],{"className":23191,"style":589},[88],[49,23193,11247],{"className":23194,"style":3554},[93,94],[49,23196,298],{"className":23197},[373],[49,23199,22177],{"className":23200,"style":1843},[93,94],[49,23202],{"className":23203,"style":193},[192],[49,23205,15232],{"className":23206},[197],[49,23208],{"className":23209,"style":193},[192],[49,23211,23213,23216,23222,23225,23228],{"className":23212},[84],[49,23214],{"className":23215,"style":589},[88],[49,23217,23219],{"className":23218},[93],[49,23220,22148],{"className":23221},[93,6607],[49,23223,298],{"className":23224},[373],[49,23226,19544],{"className":23227,"style":19569},[93,94],[49,23229,15087],{"className":23230},[406],[23059,23232,23233],{},"需要实验或因果图假设",[23038,23235,23236,23239,23242,23302],{},[23059,23237,23238],{},"3. 反事实",[23059,23240,23241],{},"想象（imagining）",[23059,23243,5733,23244,23272,23273,23301],{},[49,23245,23247,23260],{"className":23246,"translate":53},[52],[49,23248,23250],{"className":23249},[57],[59,23251,23252],{"xmlns":61},[63,23253,23254,23258],{},[66,23255,23256],{},[69,23257,19544],{},[73,23259,19544],{"encoding":75},[49,23261,23263],{"className":23262,"ariaHidden":80},[79],[49,23264,23266,23269],{"className":23265},[84],[49,23267],{"className":23268,"style":89},[88],[49,23270,19544],{"className":23271,"style":19569},[93,94]," 不同，",[49,23274,23276,23289],{"className":23275,"translate":53},[52],[49,23277,23279],{"className":23278},[57],[59,23280,23281],{"xmlns":61},[63,23282,23283,23287],{},[66,23284,23285],{},[69,23286,22177],{},[73,23288,22177],{"encoding":75},[49,23290,23292],{"className":23291,"ariaHidden":80},[79],[49,23293,23295,23298],{"className":23294},[84],[49,23296],{"className":23297,"style":89},[88],[49,23299,22177],{"className":23300,"style":1843},[93,94]," 会怎样",[23059,23303,23304],{},"需要完整 SCM，包括外生噪声的联合分布",[11,23306,23307],{},"几何和拓扑视角，本质上被限制在第一层；只有引入因果结构，才能爬上第二、第三层。这也是为什么「因果关系需要额外信息才能识别」这句话不是技术细节，而是三种视角在存在论意义上的分界线。",[39,23309,23310],{"id":23310},"机器学习视角",[11,23312,23313],{},"回到机器学习的语境，三种视角之间的差异直接决定了模型在分布偏移下的表现。可以按「随分布变化保持不变的程度」给三者排一个序：",[49,23315,23317],{"className":23316,"translate":53},[752],[49,23318,23320,23354],{"className":23319,"translate":53},[52],[49,23321,23323],{"className":23322},[57],[59,23324,23325],{"xmlns":61,"display":761},[63,23326,23327,23351],{},[66,23328,23329,23332,23334,23337,23339,23342,23344,23346,23348],{},[3314,23330,23331],{},"Geometry",[3314,23333,19798],{},[155,23335,23336],{},"≺",[3314,23338,19798],{},[3314,23340,23341],{},"Topology Invariance",[3314,23343,19798],{},[155,23345,23336],{},[3314,23347,19798],{},[3314,23349,23350],{},"Causality",[73,23352,23353],{"encoding":75},"\\text{Geometry} \\;\\prec\\; \\text{Topology Invariance} \\;\\prec\\; \\text{Causality}",[49,23355,23357,23384,23411],{"className":23356,"ariaHidden":80},[79],[49,23358,23360,23363,23369,23372,23375,23378,23381],{"className":23359},[84],[49,23361],{"className":23362,"style":22629},[88],[49,23364,23366],{"className":23365},[93,3828],[49,23367,23331],{"className":23368},[93],[49,23370],{"className":23371,"style":193},[192],[49,23373],{"className":23374,"style":193},[192],[49,23376,23336],{"className":23377},[197],[49,23379],{"className":23380,"style":193},[192],[49,23382],{"className":23383,"style":193},[192],[49,23385,23387,23390,23396,23399,23402,23405,23408],{"className":23386},[84],[49,23388],{"className":23389,"style":184},[88],[49,23391,23393],{"className":23392},[93,3828],[49,23394,23341],{"className":23395},[93],[49,23397],{"className":23398,"style":193},[192],[49,23400],{"className":23401,"style":193},[192],[49,23403,23336],{"className":23404},[197],[49,23406],{"className":23407,"style":193},[192],[49,23409],{"className":23410,"style":193},[192],[49,23412,23414,23417],{"className":23413},[84],[49,23415],{"className":23416,"style":184},[88],[49,23418,23420],{"className":23419},[93,3828],[49,23421,23350],{"className":23422},[93],[5729,23424,23425,23431,23481],{},[21,23426,23427,23430],{},[28,23428,23429],{},"几何层面的失败","：训练集和测试集哪怕只是特征做了不同的归一化，相似度排序就可能整体偏移——这是最脆弱的一层，任何非本质的表示变化都会渗透进度量数值本身。",[21,23432,23433,23436,23437,23480],{},[28,23434,23435],{},"拓扑层面的稳健性","：持久同调等方法对噪声、尺度变化天然不敏感，因为它们本来就是在滤除度量细节之后提取「形状」。但拓扑结构依然是观测分布 ",[49,23438,23440,23459],{"className":23439,"translate":53},[52],[49,23441,23443],{"className":23442},[57],[59,23444,23445],{"xmlns":61},[63,23446,23447,23457],{},[66,23448,23449,23451,23453,23455],{},[69,23450,11247],{},[155,23452,298],{"stretchy":297},[69,23454,21005],{},[155,23456,307],{"stretchy":297},[73,23458,22568],{"encoding":75},[49,23460,23462],{"className":23461,"ariaHidden":80},[79],[49,23463,23465,23468,23471,23474,23477],{"className":23464},[84],[49,23466],{"className":23467,"style":589},[88],[49,23469,11247],{"className":23470,"style":3554},[93,94],[49,23472,298],{"className":23473},[373],[49,23475,21005],{"className":23476,"style":1843},[93,94],[49,23478,307],{"className":23479},[406]," 的函数——一旦分布本身发生结构性改变（不只是尺度变化，而是变量间依赖关系变了），拓扑不变量也会随之改变，因为它仍然停留在因果之梯的第一层。",[21,23482,23483,23486,23487,23537,23538,23677,23678,23722,23723,23807],{},[28,23484,23485],{},"因果层面的稳健性","：如果 ",[49,23488,23490,23507],{"className":23489,"translate":53},[52],[49,23491,23493],{"className":23492},[57],[59,23494,23495],{"xmlns":61},[63,23496,23497,23505],{},[66,23498,23499,23501,23503],{},[69,23500,19544],{},[155,23502,169],{},[69,23504,22177],{},[73,23506,22721],{"encoding":75},[49,23508,23510,23528],{"className":23509,"ariaHidden":80},[79],[49,23511,23513,23516,23519,23522,23525],{"className":23512},[84],[49,23514],{"className":23515,"style":89},[88],[49,23517,19544],{"className":23518,"style":19569},[93,94],[49,23520],{"className":23521,"style":193},[192],[49,23523,169],{"className":23524},[197],[49,23526],{"className":23527,"style":193},[192],[49,23529,23531,23534],{"className":23530},[84],[49,23532],{"className":23533,"style":89},[88],[49,23535,22177],{"className":23536,"style":1843},[93,94]," 的机制 ",[49,23539,23541,23573],{"className":23540,"translate":53},[52],[49,23542,23544],{"className":23543},[57],[59,23545,23546],{"xmlns":61},[63,23547,23548,23570],{},[66,23549,23550,23556,23558,23560,23562,23568],{},[527,23551,23552,23554],{},[69,23553,153],{},[69,23555,22177],{},[155,23557,298],{"stretchy":297},[69,23559,19544],{},[155,23561,536],{"separator":80},[527,23563,23564,23566],{},[69,23565,20887],{},[69,23567,22177],{},[155,23569,307],{"stretchy":297},[73,23571,23572],{"encoding":75},"f_Y(X, U_Y)",[49,23574,23576],{"className":23575,"ariaHidden":80},[79],[49,23577,23579,23582,23622,23625,23628,23631,23634,23674],{"className":23578},[84],[49,23580],{"className":23581,"style":589},[88],[49,23583,23585,23588],{"className":23584},[93],[49,23586,153],{"className":23587,"style":188},[93,94],[49,23589,23591],{"className":23590},[218],[49,23592,23594,23614],{"className":23593},[222,605],[49,23595,23597,23611],{"className":23596},[226],[49,23598,23600],{"className":23599,"style":678},[230],[49,23601,23602,23605],{"style":9478},[49,23603],{"className":23604,"style":238},[237],[49,23606,23608],{"className":23607},[242,243,244,245],[49,23609,22177],{"className":23610,"style":1843},[93,94,245],[49,23612,629],{"className":23613},[628],[49,23615,23617],{"className":23616},[226],[49,23618,23620],{"className":23619,"style":636},[230],[49,23621],{},[49,23623,298],{"className":23624},[373],[49,23626,19544],{"className":23627,"style":19569},[93,94],[49,23629,536],{"className":23630},[642],[49,23632],{"className":23633,"style":646},[192],[49,23635,23637,23640],{"className":23636},[93],[49,23638,20887],{"className":23639,"style":95},[93,94],[49,23641,23643],{"className":23642},[218],[49,23644,23646,23666],{"className":23645},[222,605],[49,23647,23649,23663],{"className":23648},[226],[49,23650,23652],{"className":23651,"style":678},[230],[49,23653,23654,23657],{"style":16752},[49,23655],{"className":23656,"style":238},[237],[49,23658,23660],{"className":23659},[242,243,244,245],[49,23661,22177],{"className":23662,"style":1843},[93,94,245],[49,23664,629],{"className":23665},[628],[49,23667,23669],{"className":23668},[226],[49,23670,23672],{"className":23671,"style":636},[230],[49,23673],{},[49,23675,307],{"className":23676},[406]," 本身不随环境改变（这是很多领域里的物理假设——比如「重力加速度与颜色无关」），那么哪怕 ",[49,23679,23681,23701],{"className":23680,"translate":53},[52],[49,23682,23684],{"className":23683},[57],[59,23685,23686],{"xmlns":61},[63,23687,23688,23698],{},[66,23689,23690,23692,23694,23696],{},[69,23691,11247],{},[155,23693,298],{"stretchy":297},[69,23695,19544],{},[155,23697,307],{"stretchy":297},[73,23699,23700],{"encoding":75},"P(X)",[49,23702,23704],{"className":23703,"ariaHidden":80},[79],[49,23705,23707,23710,23713,23716,23719],{"className":23706},[84],[49,23708],{"className":23709,"style":589},[88],[49,23711,11247],{"className":23712,"style":3554},[93,94],[49,23714,298],{"className":23715},[373],[49,23717,19544],{"className":23718,"style":19569},[93,94],[49,23720,307],{"className":23721},[406]," 的边际分布因为采样环境不同而剧烈变化，",[49,23724,23726,23759],{"className":23725,"translate":53},[52],[49,23727,23729],{"className":23728},[57],[59,23730,23731],{"xmlns":61},[63,23732,23733,23757],{},[66,23734,23735,23737,23739,23741,23743,23749,23751,23753,23755],{},[69,23736,11247],{},[155,23738,298],{"stretchy":297},[69,23740,22177],{},[155,23742,15232],{},[66,23744,23745,23747],{},[69,23746,15334],{"mathvariant":768},[69,23748,3291],{"mathvariant":768},[155,23750,298],{"stretchy":297},[69,23752,19544],{},[155,23754,307],{"stretchy":297},[155,23756,307],{"stretchy":297},[73,23758,23182],{"encoding":75},[49,23760,23762,23786],{"className":23761,"ariaHidden":80},[79],[49,23763,23765,23768,23771,23774,23777,23780,23783],{"className":23764},[84],[49,23766],{"className":23767,"style":589},[88],[49,23769,11247],{"className":23770,"style":3554},[93,94],[49,23772,298],{"className":23773},[373],[49,23775,22177],{"className":23776,"style":1843},[93,94],[49,23778],{"className":23779,"style":193},[192],[49,23781,15232],{"className":23782},[197],[49,23784],{"className":23785,"style":193},[192],[49,23787,23789,23792,23798,23801,23804],{"className":23788},[84],[49,23790],{"className":23791,"style":589},[88],[49,23793,23795],{"className":23794},[93],[49,23796,22148],{"className":23797},[93,6607],[49,23799,298],{"className":23800},[373],[49,23802,19544],{"className":23803,"style":19569},[93,94],[49,23805,15087],{"className":23806},[406]," 这个机制仍然成立，模型依然可以泛化。这正是 Arjovsky 等人提出不变风险最小化方法的出发点：不去拟合在各个环境里都表现最好的相关性，而是寻找在所有环境下都保持不变的预测机制——本质上是把学习目标从「几何\u002F统计上的最优拟合」换成「因果机制上的不变性」。",[11,23809,23810],{},"用一句话概括：几何模型学的是「数据长什么样」，拓扑模型学的是「数据的骨架怎么连」，而因果模型学的是「数据为什么会长成这样、什么力量在背后生成它」。前两者是对现象的描述，后者试图逼近生成现象的机制——这也是为什么协变量偏移能摧毁前两者，却不一定能摧毁后者。",[39,23812,23813],{"id":23813},"递进的先验",[11,23815,23816,23817,23886],{},"如果用范畴论的语言简单勾勒一下三者的关系：存在遗忘函子 ",[49,23818,23820,23850],{"className":23819,"translate":53},[52],[49,23821,23823],{"className":23822},[57],[59,23824,23825],{"xmlns":61},[63,23826,23827,23847],{},[66,23828,23829,23837,23839],{},[66,23830,23831,23833,23835],{},[69,23832,16556],{"mathvariant":15255},[69,23834,20896],{"mathvariant":15255},[69,23836,14115],{"mathvariant":15255},[155,23838,169],{},[66,23840,23841,23843,23845],{},[69,23842,3252],{"mathvariant":15255},[69,23844,3291],{"mathvariant":15255},[69,23846,11],{"mathvariant":15255},[73,23848,23849],{"encoding":75},"\\mathbf{Met} \\to \\mathbf{Top}",[49,23851,23853,23874],{"className":23852,"ariaHidden":80},[79],[49,23854,23856,23859,23865,23868,23871],{"className":23855},[84],[49,23857],{"className":23858,"style":20939},[88],[49,23860,23862],{"className":23861},[93],[49,23863,20946],{"className":23864},[93,15556],[49,23866],{"className":23867,"style":193},[192],[49,23869,169],{"className":23870},[197],[49,23872],{"className":23873,"style":193},[192],[49,23875,23877,23880],{"className":23876},[84],[49,23878],{"className":23879,"style":20962},[88],[49,23881,23883],{"className":23882},[93],[49,23884,20969],{"className":23885},[93,15556],"，把度量空间的态射（等距同构）松弛成拓扑空间的态射（同胚），信息在这一步单调减少，但依然停留在「对称关系」的范畴里。因果结构则不在这条链上——它是在对象上额外加了一层有向、非对称的生成机制，这层机制无法通过对已有对称结构做进一步遗忘得到，只能通过引入新的认识论工具（干预、时间、结构假设）来「添加」。",[11,23888,23889],{},"所以更准确的图景不是一条线性的「由重到轻」的链条，而是：",[49,23891,23893],{"className":23892,"translate":53},[752],[49,23894,23896,23995],{"className":23895,"translate":53},[52],[49,23897,23899],{"className":23898},[57],[59,23900,23901],{"xmlns":61,"display":761},[63,23902,23903,23992],{},[787,23904,23905,23925,23941,23959,23974],{"rowspacing":789,"columnalign":790,"columnspacing":791},[793,23906,23907],{},[796,23908,23909],{},[799,23910,23911],{"scriptlevel":801,"displaystyle":297},[23912,23913,23914,23922],"munder",{},[23912,23915,23916,23919],{},[3314,23917,23918],{},"几何",[155,23920,23921],{"stretchy":80},"⏟",[3314,23923,23924],{},"对称，度量化，最重",[793,23926,23927],{},[796,23928,23929],{},[799,23930,23931],{"scriptlevel":801,"displaystyle":297},[66,23932,23933,23936],{},[155,23934,23935],{"fence":297,"stretchy":80,"minsize":15229,"maxsize":15229},"↓",[799,23937,23938],{"scriptlevel":533,"displaystyle":297},[3314,23939,23940],{},"遗忘度量",[793,23942,23943],{},[796,23944,23945],{},[799,23946,23947],{"scriptlevel":801,"displaystyle":297},[23912,23948,23949,23956],{},[23912,23950,23951,23954],{},[3314,23952,23953],{},"拓扑",[155,23955,23921],{"stretchy":80},[3314,23957,23958],{},"对称，非度量化，中等",[793,23960,23961],{},[796,23962,23963],{},[799,23964,23965],{"scriptlevel":801,"displaystyle":297},[66,23966,23967,23969],{},[155,23968,23935],{"fence":297,"stretchy":80,"minsize":15229,"maxsize":15229},[799,23970,23971],{"scriptlevel":533,"displaystyle":297},[3314,23972,23973],{},"额外的干预\u002F时序信息",[793,23975,23976],{},[796,23977,23978],{},[799,23979,23980],{"scriptlevel":801,"displaystyle":297},[23912,23981,23982,23989],{},[23912,23983,23984,23987],{},[3314,23985,23986],{},"因果",[155,23988,23921],{"stretchy":80},[3314,23990,23991],{},"非对称，机制化，最稳健",[73,23993,23994],{"encoding":75},"\\begin{array}{c}\n\\underbrace{\\text{几何}}_{\\text{对称，度量化，最重}} \\\\[8pt]\n\\Big\\downarrow \\scriptstyle \\text{遗忘度量} \\\\[8pt]\n\\underbrace{\\text{拓扑}}_{\\text{对称，非度量化，中等}} \\\\[8pt]\n\\Big\\downarrow \\scriptstyle \\text{额外的干预\u002F时序信息} \\\\[8pt]\n\\underbrace{\\text{因果}}_{\\text{非对称，机制化，最稳健}}\n\\end{array}",[49,23996,23998],{"className":23997,"ariaHidden":80},[79],[49,23999,24001,24005],{"className":24000},[84],[49,24002],{"className":24003,"style":24004},[88],"height:11.121em;vertical-align:-5.3105em;",[49,24006,24008],{"className":24007},[93],[49,24009,24011,24014,24585],{"className":24010},[787],[49,24012],{"className":24013,"style":1006},[1005],[49,24015,24017],{"className":24016},[1010],[49,24018,24020,24576],{"className":24019},[222,605],[49,24021,24023,24573],{"className":24022},[226],[49,24024,24027,24166,24251,24372,24452],{"className":24025,"style":24026},[230],"height:5.8105em;",[49,24028,24030,24034],{"style":24029},"top:-8.1215em;",[49,24031],{"className":24032,"style":24033},[237],"height:3.151em;",[49,24035,24037],{"className":24036},[93],[49,24038,24040],{"className":24039},[650,23912],[49,24041,24043,24157],{"className":24042},[222,605],[49,24044,24046,24154],{"className":24045},[226],[49,24047,24049,24067],{"className":24048,"style":89},[230],[49,24050,24052,24055],{"style":24051},"top:-1.6737em;",[49,24053],{"className":24054,"style":1057},[237],[49,24056,24058],{"className":24057},[242,243,244,245],[49,24059,24061],{"className":24060},[93,245],[49,24062,24064],{"className":24063},[93,3828,245],[49,24065,23924],{"className":24066},[93,11327,245],[49,24068,24069,24072],{"style":9804},[49,24070],{"className":24071,"style":1057},[237],[49,24073,24075],{"className":24074},[650,23912],[49,24076,24078,24145],{"className":24077},[222,605],[49,24079,24081,24142],{"className":24080},[226],[49,24082,24084,24128],{"className":24083,"style":89},[230],[49,24085,24088,24091],{"className":24086,"style":24087},[9803],"top:-2.352em;",[49,24089],{"className":24090,"style":1057},[237],[49,24092,24096,24108,24118],{"className":24093,"style":24095},[24094],"stretchy","height:0.548em;min-width:1.6em;",[49,24097,24101],{"className":24098,"style":24100},[24099],"brace-left","height:0.548em;",[974,24102,24105],{"xmlns":976,"width":9825,"height":24103,"viewBox":24104,"preserveAspectRatio":9828},"0.548em","0 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representation learning）想做的事情，正是把几何\u002F拓扑意义上学到的表示，进一步「提纯」出背后满足机制不变性的潜变量和因果图——这也是几何、拓扑、因果这三种视角在方法论上真正会师的地方。",{"title":19471,"searchDepth":19472,"depth":19472,"links":24595},[24596,24597,24598,24599,24600,24601],{"id":19520,"depth":19472,"text":19521},{"id":20580,"depth":19472,"text":20581},{"id":20973,"depth":19472,"text":20974},{"id":22527,"depth":19472,"text":22527},{"id":23310,"depth":19472,"text":23310},{"id":23813,"depth":19472,"text":23813},{},"2026-08-19","\u002Fblog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao",{"title":19506,"description":19511},"blog\u002F2026\u002F2026-08-19-yan-jiu-guan-xi-de-san-zhong-shi-jiao","研究「关系」有三种互补视角：几何，拓扑和因果。分别关注测量、连通和约束。它们不是并列的学科，而是看待「存在」的三个层层递进的维度。三者共同构成智能系统理解世界的先验地基。",[19502],"JouSEmv-eU7CnfBnekiG75RAlq4ZoOC_3cvO_1NWTAI",1787405167030]