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",[38,1119,1121,1139],{"className":1120,"translate":42},[41],[38,1122,1124],{"className":1123},[46],[48,1125,1126],{"xmlns":50},[52,1127,1128,1136],{},[55,1129,1130],{},[295,1131,1132,1134],{},[58,1133,703],{"mathvariant":285},[58,1135,601],{},[81,1137,1138],{"encoding":83},"\\mathbf{e}_i",[38,1140,1142],{"className":1141,"ariaHidden":89},[88],[38,1143,1145,1149],{"className":1144},[93],[38,1146],{"className":1147,"style":1148},[97],"height:0.5944em;vertical-align:-0.15em;",[38,1150,1152,1155],{"className":1151},[102],[38,1153,703],{"className":1154},[102,351],[38,1156,1158],{"className":1157},[134],[38,1159,1161,1181],{"className":1160},[138,384],[38,1162,1164,1178],{"className":1163},[142],[38,1165,1167],{"className":1166,"style":786},[146],[38,1168,1169,1172],{"style":394},[38,1170],{"className":1171,"style":155},[154],[38,1173,1175],{"className":1174},[159,160,161,162],[38,1176,601],{"className":1177},[102,103,162],[38,1179,408],{"className":1180},[407],[38,1182,1184],{"className":1183},[142],[38,1185,1187],{"className":1186,"style":415},[146],[38,1188],{}," 为第 ",[38,1191,1193,1206],{"className":1192,"translate":42},[41],[38,1194,1196],{"className":1195},[46],[48,1197,1198],{"xmlns":50},[52,1199,1200,1204],{},[55,1201,1202],{},[58,1203,601],{},[81,1205,601],{"encoding":83},[38,1207,1209],{"className":1208,"ariaHidden":89},[88],[38,1210,1212,1215],{"className":1211},[93],[38,1213],{"className":1214,"style":613},[97],[38,1216,601],{"className":1217},[102,103]," 个标准基向量，则称 ",[38,1220,1222,1235],{"className":1221,"translate":42},[41],[38,1223,1225],{"className":1224},[46],[48,1226,1227],{"xmlns":50},[52,1228,1229,1233],{},[55,1230,1231],{},[58,1232,60],{},[81,1234,60],{"encoding":83},[38,1236,1238],{"className":1237,"ariaHidden":89},[88],[38,1239,1241,1244],{"className":1240},[93],[38,1242],{"className":1243,"style":98},[97],[38,1245,60],{"className":1246,"style":104},[102,103]," 在该方向上可偏导。",[31,1249,1250],{"id":1250},"定义",[1252,1253,1254,1318,1902],"blockquote",{},[11,1255,1256,1259,1260,1288,1289,1317],{},[15,1257,1258],{},"定义 1.1（梯度）"," 若 ",[38,1261,1263,1276],{"className":1262,"translate":42},[41],[38,1264,1266],{"className":1265},[46],[48,1267,1268],{"xmlns":50},[52,1269,1270,1274],{},[55,1271,1272],{},[58,1273,60],{},[81,1275,60],{"encoding":83},[38,1277,1279],{"className":1278,"ariaHidden":89},[88],[38,1280,1282,1285],{"className":1281},[93],[38,1283],{"className":1284,"style":98},[97],[38,1286,60],{"className":1287,"style":104},[102,103]," 在 ",[38,1290,1292,1305],{"className":1291,"translate":42},[41],[38,1293,1295],{"className":1294},[46],[48,1296,1297],{"xmlns":50},[52,1298,1299,1303],{},[55,1300,1301],{},[58,1302,286],{"mathvariant":285},[81,1304,1104],{"encoding":83},[38,1306,1308],{"className":1307,"ariaHidden":89},[88],[38,1309,1311,1314],{"className":1310},[93],[38,1312],{"className":1313,"style":347},[97],[38,1315,286],{"className":1316},[102,351]," 处所有偏导数均存在，则将这些偏导数按顺序排列成的列向量",[38,1319,1321],{"className":1320,"translate":42},[621],[38,1322,1324,1430],{"className":1323,"translate":42},[41],[38,1325,1327],{"className":1326},[46],[48,1328,1329],{"xmlns":50,"display":630},[52,1330,1331,1427],{},[55,1332,1333,1336,1338,1340,1342,1344,1346,1418,1421],{},[58,1334,1335],{"mathvariant":333},"∇",[58,1337,60],{},[62,1339,293],{"stretchy":292},[58,1341,286],{"mathvariant":285},[62,1343,330],{"stretchy":292},[62,1345,289],{},[66,1347,1348,1416],{},[55,1349,1350,1352,1370,1372,1390,1392,1394,1396,1414],{},[62,1351,293],{"fence":89},[636,1353,1354,1360],{},[55,1355,1356,1358],{},[58,1357,642],{"mathvariant":333},[58,1359,60],{},[55,1361,1362,1364],{},[58,1363,642],{"mathvariant":333},[295,1365,1366,1368],{},[58,1367,286],{},[300,1369,302],{},[62,1371,305],{"separator":89},[636,1373,1374,1380],{},[55,1375,1376,1378],{},[58,1377,642],{"mathvariant":333},[58,1379,60],{},[55,1381,1382,1384],{},[58,1383,642],{"mathvariant":333},[295,1385,1386,1388],{},[58,1387,286],{},[300,1389,312],{},[62,1391,305],{"separator":89},[62,1393,317],{},[62,1395,305],{"separator":89},[636,1397,1398,1404],{},[55,1399,1400,1402],{},[58,1401,642],{"mathvariant":333},[58,1403,60],{},[55,1405,1406,1408],{},[58,1407,642],{"mathvariant":333},[295,1409,1410,1412],{},[58,1411,286],{},[58,1413,74],{},[62,1415,330],{"fence":89},[58,1417,334],{"mathvariant":333},[62,1419,1420],{},"∈",[66,1422,1423,1425],{},[58,1424,71],{"mathvariant":70},[58,1426,74],{},[81,1428,1429],{"encoding":83},"\\nabla f(\\mathbf{x}) = \\left( \\frac{\\partial f}{\\partial x_1}, \\frac{\\partial f}{\\partial x_2}, \\dots, \\frac{\\partial f}{\\partial x_n} \\right)^\\top \\in \\mathbb{R}^n",[38,1431,1433,1464,1865],{"className":1432,"ariaHidden":89},[88],[38,1434,1436,1440,1443,1446,1449,1452,1455,1458,1461],{"className":1435},[93],[38,1437],{"className":1438,"style":1439},[97],"height:1em;vertical-align:-0.25em;",[38,1441,1335],{"className":1442},[102],[38,1444,60],{"className":1445,"style":104},[102,103],[38,1447,293],{"className":1448},[371],[38,1450,286],{"className":1451},[102,351],[38,1453,330],{"className":1454},[529],[38,1456],{"className":1457,"style":109},[108],[38,1459,289],{"className":1460},[113],[38,1462],{"className":1463,"style":109},[108],[38,1465,1467,1471,1856,1859,1862],{"className":1466},[93],[38,1468],{"className":1469,"style":1470},[97],"height:2.639em;vertical-align:-0.95em;",[38,1472,1474,1831],{"className":1473},[475],[38,1475,1477,1486,1591,1594,1597,1702,1705,1708,1711,1714,1717,1720,1825],{"className":1476},[475],[38,1478,1482],{"className":1479,"style":1481},[371,1480],"delimcenter","top:0em;",[38,1483,293],{"className":1484},[1485,161],"delimsizing",[38,1487,1489,1492,1588],{"className":1488},[102],[38,1490],{"className":1491},[371,740],[38,1493,1495],{"className":1494},[636],[38,1496,1498,1580],{"className":1497},[138,384],[38,1499,1501,1577],{"className":1500},[142],[38,1502,1504,1555,1563],{"className":1503,"style":753},[146],[38,1505,1506,1509],{"style":756},[38,1507],{"className":1508,"style":760},[154],[38,1510,1512,1515],{"className":1511},[102],[38,1513,642],{"className":1514,"style":767},[102],[38,1516,1518,1521],{"className":1517},[102],[38,1519,286],{"className":1520},[102,103],[38,1522,1524],{"className":1523},[134],[38,1525,1527,1547],{"className":1526},[138,384],[38,1528,1530,1544],{"className":1529},[142],[38,1531,1533],{"className":1532,"style":391},[146],[38,1534,1535,1538],{"style":394},[38,1536],{"className":1537,"style":155},[154],[38,1539,1541],{"className":1540},[159,160,161,162],[38,1542,302],{"className":1543},[102,162],[38,1545,408],{"className":1546},[407],[38,1548,1550],{"className":1549},[142],[38,1551,1553],{"className":1552,"style":415},[146],[38,1554],{},[38,1556,1557,1560],{"style":811},[38,1558],{"className":1559,"style":760},[154],[38,1561],{"className":1562,"style":819},[818],[38,1564,1565,1568],{"style":822},[38,1566],{"className":1567,"style":760},[154],[38,1569,1571,1574],{"className":1570},[102],[38,1572,642],{"className":1573,"style":767},[102],[38,1575,60],{"className":1576,"style":104},[102,103],[38,1578,408],{"className":1579},[407],[38,1581,1583],{"className":1582},[142],[38,1584,1586],{"className":1585,"style":844},[146],[38,1587],{},[38,1589],{"className":1590},[529,740],[38,1592,305],{"className":1593},[421],[38,1595],{"className":1596,"style":425},[108],[38,1598,1600,1603,1699],{"className":1599},[102],[38,1601],{"className":1602},[371,740],[38,1604,1606],{"className":1605},[636],[38,1607,1609,1691],{"className":1608},[138,384],[38,1610,1612,1688],{"className":1611},[142],[38,1613,1615,1666,1674],{"className":1614,"style":753},[146],[38,1616,1617,1620],{"style":756},[38,1618],{"className":1619,"style":760},[154],[38,1621,1623,1626],{"className":1622},[102],[38,1624,642],{"className":1625,"style":767},[102],[38,1627,1629,1632],{"className":1628},[102],[38,1630,286],{"className":1631},[102,103],[38,1633,1635],{"className":1634},[134],[38,1636,1638,1658],{"className":1637},[138,384],[38,1639,1641,1655],{"className":1640},[142],[38,1642,1644],{"className":1643,"style":391},[146],[38,1645,1646,1649],{"style":394},[38,1647],{"className":1648,"style":155},[154],[38,1650,1652],{"className":1651},[159,160,161,162],[38,1653,312],{"className":1654},[102,162],[38,1656,408],{"className":1657},[407],[38,1659,1661],{"className":1660},[142],[38,1662,1664],{"className":1663,"style":415},[146],[38,1665],{},[38,1667,1668,1671],{"style":811},[38,1669],{"className":1670,"style":760},[154],[38,1672],{"className":1673,"style":819},[818],[38,1675,1676,1679],{"style":822},[38,1677],{"className":1678,"style":760},[154],[38,1680,1682,1685],{"className":1681},[102],[38,1683,642],{"className":1684,"style":767},[102],[38,1686,60],{"className":1687,"style":104},[102,103],[38,1689,408],{"className":1690},[407],[38,1692,1694],{"className":1693},[142],[38,1695,1697],{"className":1696,"style":844},[146],[38,1698],{},[38,1700],{"className":1701},[529,740],[38,1703,305],{"className":1704},[421],[38,1706],{"className":1707,"style":425},[108],[38,1709,317],{"className":1710},[475],[38,1712],{"className":1713,"style":425},[108],[38,1715,305],{"className":1716},[421],[38,1718],{"className":1719,"style":425},[108],[38,1721,1723,1726,1822],{"className":1722},[102],[38,1724],{"className":1725},[371,740],[38,1727,1729],{"className":1728},[636],[38,1730,1732,1814],{"className":1731},[138,384],[38,1733,1735,1811],{"className":1734},[142],[38,1736,1738,1789,1797],{"className":1737,"style":753},[146],[38,1739,1740,1743],{"style":756},[38,1741],{"className":1742,"style":760},[154],[38,1744,1746,1749],{"className":1745},[102],[38,1747,642],{"className":1748,"style":767},[102],[38,1750,1752,1755],{"className":1751},[102],[38,1753,286],{"className":1754},[102,103],[38,1756,1758],{"className":1757},[134],[38,1759,1761,1781],{"className":1760},[138,384],[38,1762,1764,1778],{"className":1763},[142],[38,1765,1767],{"className":1766,"style":503},[146],[38,1768,1769,1772],{"style":394},[38,1770],{"className":1771,"style":155},[154],[38,1773,1775],{"className":1774},[159,160,161,162],[38,1776,74],{"className":1777},[102,103,162],[38,1779,408],{"className":1780},[407],[38,1782,1784],{"className":1783},[142],[38,1785,1787],{"className":1786,"style":415},[146],[38,1788],{},[38,1790,1791,1794],{"style":811},[38,1792],{"className":1793,"style":760},[154],[38,1795],{"className":1796,"style":819},[818],[38,1798,1799,1802],{"style":822},[38,1800],{"className":1801,"style":760},[154],[38,1803,1805,1808],{"className":1804},[102],[38,1806,642],{"className":1807,"style":767},[102],[38,1809,60],{"className":1810,"style":104},[102,103],[38,1812,408],{"className":1813},[407],[38,1815,1817],{"className":1816},[142],[38,1818,1820],{"className":1819,"style":844},[146],[38,1821],{},[38,1823],{"className":1824},[529,740],[38,1826,1828],{"className":1827,"style":1481},[529,1480],[38,1829,330],{"className":1830},[1485,161],[38,1832,1834],{"className":1833},[134],[38,1835,1837],{"className":1836},[138],[38,1838,1840],{"className":1839},[142],[38,1841,1844],{"className":1842,"style":1843},[146],"height:1.689em;",[38,1845,1847,1850],{"style":1846},"top:-3.9029em;margin-right:0.05em;",[38,1848],{"className":1849,"style":155},[154],[38,1851,1853],{"className":1852},[159,160,161,162],[38,1854,334],{"className":1855},[102,162],[38,1857],{"className":1858,"style":109},[108],[38,1860,1420],{"className":1861},[113],[38,1863],{"className":1864,"style":109},[108],[38,1866,1868,1872],{"className":1867},[93],[38,1869],{"className":1870,"style":1871},[97],"height:0.7144em;",[38,1873,1875,1878],{"className":1874},[102],[38,1876,71],{"className":1877},[102,130],[38,1879,1881],{"className":1880},[134],[38,1882,1884],{"className":1883},[138],[38,1885,1887],{"className":1886},[142],[38,1888,1890],{"className":1889,"style":1871},[146],[38,1891,1893,1896],{"style":1892},"top:-3.113em;margin-right:0.05em;",[38,1894],{"className":1895,"style":155},[154],[38,1897,1899],{"className":1898},[159,160,161,162],[38,1900,74],{"className":1901},[102,103,162],[11,1903,1904,1905,1288,1933,1961,1962,1964],{},"称为 ",[38,1906,1908,1921],{"className":1907,"translate":42},[41],[38,1909,1911],{"className":1910},[46],[48,1912,1913],{"xmlns":50},[52,1914,1915,1919],{},[55,1916,1917],{},[58,1918,60],{},[81,1920,60],{"encoding":83},[38,1922,1924],{"className":1923,"ariaHidden":89},[88],[38,1925,1927,1930],{"className":1926},[93],[38,1928],{"className":1929,"style":98},[97],[38,1931,60],{"className":1932,"style":104},[102,103],[38,1934,1936,1949],{"className":1935,"translate":42},[41],[38,1937,1939],{"className":1938},[46],[48,1940,1941],{"xmlns":50},[52,1942,1943,1947],{},[55,1944,1945],{},[58,1946,286],{"mathvariant":285},[81,1948,1104],{"encoding":83},[38,1950,1952],{"className":1951,"ariaHidden":89},[88],[38,1953,1955,1958],{"className":1954},[93],[38,1956],{"className":1957,"style":347},[97],[38,1959,286],{"className":1960},[102,351]," 处的",[15,1963,17],{},"。",[11,1966,1967],{},"梯度本质上是标量值函数导数概念在多维定义域上的自然延拓：它把\"函数关于每个自变量分量的变化率\"打包为一个向量，方向与大小共同编码了函数局部变化的完整一阶信息。",[31,1969,1971],{"id":1970},"全微分方向导数","全微分、方向导数",[1252,1973,1974,2119],{},[11,1975,1976,1259,1979,1288,2007,2035,2036,2118],{},[15,1977,1978],{},"定理 1.1（一阶泰勒近似）",[38,1980,1982,1995],{"className":1981,"translate":42},[41],[38,1983,1985],{"className":1984},[46],[48,1986,1987],{"xmlns":50},[52,1988,1989,1993],{},[55,1990,1991],{},[58,1992,60],{},[81,1994,60],{"encoding":83},[38,1996,1998],{"className":1997,"ariaHidden":89},[88],[38,1999,2001,2004],{"className":2000},[93],[38,2002],{"className":2003,"style":98},[97],[38,2005,60],{"className":2006,"style":104},[102,103],[38,2008,2010,2023],{"className":2009,"translate":42},[41],[38,2011,2013],{"className":2012},[46],[48,2014,2015],{"xmlns":50},[52,2016,2017,2021],{},[55,2018,2019],{},[58,2020,286],{"mathvariant":285},[81,2022,1104],{"encoding":83},[38,2024,2026],{"className":2025,"ariaHidden":89},[88],[38,2027,2029,2032],{"className":2028},[93],[38,2030],{"className":2031,"style":347},[97],[38,2033,286],{"className":2034},[102,351]," 处可微，则对充分小的扰动 ",[38,2037,2039,2061],{"className":2038,"translate":42},[41],[38,2040,2042],{"className":2041},[46],[48,2043,2044],{"xmlns":50},[52,2045,2046,2058],{},[55,2047,2048,2050,2052],{},[58,2049,678],{"mathvariant":285},[62,2051,1420],{},[66,2053,2054,2056],{},[58,2055,71],{"mathvariant":70},[58,2057,74],{},[81,2059,2060],{"encoding":83},"\\mathbf{h} \\in \\mathbb{R}^n",[38,2062,2064,2083],{"className":2063,"ariaHidden":89},[88],[38,2065,2067,2071,2074,2077,2080],{"className":2066},[93],[38,2068],{"className":2069,"style":2070},[97],"height:0.7335em;vertical-align:-0.0391em;",[38,2072,678],{"className":2073},[102,351],[38,2075],{"className":2076,"style":109},[108],[38,2078,1420],{"className":2079},[113],[38,2081],{"className":2082,"style":109},[108],[38,2084,2086,2089],{"className":2085},[93],[38,2087],{"className":2088,"style":123},[97],[38,2090,2092,2095],{"className":2091},[102],[38,2093,71],{"className":2094},[102,130],[38,2096,2098],{"className":2097},[134],[38,2099,2101],{"className":2100},[138],[38,2102,2104],{"className":2103},[142],[38,2105,2107],{"className":2106,"style":147},[146],[38,2108,2109,2112],{"style":150},[38,2110],{"className":2111,"style":155},[154],[38,2113,2115],{"className":2114},[159,160,161,162],[38,2116,74],{"className":2117},[102,103,162],"，有",[38,2120,2122],{"className":2121,"translate":42},[621],[38,2123,2125,2196],{"className":2124,"translate":42},[41],[38,2126,2128],{"className":2127},[46],[48,2129,2130],{"xmlns":50,"display":630},[52,2131,2132,2193],{},[55,2133,2134,2136,2138,2140,2142,2144,2146,2148,2150,2152,2154,2156,2158,2160,2162,2164,2166,2172,2174,2176,2179,2181,2184,2186,2188,2190],{},[58,2135,60],{},[62,2137,293],{"stretchy":292},[58,2139,286],{"mathvariant":285},[62,2141,696],{},[58,2143,678],{"mathvariant":285},[62,2145,330],{"stretchy":292},[62,2147,289],{},[58,2149,60],{},[62,2151,293],{"stretchy":292},[58,2153,286],{"mathvariant":285},[62,2155,330],{"stretchy":292},[62,2157,696],{},[58,2159,1335],{"mathvariant":333},[58,2161,60],{},[62,2163,293],{"stretchy":292},[58,2165,286],{"mathvariant":285},[66,2167,2168,2170],{},[62,2169,330],{"stretchy":292},[58,2171,334],{"mathvariant":333},[58,2173,678],{"mathvariant":285},[62,2175,696],{},[58,2177,2178],{},"o",[62,2180,293],{"stretchy":292},[58,2182,2183],{"mathvariant":333},"∥",[58,2185,678],{"mathvariant":285},[58,2187,2183],{"mathvariant":333},[62,2189,330],{"stretchy":292},[58,2191,2192],{"mathvariant":333},".",[81,2194,2195],{"encoding":83},"f(\\mathbf{x} + \\mathbf{h}) = f(\\mathbf{x}) + \\nabla f(\\mathbf{x})^\\top \\mathbf{h} + o(\\|\\mathbf{h}\\|).",[38,2197,2199,2223,2244,2271,2332],{"className":2198,"ariaHidden":89},[88],[38,2200,2202,2205,2208,2211,2214,2217,2220],{"className":2201},[93],[38,2203],{"className":2204,"style":1439},[97],[38,2206,60],{"className":2207,"style":104},[102,103],[38,2209,293],{"className":2210},[371],[38,2212,286],{"className":2213},[102,351],[38,2215],{"className":2216,"style":995},[108],[38,2218,696],{"className":2219},[999],[38,2221],{"className":2222,"style":995},[108],[38,2224,2226,2229,2232,2235,2238,2241],{"className":2225},[93],[38,2227],{"className":2228,"style":1439},[97],[38,2230,678],{"className":2231},[102,351],[38,2233,330],{"className":2234},[529],[38,2236],{"className":2237,"style":109},[108],[38,2239,289],{"className":2240},[113],[38,2242],{"className":2243,"style":109},[108],[38,2245,2247,2250,2253,2256,2259,2262,2265,2268],{"className":2246},[93],[38,2248],{"className":2249,"style":1439},[97],[38,2251,60],{"className":2252,"style":104},[102,103],[38,2254,293],{"className":2255},[371],[38,2257,286],{"className":2258},[102,351],[38,2260,330],{"className":2261},[529],[38,2263],{"className":2264,"style":995},[108],[38,2266,696],{"className":2267},[999],[38,2269],{"className":2270,"style":995},[108],[38,2272,2274,2278,2281,2284,2287,2290,2320,2323,2326,2329],{"className":2273},[93],[38,2275],{"className":2276,"style":2277},[97],"height:1.1491em;vertical-align:-0.25em;",[38,2279,1335],{"className":2280},[102],[38,2282,60],{"className":2283,"style":104},[102,103],[38,2285,293],{"className":2286},[371],[38,2288,286],{"className":2289},[102,351],[38,2291,2293,2296],{"className":2292},[529],[38,2294,330],{"className":2295},[529],[38,2297,2299],{"className":2298},[134],[38,2300,2302],{"className":2301},[138],[38,2303,2305],{"className":2304},[142],[38,2306,2309],{"className":2307,"style":2308},[146],"height:0.8991em;",[38,2310,2311,2314],{"style":1892},[38,2312],{"className":2313,"style":155},[154],[38,2315,2317],{"className":2316},[159,160,161,162],[38,2318,334],{"className":2319},[102,162],[38,2321,678],{"className":2322},[102,351],[38,2324],{"className":2325,"style":995},[108],[38,2327,696],{"className":2328},[999],[38,2330],{"className":2331,"style":995},[108],[38,2333,2335,2338,2341,2344,2347,2350,2353,2356],{"className":2334},[93],[38,2336],{"className":2337,"style":1439},[97],[38,2339,2178],{"className":2340},[102,103],[38,2342,293],{"className":2343},[371],[38,2345,2183],{"className":2346},[102],[38,2348,678],{"className":2349},[102,351],[38,2351,2183],{"className":2352},[102],[38,2354,330],{"className":2355},[529],[38,2357,2192],{"className":2358},[102],[11,2360,2361],{},"这一定理表明，梯度与扰动向量的内积给出了函数值变化的最佳线性近似，梯度因此可被理解为将\"方向\"映射为\"变化率\"的线性泛函在标准基下的坐标表示。",[1252,2363,2364,2434,2611,2842],{},[11,2365,2366,2369,2370,2433],{},[15,2367,2368],{},"定理 1.2（最速上升方向）"," 设 ",[38,2371,2373,2396],{"className":2372,"translate":42},[41],[38,2374,2376],{"className":2375},[46],[48,2377,2378],{"xmlns":50},[52,2379,2380,2393],{},[55,2381,2382,2384,2387,2389,2391],{},[58,2383,2183],{"mathvariant":333},[58,2385,2386],{"mathvariant":285},"u",[58,2388,2183],{"mathvariant":333},[62,2390,289],{},[300,2392,302],{},[81,2394,2395],{"encoding":83},"\\|\\mathbf{u}\\| = 1",[38,2397,2399,2423],{"className":2398,"ariaHidden":89},[88],[38,2400,2402,2405,2408,2411,2414,2417,2420],{"className":2401},[93],[38,2403],{"className":2404,"style":1439},[97],[38,2406,2183],{"className":2407},[102],[38,2409,2386],{"className":2410},[102,351],[38,2412,2183],{"className":2413},[102],[38,2415],{"className":2416,"style":109},[108],[38,2418,289],{"className":2419},[113],[38,2421],{"className":2422,"style":109},[108],[38,2424,2426,2430],{"className":2425},[93],[38,2427],{"className":2428,"style":2429},[97],"height:0.6444em;",[38,2431,302],{"className":2432},[102],"，则方向导数",[38,2435,2437],{"className":2436,"translate":42},[621],[38,2438,2440,2485],{"className":2439,"translate":42},[41],[38,2441,2443],{"className":2442},[46],[48,2444,2445],{"xmlns":50,"display":630},[52,2446,2447,2482],{},[55,2448,2449,2456,2458,2460,2462,2464,2466,2468,2470,2472,2474,2480],{},[295,2450,2451,2454],{},[58,2452,2453],{},"D",[58,2455,2386],{"mathvariant":285},[58,2457,60],{},[62,2459,293],{"stretchy":292},[58,2461,286],{"mathvariant":285},[62,2463,330],{"stretchy":292},[62,2465,289],{},[58,2467,1335],{"mathvariant":333},[58,2469,60],{},[62,2471,293],{"stretchy":292},[58,2473,286],{"mathvariant":285},[66,2475,2476,2478],{},[62,2477,330],{"stretchy":292},[58,2479,334],{"mathvariant":333},[58,2481,2386],{"mathvariant":285},[81,2483,2484],{"encoding":83},"D_{\\mathbf{u}} f(\\mathbf{x}) = \\nabla f(\\mathbf{x})^\\top \\mathbf{u}",[38,2486,2488,2561],{"className":2487,"ariaHidden":89},[88],[38,2489,2491,2494,2540,2543,2546,2549,2552,2555,2558],{"className":2490},[93],[38,2492],{"className":2493,"style":1439},[97],[38,2495,2497,2501],{"className":2496},[102],[38,2498,2453],{"className":2499,"style":2500},[102,103],"margin-right:0.0278em;",[38,2502,2504],{"className":2503},[134],[38,2505,2507,2532],{"className":2506},[138,384],[38,2508,2510,2529],{"className":2509},[142],[38,2511,2514],{"className":2512,"style":2513},[146],"height:0.1611em;",[38,2515,2517,2520],{"style":2516},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[38,2518],{"className":2519,"style":155},[154],[38,2521,2523],{"className":2522},[159,160,161,162],[38,2524,2526],{"className":2525},[102,162],[38,2527,2386],{"className":2528},[102,351,162],[38,2530,408],{"className":2531},[407],[38,2533,2535],{"className":2534},[142],[38,2536,2538],{"className":2537,"style":415},[146],[38,2539],{},[38,2541,60],{"className":2542,"style":104},[102,103],[38,2544,293],{"className":2545},[371],[38,2547,286],{"className":2548},[102,351],[38,2550,330],{"className":2551},[529],[38,2553],{"className":2554,"style":109},[108],[38,2556,289],{"className":2557},[113],[38,2559],{"className":2560,"style":109},[108],[38,2562,2564,2567,2570,2573,2576,2579,2608],{"className":2563},[93],[38,2565],{"className":2566,"style":2277},[97],[38,2568,1335],{"className":2569},[102],[38,2571,60],{"className":2572,"style":104},[102,103],[38,2574,293],{"className":2575},[371],[38,2577,286],{"className":2578},[102,351],[38,2580,2582,2585],{"className":2581},[529],[38,2583,330],{"className":2584},[529],[38,2586,2588],{"className":2587},[134],[38,2589,2591],{"className":2590},[138],[38,2592,2594],{"className":2593},[142],[38,2595,2597],{"className":2596,"style":2308},[146],[38,2598,2599,2602],{"style":1892},[38,2600],{"className":2601,"style":155},[154],[38,2603,2605],{"className":2604},[159,160,161,162],[38,2606,334],{"className":2607},[102,162],[38,2609,2386],{"className":2610},[102,351],[11,2612,2613,2614,2785,2786,1964],{},"在 ",[38,2615,2617,2666],{"className":2616,"translate":42},[41],[38,2618,2620],{"className":2619},[46],[48,2621,2622],{"xmlns":50},[52,2623,2624,2663],{},[55,2625,2626,2628,2630],{},[58,2627,2386],{"mathvariant":285},[62,2629,289],{},[2631,2632,2633],"mstyle",{"scriptlevel":683,"displaystyle":89},[636,2634,2635,2647],{},[55,2636,2637,2639,2641,2643,2645],{},[58,2638,1335],{"mathvariant":333},[58,2640,60],{},[62,2642,293],{"stretchy":292},[58,2644,286],{"mathvariant":285},[62,2646,330],{"stretchy":292},[55,2648,2649,2651,2653,2655,2657,2659,2661],{},[58,2650,2183],{"mathvariant":333},[58,2652,1335],{"mathvariant":333},[58,2654,60],{},[62,2656,293],{"stretchy":292},[58,2658,286],{"mathvariant":285},[62,2660,330],{"stretchy":292},[58,2662,2183],{"mathvariant":333},[81,2664,2665],{"encoding":83},"\\mathbf{u} = \\dfrac{\\nabla f(\\mathbf{x})}{\\|\\nabla f(\\mathbf{x})\\|}",[38,2667,2669,2687],{"className":2668,"ariaHidden":89},[88],[38,2670,2672,2675,2678,2681,2684],{"className":2671},[93],[38,2673],{"className":2674,"style":347},[97],[38,2676,2386],{"className":2677},[102,351],[38,2679],{"className":2680,"style":109},[108],[38,2682,289],{"className":2683},[113],[38,2685],{"className":2686,"style":109},[108],[38,2688,2690,2694],{"className":2689},[93],[38,2691],{"className":2692,"style":2693},[97],"height:2.363em;vertical-align:-0.936em;",[38,2695,2697,2700,2782],{"className":2696},[102],[38,2698],{"className":2699},[371,740],[38,2701,2703],{"className":2702},[636],[38,2704,2706,2773],{"className":2705},[138,384],[38,2707,2709,2770],{"className":2708},[142],[38,2710,2712,2739,2747],{"className":2711,"style":955},[146],[38,2713,2714,2717],{"style":756},[38,2715],{"className":2716,"style":760},[154],[38,2718,2720,2724,2727,2730,2733,2736],{"className":2719},[102],[38,2721,2723],{"className":2722},[102],"∥∇",[38,2725,60],{"className":2726,"style":104},[102,103],[38,2728,293],{"className":2729},[371],[38,2731,286],{"className":2732},[102,351],[38,2734,330],{"className":2735},[529],[38,2737,2183],{"className":2738},[102],[38,2740,2741,2744],{"style":811},[38,2742],{"className":2743,"style":760},[154],[38,2745],{"className":2746,"style":819},[818],[38,2748,2749,2752],{"style":822},[38,2750],{"className":2751,"style":760},[154],[38,2753,2755,2758,2761,2764,2767],{"className":2754},[102],[38,2756,1335],{"className":2757},[102],[38,2759,60],{"className":2760,"style":104},[102,103],[38,2762,293],{"className":2763},[371],[38,2765,286],{"className":2766},[102,351],[38,2768,330],{"className":2769},[529],[38,2771,408],{"className":2772},[407],[38,2774,2776],{"className":2775},[142],[38,2777,2780],{"className":2778,"style":2779},[146],"height:0.936em;",[38,2781],{},[38,2783],{"className":2784},[529,740]," 时取得最大值 ",[38,2787,2789,2815],{"className":2788,"translate":42},[41],[38,2790,2792],{"className":2791},[46],[48,2793,2794],{"xmlns":50},[52,2795,2796,2812],{},[55,2797,2798,2800,2802,2804,2806,2808,2810],{},[58,2799,2183],{"mathvariant":333},[58,2801,1335],{"mathvariant":333},[58,2803,60],{},[62,2805,293],{"stretchy":292},[58,2807,286],{"mathvariant":285},[62,2809,330],{"stretchy":292},[58,2811,2183],{"mathvariant":333},[81,2813,2814],{"encoding":83},"\\|\\nabla f(\\mathbf{x})\\|",[38,2816,2818],{"className":2817,"ariaHidden":89},[88],[38,2819,2821,2824,2827,2830,2833,2836,2839],{"className":2820},[93],[38,2822],{"className":2823,"style":1439},[97],[38,2825,2723],{"className":2826},[102],[38,2828,60],{"className":2829,"style":104},[102,103],[38,2831,293],{"className":2832},[371],[38,2834,286],{"className":2835},[102,351],[38,2837,330],{"className":2838},[529],[38,2840,2183],{"className":2841},[102],[11,2843,2844,2848,2849,2996,2997,3026,3027,3076],{},[2845,2846,2847],"em",{},"证明思路："," 由柯西–施瓦茨不等式，",[38,2850,2852,2903],{"className":2851,"translate":42},[41],[38,2853,2855],{"className":2854},[46],[48,2856,2857],{"xmlns":50},[52,2858,2859,2900],{},[55,2860,2861,2863,2865,2867,2869,2875,2877,2880,2882,2884,2886,2888,2890,2892,2894,2896,2898],{},[58,2862,1335],{"mathvariant":333},[58,2864,60],{},[62,2866,293],{"stretchy":292},[58,2868,286],{"mathvariant":285},[66,2870,2871,2873],{},[62,2872,330],{"stretchy":292},[58,2874,334],{"mathvariant":333},[58,2876,2386],{"mathvariant":285},[62,2878,2879],{},"≤",[58,2881,2183],{"mathvariant":333},[58,2883,1335],{"mathvariant":333},[58,2885,60],{},[62,2887,293],{"stretchy":292},[58,2889,286],{"mathvariant":285},[62,2891,330],{"stretchy":292},[58,2893,2183],{"mathvariant":333},[58,2895,2183],{"mathvariant":333},[58,2897,2386],{"mathvariant":285},[58,2899,2183],{"mathvariant":333},[81,2901,2902],{"encoding":83},"\\nabla f(\\mathbf{x})^\\top \\mathbf{u} \\le \\|\\nabla f(\\mathbf{x})\\| \\|\\mathbf{u}\\|",[38,2904,2906,2965],{"className":2905,"ariaHidden":89},[88],[38,2907,2909,2912,2915,2918,2921,2924,2953,2956,2959,2962],{"className":2908},[93],[38,2910],{"className":2911,"style":367},[97],[38,2913,1335],{"className":2914},[102],[38,2916,60],{"className":2917,"style":104},[102,103],[38,2919,293],{"className":2920},[371],[38,2922,286],{"className":2923},[102,351],[38,2925,2927,2930],{"className":2926},[529],[38,2928,330],{"className":2929},[529],[38,2931,2933],{"className":2932},[134],[38,2934,2936],{"className":2935},[138],[38,2937,2939],{"className":2938},[142],[38,2940,2942],{"className":2941,"style":545},[146],[38,2943,2944,2947],{"style":150},[38,2945],{"className":2946,"style":155},[154],[38,2948,2950],{"className":2949},[159,160,161,162],[38,2951,334],{"className":2952},[102,162],[38,2954,2386],{"className":2955},[102,351],[38,2957],{"className":2958,"style":109},[108],[38,2960,2879],{"className":2961},[113],[38,2963],{"className":2964,"style":109},[108],[38,2966,2968,2971,2974,2977,2980,2983,2986,2990,2993],{"className":2967},[93],[38,2969],{"className":2970,"style":1439},[97],[38,2972,2723],{"className":2973},[102],[38,2975,60],{"className":2976,"style":104},[102,103],[38,2978,293],{"className":2979},[371],[38,2981,286],{"className":2982},[102,351],[38,2984,330],{"className":2985},[529],[38,2987,2989],{"className":2988},[102],"∥∥",[38,2991,2386],{"className":2992},[102,351],[38,2994,2183],{"className":2995},[102],"，等号当且仅当 ",[38,2998,3000,3014],{"className":2999,"translate":42},[41],[38,3001,3003],{"className":3002},[46],[48,3004,3005],{"xmlns":50},[52,3006,3007,3011],{},[55,3008,3009],{},[58,3010,2386],{"mathvariant":285},[81,3012,3013],{"encoding":83},"\\mathbf{u}",[38,3015,3017],{"className":3016,"ariaHidden":89},[88],[38,3018,3020,3023],{"className":3019},[93],[38,3021],{"className":3022,"style":347},[97],[38,3024,2386],{"className":3025},[102,351]," 与 ",[38,3028,3030,3052],{"className":3029,"translate":42},[41],[38,3031,3033],{"className":3032},[46],[48,3034,3035],{"xmlns":50},[52,3036,3037,3049],{},[55,3038,3039,3041,3043,3045,3047],{},[58,3040,1335],{"mathvariant":333},[58,3042,60],{},[62,3044,293],{"stretchy":292},[58,3046,286],{"mathvariant":285},[62,3048,330],{"stretchy":292},[81,3050,3051],{"encoding":83},"\\nabla f(\\mathbf{x})",[38,3053,3055],{"className":3054,"ariaHidden":89},[88],[38,3056,3058,3061,3064,3067,3070,3073],{"className":3057},[93],[38,3059],{"className":3060,"style":1439},[97],[38,3062,1335],{"className":3063},[102],[38,3065,60],{"className":3066,"style":104},[102,103],[38,3068,293],{"className":3069},[371],[38,3071,286],{"className":3072},[102,351],[38,3074,330],{"className":3075},[529]," 同向时成立。",[11,3078,3079,3080,3083],{},"这一性质赋予梯度明确的几何意义：",[15,3081,3082],{},"梯度方向指向函数值增长最快的方向，其模长表示该方向上的最大变化率","；梯度的负方向则是函数值下降最快的方向，这正是梯度下降法的理论基石。此外，梯度处处与函数的等值面（水平集）正交，这一事实在约束优化中的拉格朗日乘子法里起着核心作用。",[31,3085,3086],{"id":3086},"机器学习",[11,3088,3089,3090,3136,3137,3218,3219,3222],{},"在机器学习中，绝大多数模型的训练问题都可归结为最小化某个标量损失函数 ",[38,3091,3093,3115],{"className":3092,"translate":42},[41],[38,3094,3096],{"className":3095},[46],[48,3097,3098],{"xmlns":50},[52,3099,3100,3112],{},[55,3101,3102,3105,3107,3110],{},[58,3103,3104],{},"L",[62,3106,293],{"stretchy":292},[58,3108,3109],{},"θ",[62,3111,330],{"stretchy":292},[81,3113,3114],{"encoding":83},"L(\\theta)",[38,3116,3118],{"className":3117,"ariaHidden":89},[88],[38,3119,3121,3124,3127,3130,3133],{"className":3120},[93],[38,3122],{"className":3123,"style":1439},[97],[38,3125,3104],{"className":3126},[102,103],[38,3128,293],{"className":3129},[371],[38,3131,3109],{"className":3132,"style":2500},[102,103],[38,3134,330],{"className":3135},[529],"，其中 ",[38,3138,3140,3162],{"className":3139,"translate":42},[41],[38,3141,3143],{"className":3142},[46],[48,3144,3145],{"xmlns":50},[52,3146,3147,3159],{},[55,3148,3149,3151,3153],{},[58,3150,3109],{},[62,3152,1420],{},[66,3154,3155,3157],{},[58,3156,71],{"mathvariant":70},[58,3158,11],{},[81,3160,3161],{"encoding":83},"\\theta \\in \\mathbb{R}^p",[38,3163,3165,3183],{"className":3164,"ariaHidden":89},[88],[38,3166,3168,3171,3174,3177,3180],{"className":3167},[93],[38,3169],{"className":3170,"style":2070},[97],[38,3172,3109],{"className":3173,"style":2500},[102,103],[38,3175],{"className":3176,"style":109},[108],[38,3178,1420],{"className":3179},[113],[38,3181],{"className":3182,"style":109},[108],[38,3184,3186,3189],{"className":3185},[93],[38,3187],{"className":3188,"style":123},[97],[38,3190,3192,3195],{"className":3191},[102],[38,3193,71],{"className":3194},[102,130],[38,3196,3198],{"className":3197},[134],[38,3199,3201],{"className":3200},[138],[38,3202,3204],{"className":3203},[142],[38,3205,3207],{"className":3206,"style":147},[146],[38,3208,3209,3212],{"style":150},[38,3210],{"className":3211,"style":155},[154],[38,3213,3215],{"className":3214},[159,160,161,162],[38,3216,11],{"className":3217},[102,103,162]," 为模型参数。",[15,3220,3221],{},"梯度下降法","（gradient descent）利用梯度的最速下降性质，通过迭代",[38,3224,3226],{"className":3225,"translate":42},[621],[38,3227,3229,3285],{"className":3228,"translate":42},[41],[38,3230,3232],{"className":3231},[46],[48,3233,3234],{"xmlns":50,"display":630},[52,3235,3236,3282],{},[55,3237,3238,3251,3253,3259,3261,3264,3268,3270,3272,3274,3280],{},[295,3239,3240,3242],{},[58,3241,3109],{},[55,3243,3244,3247,3249],{},[58,3245,3246],{},"t",[62,3248,696],{},[300,3250,302],{},[62,3252,289],{},[295,3254,3255,3257],{},[58,3256,3109],{},[58,3258,3246],{},[62,3260,710],{},[58,3262,3263],{},"η",[3265,3266,3267],"mtext",{}," ",[58,3269,1335],{"mathvariant":333},[58,3271,3104],{},[62,3273,293],{"stretchy":292},[295,3275,3276,3278],{},[58,3277,3109],{},[58,3279,3246],{},[62,3281,330],{"stretchy":292},[81,3283,3284],{"encoding":83},"\\theta_{t+1} = \\theta_t - \\eta \\, \\nabla L(\\theta_t)",[38,3286,3288,3354,3411],{"className":3287,"ariaHidden":89},[88],[38,3289,3291,3295,3345,3348,3351],{"className":3290},[93],[38,3292],{"className":3293,"style":3294},[97],"height:0.9028em;vertical-align:-0.2083em;",[38,3296,3298,3301],{"className":3297},[102],[38,3299,3109],{"className":3300,"style":2500},[102,103],[38,3302,3304],{"className":3303},[134],[38,3305,3307,3336],{"className":3306},[138,384],[38,3308,3310,3333],{"className":3309},[142],[38,3311,3313],{"className":3312,"style":391},[146],[38,3314,3315,3318],{"style":2516},[38,3316],{"className":3317,"style":155},[154],[38,3319,3321],{"className":3320},[159,160,161,162],[38,3322,3324,3327,3330],{"className":3323},[102,162],[38,3325,3246],{"className":3326},[102,103,162],[38,3328,696],{"className":3329},[999,162],[38,3331,302],{"className":3332},[102,162],[38,3334,408],{"className":3335},[407],[38,3337,3339],{"className":3338},[142],[38,3340,3343],{"className":3341,"style":3342},[146],"height:0.2083em;",[38,3344],{},[38,3346],{"className":3347,"style":109},[108],[38,3349,289],{"className":3350},[113],[38,3352],{"className":3353,"style":109},[108],[38,3355,3357,3361,3402,3405,3408],{"className":3356},[93],[38,3358],{"className":3359,"style":3360},[97],"height:0.8444em;vertical-align:-0.15em;",[38,3362,3364,3367],{"className":3363},[102],[38,3365,3109],{"className":3366,"style":2500},[102,103],[38,3368,3370],{"className":3369},[134],[38,3371,3373,3394],{"className":3372},[138,384],[38,3374,3376,3391],{"className":3375},[142],[38,3377,3380],{"className":3378,"style":3379},[146],"height:0.2806em;",[38,3381,3382,3385],{"style":2516},[38,3383],{"className":3384,"style":155},[154],[38,3386,3388],{"className":3387},[159,160,161,162],[38,3389,3246],{"className":3390},[102,103,162],[38,3392,408],{"className":3393},[407],[38,3395,3397],{"className":3396},[142],[38,3398,3400],{"className":3399,"style":415},[146],[38,3401],{},[38,3403],{"className":3404,"style":995},[108],[38,3406,710],{"className":3407},[999],[38,3409],{"className":3410,"style":995},[108],[38,3412,3414,3417,3421,3424,3427,3430,3433,3473],{"className":3413},[93],[38,3415],{"className":3416,"style":1439},[97],[38,3418,3263],{"className":3419,"style":3420},[102,103],"margin-right:0.0359em;",[38,3422],{"className":3423,"style":425},[108],[38,3425,1335],{"className":3426},[102],[38,3428,3104],{"className":3429},[102,103],[38,3431,293],{"className":3432},[371],[38,3434,3436,3439],{"className":3435},[102],[38,3437,3109],{"className":3438,"style":2500},[102,103],[38,3440,3442],{"className":3441},[134],[38,3443,3445,3465],{"className":3444},[138,384],[38,3446,3448,3462],{"className":3447},[142],[38,3449,3451],{"className":3450,"style":3379},[146],[38,3452,3453,3456],{"style":2516},[38,3454],{"className":3455,"style":155},[154],[38,3457,3459],{"className":3458},[159,160,161,162],[38,3460,3246],{"className":3461},[102,103,162],[38,3463,408],{"className":3464},[407],[38,3466,3468],{"className":3467},[142],[38,3469,3471],{"className":3470,"style":415},[146],[38,3472],{},[38,3474,330],{"className":3475},[529],[11,3477,3478,3479,3509],{},"不断沿负梯度方向更新参数，其中 ",[38,3480,3482,3496],{"className":3481,"translate":42},[41],[38,3483,3485],{"className":3484},[46],[48,3486,3487],{"xmlns":50},[52,3488,3489,3493],{},[55,3490,3491],{},[58,3492,3263],{},[81,3494,3495],{"encoding":83},"\\eta",[38,3497,3499],{"className":3498,"ariaHidden":89},[88],[38,3500,3502,3506],{"className":3501},[93],[38,3503],{"className":3504,"style":3505},[97],"height:0.625em;vertical-align:-0.1944em;",[38,3507,3263],{"className":3508,"style":3420},[102,103]," 为学习率。随机梯度下降（SGD）、动量法、Adam 等现代优化器均是在此基础上对梯度信息的进一步加工与修正。",[11,3511,3512,3513,3516],{},"在深度神经网络中，参数量往往达到百万甚至千亿级别，逐分量手工推导偏导数并不可行，",[15,3514,3515],{},"反向传播算法","（backpropagation）正是利用链式法则高效计算梯度的系统性方法——其本质是对下文将要讨论的雅可比矩阵的链式复合。",[28,3518,21],{"id":21},[31,3520,3521],{"id":3521},"标量到向量值函数",[11,3523,3524,3525,3659],{},"当函数不再是标量值，而是 ",[38,3526,3528,3559],{"className":3527,"translate":42},[41],[38,3529,3531],{"className":3530},[46],[48,3532,3533],{"xmlns":50},[52,3534,3535,3556],{},[55,3536,3537,3539,3541,3547,3549],{},[58,3538,60],{"mathvariant":285},[62,3540,64],{},[66,3542,3543,3545],{},[58,3544,71],{"mathvariant":70},[58,3546,74],{},[62,3548,77],{},[66,3550,3551,3553],{},[58,3552,71],{"mathvariant":70},[58,3554,3555],{},"m",[81,3557,3558],{"encoding":83},"\\mathbf{f}: \\mathbb{R}^n \\to \\mathbb{R}^m",[38,3560,3562,3580,3624],{"className":3561,"ariaHidden":89},[88],[38,3563,3565,3568,3571,3574,3577],{"className":3564},[93],[38,3566],{"className":3567,"style":889},[97],[38,3569,60],{"className":3570,"style":225},[102,351],[38,3572],{"className":3573,"style":109},[108],[38,3575,64],{"className":3576},[113],[38,3578],{"className":3579,"style":109},[108],[38,3581,3583,3586,3615,3618,3621],{"className":3582},[93],[38,3584],{"className":3585,"style":123},[97],[38,3587,3589,3592],{"className":3588},[102],[38,3590,71],{"className":3591},[102,130],[38,3593,3595],{"className":3594},[134],[38,3596,3598],{"className":3597},[138],[38,3599,3601],{"className":3600},[142],[38,3602,3604],{"className":3603,"style":147},[146],[38,3605,3606,3609],{"style":150},[38,3607],{"className":3608,"style":155},[154],[38,3610,3612],{"className":3611},[159,160,161,162],[38,3613,74],{"className":3614},[102,103,162],[38,3616],{"className":3617,"style":109},[108],[38,3619,77],{"className":3620},[113],[38,3622],{"className":3623,"style":109},[108],[38,3625,3627,3630],{"className":3626},[93],[38,3628],{"className":3629,"style":123},[97],[38,3631,3633,3636],{"className":3632},[102],[38,3634,71],{"className":3635},[102,130],[38,3637,3639],{"className":3638},[134],[38,3640,3642],{"className":3641},[138],[38,3643,3645],{"className":3644},[142],[38,3646,3648],{"className":3647,"style":147},[146],[38,3649,3650,3653],{"style":150},[38,3651],{"className":3652,"style":155},[154],[38,3654,3656],{"className":3655},[159,160,161,162],[38,3657,3555],{"className":3658},[102,103,162],"，即",[38,3661,3663],{"className":3662,"translate":42},[621],[38,3664,3666,3743],{"className":3665,"translate":42},[41],[38,3667,3669],{"className":3668},[46],[48,3670,3671],{"xmlns":50,"display":630},[52,3672,3673,3740],{},[55,3674,3675,3677,3679,3681,3683,3685,3688,3694,3696,3698,3700,3702,3708,3710,3712,3714,3716,3718,3720,3726,3728,3730,3732,3738],{},[58,3676,60],{"mathvariant":285},[62,3678,293],{"stretchy":292},[58,3680,286],{"mathvariant":285},[62,3682,330],{"stretchy":292},[62,3684,289],{},[62,3686,293],{"fence":292,"stretchy":89,"minsize":3687,"maxsize":3687},"1.2em",[295,3689,3690,3692],{},[58,3691,60],{},[300,3693,302],{},[62,3695,293],{"stretchy":292},[58,3697,286],{"mathvariant":285},[62,3699,330],{"stretchy":292},[62,3701,305],{"separator":89},[295,3703,3704,3706],{},[58,3705,60],{},[300,3707,312],{},[62,3709,293],{"stretchy":292},[58,3711,286],{"mathvariant":285},[62,3713,330],{"stretchy":292},[62,3715,305],{"separator":89},[62,3717,317],{},[62,3719,305],{"separator":89},[295,3721,3722,3724],{},[58,3723,60],{},[58,3725,3555],{},[62,3727,293],{"stretchy":292},[58,3729,286],{"mathvariant":285},[62,3731,330],{"stretchy":292},[66,3733,3734,3736],{},[62,3735,330],{"fence":292,"stretchy":89,"minsize":3687,"maxsize":3687},[58,3737,334],{"mathvariant":333},[62,3739,305],{"separator":89},[81,3741,3742],{"encoding":83},"\\mathbf{f}(\\mathbf{x}) = \\big(f_1(\\mathbf{x}), f_2(\\mathbf{x}), \\dots, f_m(\\mathbf{x})\\big)^\\top,",[38,3744,3746,3773],{"className":3745,"ariaHidden":89},[88],[38,3747,3749,3752,3755,3758,3761,3764,3767,3770],{"className":3748},[93],[38,3750],{"className":3751,"style":1439},[97],[38,3753,60],{"className":3754,"style":225},[102,351],[38,3756,293],{"className":3757},[371],[38,3759,286],{"className":3760},[102,351],[38,3762,330],{"className":3763},[529],[38,3765],{"className":3766,"style":109},[108],[38,3768,289],{"className":3769},[113],[38,3771],{"className":3772,"style":109},[108],[38,3774,3776,3780,3787,3828,3831,3834,3837,3840,3843,3883,3886,3889,3892,3895,3898,3901,3904,3907,3910,3950,3953,3956,3959,3993],{"className":3775},[93],[38,3777],{"className":3778,"style":3779},[97],"height:1.439em;vertical-align:-0.35em;",[38,3781,3783],{"className":3782},[102],[38,3784,293],{"className":3785},[1485,3786],"size1",[38,3788,3790,3793],{"className":3789},[102],[38,3791,60],{"className":3792,"style":104},[102,103],[38,3794,3796],{"className":3795},[134],[38,3797,3799,3820],{"className":3798},[138,384],[38,3800,3802,3817],{"className":3801},[142],[38,3803,3805],{"className":3804,"style":391},[146],[38,3806,3808,3811],{"style":3807},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[38,3809],{"className":3810,"style":155},[154],[38,3812,3814],{"className":3813},[159,160,161,162],[38,3815,302],{"className":3816},[102,162],[38,3818,408],{"className":3819},[407],[38,3821,3823],{"className":3822},[142],[38,3824,3826],{"className":3825,"style":415},[146],[38,3827],{},[38,3829,293],{"className":3830},[371],[38,3832,286],{"className":3833},[102,351],[38,3835,330],{"className":3836},[529],[38,3838,305],{"className":3839},[421],[38,3841],{"className":3842,"style":425},[108],[38,3844,3846,3849],{"className":3845},[102],[38,3847,60],{"className":3848,"style":104},[102,103],[38,3850,3852],{"className":3851},[134],[38,3853,3855,3875],{"className":3854},[138,384],[38,3856,3858,3872],{"className":3857},[142],[38,3859,3861],{"className":3860,"style":391},[146],[38,3862,3863,3866],{"style":3807},[38,3864],{"className":3865,"style":155},[154],[38,3867,3869],{"className":3868},[159,160,161,162],[38,3870,312],{"className":3871},[102,162],[38,3873,408],{"className":3874},[407],[38,3876,3878],{"className":3877},[142],[38,3879,3881],{"className":3880,"style":415},[146],[38,3882],{},[38,3884,293],{"className":3885},[371],[38,3887,286],{"className":3888},[102,351],[38,3890,330],{"className":3891},[529],[38,3893,305],{"className":3894},[421],[38,3896],{"className":3897,"style":425},[108],[38,3899,317],{"className":3900},[475],[38,3902],{"className":3903,"style":425},[108],[38,3905,305],{"className":3906},[421],[38,3908],{"className":3909,"style":425},[108],[38,3911,3913,3916],{"className":3912},[102],[38,3914,60],{"className":3915,"style":104},[102,103],[38,3917,3919],{"className":3918},[134],[38,3920,3922,3942],{"className":3921},[138,384],[38,3923,3925,3939],{"className":3924},[142],[38,3926,3928],{"className":3927,"style":503},[146],[38,3929,3930,3933],{"style":3807},[38,3931],{"className":3932,"style":155},[154],[38,3934,3936],{"className":3935},[159,160,161,162],[38,3937,3555],{"className":3938},[102,103,162],[38,3940,408],{"className":3941},[407],[38,3943,3945],{"className":3944},[142],[38,3946,3948],{"className":3947,"style":415},[146],[38,3949],{},[38,3951,293],{"className":3952},[371],[38,3954,286],{"className":3955},[102,351],[38,3957,330],{"className":3958},[529],[38,3960,3962,3968],{"className":3961},[102],[38,3963,3965],{"className":3964},[102],[38,3966,330],{"className":3967},[1485,3786],[38,3969,3971],{"className":3970},[134],[38,3972,3974],{"className":3973},[138],[38,3975,3977],{"className":3976},[142],[38,3978,3981],{"className":3979,"style":3980},[146],"height:1.089em;",[38,3982,3984,3987],{"style":3983},"top:-3.3029em;margin-right:0.05em;",[38,3985],{"className":3986,"style":155},[154],[38,3988,3990],{"className":3989},[159,160,161,162],[38,3991,334],{"className":3992},[102,162],[38,3994,305],{"className":3995},[421],[11,3997,3998,3999,4143,4144,4286,4287,4316],{},"每一个分量函数 ",[38,4000,4002,4032],{"className":4001,"translate":42},[41],[38,4003,4005],{"className":4004},[46],[48,4006,4007],{"xmlns":50},[52,4008,4009,4029],{},[55,4010,4011,4017,4019,4025,4027],{},[295,4012,4013,4015],{},[58,4014,60],{},[58,4016,601],{},[62,4018,64],{},[66,4020,4021,4023],{},[58,4022,71],{"mathvariant":70},[58,4024,74],{},[62,4026,77],{},[58,4028,71],{"mathvariant":70},[81,4030,4031],{"encoding":83},"f_i: \\mathbb{R}^n \\to \\mathbb{R}",[38,4033,4035,4090,4134],{"className":4034,"ariaHidden":89},[88],[38,4036,4038,4041,4081,4084,4087],{"className":4037},[93],[38,4039],{"className":4040,"style":98},[97],[38,4042,4044,4047],{"className":4043},[102],[38,4045,60],{"className":4046,"style":104},[102,103],[38,4048,4050],{"className":4049},[134],[38,4051,4053,4073],{"className":4052},[138,384],[38,4054,4056,4070],{"className":4055},[142],[38,4057,4059],{"className":4058,"style":786},[146],[38,4060,4061,4064],{"style":3807},[38,4062],{"className":4063,"style":155},[154],[38,4065,4067],{"className":4066},[159,160,161,162],[38,4068,601],{"className":4069},[102,103,162],[38,4071,408],{"className":4072},[407],[38,4074,4076],{"className":4075},[142],[38,4077,4079],{"className":4078,"style":415},[146],[38,4080],{},[38,4082],{"className":4083,"style":109},[108],[38,4085,64],{"className":4086},[113],[38,4088],{"className":4089,"style":109},[108],[38,4091,4093,4096,4125,4128,4131],{"className":4092},[93],[38,4094],{"className":4095,"style":123},[97],[38,4097,4099,4102],{"className":4098},[102],[38,4100,71],{"className":4101},[102,130],[38,4103,4105],{"className":4104},[134],[38,4106,4108],{"className":4107},[138],[38,4109,4111],{"className":4110},[142],[38,4112,4114],{"className":4113,"style":147},[146],[38,4115,4116,4119],{"style":150},[38,4117],{"className":4118,"style":155},[154],[38,4120,4122],{"className":4121},[159,160,161,162],[38,4123,74],{"className":4124},[102,103,162],[38,4126],{"className":4127,"style":109},[108],[38,4129,77],{"className":4130},[113],[38,4132],{"className":4133,"style":109},[108],[38,4135,4137,4140],{"className":4136},[93],[38,4138],{"className":4139,"style":123},[97],[38,4141,71],{"className":4142},[102,130]," 都各自拥有一个梯度 ",[38,4145,4147,4181],{"className":4146,"translate":42},[41],[38,4148,4150],{"className":4149},[46],[48,4151,4152],{"xmlns":50},[52,4153,4154,4178],{},[55,4155,4156,4158,4164,4166,4168,4170,4172],{},[58,4157,1335],{"mathvariant":333},[295,4159,4160,4162],{},[58,4161,60],{},[58,4163,601],{},[62,4165,293],{"stretchy":292},[58,4167,286],{"mathvariant":285},[62,4169,330],{"stretchy":292},[62,4171,1420],{},[66,4173,4174,4176],{},[58,4175,71],{"mathvariant":70},[58,4177,74],{},[81,4179,4180],{"encoding":83},"\\nabla f_i(\\mathbf{x}) \\in \\mathbb{R}^n",[38,4182,4184,4251],{"className":4183,"ariaHidden":89},[88],[38,4185,4187,4190,4193,4233,4236,4239,4242,4245,4248],{"className":4186},[93],[38,4188],{"className":4189,"style":1439},[97],[38,4191,1335],{"className":4192},[102],[38,4194,4196,4199],{"className":4195},[102],[38,4197,60],{"className":4198,"style":104},[102,103],[38,4200,4202],{"className":4201},[134],[38,4203,4205,4225],{"className":4204},[138,384],[38,4206,4208,4222],{"className":4207},[142],[38,4209,4211],{"className":4210,"style":786},[146],[38,4212,4213,4216],{"style":3807},[38,4214],{"className":4215,"style":155},[154],[38,4217,4219],{"className":4218},[159,160,161,162],[38,4220,601],{"className":4221},[102,103,162],[38,4223,408],{"className":4224},[407],[38,4226,4228],{"className":4227},[142],[38,4229,4231],{"className":4230,"style":415},[146],[38,4232],{},[38,4234,293],{"className":4235},[371],[38,4237,286],{"className":4238},[102,351],[38,4240,330],{"className":4241},[529],[38,4243],{"className":4244,"style":109},[108],[38,4246,1420],{"className":4247},[113],[38,4249],{"className":4250,"style":109},[108],[38,4252,4254,4257],{"className":4253},[93],[38,4255],{"className":4256,"style":123},[97],[38,4258,4260,4263],{"className":4259},[102],[38,4261,71],{"className":4262},[102,130],[38,4264,4266],{"className":4265},[134],[38,4267,4269],{"className":4268},[138],[38,4270,4272],{"className":4271},[142],[38,4273,4275],{"className":4274,"style":147},[146],[38,4276,4277,4280],{"style":150},[38,4278],{"className":4279,"style":155},[154],[38,4281,4283],{"className":4282},[159,160,161,162],[38,4284,74],{"className":4285},[102,103,162],"。将这 ",[38,4288,4290,4303],{"className":4289,"translate":42},[41],[38,4291,4293],{"className":4292},[46],[48,4294,4295],{"xmlns":50},[52,4296,4297,4301],{},[55,4298,4299],{},[58,4300,3555],{},[81,4302,3555],{"encoding":83},[38,4304,4306],{"className":4305,"ariaHidden":89},[88],[38,4307,4309,4313],{"className":4308},[93],[38,4310],{"className":4311,"style":4312},[97],"height:0.4306em;",[38,4314,3555],{"className":4315},[102,103]," 个梯度向量按行排列，便得到雅可比矩阵。",[31,4318,1250],{"id":4319},"定义-1",[1252,4321,4322,4440,5995],{},[11,4323,4324,1259,4327,4356,4357,4385,4386,4439],{},[15,4325,4326],{},"定义 2.1（雅可比矩阵）",[38,4328,4330,4344],{"className":4329,"translate":42},[41],[38,4331,4333],{"className":4332},[46],[48,4334,4335],{"xmlns":50},[52,4336,4337,4341],{},[55,4338,4339],{},[58,4340,60],{"mathvariant":285},[81,4342,4343],{"encoding":83},"\\mathbf{f}",[38,4345,4347],{"className":4346,"ariaHidden":89},[88],[38,4348,4350,4353],{"className":4349},[93],[38,4351],{"className":4352,"style":889},[97],[38,4354,60],{"className":4355,"style":225},[102,351]," 的每个分量在 ",[38,4358,4360,4373],{"className":4359,"translate":42},[41],[38,4361,4363],{"className":4362},[46],[48,4364,4365],{"xmlns":50},[52,4366,4367,4371],{},[55,4368,4369],{},[58,4370,286],{"mathvariant":285},[81,4372,1104],{"encoding":83},[38,4374,4376],{"className":4375,"ariaHidden":89},[88],[38,4377,4379,4382],{"className":4378},[93],[38,4380],{"className":4381,"style":347},[97],[38,4383,286],{"className":4384},[102,351]," 处均可偏导，则 ",[38,4387,4389,4408],{"className":4388,"translate":42},[41],[38,4390,4392],{"className":4391},[46],[48,4393,4394],{"xmlns":50},[52,4395,4396,4405],{},[55,4397,4398,4400,4403],{},[58,4399,3555],{},[62,4401,4402],{},"×",[58,4404,74],{},[81,4406,4407],{"encoding":83},"m \\times n",[38,4409,4411,4430],{"className":4410,"ariaHidden":89},[88],[38,4412,4414,4418,4421,4424,4427],{"className":4413},[93],[38,4415],{"className":4416,"style":4417},[97],"height:0.6667em;vertical-align:-0.0833em;",[38,4419,3555],{"className":4420},[102,103],[38,4422],{"className":4423,"style":995},[108],[38,4425,4402],{"className":4426},[999],[38,4428],{"className":4429,"style":995},[108],[38,4431,4433,4436],{"className":4432},[93],[38,4434],{"className":4435,"style":4312},[97],[38,4437,74],{"className":4438},[102,103]," 矩阵",[38,4441,4443],{"className":4442,"translate":42},[621],[38,4444,4446,4731],{"className":4445,"translate":42},[41],[38,4447,4449],{"className":4448},[46],[48,4450,4451],{"xmlns":50,"display":630},[52,4452,4453,4728],{},[55,4454,4455,4462,4464,4466,4468,4470,4651,4653],{},[295,4456,4457,4460],{},[58,4458,4459],{},"J",[58,4461,60],{"mathvariant":285},[62,4463,293],{"stretchy":292},[58,4465,286],{"mathvariant":285},[62,4467,330],{"stretchy":292},[62,4469,289],{},[55,4471,4472,4474,4649],{},[62,4473,293],{"fence":89},[4475,4476,4480,4548,4585],"mtable",{"rowspacing":4477,"columnalign":4478,"columnspacing":4479},"0.16em","center center 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=\n\\begin{pmatrix}\n\\dfrac{\\partial f_1}{\\partial x_1} & \\cdots & \\dfrac{\\partial f_1}{\\partial x_n} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\dfrac{\\partial f_m}{\\partial x_1} & \\cdots & \\dfrac{\\partial f_m}{\\partial x_n}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n\\nabla f_1(\\mathbf{x})^\\top \\\\\n\\vdots \\\\\n\\nabla 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",[38,6057,6059,6083],{"className":6058,"translate":42},[41],[38,6060,6062],{"className":6061},[46],[48,6063,6064],{"xmlns":50},[52,6065,6066,6080],{},[55,6067,6068,6074,6076,6078],{},[295,6069,6070,6072],{},[58,6071,4459],{},[58,6073,60],{"mathvariant":285},[62,6075,293],{"stretchy":292},[58,6077,286],{"mathvariant":285},[62,6079,330],{"stretchy":292},[81,6081,6082],{"encoding":83},"J_{\\mathbf{f}}(\\mathbf{x})",[38,6084,6086],{"className":6085,"ariaHidden":89},[88],[38,6087,6089,6092,6135,6138,6141],{"className":6088},[93],[38,6090],{"className":6091,"style":1439},[97],[38,6093,6095,6098],{"className":6094},[102],[38,6096,4459],{"className":6097,"style":4746},[102,103],[38,6099,6101],{"className":6100},[134],[38,6102,6104,6127],{"className":6103},[138,384],[38,6105,6107,6124],{"className":6106},[142],[38,6108,6110],{"className":6109,"style":4759},[146],[38,6111,6112,6115],{"style":4762},[38,6113],{"className":6114,"style":155},[154],[38,6116,6118],{"className":6117},[159,160,161,162],[38,6119,6121],{"className":6120},[102,162],[38,6122,60],{"className":6123,"style":225},[102,351,162],[38,6125,408],{"className":6126},[407],[38,6128,6130],{"className":6129},[142],[38,6131,6133],{"className":6132,"style":415},[146],[38,6134],{},[38,6136,293],{"className":6137},[371],[38,6139,286],{"className":6140},[102,351],[38,6142,330],{"className":6143},[529]," 或 ",[38,6146,6148,6176],{"className":6147,"translate":42},[41],[38,6149,6151],{"className":6150},[46],[48,6152,6153],{"xmlns":50},[52,6154,6155,6173],{},[55,6156,6157],{},[2631,6158,6159],{"scriptlevel":683,"displaystyle":89},[636,6160,6161,6167],{},[55,6162,6163,6165],{},[58,6164,642],{"mathvariant":333},[58,6166,60],{"mathvariant":285},[55,6168,6169,6171],{},[58,6170,642],{"mathvariant":333},[58,6172,286],{"mathvariant":285},[81,6174,6175],{"encoding":83},"\\dfrac{\\partial \\mathbf{f}}{\\partial \\mathbf{x}}",[38,6177,6179],{"className":6178,"ariaHidden":89},[88],[38,6180,6182,6186],{"className":6181},[93],[38,6183],{"className":6184,"style":6185},[97],"height:2.0574em;vertical-align:-0.686em;",[38,6187,6189,6192,6251],{"className":6188},[102],[38,6190],{"className":6191},[371,740],[38,6193,6195],{"className":6194},[636],[38,6196,6198,6243],{"className":6197},[138,384],[38,6199,6201,6240],{"className":6200},[142],[38,6202,6204,6218,6226],{"className":6203,"style":753},[146],[38,6205,6206,6209],{"style":756},[38,6207],{"className":6208,"style":760},[154],[38,6210,6212,6215],{"className":6211},[102],[38,6213,642],{"className":6214,"style":767},[102],[38,6216,286],{"className":6217},[102,351],[38,6219,6220,6223],{"style":811},[38,6221],{"className":6222,"style":760},[154],[38,6224],{"className":6225,"style":819},[818],[38,6227,6228,6231],{"style":822},[38,6229],{"className":6230,"style":760},[154],[38,6232,6234,6237],{"className":6233},[102],[38,6235,642],{"className":6236,"style":767},[102],[38,6238,60],{"className":6239,"style":225},[102,351],[38,6241,408],{"className":6242},[407],[38,6244,6246],{"className":6245},[142],[38,6247,6249],{"className":6248,"style":1079},[146],[38,6250],{},[38,6252],{"className":6253},[529,740],[11,6255,6256,6257,6312],{},"由此可见，",[15,6258,6259,6260,6311],{},"梯度是雅可比矩阵在 ",[38,6261,6263,6281],{"className":6262,"translate":42},[41],[38,6264,6266],{"className":6265},[46],[48,6267,6268],{"xmlns":50},[52,6269,6270,6278],{},[55,6271,6272,6274,6276],{},[58,6273,3555],{},[62,6275,289],{},[300,6277,302],{},[81,6279,6280],{"encoding":83},"m=1",[38,6282,6284,6302],{"className":6283,"ariaHidden":89},[88],[38,6285,6287,6290,6293,6296,6299],{"className":6286},[93],[38,6288],{"className":6289,"style":4312},[97],[38,6291,3555],{"className":6292},[102,103],[38,6294],{"className":6295,"style":109},[108],[38,6297,289],{"className":6298},[113],[38,6300],{"className":6301,"style":109},[108],[38,6303,6305,6308],{"className":6304},[93],[38,6306],{"className":6307,"style":2429},[97],[38,6309,302],{"className":6310},[102]," 时的特例","（此时雅可比退化为一个行向量，其转置即为梯度）。",[31,6314,6315],{"id":6315},"线性算子",[1252,6317,6318,6380],{},[11,6319,6320,1259,6323,1288,6351,6379],{},[15,6321,6322],{},"定理 2.1（多元一阶泰勒近似）",[38,6324,6326,6339],{"className":6325,"translate":42},[41],[38,6327,6329],{"className":6328},[46],[48,6330,6331],{"xmlns":50},[52,6332,6333,6337],{},[55,6334,6335],{},[58,6336,60],{"mathvariant":285},[81,6338,4343],{"encoding":83},[38,6340,6342],{"className":6341,"ariaHidden":89},[88],[38,6343,6345,6348],{"className":6344},[93],[38,6346],{"className":6347,"style":889},[97],[38,6349,60],{"className":6350,"style":225},[102,351],[38,6352,6354,6367],{"className":6353,"translate":42},[41],[38,6355,6357],{"className":6356},[46],[48,6358,6359],{"xmlns":50},[52,6360,6361,6365],{},[55,6362,6363],{},[58,6364,286],{"mathvariant":285},[81,6366,1104],{"encoding":83},[38,6368,6370],{"className":6369,"ariaHidden":89},[88],[38,6371,6373,6376],{"className":6372},[93],[38,6374],{"className":6375,"style":347},[97],[38,6377,286],{"className":6378},[102,351]," 处可微，则",[38,6381,6383],{"className":6382,"translate":42},[621],[38,6384,6386,6454],{"className":6385,"translate":42},[41],[38,6387,6389],{"className":6388},[46],[48,6390,6391],{"xmlns":50,"display":630},[52,6392,6393,6451],{},[55,6394,6395,6397,6399,6401,6403,6405,6407,6409,6411,6413,6415,6417,6419,6425,6427,6429,6431,6433,6435,6437,6439,6441,6443,6445,6447,6449],{},[58,6396,60],{"mathvariant":285},[62,6398,293],{"stretchy":292},[58,6400,286],{"mathvariant":285},[62,6402,696],{},[58,6404,678],{"mathvariant":285},[62,6406,330],{"stretchy":292},[62,6408,289],{},[58,6410,60],{"mathvariant":285},[62,6412,293],{"stretchy":292},[58,6414,286],{"mathvariant":285},[62,6416,330],{"stretchy":292},[62,6418,696],{},[295,6420,6421,6423],{},[58,6422,4459],{},[58,6424,60],{"mathvariant":285},[62,6426,293],{"stretchy":292},[58,6428,286],{"mathvariant":285},[62,6430,330],{"stretchy":292},[3265,6432,3267],{},[58,6434,678],{"mathvariant":285},[62,6436,696],{},[58,6438,2178],{},[62,6440,293],{"stretchy":292},[58,6442,2183],{"mathvariant":333},[58,6444,678],{"mathvariant":285},[58,6446,2183],{"mathvariant":333},[62,6448,330],{"stretchy":292},[58,6450,2192],{"mathvariant":333},[81,6452,6453],{"encoding":83},"\\mathbf{f}(\\mathbf{x} + \\mathbf{h}) = \\mathbf{f}(\\mathbf{x}) + J_{\\mathbf{f}}(\\mathbf{x}) \\, \\mathbf{h} + o(\\|\\mathbf{h}\\|).",[38,6455,6457,6481,6502,6529,6602],{"className":6456,"ariaHidden":89},[88],[38,6458,6460,6463,6466,6469,6472,6475,6478],{"className":6459},[93],[38,6461],{"className":6462,"style":1439},[97],[38,6464,60],{"className":6465,"style":225},[102,351],[38,6467,293],{"className":6468},[371],[38,6470,286],{"className":6471},[102,351],[38,6473],{"className":6474,"style":995},[108],[38,6476,696],{"className":6477},[999],[38,6479],{"className":6480,"style":995},[108],[38,6482,6484,6487,6490,6493,6496,6499],{"className":6483},[93],[38,6485],{"className":6486,"style":1439},[97],[38,6488,678],{"className":6489},[102,351],[38,6491,330],{"className":6492},[529],[38,6494],{"className":6495,"style":109},[108],[38,6497,289],{"className":6498},[113],[38,6500],{"className":6501,"style":109},[108],[38,6503,6505,6508,6511,6514,6517,6520,6523,6526],{"className":6504},[93],[38,6506],{"className":6507,"style":1439},[97],[38,6509,60],{"className":6510,"style":225},[102,351],[38,6512,293],{"className":6513},[371],[38,6515,286],{"className":6516},[102,351],[38,6518,330],{"className":6519},[529],[38,6521],{"className":6522,"style":995},[108],[38,6524,696],{"className":6525},[999],[38,6527],{"className":6528,"style":995},[108],[38,6530,6532,6535,6578,6581,6584,6587,6590,6593,6596,6599],{"className":6531},[93],[38,6533],{"className":6534,"style":1439},[97],[38,6536,6538,6541],{"className":6537},[102],[38,6539,4459],{"className":6540,"style":4746},[102,103],[38,6542,6544],{"className":6543},[134],[38,6545,6547,6570],{"className":6546},[138,384],[38,6548,6550,6567],{"className":6549},[142],[38,6551,6553],{"className":6552,"style":4759},[146],[38,6554,6555,6558],{"style":4762},[38,6556],{"className":6557,"style":155},[154],[38,6559,6561],{"className":6560},[159,160,161,162],[38,6562,6564],{"className":6563},[102,162],[38,6565,60],{"className":6566,"style":225},[102,351,162],[38,6568,408],{"className":6569},[407],[38,6571,6573],{"className":6572},[142],[38,6574,6576],{"className":6575,"style":415},[146],[38,6577],{},[38,6579,293],{"className":6580},[371],[38,6582,286],{"className":6583},[102,351],[38,6585,330],{"className":6586},[529],[38,6588],{"className":6589,"style":425},[108],[38,6591,678],{"className":6592},[102,351],[38,6594],{"className":6595,"style":995},[108],[38,6597,696],{"className":6598},[999],[38,6600],{"className":6601,"style":995},[108],[38,6603,6605,6608,6611,6614,6617,6620,6623,6626],{"className":6604},[93],[38,6606],{"className":6607,"style":1439},[97],[38,6609,2178],{"className":6610},[102,103],[38,6612,293],{"className":6613},[371],[38,6615,2183],{"className":6616},[102],[38,6618,678],{"className":6619},[102,351],[38,6621,2183],{"className":6622},[102],[38,6624,330],{"className":6625},[529],[38,6627,2192],{"className":6628},[102],[11,6630,6631],{},"雅可比矩阵因此扮演着\"局部最佳线性映射\"的角色：它是全导数（Fréchet 导数）在标准基下的矩阵表示，将输入空间的扰动方向线性地映射为输出空间的扰动方向。",[31,6633,6634],{"id":6634},"链式法则",[1252,6636,6637,7094],{},[11,6638,6639,2369,6642,6778,6779,6914,6915,7093],{},[15,6640,6641],{},"定理 2.2（链式法则的矩阵形式）",[38,6643,6645,6676],{"className":6644,"translate":42},[41],[38,6646,6648],{"className":6647},[46],[48,6649,6650],{"xmlns":50},[52,6651,6652,6673],{},[55,6653,6654,6657,6659,6665,6667],{},[58,6655,6656],{"mathvariant":285},"g",[62,6658,64],{},[66,6660,6661,6663],{},[58,6662,71],{"mathvariant":70},[58,6664,74],{},[62,6666,77],{},[66,6668,6669,6671],{},[58,6670,71],{"mathvariant":70},[58,6672,3555],{},[81,6674,6675],{"encoding":83},"\\mathbf{g}: \\mathbb{R}^n \\to \\mathbb{R}^m",[38,6677,6679,6699,6743],{"className":6678,"ariaHidden":89},[88],[38,6680,6682,6686,6690,6693,6696],{"className":6681},[93],[38,6683],{"className":6684,"style":6685},[97],"height:0.6389em;vertical-align:-0.1944em;",[38,6687,6656],{"className":6688,"style":6689},[102,351],"margin-right:0.016em;",[38,6691],{"className":6692,"style":109},[108],[38,6694,64],{"className":6695},[113],[38,6697],{"className":6698,"style":109},[108],[38,6700,6702,6705,6734,6737,6740],{"className":6701},[93],[38,6703],{"className":6704,"style":123},[97],[38,6706,6708,6711],{"className":6707},[102],[38,6709,71],{"className":6710},[102,130],[38,6712,6714],{"className":6713},[134],[38,6715,6717],{"className":6716},[138],[38,6718,6720],{"className":6719},[142],[38,6721,6723],{"className":6722,"style":147},[146],[38,6724,6725,6728],{"style":150},[38,6726],{"className":6727,"style":155},[154],[38,6729,6731],{"className":6730},[159,160,161,162],[38,6732,74],{"className":6733},[102,103,162],[38,6735],{"className":6736,"style":109},[108],[38,6738,77],{"className":6739},[113],[38,6741],{"className":6742,"style":109},[108],[38,6744,6746,6749],{"className":6745},[93],[38,6747],{"className":6748,"style":123},[97],[38,6750,6752,6755],{"className":6751},[102],[38,6753,71],{"className":6754},[102,130],[38,6756,6758],{"className":6757},[134],[38,6759,6761],{"className":6760},[138],[38,6762,6764],{"className":6763},[142],[38,6765,6767],{"className":6766,"style":147},[146],[38,6768,6769,6772],{"style":150},[38,6770],{"className":6771,"style":155},[154],[38,6773,6775],{"className":6774},[159,160,161,162],[38,6776,3555],{"className":6777},[102,103,162],"，",[38,6780,6782,6813],{"className":6781,"translate":42},[41],[38,6783,6785],{"className":6784},[46],[48,6786,6787],{"xmlns":50},[52,6788,6789,6810],{},[55,6790,6791,6793,6795,6801,6803],{},[58,6792,60],{"mathvariant":285},[62,6794,64],{},[66,6796,6797,6799],{},[58,6798,71],{"mathvariant":70},[58,6800,3555],{},[62,6802,77],{},[66,6804,6805,6807],{},[58,6806,71],{"mathvariant":70},[58,6808,6809],{},"k",[81,6811,6812],{"encoding":83},"\\mathbf{f}: \\mathbb{R}^m \\to \\mathbb{R}^k",[38,6814,6816,6834,6878],{"className":6815,"ariaHidden":89},[88],[38,6817,6819,6822,6825,6828,6831],{"className":6818},[93],[38,6820],{"className":6821,"style":889},[97],[38,6823,60],{"className":6824,"style":225},[102,351],[38,6826],{"className":6827,"style":109},[108],[38,6829,64],{"className":6830},[113],[38,6832],{"className":6833,"style":109},[108],[38,6835,6837,6840,6869,6872,6875],{"className":6836},[93],[38,6838],{"className":6839,"style":123},[97],[38,6841,6843,6846],{"className":6842},[102],[38,6844,71],{"className":6845},[102,130],[38,6847,6849],{"className":6848},[134],[38,6850,6852],{"className":6851},[138],[38,6853,6855],{"className":6854},[142],[38,6856,6858],{"className":6857,"style":147},[146],[38,6859,6860,6863],{"style":150},[38,6861],{"className":6862,"style":155},[154],[38,6864,6866],{"className":6865},[159,160,161,162],[38,6867,3555],{"className":6868},[102,103,162],[38,6870],{"className":6871,"style":109},[108],[38,6873,77],{"className":6874},[113],[38,6876],{"className":6877,"style":109},[108],[38,6879,6881,6884],{"className":6880},[93],[38,6882],{"className":6883,"style":545},[97],[38,6885,6887,6890],{"className":6886},[102],[38,6888,71],{"className":6889},[102,130],[38,6891,6893],{"className":6892},[134],[38,6894,6896],{"className":6895},[138],[38,6897,6899],{"className":6898},[142],[38,6900,6902],{"className":6901,"style":545},[146],[38,6903,6904,6907],{"style":150},[38,6905],{"className":6906,"style":155},[154],[38,6908,6910],{"className":6909},[159,160,161,162],[38,6911,6809],{"className":6912,"style":6913},[102,103,162],"margin-right:0.0315em;","，复合函数 ",[38,6916,6918,6957],{"className":6917,"translate":42},[41],[38,6919,6921],{"className":6920},[46],[48,6922,6923],{"xmlns":50},[52,6924,6925,6954],{},[55,6926,6927,6929,6931,6933,6936,6938,6940,6946,6948],{},[58,6928,678],{"mathvariant":285},[62,6930,289],{},[58,6932,60],{"mathvariant":285},[62,6934,6935],{},"∘",[58,6937,6656],{"mathvariant":285},[62,6939,64],{},[66,6941,6942,6944],{},[58,6943,71],{"mathvariant":70},[58,6945,74],{},[62,6947,77],{},[66,6949,6950,6952],{},[58,6951,71],{"mathvariant":70},[58,6953,6809],{},[81,6955,6956],{"encoding":83},"\\mathbf{h} = \\mathbf{f} \\circ \\mathbf{g}: \\mathbb{R}^n \\to \\mathbb{R}^k",[38,6958,6960,6978,6996,7014,7058],{"className":6959,"ariaHidden":89},[88],[38,6961,6963,6966,6969,6972,6975],{"className":6962},[93],[38,6964],{"className":6965,"style":889},[97],[38,6967,678],{"className":6968},[102,351],[38,6970],{"className":6971,"style":109},[108],[38,6973,289],{"className":6974},[113],[38,6976],{"className":6977,"style":109},[108],[38,6979,6981,6984,6987,6990,6993],{"className":6980},[93],[38,6982],{"className":6983,"style":889},[97],[38,6985,60],{"className":6986,"style":225},[102,351],[38,6988],{"className":6989,"style":995},[108],[38,6991,6935],{"className":6992},[999],[38,6994],{"className":6995,"style":995},[108],[38,6997,6999,7002,7005,7008,7011],{"className":6998},[93],[38,7000],{"className":7001,"style":6685},[97],[38,7003,6656],{"className":7004,"style":6689},[102,351],[38,7006],{"className":7007,"style":109},[108],[38,7009,64],{"className":7010},[113],[38,7012],{"className":7013,"style":109},[108],[38,7015,7017,7020,7049,7052,7055],{"className":7016},[93],[38,7018],{"className":7019,"style":123},[97],[38,7021,7023,7026],{"className":7022},[102],[38,7024,71],{"className":7025},[102,130],[38,7027,7029],{"className":7028},[134],[38,7030,7032],{"className":7031},[138],[38,7033,7035],{"className":7034},[142],[38,7036,7038],{"className":7037,"style":147},[146],[38,7039,7040,7043],{"style":150},[38,7041],{"className":7042,"style":155},[154],[38,7044,7046],{"className":7045},[159,160,161,162],[38,7047,74],{"className":7048},[102,103,162],[38,7050],{"className":7051,"style":109},[108],[38,7053,77],{"className":7054},[113],[38,7056],{"className":7057,"style":109},[108],[38,7059,7061,7064],{"className":7060},[93],[38,7062],{"className":7063,"style":545},[97],[38,7065,7067,7070],{"className":7066},[102],[38,7068,71],{"className":7069},[102,130],[38,7071,7073],{"className":7072},[134],[38,7074,7076],{"className":7075},[138],[38,7077,7079],{"className":7078},[142],[38,7080,7082],{"className":7081,"style":545},[146],[38,7083,7084,7087],{"style":150},[38,7085],{"className":7086,"style":155},[154],[38,7088,7090],{"className":7089},[159,160,161,162],[38,7091,6809],{"className":7092,"style":6913},[102,103,162],"，则",[38,7095,7097],{"className":7096,"translate":42},[621],[38,7098,7100,7160],{"className":7099,"translate":42},[41],[38,7101,7103],{"className":7102},[46],[48,7104,7105],{"xmlns":50,"display":630},[52,7106,7107,7157],{},[55,7108,7109,7115,7117,7119,7121,7123,7129,7131,7133,7135,7137,7139,7141,7143,7149,7151,7153,7155],{},[295,7110,7111,7113],{},[58,7112,4459],{},[58,7114,678],{"mathvariant":285},[62,7116,293],{"stretchy":292},[58,7118,286],{"mathvariant":285},[62,7120,330],{"stretchy":292},[62,7122,289],{},[295,7124,7125,7127],{},[58,7126,4459],{},[58,7128,60],{"mathvariant":285},[62,7130,293],{"fence":292,"stretchy":89,"minsize":3687,"maxsize":3687},[58,7132,6656],{"mathvariant":285},[62,7134,293],{"stretchy":292},[58,7136,286],{"mathvariant":285},[62,7138,330],{"stretchy":292},[62,7140,330],{"fence":292,"stretchy":89,"minsize":3687,"maxsize":3687},[3265,7142,3267],{},[295,7144,7145,7147],{},[58,7146,4459],{},[58,7148,6656],{"mathvariant":285},[62,7150,293],{"stretchy":292},[58,7152,286],{"mathvariant":285},[62,7154,330],{"stretchy":292},[58,7156,2192],{"mathvariant":333},[81,7158,7159],{"encoding":83},"J_{\\mathbf{h}}(\\mathbf{x}) = J_{\\mathbf{f}}\\big(\\mathbf{g}(\\mathbf{x})\\big) \\, J_{\\mathbf{g}}(\\mathbf{x}).",[38,7161,7163,7230],{"className":7162,"ariaHidden":89},[88],[38,7164,7166,7169,7212,7215,7218,7221,7224,7227],{"className":7165},[93],[38,7167],{"className":7168,"style":1439},[97],[38,7170,7172,7175],{"className":7171},[102],[38,7173,4459],{"className":7174,"style":4746},[102,103],[38,7176,7178],{"className":7177},[134],[38,7179,7181,7204],{"className":7180},[138,384],[38,7182,7184,7201],{"className":7183},[142],[38,7185,7187],{"className":7186,"style":4759},[146],[38,7188,7189,7192],{"style":4762},[38,7190],{"className":7191,"style":155},[154],[38,7193,7195],{"className":7194},[159,160,161,162],[38,7196,7198],{"className":7197},[102,162],[38,7199,678],{"className":7200},[102,351,162],[38,7202,408],{"className":7203},[407],[38,7205,7207],{"className":7206},[142],[38,7208,7210],{"className":7209,"style":415},[146],[38,7211],{},[38,7213,293],{"className":7214},[371],[38,7216,286],{"className":7217},[102,351],[38,7219,330],{"className":7220},[529],[38,7222],{"className":7223,"style":109},[108],[38,7225,289],{"className":7226},[113],[38,7228],{"className":7229,"style":109},[108],[38,7231,7233,7237,7280,7286,7289,7292,7295,7298,7304,7307,7351,7354,7357,7360],{"className":7232},[93],[38,7234],{"className":7235,"style":7236},[97],"height:1.2em;vertical-align:-0.35em;",[38,7238,7240,7243],{"className":7239},[102],[38,7241,4459],{"className":7242,"style":4746},[102,103],[38,7244,7246],{"className":7245},[134],[38,7247,7249,7272],{"className":7248},[138,384],[38,7250,7252,7269],{"className":7251},[142],[38,7253,7255],{"className":7254,"style":4759},[146],[38,7256,7257,7260],{"style":4762},[38,7258],{"className":7259,"style":155},[154],[38,7261,7263],{"className":7262},[159,160,161,162],[38,7264,7266],{"className":7265},[102,162],[38,7267,60],{"className":7268,"style":225},[102,351,162],[38,7270,408],{"className":7271},[407],[38,7273,7275],{"className":7274},[142],[38,7276,7278],{"className":7277,"style":415},[146],[38,7279],{},[38,7281,7283],{"className":7282},[102],[38,7284,293],{"className":7285},[1485,3786],[38,7287,6656],{"className":7288,"style":6689},[102,351],[38,7290,293],{"className":7291},[371],[38,7293,286],{"className":7294},[102,351],[38,7296,330],{"className":7297},[529],[38,7299,7301],{"className":7300},[102],[38,7302,330],{"className":7303},[1485,3786],[38,7305],{"className":7306,"style":425},[108],[38,7308,7310,7313],{"className":7309},[102],[38,7311,4459],{"className":7312,"style":4746},[102,103],[38,7314,7316],{"className":7315},[134],[38,7317,7319,7342],{"className":7318},[138,384],[38,7320,7322,7339],{"className":7321},[142],[38,7323,7325],{"className":7324,"style":2513},[146],[38,7326,7327,7330],{"style":4762},[38,7328],{"className":7329,"style":155},[154],[38,7331,7333],{"className":7332},[159,160,161,162],[38,7334,7336],{"className":7335},[102,162],[38,7337,6656],{"className":7338,"style":6689},[102,351,162],[38,7340,408],{"className":7341},[407],[38,7343,7345],{"className":7344},[142],[38,7346,7349],{"className":7347,"style":7348},[146],"height:0.2861em;",[38,7350],{},[38,7352,293],{"className":7353},[371],[38,7355,286],{"className":7356},[102,351],[38,7358,330],{"className":7359},[529],[38,7361,2192],{"className":7362},[102],[11,7364,7365],{},"即复合函数的雅可比矩阵等于各层雅可比矩阵按复合顺序的矩阵乘积。这一定理是整个深度学习自动微分体系的数学基础。",[31,7367,3086],{"id":7368},"机器学习-1",[7370,7371,7372,7568],"ul",{},[7373,7374,7375,7378,7379,7408,7409,7567],"li",{},[15,7376,7377],{},"反向传播的本质","：一个 ",[38,7380,7382,7395],{"className":7381,"translate":42},[41],[38,7383,7385],{"className":7384},[46],[48,7386,7387],{"xmlns":50},[52,7388,7389,7393],{},[55,7390,7391],{},[58,7392,3104],{},[81,7394,3104],{"encoding":83},[38,7396,7398],{"className":7397,"ariaHidden":89},[88],[38,7399,7401,7405],{"className":7400},[93],[38,7402],{"className":7403,"style":7404},[97],"height:0.6833em;",[38,7406,3104],{"className":7407},[102,103]," 层神经网络可视为逐层复合的向量值函数 ",[38,7410,7412,7442],{"className":7411,"translate":42},[41],[38,7413,7415],{"className":7414},[46],[48,7416,7417],{"xmlns":50},[52,7418,7419,7439],{},[55,7420,7421,7427,7429,7431,7433],{},[295,7422,7423,7425],{},[58,7424,60],{"mathvariant":285},[58,7426,3104],{},[62,7428,6935],{},[62,7430,4519],{},[62,7432,6935],{},[295,7434,7435,7437],{},[58,7436,60],{"mathvariant":285},[300,7438,302],{},[81,7440,7441],{"encoding":83},"\\mathbf{f}_L \\circ \\cdots \\circ \\mathbf{f}_1",[38,7443,7445,7502,7521],{"className":7444,"ariaHidden":89},[88],[38,7446,7448,7451,7493,7496,7499],{"className":7447},[93],[38,7449],{"className":7450,"style":3360},[97],[38,7452,7454,7457],{"className":7453},[102],[38,7455,60],{"className":7456,"style":225},[102,351],[38,7458,7460],{"className":7459},[134],[38,7461,7463,7485],{"className":7462},[138,384],[38,7464,7466,7482],{"className":7465},[142],[38,7467,7470],{"className":7468,"style":7469},[146],"height:0.3283em;",[38,7471,7473,7476],{"style":7472},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[38,7474],{"className":7475,"style":155},[154],[38,7477,7479],{"className":7478},[159,160,161,162],[38,7480,3104],{"className":7481},[102,103,162],[38,7483,408],{"className":7484},[407],[38,7486,7488],{"className":7487},[142],[38,7489,7491],{"className":7490,"style":415},[146],[38,7492],{},[38,7494],{"className":7495,"style":995},[108],[38,7497,6935],{"className":7498},[999],[38,7500],{"className":7501,"style":995},[108],[38,7503,7505,7509,7512,7515,7518],{"className":7504},[93],[38,7506],{"className":7507,"style":7508},[97],"height:0.4445em;",[38,7510,4519],{"className":7511},[475],[38,7513],{"className":7514,"style":995},[108],[38,7516,6935],{"className":7517},[999],[38,7519],{"className":7520,"style":995},[108],[38,7522,7524,7527],{"className":7523},[93],[38,7525],{"className":7526,"style":3360},[97],[38,7528,7530,7533],{"className":7529},[102],[38,7531,60],{"className":7532,"style":225},[102,351],[38,7534,7536],{"className":7535},[134],[38,7537,7539,7559],{"className":7538},[138,384],[38,7540,7542,7556],{"className":7541},[142],[38,7543,7545],{"className":7544,"style":391},[146],[38,7546,7547,7550],{"style":7472},[38,7548],{"className":7549,"style":155},[154],[38,7551,7553],{"className":7552},[159,160,161,162],[38,7554,302],{"className":7555},[102,162],[38,7557,408],{"className":7558},[407],[38,7560,7562],{"className":7561},[142],[38,7563,7565],{"className":7564,"style":415},[146],[38,7566],{},"，损失函数关于参数的梯度即通过定理 2.2 反复应用链式法则、将各层雅可比矩阵逐层左乘而得。自动微分框架（如 PyTorch、TensorFlow 的计算图）本质上是在高效地计算并组合一系列雅可比–向量积（Jacobian-vector product, JVP）与向量–雅可比积（vector-Jacobian product, VJP），而无需显式构造完整的雅可比矩阵。",[7373,7569,7570,7573,7574,7641],{},[15,7571,7572],{},"归一化流（Normalizing Flows）","：在生成模型中，通过可逆变换 ",[38,7575,7577,7602],{"className":7576,"translate":42},[41],[38,7578,7580],{"className":7579},[46],[48,7581,7582],{"xmlns":50},[52,7583,7584,7599],{},[55,7585,7586,7589,7591,7593,7595,7597],{},[58,7587,7588],{"mathvariant":285},"z",[62,7590,289],{},[58,7592,60],{"mathvariant":285},[62,7594,293],{"stretchy":292},[58,7596,286],{"mathvariant":285},[62,7598,330],{"stretchy":292},[81,7600,7601],{"encoding":83},"\\mathbf{z} = \\mathbf{f}(\\mathbf{x})",[38,7603,7605,7623],{"className":7604,"ariaHidden":89},[88],[38,7606,7608,7611,7614,7617,7620],{"className":7607},[93],[38,7609],{"className":7610,"style":347},[97],[38,7612,7588],{"className":7613},[102,351],[38,7615],{"className":7616,"style":109},[108],[38,7618,289],{"className":7619},[113],[38,7621],{"className":7622,"style":109},[108],[38,7624,7626,7629,7632,7635,7638],{"className":7625},[93],[38,7627],{"className":7628,"style":1439},[97],[38,7630,60],{"className":7631,"style":225},[102,351],[38,7633,293],{"className":7634},[371],[38,7636,286],{"className":7637},[102,351],[38,7639,330],{"className":7640},[529]," 建模概率密度时，变量替换公式",[38,7643,7645],{"className":7644,"translate":42},[621],[38,7646,7648,7718],{"className":7647,"translate":42},[41],[38,7649,7651],{"className":7650},[46],[48,7652,7653],{"xmlns":50,"display":630},[52,7654,7655,7715],{},[55,7656,7657,7664,7666,7668,7670,7672,7679,7681,7683,7685,7687,7689,7691],{},[295,7658,7659,7661],{},[58,7660,11],{},[58,7662,7663],{},"X",[62,7665,293],{"stretchy":292},[58,7667,286],{"mathvariant":285},[62,7669,330],{"stretchy":292},[62,7671,289],{},[295,7673,7674,7676],{},[58,7675,11],{},[58,7677,7678],{},"Z",[62,7680,293],{"fence":292,"stretchy":89,"minsize":3687,"maxsize":3687},[58,7682,60],{"mathvariant":285},[62,7684,293],{"stretchy":292},[58,7686,286],{"mathvariant":285},[62,7688,330],{"stretchy":292},[62,7690,330],{"fence":292,"stretchy":89,"minsize":3687,"maxsize":3687},[55,7692,7693,7696,7699,7701,7707,7709,7711,7713],{},[62,7694,7695],{"fence":89},"∣",[58,7697,7698],{},"det",[62,7700,673],{},[295,7702,7703,7705],{},[58,7704,4459],{},[58,7706,60],{"mathvariant":285},[62,7708,293],{"stretchy":292},[58,7710,286],{"mathvariant":285},[62,7712,330],{"stretchy":292},[62,7714,7695],{"fence":89},[81,7716,7717],{"encoding":83},"p_X(\\mathbf{x}) = p_Z\\big(\\mathbf{f}(\\mathbf{x})\\big) \\left| \\det J_{\\mathbf{f}}(\\mathbf{x}) \\right|",[38,7719,7721,7786],{"className":7720,"ariaHidden":89},[88],[38,7722,7724,7727,7768,7771,7774,7777,7780,7783],{"className":7723},[93],[38,7725],{"className":7726,"style":1439},[97],[38,7728,7730,7733],{"className":7729},[102],[38,7731,11],{"className":7732},[102,103],[38,7734,7736],{"className":7735},[134],[38,7737,7739,7760],{"className":7738},[138,384],[38,7740,7742,7757],{"className":7741},[142],[38,7743,7745],{"className":7744,"style":7469},[146],[38,7746,7747,7750],{"style":394},[38,7748],{"className":7749,"style":155},[154],[38,7751,7753],{"className":7752},[159,160,161,162],[38,7754,7663],{"className":7755,"style":7756},[102,103,162],"margin-right:0.0785em;",[38,7758,408],{"className":7759},[407],[38,7761,7763],{"className":7762},[142],[38,7764,7766],{"className":7765,"style":415},[146],[38,7767],{},[38,7769,293],{"className":7770},[371],[38,7772,286],{"className":7773},[102,351],[38,7775,330],{"className":7776},[529],[38,7778],{"className":7779,"style":109},[108],[38,7781,289],{"className":7782},[113],[38,7784],{"className":7785,"style":109},[108],[38,7787,7789,7792,7833,7839,7842,7845,7848,7851,7857,7860],{"className":7788},[93],[38,7790],{"className":7791,"style":7236},[97],[38,7793,7795,7798],{"className":7794},[102],[38,7796,11],{"className":7797},[102,103],[38,7799,7801],{"className":7800},[134],[38,7802,7804,7825],{"className":7803},[138,384],[38,7805,7807,7822],{"className":7806},[142],[38,7808,7810],{"className":7809,"style":7469},[146],[38,7811,7812,7815],{"style":394},[38,7813],{"className":7814,"style":155},[154],[38,7816,7818],{"className":7817},[159,160,161,162],[38,7819,7678],{"className":7820,"style":7821},[102,103,162],"margin-right:0.0715em;",[38,7823,408],{"className":7824},[407],[38,7826,7828],{"className":7827},[142],[38,7829,7831],{"className":7830,"style":415},[146],[38,7832],{},[38,7834,7836],{"className":7835},[102],[38,7837,293],{"className":7838},[1485,3786],[38,7840,60],{"className":7841,"style":225},[102,351],[38,7843,293],{"className":7844},[371],[38,7846,286],{"className":7847},[102,351],[38,7849,330],{"className":7850},[529],[38,7852,7854],{"className":7853},[102],[38,7855,330],{"className":7856},[1485,3786],[38,7858],{"className":7859,"style":425},[108],[38,7861,7863,7866,7869,7872,7915,7918,7921,7924],{"className":7862},[475],[38,7864,7695],{"className":7865,"style":1481},[371,1480],[38,7867,7698],{"className":7868},[878],[38,7870],{"className":7871,"style":425},[108],[38,7873,7875,7878],{"className":7874},[102],[38,7876,4459],{"className":7877,"style":4746},[102,103],[38,7879,7881],{"className":7880},[134],[38,7882,7884,7907],{"className":7883},[138,384],[38,7885,7887,7904],{"className":7886},[142],[38,7888,7890],{"className":7889,"style":4759},[146],[38,7891,7892,7895],{"style":4762},[38,7893],{"className":7894,"style":155},[154],[38,7896,7898],{"className":7897},[159,160,161,162],[38,7899,7901],{"className":7900},[102,162],[38,7902,60],{"className":7903,"style":225},[102,351,162],[38,7905,408],{"className":7906},[407],[38,7908,7910],{"className":7909},[142],[38,7911,7913],{"className":7912,"style":415},[146],[38,7914],{},[38,7916,293],{"className":7917},[371],[38,7919,286],{"className":7920},[102,351],[38,7922,330],{"className":7923},[529],[38,7925,7695],{"className":7926,"style":1481},[529,1480],[11,7928,7929],{},"直接依赖雅可比行列式，这要求所设计的变换族拥有易于计算的雅可比矩阵（如三角结构）。",[7370,7931,7932,7938],{},[7373,7933,7934,7937],{},[15,7935,7936],{},"对抗鲁棒性与敏感性分析","：输出关于输入的雅可比矩阵刻画了模型对输入扰动的敏感程度，是对抗样本生成（如 FGSM）与鲁棒性正则化方法（雅可比范数正则化）的理论工具。",[7373,7939,7940,7943],{},[15,7941,7942],{},"多任务学习与雅可比冲突","：当模型同时输出多个预测目标时，各任务梯度构成的雅可比矩阵的行与行之间可能存在方向冲突，这催生了如 PCGrad 等专门处理\"梯度冲突\"的多任务优化方法。",[28,7945,25],{"id":25},[31,7947,7948],{"id":7948},"对梯度再求导",[11,7950,7951,7952,8054,8055,8058,8059,8197,8198,8226,8227,8255],{},"海塞矩阵刻画的是标量值函数 ",[38,7953,7955,7980],{"className":7954,"translate":42},[41],[38,7956,7958],{"className":7957},[46],[48,7959,7960],{"xmlns":50},[52,7961,7962,7978],{},[55,7963,7964,7966,7968,7974,7976],{},[58,7965,60],{},[62,7967,64],{},[66,7969,7970,7972],{},[58,7971,71],{"mathvariant":70},[58,7973,74],{},[62,7975,77],{},[58,7977,71],{"mathvariant":70},[81,7979,84],{"encoding":83},[38,7981,7983,8001,8045],{"className":7982,"ariaHidden":89},[88],[38,7984,7986,7989,7992,7995,7998],{"className":7985},[93],[38,7987],{"className":7988,"style":98},[97],[38,7990,60],{"className":7991,"style":104},[102,103],[38,7993],{"className":7994,"style":109},[108],[38,7996,64],{"className":7997},[113],[38,7999],{"className":8000,"style":109},[108],[38,8002,8004,8007,8036,8039,8042],{"className":8003},[93],[38,8005],{"className":8006,"style":123},[97],[38,8008,8010,8013],{"className":8009},[102],[38,8011,71],{"className":8012},[102,130],[38,8014,8016],{"className":8015},[134],[38,8017,8019],{"className":8018},[138],[38,8020,8022],{"className":8021},[142],[38,8023,8025],{"className":8024,"style":147},[146],[38,8026,8027,8030],{"style":150},[38,8028],{"className":8029,"style":155},[154],[38,8031,8033],{"className":8032},[159,160,161,162],[38,8034,74],{"className":8035},[102,103,162],[38,8037],{"className":8038,"style":109},[108],[38,8040,77],{"className":8041},[113],[38,8043],{"className":8044,"style":109},[108],[38,8046,8048,8051],{"className":8047},[93],[38,8049],{"className":8050,"style":123},[97],[38,8052,71],{"className":8053},[102,130]," 的",[15,8056,8057],{},"二阶","变化信息。其构造方式恰好揭示了本文的核心论点：既然梯度 ",[38,8060,8062,8094],{"className":8061,"translate":42},[41],[38,8063,8065],{"className":8064},[46],[48,8066,8067],{"xmlns":50},[52,8068,8069,8091],{},[55,8070,8071,8073,8075,8077,8083,8085],{},[58,8072,1335],{"mathvariant":333},[58,8074,60],{},[62,8076,64],{},[66,8078,8079,8081],{},[58,8080,71],{"mathvariant":70},[58,8082,74],{},[62,8084,77],{},[66,8086,8087,8089],{},[58,8088,71],{"mathvariant":70},[58,8090,74],{},[81,8092,8093],{"encoding":83},"\\nabla f: \\mathbb{R}^n \\to \\mathbb{R}^n",[38,8095,8097,8118,8162],{"className":8096,"ariaHidden":89},[88],[38,8098,8100,8103,8106,8109,8112,8115],{"className":8099},[93],[38,8101],{"className":8102,"style":98},[97],[38,8104,1335],{"className":8105},[102],[38,8107,60],{"className":8108,"style":104},[102,103],[38,8110],{"className":8111,"style":109},[108],[38,8113,64],{"className":8114},[113],[38,8116],{"className":8117,"style":109},[108],[38,8119,8121,8124,8153,8156,8159],{"className":8120},[93],[38,8122],{"className":8123,"style":123},[97],[38,8125,8127,8130],{"className":8126},[102],[38,8128,71],{"className":8129},[102,130],[38,8131,8133],{"className":8132},[134],[38,8134,8136],{"className":8135},[138],[38,8137,8139],{"className":8138},[142],[38,8140,8142],{"className":8141,"style":147},[146],[38,8143,8144,8147],{"style":150},[38,8145],{"className":8146,"style":155},[154],[38,8148,8150],{"className":8149},[159,160,161,162],[38,8151,74],{"className":8152},[102,103,162],[38,8154],{"className":8155,"style":109},[108],[38,8157,77],{"className":8158},[113],[38,8160],{"className":8161,"style":109},[108],[38,8163,8165,8168],{"className":8164},[93],[38,8166],{"className":8167,"style":123},[97],[38,8169,8171,8174],{"className":8170},[102],[38,8172,71],{"className":8173},[102,130],[38,8175,8177],{"className":8176},[134],[38,8178,8180],{"className":8179},[138],[38,8181,8183],{"className":8182},[142],[38,8184,8186],{"className":8185,"style":147},[146],[38,8187,8188,8191],{"style":150},[38,8189],{"className":8190,"style":155},[154],[38,8192,8194],{"className":8193},[159,160,161,162],[38,8195,74],{"className":8196},[102,103,162]," 本身是一个向量值函数（输入维度 ",[38,8199,8201,8214],{"className":8200,"translate":42},[41],[38,8202,8204],{"className":8203},[46],[48,8205,8206],{"xmlns":50},[52,8207,8208,8212],{},[55,8209,8210],{},[58,8211,74],{},[81,8213,74],{"encoding":83},[38,8215,8217],{"className":8216,"ariaHidden":89},[88],[38,8218,8220,8223],{"className":8219},[93],[38,8221],{"className":8222,"style":4312},[97],[38,8224,74],{"className":8225},[102,103],"，输出维度也是 ",[38,8228,8230,8243],{"className":8229,"translate":42},[41],[38,8231,8233],{"className":8232},[46],[48,8234,8235],{"xmlns":50},[52,8236,8237,8241],{},[55,8238,8239],{},[58,8240,74],{},[81,8242,74],{"encoding":83},[38,8244,8246],{"className":8245,"ariaHidden":89},[88],[38,8247,8249,8252],{"className":8248},[93],[38,8250],{"className":8251,"style":4312},[97],[38,8253,74],{"className":8254},[102,103],"），那么按照 2.2 节的定义，我们完全可以对它求雅可比矩阵。",[31,8257,1250],{"id":8258},"定义-2",[1252,8260,8261,8420,9731],{},[11,8262,8263,2369,8266,8368,8369,4439],{},[15,8264,8265],{},"定义 3.1（海塞矩阵）",[38,8267,8269,8294],{"className":8268,"translate":42},[41],[38,8270,8272],{"className":8271},[46],[48,8273,8274],{"xmlns":50},[52,8275,8276,8292],{},[55,8277,8278,8280,8282,8288,8290],{},[58,8279,60],{},[62,8281,64],{},[66,8283,8284,8286],{},[58,8285,71],{"mathvariant":70},[58,8287,74],{},[62,8289,77],{},[58,8291,71],{"mathvariant":70},[81,8293,84],{"encoding":83},[38,8295,8297,8315,8359],{"className":8296,"ariaHidden":89},[88],[38,8298,8300,8303,8306,8309,8312],{"className":8299},[93],[38,8301],{"className":8302,"style":98},[97],[38,8304,60],{"className":8305,"style":104},[102,103],[38,8307],{"className":8308,"style":109},[108],[38,8310,64],{"className":8311},[113],[38,8313],{"className":8314,"style":109},[108],[38,8316,8318,8321,8350,8353,8356],{"className":8317},[93],[38,8319],{"className":8320,"style":123},[97],[38,8322,8324,8327],{"className":8323},[102],[38,8325,71],{"className":8326},[102,130],[38,8328,8330],{"className":8329},[134],[38,8331,8333],{"className":8332},[138],[38,8334,8336],{"className":8335},[142],[38,8337,8339],{"className":8338,"style":147},[146],[38,8340,8341,8344],{"style":150},[38,8342],{"className":8343,"style":155},[154],[38,8345,8347],{"className":8346},[159,160,161,162],[38,8348,74],{"className":8349},[102,103,162],[38,8351],{"className":8352,"style":109},[108],[38,8354,77],{"className":8355},[113],[38,8357],{"className":8358,"style":109},[108],[38,8360,8362,8365],{"className":8361},[93],[38,8363],{"className":8364,"style":123},[97],[38,8366,71],{"className":8367},[102,130]," 二阶可偏导，则 ",[38,8370,8372,8390],{"className":8371,"translate":42},[41],[38,8373,8375],{"className":8374},[46],[48,8376,8377],{"xmlns":50},[52,8378,8379,8387],{},[55,8380,8381,8383,8385],{},[58,8382,74],{},[62,8384,4402],{},[58,8386,74],{},[81,8388,8389],{"encoding":83},"n \\times n",[38,8391,8393,8411],{"className":8392,"ariaHidden":89},[88],[38,8394,8396,8399,8402,8405,8408],{"className":8395},[93],[38,8397],{"className":8398,"style":4417},[97],[38,8400,74],{"className":8401},[102,103],[38,8403],{"className":8404,"style":995},[108],[38,8406,4402],{"className":8407},[999],[38,8409],{"className":8410,"style":995},[108],[38,8412,8414,8417],{"className":8413},[93],[38,8415],{"className":8416,"style":4312},[97],[38,8418,74],{"className":8419},[102,103],[38,8421,8423],{"className":8422,"translate":42},[621],[38,8424,8426,8660],{"className":8425,"translate":42},[41],[38,8427,8429],{"className":8428},[46],[48,8430,8431],{"xmlns":50,"display":630},[52,8432,8433,8657],{},[55,8434,8435,8442,8444,8446,8448,8450,8460,8462,8464,8466,8468],{},[295,8436,8437,8440],{},[58,8438,8439],{},"H",[58,8441,60],{},[62,8443,293],{"stretchy":292},[58,8445,286],{"mathvariant":285},[62,8447,330],{"stretchy":292},[62,8449,289],{},[295,8451,8452,8454],{},[58,8453,4459],{},[55,8455,8456,8458],{},[58,8457,1335],{"mathvariant":333},[58,8459,60],{},[62,8461,293],{"stretchy":292},[58,8463,286],{"mathvariant":285},[62,8465,330],{"stretchy":292},[62,8467,289],{},[55,8469,8470,8472,8655],{},[62,8471,293],{"fence":89},[4475,8473,8474,8549,8581],{"rowspacing":4477,"columnalign":4478,"columnspacing":4479},[4481,8475,8476,8507,8513],{},[4484,8477,8478],{},[2631,8479,8480],{"scriptlevel":683,"displaystyle":292},[2631,8481,8482],{"scriptlevel":683,"displaystyle":89},[636,8483,8484,8494],{},[55,8485,8486,8492],{},[66,8487,8488,8490],{},[58,8489,642],{"mathvariant":333},[300,8491,312],{},[58,8493,60],{},[55,8495,8496,8498],{},[58,8497,642],{"mathvariant":333},[8499,8500,8501,8503,8505],"msubsup",{},[58,8502,286],{},[300,8504,302],{},[300,8506,312],{},[4484,8508,8509],{},[2631,8510,8511],{"scriptlevel":683,"displaystyle":292},[62,8512,4519],{"lspace":4518,"rspace":4518},[4484,8514,8515],{},[2631,8516,8517],{"scriptlevel":683,"displaystyle":292},[2631,8518,8519],{"scriptlevel":683,"displaystyle":89},[636,8520,8521,8531],{},[55,8522,8523,8529],{},[66,8524,8525,8527],{},[58,8526,642],{"mathvariant":333},[300,8528,312],{},[58,8530,60],{},[55,8532,8533,8535,8541,8543],{},[58,8534,642],{"mathvariant":333},[295,8536,8537,8539],{},[58,8538,286],{},[300,8540,302],{},[58,8542,642],{"mathvariant":333},[295,8544,8545,8547],{},[58,8546,286],{},[58,8548,74],{},[4481,8550,8551,8563,8569],{},[4484,8552,8553],{},[2631,8554,8555],{"scriptlevel":683,"displaystyle":292},[55,8556,8557,8559],{},[58,8558,4558],{"mathvariant":333},[4560,8560,8561],{"height":4518,"voffset":4518},[108,8562],{"mathbackground":4564,"width":4518,"height":4565},[4484,8564,8565],{},[2631,8566,8567],{"scriptlevel":683,"displaystyle":292},[62,8568,4572],{"lspace":4518,"rspace":4518},[4484,8570,8571],{},[2631,8572,8573],{"scriptlevel":683,"displaystyle":292},[55,8574,8575,8577],{},[58,8576,4558],{"mathvariant":333},[4560,8578,8579],{"height":4518,"voffset":4518},[108,8580],{"mathbackground":4564,"width":4518,"height":4565},[4481,8582,8583,8619,8625],{},[4484,8584,8585],{},[2631,8586,8587],{"scriptlevel":683,"displaystyle":292},[2631,8588,8589],{"scriptlevel":683,"displaystyle":89},[636,8590,8591,8601],{},[55,8592,8593,8599],{},[66,8594,8595,8597],{},[58,8596,642],{"mathvariant":333},[300,8598,312],{},[58,8600,60],{},[55,8602,8603,8605,8611,8613],{},[58,8604,642],{"mathvariant":333},[295,8606,8607,8609],{},[58,8608,286],{},[58,8610,74],{},[58,8612,642],{"mathvariant":333},[295,8614,8615,8617],{},[58,8616,286],{},[300,8618,302],{},[4484,8620,8621],{},[2631,8622,8623],{"scriptlevel":683,"displaystyle":292},[62,8624,4519],{"lspace":4518,"rspace":4518},[4484,8626,8627],{},[2631,8628,8629],{"scriptlevel":683,"displaystyle":292},[2631,8630,8631],{"scriptlevel":683,"displaystyle":89},[636,8632,8633,8643],{},[55,8634,8635,8641],{},[66,8636,8637,8639],{},[58,8638,642],{"mathvariant":333},[300,8640,312],{},[58,8642,60],{},[55,8644,8645,8647],{},[58,8646,642],{"mathvariant":333},[8499,8648,8649,8651,8653],{},[58,8650,286],{},[58,8652,74],{},[300,8654,312],{},[62,8656,330],{"fence":89},[81,8658,8659],{"encoding":83},"H_f(\\mathbf{x}) = J_{\\nabla f}(\\mathbf{x}) =\n\\begin{pmatrix}\n\\dfrac{\\partial^2 f}{\\partial x_1^2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_1 \\partial x_n} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\dfrac{\\partial^2 f}{\\partial x_n \\partial x_1} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_n^2}\n\\end{pmatrix}",[38,8661,8663,8730,8800],{"className":8662,"ariaHidden":89},[88],[38,8664,8666,8670,8712,8715,8718,8721,8724,8727],{"className":8665},[93],[38,8667],{"className":8668,"style":8669},[97],"height:1.0361em;vertical-align:-0.2861em;",[38,8671,8673,8677],{"className":8672},[102],[38,8674,8439],{"className":8675,"style":8676},[102,103],"margin-right:0.0813em;",[38,8678,8680],{"className":8679},[134],[38,8681,8683,8704],{"className":8682},[138,384],[38,8684,8686,8701],{"className":8685},[142],[38,8687,8689],{"className":8688,"style":4759},[146],[38,8690,8692,8695],{"style":8691},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[38,8693],{"className":8694,"style":155},[154],[38,8696,8698],{"className":8697},[159,160,161,162],[38,8699,60],{"className":8700,"style":104},[102,103,162],[38,8702,408],{"className":8703},[407],[38,8705,8707],{"className":8706},[142],[38,8708,8710],{"className":8709,"style":7348},[146],[38,8711],{},[38,8713,293],{"className":8714},[371],[38,8716,286],{"className":8717},[102,351],[38,8719,330],{"className":8720},[529],[38,8722],{"className":8723,"style":109},[108],[38,8725,289],{"className":8726},[113],[38,8728],{"className":8729,"style":109},[108],[38,8731,8733,8736,8782,8785,8788,8791,8794,8797],{"className":8732},[93],[38,8734],{"className":8735,"style":8669},[97],[38,8737,8739,8742],{"className":8738},[102],[38,8740,4459],{"className":8741,"style":4746},[102,103],[38,8743,8745],{"className":8744},[134],[38,8746,8748,8774],{"className":8747},[138,384],[38,8749,8751,8771],{"className":8750},[142],[38,8752,8754],{"className":8753,"style":4759},[146],[38,8755,8756,8759],{"style":4762},[38,8757],{"className":8758,"style":155},[154],[38,8760,8762],{"className":8761},[159,160,161,162],[38,8763,8765,8768],{"className":8764},[102,162],[38,8766,1335],{"className":8767},[102,162],[38,8769,60],{"className":8770,"style":104},[102,103,162],[38,8772,408],{"className":8773},[407],[38,8775,8777],{"className":8776},[142],[38,8778,8780],{"className":8779,"style":7348},[146],[38,8781],{},[38,8783,293],{"className":8784},[371],[38,8786,286],{"className":8787},[102,351],[38,8789,330],{"className":8790},[529],[38,8792],{"className":8793,"style":109},[108],[38,8795,289],{"className":8796},[113],[38,8798],{"className":8799,"style":109},[108],[38,8801,8803,8807],{"className":8802},[93],[38,8804],{"className":8805,"style":8806},[97],"height:6.7275em;vertical-align:-3.1138em;",[38,8808,8810,8855,9693],{"className":8809},[475],[38,8811,8813],{"className":8812},[371],[38,8814,8816],{"className":8815},[1485,4820],[38,8817,8819,8846],{"className":8818},[138,384],[38,8820,8822,8843],{"className":8821},[142],[38,8823,8826],{"className":8824,"style":8825},[146],"height:3.55em;",[38,8827,8829,8833],{"style":8828},"top:-5.55em;",[38,8830],{"className":8831,"style":8832},[154],"height:8.6em;",[38,8834,8836],{"style":8835},"width:0.875em;height:6.6em;",[4842,8837,8840],{"xmlns":4844,"width":4845,"height":8838,"viewBox":8839},"6.6em","0 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",[38,9800,9802,9833],{"className":9801,"translate":42},[41],[38,9803,9805],{"className":9804},[46],[48,9806,9807],{"xmlns":50},[52,9808,9809,9831],{},[55,9810,9811,9813,9815,9817,9823,9825],{},[58,9812,1335],{"mathvariant":333},[58,9814,60],{},[62,9816,64],{},[66,9818,9819,9821],{},[58,9820,71],{"mathvariant":70},[58,9822,74],{},[62,9824,77],{},[66,9826,9827,9829],{},[58,9828,71],{"mathvariant":70},[58,9830,74],{},[81,9832,8093],{"encoding":83},[38,9834,9836,9857,9901],{"className":9835,"ariaHidden":89},[88],[38,9837,9839,9842,9845,9848,9851,9854],{"className":9838},[93],[38,9840],{"className":9841,"style":98},[97],[38,9843,1335],{"className":9844},[102],[38,9846,60],{"className":9847,"style":104},[102,103],[38,9849],{"className":9850,"style":109},[108],[38,9852,64],{"className":9853},[113],[38,9855],{"className":9856,"style":109},[108],[38,9858,9860,9863,9892,9895,9898],{"className":9859},[93],[38,9861],{"className":9862,"style":123},[97],[38,9864,9866,9869],{"className":9865},[102],[38,9867,71],{"className":9868},[102,130],[38,9870,9872],{"className":9871},[134],[38,9873,9875],{"className":9874},[138],[38,9876,9878],{"className":9877},[142],[38,9879,9881],{"className":9880,"style":147},[146],[38,9882,9883,9886],{"style":150},[38,9884],{"className":9885,"style":155},[154],[38,9887,9889],{"className":9888},[159,160,161,162],[38,9890,74],{"className":9891},[102,103,162],[38,9893],{"className":9894,"style":109},[108],[38,9896,77],{"className":9897},[113],[38,9899],{"className":9900,"style":109},[108],[38,9902,9904,9907],{"className":9903},[93],[38,9905],{"className":9906,"style":123},[97],[38,9908,9910,9913],{"className":9909},[102],[38,9911,71],{"className":9912},[102,130],[38,9914,9916],{"className":9915},[134],[38,9917,9919],{"className":9918},[138],[38,9920,9922],{"className":9921},[142],[38,9923,9925],{"className":9924,"style":147},[146],[38,9926,9927,9930],{"style":150},[38,9928],{"className":9929,"style":155},[154],[38,9931,9933],{"className":9932},[159,160,161,162],[38,9934,74],{"className":9935},[102,103,162]," 的雅可比矩阵，即 ",[38,9938,9940,9982],{"className":9939,"translate":42},[41],[38,9941,9943],{"className":9942},[46],[48,9944,9945],{"xmlns":50},[52,9946,9947,9979],{},[55,9948,9949,9955,9957,9959,9961,9963,9973,9975,9977],{},[295,9950,9951,9953],{},[58,9952,8439],{},[58,9954,60],{},[62,9956,293],{"stretchy":292},[58,9958,286],{"mathvariant":285},[62,9960,330],{"stretchy":292},[62,9962,289],{},[295,9964,9965,9967],{},[58,9966,4459],{},[55,9968,9969,9971],{},[58,9970,1335],{"mathvariant":333},[58,9972,60],{},[62,9974,293],{"stretchy":292},[58,9976,286],{"mathvariant":285},[62,9978,330],{"stretchy":292},[81,9980,9981],{"encoding":83},"H_f(\\mathbf{x}) = J_{\\nabla f}(\\mathbf{x})",[38,9983,9985,10049],{"className":9984,"ariaHidden":89},[88],[38,9986,9988,9991,10031,10034,10037,10040,10043,10046],{"className":9987},[93],[38,9989],{"className":9990,"style":8669},[97],[38,9992,9994,9997],{"className":9993},[102],[38,9995,8439],{"className":9996,"style":8676},[102,103],[38,9998,10000],{"className":9999},[134],[38,10001,10003,10023],{"className":10002},[138,384],[38,10004,10006,10020],{"className":10005},[142],[38,10007,10009],{"className":10008,"style":4759},[146],[38,10010,10011,10014],{"style":8691},[38,10012],{"className":10013,"style":155},[154],[38,10015,10017],{"className":10016},[159,160,161,162],[38,10018,60],{"className":10019,"style":104},[102,103,162],[38,10021,408],{"className":10022},[407],[38,10024,10026],{"className":10025},[142],[38,10027,10029],{"className":10028,"style":7348},[146],[38,10030],{},[38,10032,293],{"className":10033},[371],[38,10035,286],{"className":10036},[102,351],[38,10038,330],{"className":10039},[529],[38,10041],{"className":10042,"style":109},[108],[38,10044,289],{"className":10045},[113],[38,10047],{"className":10048,"style":109},[108],[38,10050,10052,10055,10101,10104,10107],{"className":10051},[93],[38,10053],{"className":10054,"style":8669},[97],[38,10056,10058,10061],{"className":10057},[102],[38,10059,4459],{"className":10060,"style":4746},[102,103],[38,10062,10064],{"className":10063},[134],[38,10065,10067,10093],{"className":10066},[138,384],[38,10068,10070,10090],{"className":10069},[142],[38,10071,10073],{"className":10072,"style":4759},[146],[38,10074,10075,10078],{"style":4762},[38,10076],{"className":10077,"style":155},[154],[38,10079,10081],{"className":10080},[159,160,161,162],[38,10082,10084,10087],{"className":10083},[102,162],[38,10085,1335],{"className":10086},[102,162],[38,10088,60],{"className":10089,"style":104},[102,103,162],[38,10091,408],{"className":10092},[407],[38,10094,10096],{"className":10095},[142],[38,10097,10099],{"className":10098,"style":7348},[146],[38,10100],{},[38,10102,293],{"className":10103},[371],[38,10105,286],{"className":10106},[102,351],[38,10108,330],{"className":10109},[529],[11,10111,10112],{},"这一命题并非巧合，而是定义使然：它精确地表述了本文开篇提出的递进关系——梯度是一阶导数算子作用于标量函数的结果，而海塞矩阵是该导数算子的二次复合，或者说，是\"对雅可比矩阵这一构造方法本身再应用一次\"的产物。",[31,10114,10115],{"id":10115},"对称性",[1252,10117,10118,10263],{},[11,10119,10120,1259,10123,1288,10151,10179,10180,10262],{},[15,10121,10122],{},"定理 3.1（施瓦茨定理 \u002F 克莱罗定理）",[38,10124,10126,10139],{"className":10125,"translate":42},[41],[38,10127,10129],{"className":10128},[46],[48,10130,10131],{"xmlns":50},[52,10132,10133,10137],{},[55,10134,10135],{},[58,10136,60],{},[81,10138,60],{"encoding":83},[38,10140,10142],{"className":10141,"ariaHidden":89},[88],[38,10143,10145,10148],{"className":10144},[93],[38,10146],{"className":10147,"style":98},[97],[38,10149,60],{"className":10150,"style":104},[102,103],[38,10152,10154,10167],{"className":10153,"translate":42},[41],[38,10155,10157],{"className":10156},[46],[48,10158,10159],{"xmlns":50},[52,10160,10161,10165],{},[55,10162,10163],{},[58,10164,286],{"mathvariant":285},[81,10166,1104],{"encoding":83},[38,10168,10170],{"className":10169,"ariaHidden":89},[88],[38,10171,10173,10176],{"className":10172},[93],[38,10174],{"className":10175,"style":347},[97],[38,10177,286],{"className":10178},[102,351]," 的某邻域内二阶偏导数连续（即 ",[38,10181,10183,10206],{"className":10182,"translate":42},[41],[38,10184,10186],{"className":10185},[46],[48,10187,10188],{"xmlns":50},[52,10189,10190,10203],{},[55,10191,10192,10194,10196],{},[58,10193,60],{},[62,10195,1420],{},[66,10197,10198,10201],{},[58,10199,10200],{},"C",[300,10202,312],{},[81,10204,10205],{"encoding":83},"f \\in C^2",[38,10207,10209,10227],{"className":10208,"ariaHidden":89},[88],[38,10210,10212,10215,10218,10221,10224],{"className":10211},[93],[38,10213],{"className":10214,"style":98},[97],[38,10216,60],{"className":10217,"style":104},[102,103],[38,10219],{"className":10220,"style":109},[108],[38,10222,1420],{"className":10223},[113],[38,10225],{"className":10226,"style":109},[108],[38,10228,10230,10233],{"className":10229},[93],[38,10231],{"className":10232,"style":9002},[97],[38,10234,10236,10239],{"className":10235},[102],[38,10237,10200],{"className":10238,"style":7821},[102,103],[38,10240,10242],{"className":10241},[134],[38,10243,10245],{"className":10244},[138],[38,10246,10248],{"className":10247},[142],[38,10249,10251],{"className":10250,"style":9002},[146],[38,10252,10253,10256],{"style":150},[38,10254],{"className":10255,"style":155},[154],[38,10257,10259],{"className":10258},[159,160,161,162],[38,10260,312],{"className":10261},[102,162],"），则混合偏导数与求导顺序无关：",[38,10264,10266],{"className":10265,"translate":42},[621],[38,10267,10269,10359],{"className":10268,"translate":42},[41],[38,10270,10272],{"className":10271},[46],[48,10273,10274],{"xmlns":50,"display":630},[52,10275,10276,10356],{},[55,10277,10278,10309,10311,10341,10343,10345,10348,10350,10352,10354],{},[636,10279,10280,10290],{},[55,10281,10282,10288],{},[66,10283,10284,10286],{},[58,10285,642],{"mathvariant":333},[300,10287,312],{},[58,10289,60],{},[55,10291,10292,10294,10300,10302],{},[58,10293,642],{"mathvariant":333},[295,10295,10296,10298],{},[58,10297,286],{},[58,10299,601],{},[58,10301,642],{"mathvariant":333},[295,10303,10304,10306],{},[58,10305,286],{},[58,10307,10308],{},"j",[62,10310,289],{},[636,10312,10313,10323],{},[55,10314,10315,10321],{},[66,10316,10317,10319],{},[58,10318,642],{"mathvariant":333},[300,10320,312],{},[58,10322,60],{},[55,10324,10325,10327,10333,10335],{},[58,10326,642],{"mathvariant":333},[295,10328,10329,10331],{},[58,10330,286],{},[58,10332,10308],{},[58,10334,642],{"mathvariant":333},[295,10336,10337,10339],{},[58,10338,286],{},[58,10340,601],{},[62,10342,305],{"separator":89},[108,10344],{"width":4479},[58,10346,10347],{"mathvariant":333},"∀",[58,10349,601],{},[62,10351,305],{"separator":89},[58,10353,10308],{},[58,10355,2192],{"mathvariant":333},[81,10357,10358],{"encoding":83},"\\frac{\\partial^2 f}{\\partial x_i \\partial x_j} = \\frac{\\partial^2 f}{\\partial x_j \\partial x_i}, \\quad \\forall i, j.",[38,10360,10362,10554],{"className":10361,"ariaHidden":89},[88],[38,10363,10365,10369,10545,10548,10551],{"className":10364},[93],[38,10366],{"className":10367,"style":10368},[97],"height:2.4632em;vertical-align:-0.9721em;",[38,10370,10372,10375,10542],{"className":10371},[102],[38,10373],{"className":10374},[371,740],[38,10376,10378],{"className":10377},[636],[38,10379,10381,10533],{"className":10380},[138,384],[38,10382,10384,10530],{"className":10383},[142],[38,10385,10387,10482,10490],{"className":10386,"style":8901},[146],[38,10388,10389,10392],{"style":756},[38,10390],{"className":10391,"style":760},[154],[38,10393,10395,10398,10438,10441],{"className":10394},[102],[38,10396,642],{"className":10397,"style":767},[102],[38,10399,10401,10404],{"className":10400},[102],[38,10402,286],{"className":10403},[102,103],[38,10405,10407],{"className":10406},[134],[38,10408,10410,10430],{"className":10409},[138,384],[38,10411,10413,10427],{"className":10412},[142],[38,10414,10416],{"className":10415,"style":786},[146],[38,10417,10418,10421],{"style":394},[38,10419],{"className":10420,"style":155},[154],[38,10422,10424],{"className":10423},[159,160,161,162],[38,10425,601],{"className":10426},[102,103,162],[38,10428,408],{"className":10429},[407],[38,10431,10433],{"className":10432},[142],[38,10434,10436],{"className":10435,"style":415},[146],[38,10437],{},[38,10439,642],{"className":10440,"style":767},[102],[38,10442,10444,10447],{"className":10443},[102],[38,10445,286],{"className":10446},[102,103],[38,10448,10450],{"className":10449},[134],[38,10451,10453,10474],{"className":10452},[138,384],[38,10454,10456,10471],{"className":10455},[142],[38,10457,10459],{"className":10458,"style":786},[146],[38,10460,10461,10464],{"style":394},[38,10462],{"className":10463,"style":155},[154],[38,10465,10467],{"className":10466},[159,160,161,162],[38,10468,10308],{"className":10469,"style":10470},[102,103,162],"margin-right:0.0572em;",[38,10472,408],{"className":10473},[407],[38,10475,10477],{"className":10476},[142],[38,10478,10480],{"className":10479,"style":7348},[146],[38,10481],{},[38,10483,10484,10487],{"style":811},[38,10485],{"className":10486,"style":760},[154],[38,10488],{"className":10489,"style":819},[818],[38,10491,10492,10495],{"style":822},[38,10493],{"className":10494,"style":760},[154],[38,10496,10498,10527],{"className":10497},[102],[38,10499,10501,10504],{"className":10500},[102],[38,10502,642],{"className":10503,"style":767},[102],[38,10505,10507],{"className":10506},[134],[38,10508,10510],{"className":10509},[138],[38,10511,10513],{"className":10512},[142],[38,10514,10516],{"className":10515,"style":9002},[146],[38,10517,10518,10521],{"style":150},[38,10519],{"className":10520,"style":155},[154],[38,10522,10524],{"className":10523},[159,160,161,162],[38,10525,312],{"className":10526},[102,162],[38,10528,60],{"className":10529,"style":104},[102,103],[38,10531,408],{"className":10532},[407],[38,10534,10536],{"className":10535},[142],[38,10537,10540],{"className":10538,"style":10539},[146],"height:0.9721em;",[38,10541],{},[38,10543],{"className":10544},[529,740],[38,10546],{"className":10547,"style":109},[108],[38,10549,289],{"className":10550},[113],[38,10552],{"className":10553,"style":109},[108],[38,10555,10557,10560,10734,10737,10741,10744,10747,10750,10753,10756,10759],{"className":10556},[93],[38,10558],{"className":10559,"style":10368},[97],[38,10561,10563,10566,10731],{"className":10562},[102],[38,10564],{"className":10565},[371,740],[38,10567,10569],{"className":10568},[636],[38,10570,10572,10723],{"className":10571},[138,384],[38,10573,10575,10720],{"className":10574},[142],[38,10576,10578,10672,10680],{"className":10577,"style":8901},[146],[38,10579,10580,10583],{"style":756},[38,10581],{"className":10582,"style":760},[154],[38,10584,10586,10589,10629,10632],{"className":10585},[102],[38,10587,642],{"className":10588,"style":767},[102],[38,10590,10592,10595],{"className":10591},[102],[38,10593,286],{"className":10594},[102,103],[38,10596,10598],{"className":10597},[134],[38,10599,10601,10621],{"className":10600},[138,384],[38,10602,10604,10618],{"className":10603},[142],[38,10605,10607],{"className":10606,"style":786},[146],[38,10608,10609,10612],{"style":394},[38,10610],{"className":10611,"style":155},[154],[38,10613,10615],{"className":10614},[159,160,161,162],[38,10616,10308],{"className":10617,"style":10470},[102,103,162],[38,10619,408],{"className":10620},[407],[38,10622,10624],{"className":10623},[142],[38,10625,10627],{"className":10626,"style":7348},[146],[38,10628],{},[38,10630,642],{"className":10631,"style":767},[102],[38,10633,10635,10638],{"className":10634},[102],[38,10636,286],{"className":10637},[102,103],[38,10639,10641],{"className":10640},[134],[38,10642,10644,10664],{"className":10643},[138,384],[38,10645,10647,10661],{"className":10646},[142],[38,10648,10650],{"className":10649,"style":786},[146],[38,10651,10652,10655],{"style":394},[38,10653],{"className":10654,"style":155},[154],[38,10656,10658],{"className":10657},[159,160,161,162],[38,10659,601],{"className":10660},[102,103,162],[38,10662,408],{"className":10663},[407],[38,10665,10667],{"className":10666},[142],[38,10668,10670],{"className":10669,"style":415},[146],[38,10671],{},[38,10673,10674,10677],{"style":811},[38,10675],{"className":10676,"style":760},[154],[38,10678],{"className":10679,"style":819},[818],[38,10681,10682,10685],{"style":822},[38,10683],{"className":10684,"style":760},[154],[38,10686,10688,10717],{"className":10687},[102],[38,10689,10691,10694],{"className":10690},[102],[38,10692,642],{"className":10693,"style":767},[102],[38,10695,10697],{"className":10696},[134],[38,10698,10700],{"className":10699},[138],[38,10701,10703],{"className":10702},[142],[38,10704,10706],{"className":10705,"style":9002},[146],[38,10707,10708,10711],{"style":150},[38,10709],{"className":10710,"style":155},[154],[38,10712,10714],{"className":10713},[159,160,161,162],[38,10715,312],{"className":10716},[102,162],[38,10718,60],{"className":10719,"style":104},[102,103],[38,10721,408],{"className":10722},[407],[38,10724,10726],{"className":10725},[142],[38,10727,10729],{"className":10728,"style":10539},[146],[38,10730],{},[38,10732],{"className":10733},[529,740],[38,10735,305],{"className":10736},[421],[38,10738],{"className":10739,"style":10740},[108],"margin-right:1em;",[38,10742],{"className":10743,"style":425},[108],[38,10745,10347],{"className":10746},[102],[38,10748,601],{"className":10749},[102,103],[38,10751,305],{"className":10752},[421],[38,10754],{"className":10755,"style":425},[108],[38,10757,10308],{"className":10758,"style":10470},[102,103],[38,10760,2192],{"className":10761},[102],[1252,10763,10764],{},[11,10765,10766,10769,10770,1964],{},[15,10767,10768],{},"推论 3.1"," 在上述条件下，海塞矩阵是对称矩阵，即 ",[38,10771,10773,10815],{"className":10772,"translate":42},[41],[38,10774,10776],{"className":10775},[46],[48,10777,10778],{"xmlns":50},[52,10779,10780,10812],{},[55,10781,10782,10788,10790,10792,10794,10796,10802,10804,10806],{},[295,10783,10784,10786],{},[58,10785,8439],{},[58,10787,60],{},[62,10789,293],{"stretchy":292},[58,10791,286],{"mathvariant":285},[62,10793,330],{"stretchy":292},[62,10795,289],{},[295,10797,10798,10800],{},[58,10799,8439],{},[58,10801,60],{},[62,10803,293],{"stretchy":292},[58,10805,286],{"mathvariant":285},[66,10807,10808,10810],{},[62,10809,330],{"stretchy":292},[58,10811,334],{"mathvariant":333},[81,10813,10814],{"encoding":83},"H_f(\\mathbf{x}) = H_f(\\mathbf{x})^\\top",[38,10816,10818,10882],{"className":10817,"ariaHidden":89},[88],[38,10819,10821,10824,10864,10867,10870,10873,10876,10879],{"className":10820},[93],[38,10822],{"className":10823,"style":8669},[97],[38,10825,10827,10830],{"className":10826},[102],[38,10828,8439],{"className":10829,"style":8676},[102,103],[38,10831,10833],{"className":10832},[134],[38,10834,10836,10856],{"className":10835},[138,384],[38,10837,10839,10853],{"className":10838},[142],[38,10840,10842],{"className":10841,"style":4759},[146],[38,10843,10844,10847],{"style":8691},[38,10845],{"className":10846,"style":155},[154],[38,10848,10850],{"className":10849},[159,160,161,162],[38,10851,60],{"className":10852,"style":104},[102,103,162],[38,10854,408],{"className":10855},[407],[38,10857,10859],{"className":10858},[142],[38,10860,10862],{"className":10861,"style":7348},[146],[38,10863],{},[38,10865,293],{"className":10866},[371],[38,10868,286],{"className":10869},[102,351],[38,10871,330],{"className":10872},[529],[38,10874],{"className":10875,"style":109},[108],[38,10877,289],{"className":10878},[113],[38,10880],{"className":10881,"style":109},[108],[38,10883,10885,10889,10929,10932,10935],{"className":10884},[93],[38,10886],{"className":10887,"style":10888},[97],"height:1.1352em;vertical-align:-0.2861em;",[38,10890,10892,10895],{"className":10891},[102],[38,10893,8439],{"className":10894,"style":8676},[102,103],[38,10896,10898],{"className":10897},[134],[38,10899,10901,10921],{"className":10900},[138,384],[38,10902,10904,10918],{"className":10903},[142],[38,10905,10907],{"className":10906,"style":4759},[146],[38,10908,10909,10912],{"style":8691},[38,10910],{"className":10911,"style":155},[154],[38,10913,10915],{"className":10914},[159,160,161,162],[38,10916,60],{"className":10917,"style":104},[102,103,162],[38,10919,408],{"className":10920},[407],[38,10922,10924],{"className":10923},[142],[38,10925,10927],{"className":10926,"style":7348},[146],[38,10928],{},[38,10930,293],{"className":10931},[371],[38,10933,286],{"className":10934},[102,351],[38,10936,10938,10941],{"className":10937},[529],[38,10939,330],{"className":10940},[529],[38,10942,10944],{"className":10943},[134],[38,10945,10947],{"className":10946},[138],[38,10948,10950],{"className":10949},[142],[38,10951,10953],{"className":10952,"style":545},[146],[38,10954,10955,10958],{"style":150},[38,10956],{"className":10957,"style":155},[154],[38,10959,10961],{"className":10960},[159,160,161,162],[38,10962,334],{"className":10963},[102,162],[11,10965,10966],{},"对称性使得海塞矩阵拥有实特征值与正交特征向量分解，这是后续凸性判定与优化算法设计的关键代数基础。",[31,10968,10969],{"id":10969},"二阶泰勒展开与凸性判定",[1252,10971,10972,11057],{},[11,10973,10974,1259,10977,7093],{},[15,10975,10976],{},"定理 3.2（二阶泰勒近似）",[38,10978,10980,11001],{"className":10979,"translate":42},[41],[38,10981,10983],{"className":10982},[46],[48,10984,10985],{"xmlns":50},[52,10986,10987,10999],{},[55,10988,10989,10991,10993],{},[58,10990,60],{},[62,10992,1420],{},[66,10994,10995,10997],{},[58,10996,10200],{},[300,10998,312],{},[81,11000,10205],{"encoding":83},[38,11002,11004,11022],{"className":11003,"ariaHidden":89},[88],[38,11005,11007,11010,11013,11016,11019],{"className":11006},[93],[38,11008],{"className":11009,"style":98},[97],[38,11011,60],{"className":11012,"style":104},[102,103],[38,11014],{"className":11015,"style":109},[108],[38,11017,1420],{"className":11018},[113],[38,11020],{"className":11021,"style":109},[108],[38,11023,11025,11028],{"className":11024},[93],[38,11026],{"className":11027,"style":9002},[97],[38,11029,11031,11034],{"className":11030},[102],[38,11032,10200],{"className":11033,"style":7821},[102,103],[38,11035,11037],{"className":11036},[134],[38,11038,11040],{"className":11039},[138],[38,11041,11043],{"className":11042},[142],[38,11044,11046],{"className":11045,"style":9002},[146],[38,11047,11048,11051],{"style":150},[38,11049],{"className":11050,"style":155},[154],[38,11052,11054],{"className":11053},[159,160,161,162],[38,11055,312],{"className":11056},[102,162],[38,11058,11060],{"className":11059,"translate":42},[621],[38,11061,11063,11165],{"className":11062,"translate":42},[41],[38,11064,11066],{"className":11065},[46],[48,11067,11068],{"xmlns":50,"display":630},[52,11069,11070,11162],{},[55,11071,11072,11074,11076,11078,11080,11082,11084,11086,11088,11090,11092,11094,11096,11098,11100,11102,11104,11110,11112,11114,11120,11126,11132,11134,11136,11138,11140,11142,11144,11146,11148,11150,11152,11158,11160],{},[58,11073,60],{},[62,11075,293],{"stretchy":292},[58,11077,286],{"mathvariant":285},[62,11079,696],{},[58,11081,678],{"mathvariant":285},[62,11083,330],{"stretchy":292},[62,11085,289],{},[58,11087,60],{},[62,11089,293],{"stretchy":292},[58,11091,286],{"mathvariant":285},[62,11093,330],{"stretchy":292},[62,11095,696],{},[58,11097,1335],{"mathvariant":333},[58,11099,60],{},[62,11101,293],{"stretchy":292},[58,11103,286],{"mathvariant":285},[66,11105,11106,11108],{},[62,11107,330],{"stretchy":292},[58,11109,334],{"mathvariant":333},[58,11111,678],{"mathvariant":285},[62,11113,696],{},[636,11115,11116,11118],{},[300,11117,302],{},[300,11119,312],{},[66,11121,11122,11124],{},[58,11123,678],{"mathvariant":285},[58,11125,334],{"mathvariant":333},[295,11127,11128,11130],{},[58,11129,8439],{},[58,11131,60],{},[62,11133,293],{"stretchy":292},[58,11135,286],{"mathvariant":285},[62,11137,330],{"stretchy":292},[3265,11139,3267],{},[58,11141,678],{"mathvariant":285},[62,11143,696],{},[58,11145,2178],{},[62,11147,293],{"stretchy":292},[58,11149,2183],{"mathvariant":333},[58,11151,678],{"mathvariant":285},[66,11153,11154,11156],{},[58,11155,2183],{"mathvariant":333},[300,11157,312],{},[62,11159,330],{"stretchy":292},[58,11161,2192],{"mathvariant":333},[81,11163,11164],{"encoding":83},"f(\\mathbf{x} + \\mathbf{h}) = f(\\mathbf{x}) + \\nabla f(\\mathbf{x})^\\top \\mathbf{h} + \\frac{1}{2} \\mathbf{h}^\\top H_f(\\mathbf{x}) \\, \\mathbf{h} + o(\\|\\mathbf{h}\\|^2).",[38,11166,11168,11192,11213,11240,11299,11462],{"className":11167,"ariaHidden":89},[88],[38,11169,11171,11174,11177,11180,11183,11186,11189],{"className":11170},[93],[38,11172],{"className":11173,"style":1439},[97],[38,11175,60],{"className":11176,"style":104},[102,103],[38,11178,293],{"className":11179},[371],[38,11181,286],{"className":11182},[102,351],[38,11184],{"className":11185,"style":995},[108],[38,11187,696],{"className":11188},[999],[38,11190],{"className":11191,"style":995},[108],[38,11193,11195,11198,11201,11204,11207,11210],{"className":11194},[93],[38,11196],{"className":11197,"style":1439},[97],[38,11199,678],{"className":11200},[102,351],[38,11202,330],{"className":11203},[529],[38,11205],{"className":11206,"style":109},[108],[38,11208,289],{"className":11209},[113],[38,11211],{"className":11212,"style":109},[108],[38,11214,11216,11219,11222,11225,11228,11231,11234,11237],{"className":11215},[93],[38,11217],{"className":11218,"style":1439},[97],[38,11220,60],{"className":11221,"style":104},[102,103],[38,11223,293],{"className":11224},[371],[38,11226,286],{"className":11227},[102,351],[38,11229,330],{"className":11230},[529],[38,11232],{"className":11233,"style":995},[108],[38,11235,696],{"className":11236},[999],[38,11238],{"className":11239,"style":995},[108],[38,11241,11243,11246,11249,11252,11255,11258,11287,11290,11293,11296],{"className":11242},[93],[38,11244],{"className":11245,"style":2277},[97],[38,11247,1335],{"className":11248},[102],[38,11250,60],{"className":11251,"style":104},[102,103],[38,11253,293],{"className":11254},[371],[38,11256,286],{"className":11257},[102,351],[38,11259,11261,11264],{"className":11260},[529],[38,11262,330],{"className":11263},[529],[38,11265,11267],{"className":11266},[134],[38,11268,11270],{"className":11269},[138],[38,11271,11273],{"className":11272},[142],[38,11274,11276],{"className":11275,"style":2308},[146],[38,11277,11278,11281],{"style":1892},[38,11279],{"className":11280,"style":155},[154],[38,11282,11284],{"className":11283},[159,160,161,162],[38,11285,334],{"className":11286},[102,162],[38,11288,678],{"className":11289},[102,351],[38,11291],{"className":11292,"style":995},[108],[38,11294,696],{"className":11295},[999],[38,11297],{"className":11298,"style":995},[108],[38,11300,11302,11306,11369,11398,11438,11441,11444,11447,11450,11453,11456,11459],{"className":11301},[93],[38,11303],{"className":11304,"style":11305},[97],"height:2.0074em;vertical-align:-0.686em;",[38,11307,11309,11312,11366],{"className":11308},[102],[38,11310],{"className":11311},[371,740],[38,11313,11315],{"className":11314},[636],[38,11316,11318,11358],{"className":11317},[138,384],[38,11319,11321,11355],{"className":11320},[142],[38,11322,11325,11336,11344],{"className":11323,"style":11324},[146],"height:1.3214em;",[38,11326,11327,11330],{"style":756},[38,11328],{"className":11329,"style":760},[154],[38,11331,11333],{"className":11332},[102],[38,11334,312],{"className":11335},[102],[38,11337,11338,11341],{"style":811},[38,11339],{"className":11340,"style":760},[154],[38,11342],{"className":11343,"style":819},[818],[38,11345,11346,11349],{"style":822},[38,11347],{"className":11348,"style":760},[154],[38,11350,11352],{"className":11351},[102],[38,11353,302],{"className":11354},[102],[38,11356,408],{"className":11357},[407],[38,11359,11361],{"className":11360},[142],[38,11362,11364],{"className":11363,"style":1079},[146],[38,11365],{},[38,11367],{"className":11368},[529,740],[38,11370,11372,11375],{"className":11371},[102],[38,11373,678],{"className":11374},[102,351],[38,11376,11378],{"className":11377},[134],[38,11379,11381],{"className":11380},[138],[38,11382,11384],{"className":11383},[142],[38,11385,11387],{"className":11386,"style":2308},[146],[38,11388,11389,11392],{"style":1892},[38,11390],{"className":11391,"style":155},[154],[38,11393,11395],{"className":11394},[159,160,161,162],[38,11396,334],{"className":11397},[102,162],[38,11399,11401,11404],{"className":11400},[102],[38,11402,8439],{"className":11403,"style":8676},[102,103],[38,11405,11407],{"className":11406},[134],[38,11408,11410,11430],{"className":11409},[138,384],[38,11411,11413,11427],{"className":11412},[142],[38,11414,11416],{"className":11415,"style":4759},[146],[38,11417,11418,11421],{"style":8691},[38,11419],{"className":11420,"style":155},[154],[38,11422,11424],{"className":11423},[159,160,161,162],[38,11425,60],{"className":11426,"style":104},[102,103,162],[38,11428,408],{"className":11429},[407],[38,11431,11433],{"className":11432},[142],[38,11434,11436],{"className":11435,"style":7348},[146],[38,11437],{},[38,11439,293],{"className":11440},[371],[38,11442,286],{"className":11443},[102,351],[38,11445,330],{"className":11446},[529],[38,11448],{"className":11449,"style":425},[108],[38,11451,678],{"className":11452},[102,351],[38,11454],{"className":11455,"style":995},[108],[38,11457,696],{"className":11458},[999],[38,11460],{"className":11461,"style":995},[108],[38,11463,11465,11469,11472,11475,11478,11481,11511,11514],{"className":11464},[93],[38,11466],{"className":11467,"style":11468},[97],"height:1.1141em;vertical-align:-0.25em;",[38,11470,2178],{"className":11471},[102,103],[38,11473,293],{"className":11474},[371],[38,11476,2183],{"className":11477},[102],[38,11479,678],{"className":11480},[102,351],[38,11482,11484,11487],{"className":11483},[102],[38,11485,2183],{"className":11486},[102],[38,11488,11490],{"className":11489},[134],[38,11491,11493],{"className":11492},[138],[38,11494,11496],{"className":11495},[142],[38,11497,11500],{"className":11498,"style":11499},[146],"height:0.8641em;",[38,11501,11502,11505],{"style":1892},[38,11503],{"className":11504,"style":155},[154],[38,11506,11508],{"className":11507},[159,160,161,162],[38,11509,312],{"className":11510},[102,162],[38,11512,330],{"className":11513},[529],[38,11515,2192],{"className":11516},[102],[11,11518,11519,11520,11630],{},"海塞矩阵通过二次型 ",[38,11521,11523,11549],{"className":11522,"translate":42},[41],[38,11524,11526],{"className":11525},[46],[48,11527,11528],{"xmlns":50},[52,11529,11530,11546],{},[55,11531,11532,11538,11544],{},[66,11533,11534,11536],{},[58,11535,678],{"mathvariant":285},[58,11537,334],{"mathvariant":333},[295,11539,11540,11542],{},[58,11541,8439],{},[58,11543,60],{},[58,11545,678],{"mathvariant":285},[81,11547,11548],{"encoding":83},"\\mathbf{h}^\\top H_f \\mathbf{h}",[38,11550,11552],{"className":11551,"ariaHidden":89},[88],[38,11553,11555,11558,11587,11627],{"className":11554},[93],[38,11556],{"className":11557,"style":10888},[97],[38,11559,11561,11564],{"className":11560},[102],[38,11562,678],{"className":11563},[102,351],[38,11565,11567],{"className":11566},[134],[38,11568,11570],{"className":11569},[138],[38,11571,11573],{"className":11572},[142],[38,11574,11576],{"className":11575,"style":545},[146],[38,11577,11578,11581],{"style":150},[38,11579],{"className":11580,"style":155},[154],[38,11582,11584],{"className":11583},[159,160,161,162],[38,11585,334],{"className":11586},[102,162],[38,11588,11590,11593],{"className":11589},[102],[38,11591,8439],{"className":11592,"style":8676},[102,103],[38,11594,11596],{"className":11595},[134],[38,11597,11599,11619],{"className":11598},[138,384],[38,11600,11602,11616],{"className":11601},[142],[38,11603,11605],{"className":11604,"style":4759},[146],[38,11606,11607,11610],{"style":8691},[38,11608],{"className":11609,"style":155},[154],[38,11611,11613],{"className":11612},[159,160,161,162],[38,11614,60],{"className":11615,"style":104},[102,103,162],[38,11617,408],{"className":11618},[407],[38,11620,11622],{"className":11621},[142],[38,11623,11625],{"className":11624,"style":7348},[146],[38,11626],{},[38,11628,678],{"className":11629},[102,351]," 描述了函数偏离其切平面的\"弯曲程度\"，也即局部凸凹性。",[1252,11632,11633,11831],{},[11,11634,11635,2369,11638,11699,11700,11728,11729,11830],{},[15,11636,11637],{},"定理 3.3（二阶最优性判据）",[38,11639,11641,11660],{"className":11640,"translate":42},[41],[38,11642,11644],{"className":11643},[46],[48,11645,11646],{"xmlns":50},[52,11647,11648,11657],{},[55,11649,11650],{},[66,11651,11652,11654],{},[58,11653,286],{"mathvariant":285},[62,11655,11656],{},"∗",[81,11658,11659],{"encoding":83},"\\mathbf{x}^*",[38,11661,11663],{"className":11662,"ariaHidden":89},[88],[38,11664,11666,11670],{"className":11665},[93],[38,11667],{"className":11668,"style":11669},[97],"height:0.6887em;",[38,11671,11673,11676],{"className":11672},[102],[38,11674,286],{"className":11675},[102,351],[38,11677,11679],{"className":11678},[134],[38,11680,11682],{"className":11681},[138],[38,11683,11685],{"className":11684},[142],[38,11686,11688],{"className":11687,"style":11669},[146],[38,11689,11690,11693],{"style":150},[38,11691],{"className":11692,"style":155},[154],[38,11694,11696],{"className":11695},[159,160,161,162],[38,11697,11656],{"className":11698},[999,162]," 为 ",[38,11701,11703,11716],{"className":11702,"translate":42},[41],[38,11704,11706],{"className":11705},[46],[48,11707,11708],{"xmlns":50},[52,11709,11710,11714],{},[55,11711,11712],{},[58,11713,60],{},[81,11715,60],{"encoding":83},[38,11717,11719],{"className":11718,"ariaHidden":89},[88],[38,11720,11722,11725],{"className":11721},[93],[38,11723],{"className":11724,"style":98},[97],[38,11726,60],{"className":11727,"style":104},[102,103]," 的驻点（即 ",[38,11730,11732,11762],{"className":11731,"translate":42},[41],[38,11733,11735],{"className":11734},[46],[48,11736,11737],{"xmlns":50},[52,11738,11739,11759],{},[55,11740,11741,11743,11745,11747,11753,11755,11757],{},[58,11742,1335],{"mathvariant":333},[58,11744,60],{},[62,11746,293],{"stretchy":292},[66,11748,11749,11751],{},[58,11750,286],{"mathvariant":285},[62,11752,11656],{},[62,11754,330],{"stretchy":292},[62,11756,289],{},[300,11758,683],{"mathvariant":285},[81,11760,11761],{"encoding":83},"\\nabla f(\\mathbf{x}^*) = \\mathbf{0}",[38,11763,11765,11821],{"className":11764,"ariaHidden":89},[88],[38,11766,11768,11771,11774,11777,11780,11809,11812,11815,11818],{"className":11767},[93],[38,11769],{"className":11770,"style":1439},[97],[38,11772,1335],{"className":11773},[102],[38,11775,60],{"className":11776,"style":104},[102,103],[38,11778,293],{"className":11779},[371],[38,11781,11783,11786],{"className":11782},[102],[38,11784,286],{"className":11785},[102,351],[38,11787,11789],{"className":11788},[134],[38,11790,11792],{"className":11791},[138],[38,11793,11795],{"className":11794},[142],[38,11796,11798],{"className":11797,"style":11669},[146],[38,11799,11800,11803],{"style":150},[38,11801],{"className":11802,"style":155},[154],[38,11804,11806],{"className":11805},[159,160,161,162],[38,11807,11656],{"className":11808},[999,162],[38,11810,330],{"className":11811},[529],[38,11813],{"className":11814,"style":109},[108],[38,11816,289],{"className":11817},[113],[38,11819],{"className":11820,"style":109},[108],[38,11822,11824,11827],{"className":11823},[93],[38,11825],{"className":11826,"style":2429},[97],[38,11828,683],{"className":11829},[102,351],"）。",[11832,11833,11834,12012,12188],"ol",{},[7373,11835,11836,11837,11952,11953,12011],{},"若 ",[38,11838,11840,11868],{"className":11839,"translate":42},[41],[38,11841,11843],{"className":11842},[46],[48,11844,11845],{"xmlns":50},[52,11846,11847,11865],{},[55,11848,11849,11855,11857,11863],{},[295,11850,11851,11853],{},[58,11852,8439],{},[58,11854,60],{},[62,11856,293],{"stretchy":292},[66,11858,11859,11861],{},[58,11860,286],{"mathvariant":285},[62,11862,11656],{},[62,11864,330],{"stretchy":292},[81,11866,11867],{"encoding":83},"H_f(\\mathbf{x}^*)",[38,11869,11871],{"className":11870,"ariaHidden":89},[88],[38,11872,11874,11877,11917,11920,11949],{"className":11873},[93],[38,11875],{"className":11876,"style":8669},[97],[38,11878,11880,11883],{"className":11879},[102],[38,11881,8439],{"className":11882,"style":8676},[102,103],[38,11884,11886],{"className":11885},[134],[38,11887,11889,11909],{"className":11888},[138,384],[38,11890,11892,11906],{"className":11891},[142],[38,11893,11895],{"className":11894,"style":4759},[146],[38,11896,11897,11900],{"style":8691},[38,11898],{"className":11899,"style":155},[154],[38,11901,11903],{"className":11902},[159,160,161,162],[38,11904,60],{"className":11905,"style":104},[102,103,162],[38,11907,408],{"className":11908},[407],[38,11910,11912],{"className":11911},[142],[38,11913,11915],{"className":11914,"style":7348},[146],[38,11916],{},[38,11918,293],{"className":11919},[371],[38,11921,11923,11926],{"className":11922},[102],[38,11924,286],{"className":11925},[102,351],[38,11927,11929],{"className":11928},[134],[38,11930,11932],{"className":11931},[138],[38,11933,11935],{"className":11934},[142],[38,11936,11938],{"className":11937,"style":11669},[146],[38,11939,11940,11943],{"style":150},[38,11941],{"className":11942,"style":155},[154],[38,11944,11946],{"className":11945},[159,160,161,162],[38,11947,11656],{"className":11948},[999,162],[38,11950,330],{"className":11951},[529]," 正定，则 ",[38,11954,11956,11973],{"className":11955,"translate":42},[41],[38,11957,11959],{"className":11958},[46],[48,11960,11961],{"xmlns":50},[52,11962,11963,11971],{},[55,11964,11965],{},[66,11966,11967,11969],{},[58,11968,286],{"mathvariant":285},[62,11970,11656],{},[81,11972,11659],{"encoding":83},[38,11974,11976],{"className":11975,"ariaHidden":89},[88],[38,11977,11979,11982],{"className":11978},[93],[38,11980],{"className":11981,"style":11669},[97],[38,11983,11985,11988],{"className":11984},[102],[38,11986,286],{"className":11987},[102,351],[38,11989,11991],{"className":11990},[134],[38,11992,11994],{"className":11993},[138],[38,11995,11997],{"className":11996},[142],[38,11998,12000],{"className":11999,"style":11669},[146],[38,12001,12002,12005],{"style":150},[38,12003],{"className":12004,"style":155},[154],[38,12006,12008],{"className":12007},[159,160,161,162],[38,12009,11656],{"className":12010},[999,162]," 是严格局部极小点；",[7373,12013,11836,12014,12128,12129,12187],{},[38,12015,12017,12044],{"className":12016,"translate":42},[41],[38,12018,12020],{"className":12019},[46],[48,12021,12022],{"xmlns":50},[52,12023,12024,12042],{},[55,12025,12026,12032,12034,12040],{},[295,12027,12028,12030],{},[58,12029,8439],{},[58,12031,60],{},[62,12033,293],{"stretchy":292},[66,12035,12036,12038],{},[58,12037,286],{"mathvariant":285},[62,12039,11656],{},[62,12041,330],{"stretchy":292},[81,12043,11867],{"encoding":83},[38,12045,12047],{"className":12046,"ariaHidden":89},[88],[38,12048,12050,12053,12093,12096,12125],{"className":12049},[93],[38,12051],{"className":12052,"style":8669},[97],[38,12054,12056,12059],{"className":12055},[102],[38,12057,8439],{"className":12058,"style":8676},[102,103],[38,12060,12062],{"className":12061},[134],[38,12063,12065,12085],{"className":12064},[138,384],[38,12066,12068,12082],{"className":12067},[142],[38,12069,12071],{"className":12070,"style":4759},[146],[38,12072,12073,12076],{"style":8691},[38,12074],{"className":12075,"style":155},[154],[38,12077,12079],{"className":12078},[159,160,161,162],[38,12080,60],{"className":12081,"style":104},[102,103,162],[38,12083,408],{"className":12084},[407],[38,12086,12088],{"className":12087},[142],[38,12089,12091],{"className":12090,"style":7348},[146],[38,12092],{},[38,12094,293],{"className":12095},[371],[38,12097,12099,12102],{"className":12098},[102],[38,12100,286],{"className":12101},[102,351],[38,12103,12105],{"className":12104},[134],[38,12106,12108],{"className":12107},[138],[38,12109,12111],{"className":12110},[142],[38,12112,12114],{"className":12113,"style":11669},[146],[38,12115,12116,12119],{"style":150},[38,12117],{"className":12118,"style":155},[154],[38,12120,12122],{"className":12121},[159,160,161,162],[38,12123,11656],{"className":12124},[999,162],[38,12126,330],{"className":12127},[529]," 负定，则 ",[38,12130,12132,12149],{"className":12131,"translate":42},[41],[38,12133,12135],{"className":12134},[46],[48,12136,12137],{"xmlns":50},[52,12138,12139,12147],{},[55,12140,12141],{},[66,12142,12143,12145],{},[58,12144,286],{"mathvariant":285},[62,12146,11656],{},[81,12148,11659],{"encoding":83},[38,12150,12152],{"className":12151,"ariaHidden":89},[88],[38,12153,12155,12158],{"className":12154},[93],[38,12156],{"className":12157,"style":11669},[97],[38,12159,12161,12164],{"className":12160},[102],[38,12162,286],{"className":12163},[102,351],[38,12165,12167],{"className":12166},[134],[38,12168,12170],{"className":12169},[138],[38,12171,12173],{"className":12172},[142],[38,12174,12176],{"className":12175,"style":11669},[146],[38,12177,12178,12181],{"style":150},[38,12179],{"className":12180,"style":155},[154],[38,12182,12184],{"className":12183},[159,160,161,162],[38,12185,11656],{"className":12186},[999,162]," 是严格局部极大点；",[7373,12189,11836,12190,12304,12305,12363,12364,12367],{},[38,12191,12193,12220],{"className":12192,"translate":42},[41],[38,12194,12196],{"className":12195},[46],[48,12197,12198],{"xmlns":50},[52,12199,12200,12218],{},[55,12201,12202,12208,12210,12216],{},[295,12203,12204,12206],{},[58,12205,8439],{},[58,12207,60],{},[62,12209,293],{"stretchy":292},[66,12211,12212,12214],{},[58,12213,286],{"mathvariant":285},[62,12215,11656],{},[62,12217,330],{"stretchy":292},[81,12219,11867],{"encoding":83},[38,12221,12223],{"className":12222,"ariaHidden":89},[88],[38,12224,12226,12229,12269,12272,12301],{"className":12225},[93],[38,12227],{"className":12228,"style":8669},[97],[38,12230,12232,12235],{"className":12231},[102],[38,12233,8439],{"className":12234,"style":8676},[102,103],[38,12236,12238],{"className":12237},[134],[38,12239,12241,12261],{"className":12240},[138,384],[38,12242,12244,12258],{"className":12243},[142],[38,12245,12247],{"className":12246,"style":4759},[146],[38,12248,12249,12252],{"style":8691},[38,12250],{"className":12251,"style":155},[154],[38,12253,12255],{"className":12254},[159,160,161,162],[38,12256,60],{"className":12257,"style":104},[102,103,162],[38,12259,408],{"className":12260},[407],[38,12262,12264],{"className":12263},[142],[38,12265,12267],{"className":12266,"style":7348},[146],[38,12268],{},[38,12270,293],{"className":12271},[371],[38,12273,12275,12278],{"className":12274},[102],[38,12276,286],{"className":12277},[102,351],[38,12279,12281],{"className":12280},[134],[38,12282,12284],{"className":12283},[138],[38,12285,12287],{"className":12286},[142],[38,12288,12290],{"className":12289,"style":11669},[146],[38,12291,12292,12295],{"style":150},[38,12293],{"className":12294,"style":155},[154],[38,12296,12298],{"className":12297},[159,160,161,162],[38,12299,11656],{"className":12300},[999,162],[38,12302,330],{"className":12303},[529]," 不定（既有正特征值又有负特征值），则 ",[38,12306,12308,12325],{"className":12307,"translate":42},[41],[38,12309,12311],{"className":12310},[46],[48,12312,12313],{"xmlns":50},[52,12314,12315,12323],{},[55,12316,12317],{},[66,12318,12319,12321],{},[58,12320,286],{"mathvariant":285},[62,12322,11656],{},[81,12324,11659],{"encoding":83},[38,12326,12328],{"className":12327,"ariaHidden":89},[88],[38,12329,12331,12334],{"className":12330},[93],[38,12332],{"className":12333,"style":11669},[97],[38,12335,12337,12340],{"className":12336},[102],[38,12338,286],{"className":12339},[102,351],[38,12341,12343],{"className":12342},[134],[38,12344,12346],{"className":12345},[138],[38,12347,12349],{"className":12348},[142],[38,12350,12352],{"className":12351,"style":11669},[146],[38,12353,12354,12357],{"style":150},[38,12355],{"className":12356,"style":155},[154],[38,12358,12360],{"className":12359},[159,160,161,162],[38,12361,11656],{"className":12362},[999,162]," 是",[15,12365,12366],{},"鞍点","（saddle point）。",[1252,12369,12370],{},[11,12371,12372,12375,12376,12404,12405,12433,12434,12542,12543,12595],{},[15,12373,12374],{},"定理 3.4（凸性的二阶判据）"," 函数 ",[38,12377,12379,12392],{"className":12378,"translate":42},[41],[38,12380,12382],{"className":12381},[46],[48,12383,12384],{"xmlns":50},[52,12385,12386,12390],{},[55,12387,12388],{},[58,12389,60],{},[81,12391,60],{"encoding":83},[38,12393,12395],{"className":12394,"ariaHidden":89},[88],[38,12396,12398,12401],{"className":12397},[93],[38,12399],{"className":12400,"style":98},[97],[38,12402,60],{"className":12403,"style":104},[102,103]," 在凸集 ",[38,12406,12408,12421],{"className":12407,"translate":42},[41],[38,12409,12411],{"className":12410},[46],[48,12412,12413],{"xmlns":50},[52,12414,12415,12419],{},[55,12416,12417],{},[58,12418,199],{},[81,12420,199],{"encoding":83},[38,12422,12424],{"className":12423,"ariaHidden":89},[88],[38,12425,12427,12430],{"className":12426},[93],[38,12428],{"className":12429,"style":7404},[97],[38,12431,199],{"className":12432,"style":225},[102,103]," 上为凸函数，当且仅当 ",[38,12435,12437,12466],{"className":12436,"translate":42},[41],[38,12438,12440],{"className":12439},[46],[48,12441,12442],{"xmlns":50},[52,12443,12444,12463],{},[55,12445,12446,12452,12454,12456,12458,12461],{},[295,12447,12448,12450],{},[58,12449,8439],{},[58,12451,60],{},[62,12453,293],{"stretchy":292},[58,12455,286],{"mathvariant":285},[62,12457,330],{"stretchy":292},[62,12459,12460],{},"⪰",[300,12462,683],{},[81,12464,12465],{"encoding":83},"H_f(\\mathbf{x}) \\succeq 0",[38,12467,12469,12533],{"className":12468,"ariaHidden":89},[88],[38,12470,12472,12475,12515,12518,12521,12524,12527,12530],{"className":12471},[93],[38,12473],{"className":12474,"style":8669},[97],[38,12476,12478,12481],{"className":12477},[102],[38,12479,8439],{"className":12480,"style":8676},[102,103],[38,12482,12484],{"className":12483},[134],[38,12485,12487,12507],{"className":12486},[138,384],[38,12488,12490,12504],{"className":12489},[142],[38,12491,12493],{"className":12492,"style":4759},[146],[38,12494,12495,12498],{"style":8691},[38,12496],{"className":12497,"style":155},[154],[38,12499,12501],{"className":12500},[159,160,161,162],[38,12502,60],{"className":12503,"style":104},[102,103,162],[38,12505,408],{"className":12506},[407],[38,12508,12510],{"className":12509},[142],[38,12511,12513],{"className":12512,"style":7348},[146],[38,12514],{},[38,12516,293],{"className":12517},[371],[38,12519,286],{"className":12520},[102,351],[38,12522,330],{"className":12523},[529],[38,12525],{"className":12526,"style":109},[108],[38,12528,12460],{"className":12529},[113],[38,12531],{"className":12532,"style":109},[108],[38,12534,12536,12539],{"className":12535},[93],[38,12537],{"className":12538,"style":2429},[97],[38,12540,683],{"className":12541},[102],"（半正定）对所有 ",[38,12544,12546,12564],{"className":12545,"translate":42},[41],[38,12547,12549],{"className":12548},[46],[48,12550,12551],{"xmlns":50},[52,12552,12553,12561],{},[55,12554,12555,12557,12559],{},[58,12556,286],{"mathvariant":285},[62,12558,1420],{},[58,12560,199],{},[81,12562,12563],{"encoding":83},"\\mathbf{x} \\in U",[38,12565,12567,12586],{"className":12566,"ariaHidden":89},[88],[38,12568,12570,12574,12577,12580,12583],{"className":12569},[93],[38,12571],{"className":12572,"style":12573},[97],"height:0.5782em;vertical-align:-0.0391em;",[38,12575,286],{"className":12576},[102,351],[38,12578],{"className":12579,"style":109},[108],[38,12581,1420],{"className":12582},[113],[38,12584],{"className":12585,"style":109},[108],[38,12587,12589,12592],{"className":12588},[93],[38,12590],{"className":12591,"style":7404},[97],[38,12593,199],{"className":12594,"style":225},[102,103]," 成立。",[31,12597,3086],{"id":12598},"机器学习-2",[7370,12600,12601],{},[7373,12602,12603,12606],{},[15,12604,12605],{},"牛顿法与二阶优化","：相较于仅利用一阶信息的梯度下降，牛顿法利用海塞矩阵构造局部二次模型并直接求其极小点：",[38,12608,12610],{"className":12609,"translate":42},[621],[38,12611,12613,12687],{"className":12612,"translate":42},[41],[38,12614,12616],{"className":12615},[46],[48,12617,12618],{"xmlns":50,"display":630},[52,12619,12620,12684],{},[55,12621,12622,12634,12636,12642,12644,12650,12652,12658,12668,12670,12672,12674,12680,12682],{},[295,12623,12624,12626],{},[58,12625,3109],{},[55,12627,12628,12630,12632],{},[58,12629,3246],{},[62,12631,696],{},[300,12633,302],{},[62,12635,289],{},[295,12637,12638,12640],{},[58,12639,3109],{},[58,12641,3246],{},[62,12643,710],{},[295,12645,12646,12648],{},[58,12647,8439],{},[58,12649,60],{},[62,12651,293],{"stretchy":292},[295,12653,12654,12656],{},[58,12655,3109],{},[58,12657,3246],{},[66,12659,12660,12662],{},[62,12661,330],{"stretchy":292},[55,12663,12664,12666],{},[62,12665,710],{},[300,12667,302],{},[58,12669,1335],{"mathvariant":333},[58,12671,60],{},[62,12673,293],{"stretchy":292},[295,12675,12676,12678],{},[58,12677,3109],{},[58,12679,3246],{},[62,12681,330],{"stretchy":292},[58,12683,2192],{"mathvariant":333},[81,12685,12686],{"encoding":83},"\\theta_{t+1} = \\theta_t - H_f(\\theta_t)^{-1} \\nabla f(\\theta_t).",[38,12688,12690,12754,12809],{"className":12689,"ariaHidden":89},[88],[38,12691,12693,12696,12745,12748,12751],{"className":12692},[93],[38,12694],{"className":12695,"style":3294},[97],[38,12697,12699,12702],{"className":12698},[102],[38,12700,3109],{"className":12701,"style":2500},[102,103],[38,12703,12705],{"className":12704},[134],[38,12706,12708,12737],{"className":12707},[138,384],[38,12709,12711,12734],{"className":12710},[142],[38,12712,12714],{"className":12713,"style":391},[146],[38,12715,12716,12719],{"style":2516},[38,12717],{"className":12718,"style":155},[154],[38,12720,12722],{"className":12721},[159,160,161,162],[38,12723,12725,12728,12731],{"className":12724},[102,162],[38,12726,3246],{"className":12727},[102,103,162],[38,12729,696],{"className":12730},[999,162],[38,12732,302],{"className":12733},[102,162],[38,12735,408],{"className":12736},[407],[38,12738,12740],{"className":12739},[142],[38,12741,12743],{"className":12742,"style":3342},[146],[38,12744],{},[38,12746],{"className":12747,"style":109},[108],[38,12749,289],{"className":12750},[113],[38,12752],{"className":12753,"style":109},[108],[38,12755,12757,12760,12800,12803,12806],{"className":12756},[93],[38,12758],{"className":12759,"style":3360},[97],[38,12761,12763,12766],{"className":12762},[102],[38,12764,3109],{"className":12765,"style":2500},[102,103],[38,12767,12769],{"className":12768},[134],[38,12770,12772,12792],{"className":12771},[138,384],[38,12773,12775,12789],{"className":12774},[142],[38,12776,12778],{"className":12777,"style":3379},[146],[38,12779,12780,12783],{"style":2516},[38,12781],{"className":12782,"style":155},[154],[38,12784,12786],{"className":12785},[159,160,161,162],[38,12787,3246],{"className":12788},[102,103,162],[38,12790,408],{"className":12791},[407],[38,12793,12795],{"className":12794},[142],[38,12796,12798],{"className":12797,"style":415},[146],[38,12799],{},[38,12801],{"className":12802,"style":995},[108],[38,12804,710],{"className":12805},[999],[38,12807],{"className":12808,"style":995},[108],[38,12810,12812,12816,12856,12859,12899,12934,12937,12940,12943,12983,12986],{"className":12811},[93],[38,12813],{"className":12814,"style":12815},[97],"height:1.1502em;vertical-align:-0.2861em;",[38,12817,12819,12822],{"className":12818},[102],[38,12820,8439],{"className":12821,"style":8676},[102,103],[38,12823,12825],{"className":12824},[134],[38,12826,12828,12848],{"className":12827},[138,384],[38,12829,12831,12845],{"className":12830},[142],[38,12832,12834],{"className":12833,"style":4759},[146],[38,12835,12836,12839],{"style":8691},[38,12837],{"className":12838,"style":155},[154],[38,12840,12842],{"className":12841},[159,160,161,162],[38,12843,60],{"className":12844,"style":104},[102,103,162],[38,12846,408],{"className":12847},[407],[38,12849,12851],{"className":12850},[142],[38,12852,12854],{"className":12853,"style":7348},[146],[38,12855],{},[38,12857,293],{"className":12858},[371],[38,12860,12862,12865],{"className":12861},[102],[38,12863,3109],{"className":12864,"style":2500},[102,103],[38,12866,12868],{"className":12867},[134],[38,12869,12871,12891],{"className":12870},[138,384],[38,12872,12874,12888],{"className":12873},[142],[38,12875,12877],{"className":12876,"style":3379},[146],[38,12878,12879,12882],{"style":2516},[38,12880],{"className":12881,"style":155},[154],[38,12883,12885],{"className":12884},[159,160,161,162],[38,12886,3246],{"className":12887},[102,103,162],[38,12889,408],{"className":12890},[407],[38,12892,12894],{"className":12893},[142],[38,12895,12897],{"className":12896,"style":415},[146],[38,12898],{},[38,12900,12902,12905],{"className":12901},[529],[38,12903,330],{"className":12904},[529],[38,12906,12908],{"className":12907},[134],[38,12909,12911],{"className":12910},[138],[38,12912,12914],{"className":12913},[142],[38,12915,12917],{"className":12916,"style":11499},[146],[38,12918,12919,12922],{"style":1892},[38,12920],{"className":12921,"style":155},[154],[38,12923,12925],{"className":12924},[159,160,161,162],[38,12926,12928,12931],{"className":12927},[102,162],[38,12929,710],{"className":12930},[102,162],[38,12932,302],{"className":12933},[102,162],[38,12935,1335],{"className":12936},[102],[38,12938,60],{"className":12939,"style":104},[102,103],[38,12941,293],{"className":12942},[371],[38,12944,12946,12949],{"className":12945},[102],[38,12947,3109],{"className":12948,"style":2500},[102,103],[38,12950,12952],{"className":12951},[134],[38,12953,12955,12975],{"className":12954},[138,384],[38,12956,12958,12972],{"className":12957},[142],[38,12959,12961],{"className":12960,"style":3379},[146],[38,12962,12963,12966],{"style":2516},[38,12964],{"className":12965,"style":155},[154],[38,12967,12969],{"className":12968},[159,160,161,162],[38,12970,3246],{"className":12971},[102,103,162],[38,12973,408],{"className":12974},[407],[38,12976,12978],{"className":12977},[142],[38,12979,12981],{"className":12980,"style":415},[146],[38,12982],{},[38,12984,330],{"className":12985},[529],[38,12987,2192],{"className":12988},[102],[11,12990,12991,12992,13090,13091,3026,13168,13242,13243,13246],{},"牛顿法在二次逼近良好的区域内具有二阶收敛速度，但对于深度学习中动辄百万级参数的模型，显式求逆 ",[38,12993,12995,13019],{"className":12994,"translate":42},[41],[38,12996,12998],{"className":12997},[46],[48,12999,13000],{"xmlns":50},[52,13001,13002,13016],{},[55,13003,13004],{},[8499,13005,13006,13008,13010],{},[58,13007,8439],{},[58,13009,60],{},[55,13011,13012,13014],{},[62,13013,710],{},[300,13015,302],{},[81,13017,13018],{"encoding":83},"H_f^{-1}",[38,13020,13022],{"className":13021,"ariaHidden":89},[88],[38,13023,13025,13029],{"className":13024},[93],[38,13026],{"className":13027,"style":13028},[97],"height:1.2917em;vertical-align:-0.4374em;",[38,13030,13032,13035],{"className":13031},[102],[38,13033,8439],{"className":13034,"style":8676},[102,103],[38,13036,13038],{"className":13037},[134],[38,13039,13041,13081],{"className":13040},[138,384],[38,13042,13044,13078],{"className":13043},[142],[38,13045,13048,13060],{"className":13046,"style":13047},[146],"height:0.8542em;",[38,13049,13051,13054],{"style":13050},"top:-2.3987em;margin-left:-0.0813em;margin-right:0.05em;",[38,13052],{"className":13053,"style":155},[154],[38,13055,13057],{"className":13056},[159,160,161,162],[38,13058,60],{"className":13059,"style":104},[102,103,162],[38,13061,13063,13066],{"style":13062},"top:-3.1031em;margin-right:0.05em;",[38,13064],{"className":13065,"style":155},[154],[38,13067,13069],{"className":13068},[159,160,161,162],[38,13070,13072,13075],{"className":13071},[102,162],[38,13073,710],{"className":13074},[102,162],[38,13076,302],{"className":13077},[102,162],[38,13079,408],{"className":13080},[407],[38,13082,13084],{"className":13083},[142],[38,13085,13088],{"className":13086,"style":13087},[146],"height:0.4374em;",[38,13089],{}," 的计算与存储代价高达 ",[38,13092,13094,13120],{"className":13093,"translate":42},[41],[38,13095,13097],{"className":13096},[46],[48,13098,13099],{"xmlns":50},[52,13100,13101,13117],{},[55,13102,13103,13106,13108,13115],{},[58,13104,13105],{},"O",[62,13107,293],{"stretchy":292},[66,13109,13110,13112],{},[58,13111,11],{},[300,13113,13114],{},"3",[62,13116,330],{"stretchy":292},[81,13118,13119],{"encoding":83},"O(p^3)",[38,13121,13123],{"className":13122,"ariaHidden":89},[88],[38,13124,13126,13130,13133,13136,13165],{"className":13125},[93],[38,13127],{"className":13128,"style":13129},[97],"height:1.0641em;vertical-align:-0.25em;",[38,13131,13105],{"className":13132,"style":2500},[102,103],[38,13134,293],{"className":13135},[371],[38,13137,13139,13142],{"className":13138},[102],[38,13140,11],{"className":13141},[102,103],[38,13143,13145],{"className":13144},[134],[38,13146,13148],{"className":13147},[138],[38,13149,13151],{"className":13150},[142],[38,13152,13154],{"className":13153,"style":9002},[146],[38,13155,13156,13159],{"style":150},[38,13157],{"className":13158,"style":155},[154],[38,13160,13162],{"className":13161},[159,160,161,162],[38,13163,13114],{"className":13164},[102,162],[38,13166,330],{"className":13167},[529],[38,13169,13171,13195],{"className":13170,"translate":42},[41],[38,13172,13174],{"className":13173},[46],[48,13175,13176],{"xmlns":50},[52,13177,13178,13192],{},[55,13179,13180,13182,13184,13190],{},[58,13181,13105],{},[62,13183,293],{"stretchy":292},[66,13185,13186,13188],{},[58,13187,11],{},[300,13189,312],{},[62,13191,330],{"stretchy":292},[81,13193,13194],{"encoding":83},"O(p^2)",[38,13196,13198],{"className":13197,"ariaHidden":89},[88],[38,13199,13201,13204,13207,13210,13239],{"className":13200},[93],[38,13202],{"className":13203,"style":13129},[97],[38,13205,13105],{"className":13206,"style":2500},[102,103],[38,13208,293],{"className":13209},[371],[38,13211,13213,13216],{"className":13212},[102],[38,13214,11],{"className":13215},[102,103],[38,13217,13219],{"className":13218},[134],[38,13220,13222],{"className":13221},[138],[38,13223,13225],{"className":13224},[142],[38,13226,13228],{"className":13227,"style":9002},[146],[38,13229,13230,13233],{"style":150},[38,13231],{"className":13232,"style":155},[154],[38,13234,13236],{"className":13235},[159,160,161,162],[38,13237,312],{"className":13238},[102,162],[38,13240,330],{"className":13241},[529],"，因而催生了拟牛顿法（如 BFGS、L-BFGS）与仅利用海塞矩阵–向量积（Hessian-vector product）而不显式构造海塞矩阵的",[15,13244,13245],{},"Hessian-free 优化","方法。",[7370,13248,13249,13255,13265],{},[7373,13250,13251,13254],{},[15,13252,13253],{},"损失曲面的几何分析","：深度网络损失函数高度非凸，海塞矩阵的特征值谱是分析损失曲面局部几何（尖锐极小值 vs. 平坦极小值、鞍点密度）的重要工具，这与模型的泛化能力密切相关——经验研究表明，位于\"平坦极小值\"（海塞矩阵特征值普遍较小）附近的解通常具有更好的泛化性能。",[7373,13256,13257,13260,13261,13264],{},[15,13258,13259],{},"自然梯度法与费舍尔信息矩阵","：在概率模型中，损失函数关于参数的",[15,13262,13263],{},"费舍尔信息矩阵","（Fisher Information Matrix）在特定条件下与负对数似然的海塞矩阵的期望相一致，自然梯度法利用该矩阵对参数空间做黎曼度量下的校正，从而获得优化路径的参数化不变性。",[7373,13266,13267,13270],{},[15,13268,13269],{},"二阶正则化与鲁棒性","：对海塞矩阵特征值谱施加正则化（如惩罚最大特征值）已被用于提升模型抗对抗攻击的鲁棒性与训练稳定性，因为较小的海塞矩阵范数意味着损失函数在参数空间中变化更为平缓。",[28,13272,13273],{"id":13273},"统一视角",[11,13275,13276],{},"综合前三部分的讨论，可以将梯度、雅可比矩阵与海塞矩阵纳入同一个求导算子递进作用的框架中：",[13278,13279,13280,13299],"table",{},[13281,13282,13283],"thead",{},[13284,13285,13286,13290,13293,13296],"tr",{},[13287,13288,13289],"th",{},"概念",[13287,13291,13292],{},"作用对象",[13287,13294,13295],{},"结果类型",[13287,13297,13298],{},"与雅可比的关系",[13300,13301,13302,13575,13883],"tbody",{},[13284,13303,13304,13342,13446,13522],{},[13305,13306,13307,13308],"td",{},"梯度 ",[38,13309,13311,13327],{"className":13310,"translate":42},[41],[38,13312,13314],{"className":13313},[46],[48,13315,13316],{"xmlns":50},[52,13317,13318,13324],{},[55,13319,13320,13322],{},[58,13321,1335],{"mathvariant":333},[58,13323,60],{},[81,13325,13326],{"encoding":83},"\\nabla f",[38,13328,13330],{"className":13329,"ariaHidden":89},[88],[38,13331,13333,13336,13339],{"className":13332},[93],[38,13334],{"className":13335,"style":98},[97],[38,13337,1335],{"className":13338},[102],[38,13340,60],{"className":13341,"style":104},[102,103],[13305,13343,13344],{},[38,13345,13347,13372],{"className":13346,"translate":42},[41],[38,13348,13350],{"className":13349},[46],[48,13351,13352],{"xmlns":50},[52,13353,13354,13370],{},[55,13355,13356,13358,13360,13366,13368],{},[58,13357,60],{},[62,13359,64],{},[66,13361,13362,13364],{},[58,13363,71],{"mathvariant":70},[58,13365,74],{},[62,13367,77],{},[58,13369,71],{"mathvariant":70},[81,13371,84],{"encoding":83},[38,13373,13375,13393,13437],{"className":13374,"ariaHidden":89},[88],[38,13376,13378,13381,13384,13387,13390],{"className":13377},[93],[38,13379],{"className":13380,"style":98},[97],[38,13382,60],{"className":13383,"style":104},[102,103],[38,13385],{"className":13386,"style":109},[108],[38,13388,64],{"className":13389},[113],[38,13391],{"className":13392,"style":109},[108],[38,13394,13396,13399,13428,13431,13434],{"className":13395},[93],[38,13397],{"className":13398,"style":123},[97],[38,13400,13402,13405],{"className":13401},[102],[38,13403,71],{"className":13404},[102,130],[38,13406,13408],{"className":13407},[134],[38,13409,13411],{"className":13410},[138],[38,13412,13414],{"className":13413},[142],[38,13415,13417],{"className":13416,"style":147},[146],[38,13418,13419,13422],{"style":150},[38,13420],{"className":13421,"style":155},[154],[38,13423,13425],{"className":13424},[159,160,161,162],[38,13426,74],{"className":13427},[102,103,162],[38,13429],{"className":13430,"style":109},[108],[38,13432,77],{"className":13433},[113],[38,13435],{"className":13436,"style":109},[108],[38,13438,13440,13443],{"className":13439},[93],[38,13441],{"className":13442,"style":123},[97],[38,13444,71],{"className":13445},[102,130],[13305,13447,13448,13449],{},"向量 ",[38,13450,13452,13472],{"className":13451,"translate":42},[41],[38,13453,13455],{"className":13454},[46],[48,13456,13457],{"xmlns":50},[52,13458,13459,13469],{},[55,13460,13461,13463],{},[62,13462,1420],{},[66,13464,13465,13467],{},[58,13466,71],{"mathvariant":70},[58,13468,74],{},[81,13470,13471],{"encoding":83},"\\in \\mathbb{R}^n",[38,13473,13475,13487],{"className":13474,"ariaHidden":89},[88],[38,13476,13478,13481,13484],{"className":13477},[93],[38,13479],{"className":13480,"style":12573},[97],[38,13482,1420],{"className":13483},[113],[38,13485],{"className":13486,"style":109},[108],[38,13488,13490,13493],{"className":13489},[93],[38,13491],{"className":13492,"style":123},[97],[38,13494,13496,13499],{"className":13495},[102],[38,13497,71],{"className":13498},[102,130],[38,13500,13502],{"className":13501},[134],[38,13503,13505],{"className":13504},[138],[38,13506,13508],{"className":13507},[142],[38,13509,13511],{"className":13510,"style":147},[146],[38,13512,13513,13516],{"style":150},[38,13514],{"className":13515,"style":155},[154],[38,13517,13519],{"className":13518},[159,160,161,162],[38,13520,74],{"className":13521},[102,103,162],[13305,13523,13524,13574],{},[38,13525,13527,13544],{"className":13526,"translate":42},[41],[38,13528,13530],{"className":13529},[46],[48,13531,13532],{"xmlns":50},[52,13533,13534,13542],{},[55,13535,13536,13538,13540],{},[58,13537,3555],{},[62,13539,289],{},[300,13541,302],{},[81,13543,6280],{"encoding":83},[38,13545,13547,13565],{"className":13546,"ariaHidden":89},[88],[38,13548,13550,13553,13556,13559,13562],{"className":13549},[93],[38,13551],{"className":13552,"style":4312},[97],[38,13554,3555],{"className":13555},[102,103],[38,13557],{"className":13558,"style":109},[108],[38,13560,289],{"className":13561},[113],[38,13563],{"className":13564,"style":109},[108],[38,13566,13568,13571],{"className":13567},[93],[38,13569],{"className":13570,"style":2429},[97],[38,13572,302],{"className":13573},[102]," 情形下雅可比矩阵的转置",[13284,13576,13577,13654,13788,13880],{},[13305,13578,13579,13580],{},"雅可比矩阵 ",[38,13581,13583,13601],{"className":13582,"translate":42},[41],[38,13584,13586],{"className":13585},[46],[48,13587,13588],{"xmlns":50},[52,13589,13590,13598],{},[55,13591,13592],{},[295,13593,13594,13596],{},[58,13595,4459],{},[58,13597,60],{"mathvariant":285},[81,13599,13600],{"encoding":83},"J_{\\mathbf{f}}",[38,13602,13604],{"className":13603,"ariaHidden":89},[88],[38,13605,13607,13611],{"className":13606},[93],[38,13608],{"className":13609,"style":13610},[97],"height:0.8333em;vertical-align:-0.15em;",[38,13612,13614,13617],{"className":13613},[102],[38,13615,4459],{"className":13616,"style":4746},[102,103],[38,13618,13620],{"className":13619},[134],[38,13621,13623,13646],{"className":13622},[138,384],[38,13624,13626,13643],{"className":13625},[142],[38,13627,13629],{"className":13628,"style":4759},[146],[38,13630,13631,13634],{"style":4762},[38,13632],{"className":13633,"style":155},[154],[38,13635,13637],{"className":13636},[159,160,161,162],[38,13638,13640],{"className":13639},[102,162],[38,13641,60],{"className":13642,"style":225},[102,351,162],[38,13644,408],{"className":13645},[407],[38,13647,13649],{"className":13648},[142],[38,13650,13652],{"className":13651,"style":415},[146],[38,13653],{},[13305,13655,13656],{},[38,13657,13659,13688],{"className":13658,"translate":42},[41],[38,13660,13662],{"className":13661},[46],[48,13663,13664],{"xmlns":50},[52,13665,13666,13686],{},[55,13667,13668,13670,13672,13678,13680],{},[58,13669,60],{"mathvariant":285},[62,13671,64],{},[66,13673,13674,13676],{},[58,13675,71],{"mathvariant":70},[58,13677,74],{},[62,13679,77],{},[66,13681,13682,13684],{},[58,13683,71],{"mathvariant":70},[58,13685,3555],{},[81,13687,3558],{"encoding":83},[38,13689,13691,13709,13753],{"className":13690,"ariaHidden":89},[88],[38,13692,13694,13697,13700,13703,13706],{"className":13693},[93],[38,13695],{"className":13696,"style":889},[97],[38,13698,60],{"className":13699,"style":225},[102,351],[38,13701],{"className":13702,"style":109},[108],[38,13704,64],{"className":13705},[113],[38,13707],{"className":13708,"style":109},[108],[38,13710,13712,13715,13744,13747,13750],{"className":13711},[93],[38,13713],{"className":13714,"style":123},[97],[38,13716,13718,13721],{"className":13717},[102],[38,13719,71],{"className":13720},[102,130],[38,13722,13724],{"className":13723},[134],[38,13725,13727],{"className":13726},[138],[38,13728,13730],{"className":13729},[142],[38,13731,13733],{"className":13732,"style":147},[146],[38,13734,13735,13738],{"style":150},[38,13736],{"className":13737,"style":155},[154],[38,13739,13741],{"className":13740},[159,160,161,162],[38,13742,74],{"className":13743},[102,103,162],[38,13745],{"className":13746,"style":109},[108],[38,13748,77],{"className":13749},[113],[38,13751],{"className":13752,"style":109},[108],[38,13754,13756,13759],{"className":13755},[93],[38,13757],{"className":13758,"style":123},[97],[38,13760,13762,13765],{"className":13761},[102],[38,13763,71],{"className":13764},[102,130],[38,13766,13768],{"className":13767},[134],[38,13769,13771],{"className":13770},[138],[38,13772,13774],{"className":13773},[142],[38,13775,13777],{"className":13776,"style":147},[146],[38,13778,13779,13782],{"style":150},[38,13780],{"className":13781,"style":155},[154],[38,13783,13785],{"className":13784},[159,160,161,162],[38,13786,3555],{"className":13787},[102,103,162],[13305,13789,13790,13791],{},"矩阵 ",[38,13792,13794,13820],{"className":13793,"translate":42},[41],[38,13795,13797],{"className":13796},[46],[48,13798,13799],{"xmlns":50},[52,13800,13801,13817],{},[55,13802,13803,13805],{},[62,13804,1420],{},[66,13806,13807,13809],{},[58,13808,71],{"mathvariant":70},[55,13810,13811,13813,13815],{},[58,13812,3555],{},[62,13814,4402],{},[58,13816,74],{},[81,13818,13819],{"encoding":83},"\\in \\mathbb{R}^{m \\times n}",[38,13821,13823,13835],{"className":13822,"ariaHidden":89},[88],[38,13824,13826,13829,13832],{"className":13825},[93],[38,13827],{"className":13828,"style":12573},[97],[38,13830,1420],{"className":13831},[113],[38,13833],{"className":13834,"style":109},[108],[38,13836,13838,13842],{"className":13837},[93],[38,13839],{"className":13840,"style":13841},[97],"height:0.7713em;",[38,13843,13845,13848],{"className":13844},[102],[38,13846,71],{"className":13847},[102,130],[38,13849,13851],{"className":13850},[134],[38,13852,13854],{"className":13853},[138],[38,13855,13857],{"className":13856},[142],[38,13858,13860],{"className":13859,"style":13841},[146],[38,13861,13862,13865],{"style":150},[38,13863],{"className":13864,"style":155},[154],[38,13866,13868],{"className":13867},[159,160,161,162],[38,13869,13871,13874,13877],{"className":13870},[102,162],[38,13872,3555],{"className":13873},[102,103,162],[38,13875,4402],{"className":13876},[999,162],[38,13878,74],{"className":13879},[102,103,162],[13305,13881,13882],{},"一般情形，本身即为定义",[13284,13884,13885,13959,14098,14188],{},[13305,13886,13887,13888],{},"海塞矩阵 ",[38,13889,13891,13909],{"className":13890,"translate":42},[41],[38,13892,13894],{"className":13893},[46],[48,13895,13896],{"xmlns":50},[52,13897,13898,13906],{},[55,13899,13900],{},[295,13901,13902,13904],{},[58,13903,8439],{},[58,13905,60],{},[81,13907,13908],{"encoding":83},"H_f",[38,13910,13912],{"className":13911,"ariaHidden":89},[88],[38,13913,13915,13919],{"className":13914},[93],[38,13916],{"className":13917,"style":13918},[97],"height:0.9694em;vertical-align:-0.2861em;",[38,13920,13922,13925],{"className":13921},[102],[38,13923,8439],{"className":13924,"style":8676},[102,103],[38,13926,13928],{"className":13927},[134],[38,13929,13931,13951],{"className":13930},[138,384],[38,13932,13934,13948],{"className":13933},[142],[38,13935,13937],{"className":13936,"style":4759},[146],[38,13938,13939,13942],{"style":8691},[38,13940],{"className":13941,"style":155},[154],[38,13943,13945],{"className":13944},[159,160,161,162],[38,13946,60],{"className":13947,"style":104},[102,103,162],[38,13949,408],{"className":13950},[407],[38,13952,13954],{"className":13953},[142],[38,13955,13957],{"className":13956,"style":7348},[146],[38,13958],{},[13305,13960,13961],{},[38,13962,13964,13995],{"className":13963,"translate":42},[41],[38,13965,13967],{"className":13966},[46],[48,13968,13969],{"xmlns":50},[52,13970,13971,13993],{},[55,13972,13973,13975,13977,13979,13985,13987],{},[58,13974,1335],{"mathvariant":333},[58,13976,60],{},[62,13978,64],{},[66,13980,13981,13983],{},[58,13982,71],{"mathvariant":70},[58,13984,74],{},[62,13986,77],{},[66,13988,13989,13991],{},[58,13990,71],{"mathvariant":70},[58,13992,74],{},[81,13994,8093],{"encoding":83},[38,13996,13998,14019,14063],{"className":13997,"ariaHidden":89},[88],[38,13999,14001,14004,14007,14010,14013,14016],{"className":14000},[93],[38,14002],{"className":14003,"style":98},[97],[38,14005,1335],{"className":14006},[102],[38,14008,60],{"className":14009,"style":104},[102,103],[38,14011],{"className":14012,"style":109},[108],[38,14014,64],{"className":14015},[113],[38,14017],{"className":14018,"style":109},[108],[38,14020,14022,14025,14054,14057,14060],{"className":14021},[93],[38,14023],{"className":14024,"style":123},[97],[38,14026,14028,14031],{"className":14027},[102],[38,14029,71],{"className":14030},[102,130],[38,14032,14034],{"className":14033},[134],[38,14035,14037],{"className":14036},[138],[38,14038,14040],{"className":14039},[142],[38,14041,14043],{"className":14042,"style":147},[146],[38,14044,14045,14048],{"style":150},[38,14046],{"className":14047,"style":155},[154],[38,14049,14051],{"className":14050},[159,160,161,162],[38,14052,74],{"className":14053},[102,103,162],[38,14055],{"className":14056,"style":109},[108],[38,14058,77],{"className":14059},[113],[38,14061],{"className":14062,"style":109},[108],[38,14064,14066,14069],{"className":14065},[93],[38,14067],{"className":14068,"style":123},[97],[38,14070,14072,14075],{"className":14071},[102],[38,14073,71],{"className":14074},[102,130],[38,14076,14078],{"className":14077},[134],[38,14079,14081],{"className":14080},[138],[38,14082,14084],{"className":14083},[142],[38,14085,14087],{"className":14086,"style":147},[146],[38,14088,14089,14092],{"style":150},[38,14090],{"className":14091,"style":155},[154],[38,14093,14095],{"className":14094},[159,160,161,162],[38,14096,74],{"className":14097},[102,103,162],[13305,14099,13790,14100],{},[38,14101,14103,14129],{"className":14102,"translate":42},[41],[38,14104,14106],{"className":14105},[46],[48,14107,14108],{"xmlns":50},[52,14109,14110,14126],{},[55,14111,14112,14114],{},[62,14113,1420],{},[66,14115,14116,14118],{},[58,14117,71],{"mathvariant":70},[55,14119,14120,14122,14124],{},[58,14121,74],{},[62,14123,4402],{},[58,14125,74],{},[81,14127,14128],{"encoding":83},"\\in \\mathbb{R}^{n \\times n}",[38,14130,14132,14144],{"className":14131,"ariaHidden":89},[88],[38,14133,14135,14138,14141],{"className":14134},[93],[38,14136],{"className":14137,"style":12573},[97],[38,14139,1420],{"className":14140},[113],[38,14142],{"className":14143,"style":109},[108],[38,14145,14147,14150],{"className":14146},[93],[38,14148],{"className":14149,"style":13841},[97],[38,14151,14153,14156],{"className":14152},[102],[38,14154,71],{"className":14155},[102,130],[38,14157,14159],{"className":14158},[134],[38,14160,14162],{"className":14161},[138],[38,14163,14165],{"className":14164},[142],[38,14166,14168],{"className":14167,"style":13841},[146],[38,14169,14170,14173],{"style":150},[38,14171],{"className":14172,"style":155},[154],[38,14174,14176],{"className":14175},[159,160,161,162],[38,14177,14179,14182,14185],{"className":14178},[102,162],[38,14180,74],{"className":14181},[102,103,162],[38,14183,4402],{"className":14184},[999,162],[38,14186,74],{"className":14187},[102,103,162],[13305,14189,14190],{},"梯度这一向量值函数的雅可比矩阵",[1252,14192,14193],{},[11,14194,14195,14198,14199,14227,14228,14256,14257,14285,14286,14319,14320,14348,14349,14534,14535,14563,14564,14623],{},[15,14196,14197],{},"命题 4.1（算子复合观点）"," 记 ",[38,14200,14202,14215],{"className":14201,"translate":42},[41],[38,14203,14205],{"className":14204},[46],[48,14206,14207],{"xmlns":50},[52,14208,14209,14213],{},[55,14210,14211],{},[58,14212,2453],{},[81,14214,2453],{"encoding":83},[38,14216,14218],{"className":14217,"ariaHidden":89},[88],[38,14219,14221,14224],{"className":14220},[93],[38,14222],{"className":14223,"style":7404},[97],[38,14225,2453],{"className":14226,"style":2500},[102,103]," 为将函数映射到其（全）导数的算子。对标量函数 ",[38,14229,14231,14244],{"className":14230,"translate":42},[41],[38,14232,14234],{"className":14233},[46],[48,14235,14236],{"xmlns":50},[52,14237,14238,14242],{},[55,14239,14240],{},[58,14241,60],{},[81,14243,60],{"encoding":83},[38,14245,14247],{"className":14246,"ariaHidden":89},[88],[38,14248,14250,14253],{"className":14249},[93],[38,14251],{"className":14252,"style":98},[97],[38,14254,60],{"className":14255,"style":104},[102,103],"，一阶应用 ",[38,14258,14260,14273],{"className":14259,"translate":42},[41],[38,14261,14263],{"className":14262},[46],[48,14264,14265],{"xmlns":50},[52,14266,14267,14271],{},[55,14268,14269],{},[58,14270,2453],{},[81,14272,2453],{"encoding":83},[38,14274,14276],{"className":14275,"ariaHidden":89},[88],[38,14277,14279,14282],{"className":14278},[93],[38,14280],{"className":14281,"style":7404},[97],[38,14283,2453],{"className":14284,"style":2500},[102,103]," 得到梯度 ",[38,14287,14289,14304],{"className":14288,"translate":42},[41],[38,14290,14292],{"className":14291},[46],[48,14293,14294],{"xmlns":50},[52,14295,14296,14302],{},[55,14297,14298,14300],{},[58,14299,1335],{"mathvariant":333},[58,14301,60],{},[81,14303,13326],{"encoding":83},[38,14305,14307],{"className":14306,"ariaHidden":89},[88],[38,14308,14310,14313,14316],{"className":14309},[93],[38,14311],{"className":14312,"style":98},[97],[38,14314,1335],{"className":14315},[102],[38,14317,60],{"className":14318,"style":104},[102,103],"（一个向量值函数）；将 ",[38,14321,14323,14336],{"className":14322,"translate":42},[41],[38,14324,14326],{"className":14325},[46],[48,14327,14328],{"xmlns":50},[52,14329,14330,14334],{},[55,14331,14332],{},[58,14333,2453],{},[81,14335,2453],{"encoding":83},[38,14337,14339],{"className":14338,"ariaHidden":89},[88],[38,14340,14342,14345],{"className":14341},[93],[38,14343],{"className":14344,"style":7404},[97],[38,14346,2453],{"className":14347,"style":2500},[102,103]," 再次作用于这一向量值函数，即得到海塞矩阵 ",[38,14350,14352,14394],{"className":14351,"translate":42},[41],[38,14353,14355],{"className":14354},[46],[48,14356,14357],{"xmlns":50},[52,14358,14359,14391],{},[55,14360,14361,14367,14369,14371,14373,14375,14377,14379,14381],{},[295,14362,14363,14365],{},[58,14364,8439],{},[58,14366,60],{},[62,14368,289],{},[58,14370,2453],{},[62,14372,293],{"stretchy":292},[58,14374,1335],{"mathvariant":333},[58,14376,60],{},[62,14378,330],{"stretchy":292},[62,14380,289],{},[295,14382,14383,14385],{},[58,14384,4459],{},[55,14386,14387,14389],{},[58,14388,1335],{"mathvariant":333},[58,14390,60],{},[81,14392,14393],{"encoding":83},"H_f = D(\\nabla f) = J_{\\nabla f}",[38,14395,14397,14452,14482],{"className":14396,"ariaHidden":89},[88],[38,14398,14400,14403,14443,14446,14449],{"className":14399},[93],[38,14401],{"className":14402,"style":13918},[97],[38,14404,14406,14409],{"className":14405},[102],[38,14407,8439],{"className":14408,"style":8676},[102,103],[38,14410,14412],{"className":14411},[134],[38,14413,14415,14435],{"className":14414},[138,384],[38,14416,14418,14432],{"className":14417},[142],[38,14419,14421],{"className":14420,"style":4759},[146],[38,14422,14423,14426],{"style":8691},[38,14424],{"className":14425,"style":155},[154],[38,14427,14429],{"className":14428},[159,160,161,162],[38,14430,60],{"className":14431,"style":104},[102,103,162],[38,14433,408],{"className":14434},[407],[38,14436,14438],{"className":14437},[142],[38,14439,14441],{"className":14440,"style":7348},[146],[38,14442],{},[38,14444],{"className":14445,"style":109},[108],[38,14447,289],{"className":14448},[113],[38,14450],{"className":14451,"style":109},[108],[38,14453,14455,14458,14461,14464,14467,14470,14473,14476,14479],{"className":14454},[93],[38,14456],{"className":14457,"style":1439},[97],[38,14459,2453],{"className":14460,"style":2500},[102,103],[38,14462,293],{"className":14463},[371],[38,14465,1335],{"className":14466},[102],[38,14468,60],{"className":14469,"style":104},[102,103],[38,14471,330],{"className":14472},[529],[38,14474],{"className":14475,"style":109},[108],[38,14477,289],{"className":14478},[113],[38,14480],{"className":14481,"style":109},[108],[38,14483,14485,14488],{"className":14484},[93],[38,14486],{"className":14487,"style":13918},[97],[38,14489,14491,14494],{"className":14490},[102],[38,14492,4459],{"className":14493,"style":4746},[102,103],[38,14495,14497],{"className":14496},[134],[38,14498,14500,14526],{"className":14499},[138,384],[38,14501,14503,14523],{"className":14502},[142],[38,14504,14506],{"className":14505,"style":4759},[146],[38,14507,14508,14511],{"style":4762},[38,14509],{"className":14510,"style":155},[154],[38,14512,14514],{"className":14513},[159,160,161,162],[38,14515,14517,14520],{"className":14516},[102,162],[38,14518,1335],{"className":14519},[102,162],[38,14521,60],{"className":14522,"style":104},[102,103,162],[38,14524,408],{"className":14525},[407],[38,14527,14529],{"className":14528},[142],[38,14530,14532],{"className":14531,"style":7348},[146],[38,14533],{},"。因此海塞矩阵可视为算子 ",[38,14536,14538,14551],{"className":14537,"translate":42},[41],[38,14539,14541],{"className":14540},[46],[48,14542,14543],{"xmlns":50},[52,14544,14545,14549],{},[55,14546,14547],{},[58,14548,2453],{},[81,14550,2453],{"encoding":83},[38,14552,14554],{"className":14553,"ariaHidden":89},[88],[38,14555,14557,14560],{"className":14556},[93],[38,14558],{"className":14559,"style":7404},[97],[38,14561,2453],{"className":14562,"style":2500},[102,103]," 的二次幂 ",[38,14565,14567,14585],{"className":14566,"translate":42},[41],[38,14568,14570],{"className":14569},[46],[48,14571,14572],{"xmlns":50},[52,14573,14574,14582],{},[55,14575,14576],{},[66,14577,14578,14580],{},[58,14579,2453],{},[300,14581,312],{},[81,14583,14584],{"encoding":83},"D^2",[38,14586,14588],{"className":14587,"ariaHidden":89},[88],[38,14589,14591,14594],{"className":14590},[93],[38,14592],{"className":14593,"style":9002},[97],[38,14595,14597,14600],{"className":14596},[102],[38,14598,2453],{"className":14599,"style":2500},[102,103],[38,14601,14603],{"className":14602},[134],[38,14604,14606],{"className":14605},[138],[38,14607,14609],{"className":14608},[142],[38,14610,14612],{"className":14611,"style":9002},[146],[38,14613,14614,14617],{"style":150},[38,14615],{"className":14616,"style":155},[154],[38,14618,14620],{"className":14619},[159,160,161,162],[38,14621,312],{"className":14622},[102,162]," 在标量函数上的矩阵化表示。",[11,14625,14626],{},"这一视角同时揭示了为何海塞矩阵天然对称（由施瓦茨定理保证），而一般的雅可比矩阵未必是方阵、更谈不上对称——对称性正是\"对同一个梯度向量场求导\"这一特殊结构所赋予的额外代数性质。",[11,14628,14629,14630,14633],{},"在实际的机器学习系统实现中，这一递进关系也直接体现在自动微分框架的设计上：一阶自动微分（计算梯度或雅可比–向量积）是绝大多数训练流程的标准配置；而计算海塞矩阵或海塞–向量积，则通常通过对一阶自动微分的计算图",[15,14631,14632],{},"再套一层","自动微分来实现（即所谓的\"双重反向传播\"，double backpropagation），这与命题 4.1 中对梯度再求雅可比的数学结构完全对应。",[11,14635,14636],{},"梯度、雅可比矩阵与海塞矩阵是同一套微分学思想在不同阶数与不同维度组合下的具体呈现在机器学习的实践中，从梯度下降到反向传播，从鞍点分析到二阶优化与自然梯度，三者共同构成了理解模型训练动力学与损失曲面几何结构的数学骨架。",{"title":14638,"searchDepth":14639,"depth":14639,"links":14640},"",2,[14641,14648,14655,14662],{"id":17,"depth":14639,"text":17,"children":14642},[14643,14645,14646,14647],{"id":33,"depth":14644,"text":33},3,{"id":1250,"depth":14644,"text":1250},{"id":1970,"depth":14644,"text":1971},{"id":3086,"depth":14644,"text":3086},{"id":21,"depth":14639,"text":21,"children":14649},[14650,14651,14652,14653,14654],{"id":3521,"depth":14644,"text":3521},{"id":4319,"depth":14644,"text":1250},{"id":6315,"depth":14644,"text":6315},{"id":6634,"depth":14644,"text":6634},{"id":7368,"depth":14644,"text":3086},{"id":25,"depth":14639,"text":25,"children":14656},[14657,14658,14659,14660,14661],{"id":7948,"depth":14644,"text":7948},{"id":8258,"depth":14644,"text":1250},{"id":10115,"depth":14644,"text":10115},{"id":10969,"depth":14644,"text":10969},{"id":12598,"depth":14644,"text":3086},{"id":13273,"depth":14639,"text":13273},"在多元微积分与机器学习理论中，我们时常需要刻画一个函数在某一点附近如何随自变量的微小扰动而变化。当函数的定义域与值域维数超过一，标量微积分中\"导数\"这一单一概念便不再足够，需要被推广为一系列具有不同结构的对象：梯度（gradient）、雅可比矩阵（Jacobian matrix）与海塞矩阵（Hessian matrix）。这三者并非孤立的定义，而是在同一套微分框架下按维度与阶数递进衍生出来的结构。",false,true,"md","陕西西安",{},null,"\u002Fblog\u002F2026\u002F2026-04-23-shu-de-bian-hua-ti-du-ya-ke-bi-ju-zhen-he-hai-sai-ju-zhen",{"title":6,"description":14663},"blog\u002F2026\u002F2026-04-23-shu-de-bian-hua-ti-du-ya-ke-bi-ju-zhen-he-hai-sai-ju-zhen","三者递进关系体现在：海塞矩阵是梯度的雅可比矩阵。即先对函数求一阶得梯度，再将梯度视为新的向量函数求一阶雅可比，自然就得到了二阶海塞。从这个角度看，雅可比是求导算子的矩阵化，而海塞是这一过程的二次应用。",[14675,14676],"mathematics","machine-learning","2026-04-23T20:45:13+08:00","cD4XHjRRAqdSPDyIL4dOpHCPuZe33vVfsEJTNNkwcZY",{"id":14680,"title":14681,"body":14682,"description":25475,"draft":14664,"enableComment":14665,"extension":14666,"image":14638,"important":14664,"location":25476,"meta":25477,"navigation":14665,"ogImage":14669,"path":25478,"seo":25479,"stem":25480,"summary":25481,"tags":25482,"time":25483,"__hash__":25484},"blog\u002Fblog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype.md","Fréchet 均值原型",{"type":8,"value":14683,"toc":25467},[14684,14717,14720,14723,14726,14787,14986,15285,15327,15593,15596,16352,16355,16358,16361,16646,17020,17152,17155,17158,17161,17902,17976,17979,17982,17985,18059,18161,18421,18803,18957,19442,19888,20014,20618,21243,21585,21588,21591,21844,22140,22370,22763,22766,22769,22884,22887,22890,22945,23401,23493,23595,23598,23601,23674,24050,24122,24493,24565,24935,24938,25317,25464],[11,14685,14686,14687,14716],{},"当处理的数据不是简单向量（例如复杂矩阵、概率分布、曲面上的点），无法直接求和再除以 ",[38,14688,14690,14704],{"className":14689,"translate":42},[41],[38,14691,14693],{"className":14692},[46],[48,14694,14695],{"xmlns":50},[52,14696,14697,14702],{},[55,14698,14699],{},[58,14700,14701],{},"N",[81,14703,14701],{"encoding":83},[38,14705,14707],{"className":14706,"ariaHidden":89},[88],[38,14708,14710,14713],{"className":14709},[93],[38,14711],{"className":14712,"style":7404},[97],[38,14714,14701],{"className":14715,"style":225},[102,103]," 时，Fréchet 均值就变得必不可少。",[11,14718,14719],{},"Fréchet 均值原型（Fréchet Mean Prototype）的核心价值在于，它将「求平均」这一运算从平坦向量空间推广到任意弯曲空间或复杂对象，使许多经典统计方法与机器学习算法能够处理非向量数据。",[28,14721,14722],{"id":14722},"均值原型",[11,14724,14725],{},"一组向量的均值原型（Mean Prototype），是指对向量的各个对应分量分别求算术平均所得到的新向量。",[11,14727,14728,14729,14757,14758,14786],{},"更具体地说：假设有 ",[38,14730,14732,14745],{"className":14731,"translate":42},[41],[38,14733,14735],{"className":14734},[46],[48,14736,14737],{"xmlns":50},[52,14738,14739,14743],{},[55,14740,14741],{},[58,14742,3555],{},[81,14744,3555],{"encoding":83},[38,14746,14748],{"className":14747,"ariaHidden":89},[88],[38,14749,14751,14754],{"className":14750},[93],[38,14752],{"className":14753,"style":4312},[97],[38,14755,3555],{"className":14756},[102,103]," 个 ",[38,14759,14761,14774],{"className":14760,"translate":42},[41],[38,14762,14764],{"className":14763},[46],[48,14765,14766],{"xmlns":50},[52,14767,14768,14772],{},[55,14769,14770],{},[58,14771,74],{},[81,14773,74],{"encoding":83},[38,14775,14777],{"className":14776,"ariaHidden":89},[88],[38,14778,14780,14783],{"className":14779},[93],[38,14781],{"className":14782,"style":4312},[97],[38,14784,74],{"className":14785},[102,103]," 维向量：",[38,14788,14790],{"className":14789,"translate":42},[621],[38,14791,14793,14832],{"className":14792,"translate":42},[41],[38,14794,14796],{"className":14795},[46],[48,14797,14798],{"xmlns":50,"display":630},[52,14799,14800,14829],{},[55,14801,14802,14809,14811,14817,14819,14821,14823],{},[295,14803,14804,14807],{},[58,14805,14806],{"mathvariant":285},"v",[300,14808,302],{},[62,14810,305],{"separator":89},[295,14812,14813,14815],{},[58,14814,14806],{"mathvariant":285},[300,14816,312],{},[62,14818,305],{"separator":89},[62,14820,317],{},[62,14822,305],{"separator":89},[295,14824,14825,14827],{},[58,14826,14806],{"mathvariant":285},[58,14828,3555],{},[81,14830,14831],{"encoding":83},"  \\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_m",[38,14833,14835],{"className":14834,"ariaHidden":89},[88],[38,14836,14838,14841,14882,14885,14888,14928,14931,14934,14937,14940,14943,14946],{"className":14837},[93],[38,14839],{"className":14840,"style":6685},[97],[38,14842,14844,14847],{"className":14843},[102],[38,14845,14806],{"className":14846,"style":6689},[102,351],[38,14848,14850],{"className":14849},[134],[38,14851,14853,14874],{"className":14852},[138,384],[38,14854,14856,14871],{"className":14855},[142],[38,14857,14859],{"className":14858,"style":391},[146],[38,14860,14862,14865],{"style":14861},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[38,14863],{"className":14864,"style":155},[154],[38,14866,14868],{"className":14867},[159,160,161,162],[38,14869,302],{"className":14870},[102,162],[38,14872,408],{"className":14873},[407],[38,14875,14877],{"className":14876},[142],[38,14878,14880],{"className":14879,"style":415},[146],[38,14881],{},[38,14883,305],{"className":14884},[421],[38,14886],{"className":14887,"style":425},[108],[38,14889,14891,14894],{"className":14890},[102],[38,14892,14806],{"className":14893,"style":6689},[102,351],[38,14895,14897],{"className":14896},[134],[38,14898,14900,14920],{"className":14899},[138,384],[38,14901,14903,14917],{"className":14902},[142],[38,14904,14906],{"className":14905,"style":391},[146],[38,14907,14908,14911],{"style":14861},[38,14909],{"className":14910,"style":155},[154],[38,14912,14914],{"className":14913},[159,160,161,162],[38,14915,312],{"className":14916},[102,162],[38,14918,408],{"className":14919},[407],[38,14921,14923],{"className":14922},[142],[38,14924,14926],{"className":14925,"style":415},[146],[38,14927],{},[38,14929,305],{"className":14930},[421],[38,14932],{"className":14933,"style":425},[108],[38,14935,317],{"className":14936},[475],[38,14938],{"className":14939,"style":425},[108],[38,14941,305],{"className":14942},[421],[38,14944],{"className":14945,"style":425},[108],[38,14947,14949,14952],{"className":14948},[102],[38,14950,14806],{"className":14951,"style":6689},[102,351],[38,14953,14955],{"className":14954},[134],[38,14956,14958,14978],{"className":14957},[138,384],[38,14959,14961,14975],{"className":14960},[142],[38,14962,14964],{"className":14963,"style":503},[146],[38,14965,14966,14969],{"style":14861},[38,14967],{"className":14968,"style":155},[154],[38,14970,14972],{"className":14971},[159,160,161,162],[38,14973,3555],{"className":14974},[102,103,162],[38,14976,408],{"className":14977},[407],[38,14979,14981],{"className":14980},[142],[38,14982,14984],{"className":14983,"style":415},[146],[38,14985],{},[11,14987,14988,14989,1964],{},"其中每个向量 ",[38,14990,14992,15054],{"className":14991,"translate":42},[41],[38,14993,14995],{"className":14994},[46],[48,14996,14997],{"xmlns":50},[52,14998,14999,15051],{},[55,15000,15001,15007,15009,15011,15021,15023,15033,15035,15037,15039,15049],{},[295,15002,15003,15005],{},[58,15004,14806],{"mathvariant":285},[58,15006,601],{},[62,15008,289],{},[62,15010,293],{"stretchy":292},[295,15012,15013,15015],{},[58,15014,14806],{},[55,15016,15017,15019],{},[58,15018,601],{},[300,15020,302],{},[62,15022,305],{"separator":89},[295,15024,15025,15027],{},[58,15026,14806],{},[55,15028,15029,15031],{},[58,15030,601],{},[300,15032,312],{},[62,15034,305],{"separator":89},[62,15036,317],{},[62,15038,305],{"separator":89},[295,15040,15041,15043],{},[58,15042,14806],{},[55,15044,15045,15047],{},[58,15046,601],{},[58,15048,74],{},[62,15050,330],{"stretchy":292},[81,15052,15053],{"encoding":83},"\\mathbf{v}_i = (v_{i1}, v_{i2}, \\dots, v_{in})",[38,15055,15057,15112],{"className":15056,"ariaHidden":89},[88],[38,15058,15060,15063,15103,15106,15109],{"className":15059},[93],[38,15061],{"className":15062,"style":1148},[97],[38,15064,15066,15069],{"className":15065},[102],[38,15067,14806],{"className":15068,"style":6689},[102,351],[38,15070,15072],{"className":15071},[134],[38,15073,15075,15095],{"className":15074},[138,384],[38,15076,15078,15092],{"className":15077},[142],[38,15079,15081],{"className":15080,"style":786},[146],[38,15082,15083,15086],{"style":14861},[38,15084],{"className":15085,"style":155},[154],[38,15087,15089],{"className":15088},[159,160,161,162],[38,15090,601],{"className":15091},[102,103,162],[38,15093,408],{"className":15094},[407],[38,15096,15098],{"className":15097},[142],[38,15099,15101],{"className":15100,"style":415},[146],[38,15102],{},[38,15104],{"className":15105,"style":109},[108],[38,15107,289],{"className":15108},[113],[38,15110],{"className":15111,"style":109},[108],[38,15113,15115,15118,15121,15168,15171,15174,15220,15223,15226,15229,15232,15235,15238,15282],{"className":15114},[93],[38,15116],{"className":15117,"style":1439},[97],[38,15119,293],{"className":15120},[371],[38,15122,15124,15127],{"className":15123},[102],[38,15125,14806],{"className":15126,"style":3420},[102,103],[38,15128,15130],{"className":15129},[134],[38,15131,15133,15160],{"className":15132},[138,384],[38,15134,15136,15157],{"className":15135},[142],[38,15137,15139],{"className":15138,"style":786},[146],[38,15140,15142,15145],{"style":15141},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[38,15143],{"className":15144,"style":155},[154],[38,15146,15148],{"className":15147},[159,160,161,162],[38,15149,15151,15154],{"className":15150},[102,162],[38,15152,601],{"className":15153},[102,103,162],[38,15155,302],{"className":15156},[102,162],[38,15158,408],{"className":15159},[407],[38,15161,15163],{"className":15162},[142],[38,15164,15166],{"className":15165,"style":415},[146],[38,15167],{},[38,15169,305],{"className":15170},[421],[38,15172],{"className":15173,"style":425},[108],[38,15175,15177,15180],{"className":15176},[102],[38,15178,14806],{"className":15179,"style":3420},[102,103],[38,15181,15183],{"className":15182},[134],[38,15184,15186,15212],{"className":15185},[138,384],[38,15187,15189,15209],{"className":15188},[142],[38,15190,15192],{"className":15191,"style":786},[146],[38,15193,15194,15197],{"style":15141},[38,15195],{"className":15196,"style":155},[154],[38,15198,15200],{"className":15199},[159,160,161,162],[38,15201,15203,15206],{"className":15202},[102,162],[38,15204,601],{"className":15205},[102,103,162],[38,15207,312],{"className":15208},[102,162],[38,15210,408],{"className":15211},[407],[38,15213,15215],{"className":15214},[142],[38,15216,15218],{"className":15217,"style":415},[146],[38,15219],{},[38,15221,305],{"className":15222},[421],[38,15224],{"className":15225,"style":425},[108],[38,15227,317],{"className":15228},[475],[38,15230],{"className":15231,"style":425},[108],[38,15233,305],{"className":15234},[421],[38,15236],{"className":15237,"style":425},[108],[38,15239,15241,15244],{"className":15240},[102],[38,15242,14806],{"className":15243,"style":3420},[102,103],[38,15245,15247],{"className":15246},[134],[38,15248,15250,15274],{"className":15249},[138,384],[38,15251,15253,15271],{"className":15252},[142],[38,15254,15256],{"className":15255,"style":786},[146],[38,15257,15258,15261],{"style":15141},[38,15259],{"className":15260,"style":155},[154],[38,15262,15264],{"className":15263},[159,160,161,162],[38,15265,15267],{"className":15266},[102,162],[38,15268,15270],{"className":15269},[102,103,162],"in",[38,15272,408],{"className":15273},[407],[38,15275,15277],{"className":15276},[142],[38,15278,15280],{"className":15279,"style":415},[146],[38,15281],{},[38,15283,330],{"className":15284},[529],[11,15286,15287,15288,15326],{},"均值原型 ",[38,15289,15291,15307],{"className":15290,"translate":42},[41],[38,15292,15294],{"className":15293},[46],[48,15295,15296],{"xmlns":50},[52,15297,15298,15304],{},[55,15299,15300],{},[58,15301,15303],{"mathvariant":15302},"bold-italic","μ",[81,15305,15306],{"encoding":83},"\\boldsymbol{\\mu}",[38,15308,15310],{"className":15309,"ariaHidden":89},[88],[38,15311,15313,15316],{"className":15312},[93],[38,15314],{"className":15315,"style":6685},[97],[38,15317,15319],{"className":15318},[102],[38,15320,15322],{"className":15321},[102],[38,15323,15303],{"className":15324},[102,15325],"boldsymbol"," 简单地定义为：",[38,15328,15330],{"className":15329,"translate":42},[621],[38,15331,15333,15377],{"className":15332,"translate":42},[41],[38,15334,15336],{"className":15335},[46],[48,15337,15338],{"xmlns":50,"display":630},[52,15339,15340,15374],{},[55,15341,15342,15344,15346,15352,15368],{},[58,15343,15303],{"mathvariant":15302},[62,15345,289],{},[636,15347,15348,15350],{},[300,15349,302],{},[58,15351,3555],{},[15353,15354,15355,15358,15366],"munderover",{},[62,15356,15357],{},"∑",[55,15359,15360,15362,15364],{},[58,15361,601],{},[62,15363,289],{},[300,15365,302],{},[58,15367,3555],{},[295,15369,15370,15372],{},[58,15371,14806],{"mathvariant":285},[58,15373,601],{},[81,15375,15376],{"encoding":83},"  \\boldsymbol{\\mu} = \\frac{1}{m} \\sum_{i=1}^{m} \\mathbf{v}_i",[38,15378,15380,15404],{"className":15379,"ariaHidden":89},[88],[38,15381,15383,15386,15395,15398,15401],{"className":15382},[93],[38,15384],{"className":15385,"style":6685},[97],[38,15387,15389],{"className":15388},[102],[38,15390,15392],{"className":15391},[102],[38,15393,15303],{"className":15394},[102,15325],[38,15396],{"className":15397,"style":109},[108],[38,15399,289],{"className":15400},[113],[38,15402],{"className":15403,"style":109},[108],[38,15405,15407,15411,15473,15476,15550,15553],{"className":15406},[93],[38,15408],{"className":15409,"style":15410},[97],"height:2.9291em;vertical-align:-1.2777em;",[38,15412,15414,15417,15470],{"className":15413},[102],[38,15415],{"className":15416},[371,740],[38,15418,15420],{"className":15419},[636],[38,15421,15423,15462],{"className":15422},[138,384],[38,15424,15426,15459],{"className":15425},[142],[38,15427,15429,15440,15448],{"className":15428,"style":11324},[146],[38,15430,15431,15434],{"style":756},[38,15432],{"className":15433,"style":760},[154],[38,15435,15437],{"className":15436},[102],[38,15438,3555],{"className":15439},[102,103],[38,15441,15442,15445],{"style":811},[38,15443],{"className":15444,"style":760},[154],[38,15446],{"className":15447,"style":819},[818],[38,15449,15450,15453],{"style":822},[38,15451],{"className":15452,"style":760},[154],[38,15454,15456],{"className":15455},[102],[38,15457,302],{"className":15458},[102],[38,15460,408],{"className":15461},[407],[38,15463,15465],{"className":15464},[142],[38,15466,15468],{"className":15467,"style":1079},[146],[38,15469],{},[38,15471],{"className":15472},[529,740],[38,15474],{"className":15475,"style":425},[108],[38,15477,15479],{"className":15478},[878,879],[38,15480,15482,15541],{"className":15481},[138,384],[38,15483,15485,15538],{"className":15484},[142],[38,15486,15489,15510,15523],{"className":15487,"style":15488},[146],"height:1.6514em;",[38,15490,15492,15495],{"style":15491},"top:-1.8723em;margin-left:0em;",[38,15493],{"className":15494,"style":8852},[154],[38,15496,15498],{"className":15497},[159,160,161,162],[38,15499,15501,15504,15507],{"className":15500},[102,162],[38,15502,601],{"className":15503},[102,103,162],[38,15505,289],{"className":15506},[113,162],[38,15508,302],{"className":15509},[102,162],[38,15511,15513,15516],{"style":15512},"top:-3.05em;",[38,15514],{"className":15515,"style":8852},[154],[38,15517,15518],{},[38,15519,15357],{"className":15520},[878,15521,15522],"op-symbol","large-op",[38,15524,15526,15529],{"style":15525},"top:-4.3em;margin-left:0em;",[38,15527],{"className":15528,"style":8852},[154],[38,15530,15532],{"className":15531},[159,160,161,162],[38,15533,15535],{"className":15534},[102,162],[38,15536,3555],{"className":15537},[102,103,162],[38,15539,408],{"className":15540},[407],[38,15542,15544],{"className":15543},[142],[38,15545,15548],{"className":15546,"style":15547},[146],"height:1.2777em;",[38,15549],{},[38,15551],{"className":15552,"style":425},[108],[38,15554,15556,15559],{"className":15555},[102],[38,15557,14806],{"className":15558,"style":6689},[102,351],[38,15560,15562],{"className":15561},[134],[38,15563,15565,15585],{"className":15564},[138,384],[38,15566,15568,15582],{"className":15567},[142],[38,15569,15571],{"className":15570,"style":786},[146],[38,15572,15573,15576],{"style":14861},[38,15574],{"className":15575,"style":155},[154],[38,15577,15579],{"className":15578},[159,160,161,162],[38,15580,601],{"className":15581},[102,103,162],[38,15583,408],{"className":15584},[407],[38,15586,15588],{"className":15587},[142],[38,15589,15591],{"className":15590,"style":415},[146],[38,15592],{},[11,15594,15595],{},"即：",[38,15597,15599],{"className":15598,"translate":42},[621],[38,15600,15602,15729],{"className":15601,"translate":42},[41],[38,15603,15605],{"className":15604},[46],[48,15606,15607],{"xmlns":50,"display":630},[52,15608,15609,15726],{},[55,15610,15611,15613,15615],{},[58,15612,15303],{"mathvariant":15302},[62,15614,289],{},[55,15616,15617,15619,15625,15639,15649,15651,15654,15660,15674,15684,15686,15688,15690,15692,15694,15700,15714,15724],{},[62,15618,293],{"fence":89},[636,15620,15621,15623],{},[300,15622,302],{},[58,15624,3555],{},[15353,15626,15627,15629,15637],{},[62,15628,15357],{},[55,15630,15631,15633,15635],{},[58,15632,601],{},[62,15634,289],{},[300,15636,302],{},[58,15638,3555],{},[295,15640,15641,15643],{},[58,15642,14806],{},[55,15644,15645,15647],{},[58,15646,601],{},[300,15648,302],{},[62,15650,305],{"separator":89},[3265,15652,15653],{}," ",[636,15655,15656,15658],{},[300,15657,302],{},[58,15659,3555],{},[15353,15661,15662,15664,15672],{},[62,15663,15357],{},[55,15665,15666,15668,15670],{},[58,15667,601],{},[62,15669,289],{},[300,15671,302],{},[58,15673,3555],{},[295,15675,15676,15678],{},[58,15677,14806],{},[55,15679,15680,15682],{},[58,15681,601],{},[300,15683,312],{},[62,15685,305],{"separator":89},[3265,15687,15653],{},[62,15689,317],{},[62,15691,305],{"separator":89},[3265,15693,15653],{},[636,15695,15696,15698],{},[300,15697,302],{},[58,15699,3555],{},[15353,15701,15702,15704,15712],{},[62,15703,15357],{},[55,15705,15706,15708,15710],{},[58,15707,601],{},[62,15709,289],{},[300,15711,302],{},[58,15713,3555],{},[295,15715,15716,15718],{},[58,15717,14806],{},[55,15719,15720,15722],{},[58,15721,601],{},[58,15723,74],{},[62,15725,330],{"fence":89},[81,15727,15728],{"encoding":83},"  \\boldsymbol{\\mu} = \\left( \\frac{1}{m}\\sum_{i=1}^{m} v_{i1},\\ \\frac{1}{m}\\sum_{i=1}^{m} v_{i2},\\ \\dots,\\ \\frac{1}{m}\\sum_{i=1}^{m} v_{in} \\right)",[38,15730,15732,15756],{"className":15731,"ariaHidden":89},[88],[38,15733,15735,15738,15747,15750,15753],{"className":15734},[93],[38,15736],{"className":15737,"style":6685},[97],[38,15739,15741],{"className":15740},[102],[38,15742,15744],{"className":15743},[102],[38,15745,15303],{"className":15746},[102,15325],[38,15748],{"className":15749,"style":109},[108],[38,15751,289],{"className":15752},[113],[38,15754],{"className":15755,"style":109},[108],[38,15757,15759,15763],{"className":15758},[93],[38,15760],{"className":15761,"style":15762},[97],"height:3.0277em;vertical-align:-1.2777em;",[38,15764,15766,15773,15835,15838,15905,15908,15954,15957,15960,15963,16025,16028,16095,16098,16144,16147,16150,16153,16156,16159,16162,16165,16168,16230,16233,16300,16303,16346],{"className":15765},[475],[38,15767,15769],{"className":15768,"style":1481},[371,1480],[38,15770,293],{"className":15771},[1485,15772],"size4",[38,15774,15776,15779,15832],{"className":15775},[102],[38,15777],{"className":15778},[371,740],[38,15780,15782],{"className":15781},[636],[38,15783,15785,15824],{"className":15784},[138,384],[38,15786,15788,15821],{"className":15787},[142],[38,15789,15791,15802,15810],{"className":15790,"style":11324},[146],[38,15792,15793,15796],{"style":756},[38,15794],{"className":15795,"style":760},[154],[38,15797,15799],{"className":15798},[102],[38,15800,3555],{"className":15801},[102,103],[38,15803,15804,15807],{"style":811},[38,15805],{"className":15806,"style":760},[154],[38,15808],{"className":15809,"style":819},[818],[38,15811,15812,15815],{"style":822},[38,15813],{"className":15814,"style":760},[154],[38,15816,15818],{"className":15817},[102],[38,15819,302],{"className":15820},[102],[38,15822,408],{"className":15823},[407],[38,15825,15827],{"className":15826},[142],[38,15828,15830],{"className":15829,"style":1079},[146],[38,15831],{},[38,15833],{"className":15834},[529,740],[38,15836],{"className":15837,"style":425},[108],[38,15839,15841],{"className":15840},[878,879],[38,15842,15844,15897],{"className":15843},[138,384],[38,15845,15847,15894],{"className":15846},[142],[38,15848,15850,15870,15880],{"className":15849,"style":15488},[146],[38,15851,15852,15855],{"style":15491},[38,15853],{"className":15854,"style":8852},[154],[38,15856,15858],{"className":15857},[159,160,161,162],[38,15859,15861,15864,15867],{"className":15860},[102,162],[38,15862,601],{"className":15863},[102,103,162],[38,15865,289],{"className":15866},[113,162],[38,15868,302],{"className":15869},[102,162],[38,15871,15872,15875],{"style":15512},[38,15873],{"className":15874,"style":8852},[154],[38,15876,15877],{},[38,15878,15357],{"className":15879},[878,15521,15522],[38,15881,15882,15885],{"style":15525},[38,15883],{"className":15884,"style":8852},[154],[38,15886,15888],{"className":15887},[159,160,161,162],[38,15889,15891],{"className":15890},[102,162],[38,15892,3555],{"className":15893},[102,103,162],[38,15895,408],{"className":15896},[407],[38,15898,15900],{"className":15899},[142],[38,15901,15903],{"className":15902,"style":15547},[146],[38,15904],{},[38,15906],{"className":15907,"style":425},[108],[38,15909,15911,15914],{"className":15910},[102],[38,15912,14806],{"className":15913,"style":3420},[102,103],[38,15915,15917],{"className":15916},[134],[38,15918,15920,15946],{"className":15919},[138,384],[38,15921,15923,15943],{"className":15922},[142],[38,15924,15926],{"className":15925,"style":786},[146],[38,15927,15928,15931],{"style":15141},[38,15929],{"className":15930,"style":155},[154],[38,15932,15934],{"className":15933},[159,160,161,162],[38,15935,15937,15940],{"className":15936},[102,162],[38,15938,601],{"className":15939},[102,103,162],[38,15941,302],{"className":15942},[102,162],[38,15944,408],{"className":15945},[407],[38,15947,15949],{"className":15948},[142],[38,15950,15952],{"className":15951,"style":415},[146],[38,15953],{},[38,15955,305],{"className":15956},[421],[38,15958,15653],{"className":15959},[108],[38,15961],{"className":15962,"style":425},[108],[38,15964,15966,15969,16022],{"className":15965},[102],[38,15967],{"className":15968},[371,740],[38,15970,15972],{"className":15971},[636],[38,15973,15975,16014],{"className":15974},[138,384],[38,15976,15978,16011],{"className":15977},[142],[38,15979,15981,15992,16000],{"className":15980,"style":11324},[146],[38,15982,15983,15986],{"style":756},[38,15984],{"className":15985,"style":760},[154],[38,15987,15989],{"className":15988},[102],[38,15990,3555],{"className":15991},[102,103],[38,15993,15994,15997],{"style":811},[38,15995],{"className":15996,"style":760},[154],[38,15998],{"className":15999,"style":819},[818],[38,16001,16002,16005],{"style":822},[38,16003],{"className":16004,"style":760},[154],[38,16006,16008],{"className":16007},[102],[38,16009,302],{"className":16010},[102],[38,16012,408],{"className":16013},[407],[38,16015,16017],{"className":16016},[142],[38,16018,16020],{"className":16019,"style":1079},[146],[38,16021],{},[38,16023],{"className":16024},[529,740],[38,16026],{"className":16027,"style":425},[108],[38,16029,16031],{"className":16030},[878,879],[38,16032,16034,16087],{"className":16033},[138,384],[38,16035,16037,16084],{"className":16036},[142],[38,16038,16040,16060,16070],{"className":16039,"style":15488},[146],[38,16041,16042,16045],{"style":15491},[38,16043],{"className":16044,"style":8852},[154],[38,16046,16048],{"className":16047},[159,160,161,162],[38,16049,16051,16054,16057],{"className":16050},[102,162],[38,16052,601],{"className":16053},[102,103,162],[38,16055,289],{"className":16056},[113,162],[38,16058,302],{"className":16059},[102,162],[38,16061,16062,16065],{"style":15512},[38,16063],{"className":16064,"style":8852},[154],[38,16066,16067],{},[38,16068,15357],{"className":16069},[878,15521,15522],[38,16071,16072,16075],{"style":15525},[38,16073],{"className":16074,"style":8852},[154],[38,16076,16078],{"className":16077},[159,160,161,162],[38,16079,16081],{"className":16080},[102,162],[38,16082,3555],{"className":16083},[102,103,162],[38,16085,408],{"className":16086},[407],[38,16088,16090],{"className":16089},[142],[38,16091,16093],{"className":16092,"style":15547},[146],[38,16094],{},[38,16096],{"className":16097,"style":425},[108],[38,16099,16101,16104],{"className":16100},[102],[38,16102,14806],{"className":16103,"style":3420},[102,103],[38,16105,16107],{"className":16106},[134],[38,16108,16110,16136],{"className":16109},[138,384],[38,16111,16113,16133],{"className":16112},[142],[38,16114,16116],{"className":16115,"style":786},[146],[38,16117,16118,16121],{"style":15141},[38,16119],{"className":16120,"style":155},[154],[38,16122,16124],{"className":16123},[159,160,161,162],[38,16125,16127,16130],{"className":16126},[102,162],[38,16128,601],{"className":16129},[102,103,162],[38,16131,312],{"className":16132},[102,162],[38,16134,408],{"className":16135},[407],[38,16137,16139],{"className":16138},[142],[38,16140,16142],{"className":16141,"style":415},[146],[38,16143],{},[38,16145,305],{"className":16146},[421],[38,16148,15653],{"className":16149},[108],[38,16151],{"className":16152,"style":425},[108],[38,16154,317],{"className":16155},[475],[38,16157],{"className":16158,"style":425},[108],[38,16160,305],{"className":16161},[421],[38,16163,15653],{"className":16164},[108],[38,16166],{"className":16167,"style":425},[108],[38,16169,16171,16174,16227],{"className":16170},[102],[38,16172],{"className":16173},[371,740],[38,16175,16177],{"className":16176},[636],[38,16178,16180,16219],{"className":16179},[138,384],[38,16181,16183,16216],{"className":16182},[142],[38,16184,16186,16197,16205],{"className":16185,"style":11324},[146],[38,16187,16188,16191],{"style":756},[38,16189],{"className":16190,"style":760},[154],[38,16192,16194],{"className":16193},[102],[38,16195,3555],{"className":16196},[102,103],[38,16198,16199,16202],{"style":811},[38,16200],{"className":16201,"style":760},[154],[38,16203],{"className":16204,"style":819},[818],[38,16206,16207,16210],{"style":822},[38,16208],{"className":16209,"style":760},[154],[38,16211,16213],{"className":16212},[102],[38,16214,302],{"className":16215},[102],[38,16217,408],{"className":16218},[407],[38,16220,16222],{"className":16221},[142],[38,16223,16225],{"className":16224,"style":1079},[146],[38,16226],{},[38,16228],{"className":16229},[529,740],[38,16231],{"className":16232,"style":425},[108],[38,16234,16236],{"className":16235},[878,879],[38,16237,16239,16292],{"className":16238},[138,384],[38,16240,16242,16289],{"className":16241},[142],[38,16243,16245,16265,16275],{"className":16244,"style":15488},[146],[38,16246,16247,16250],{"style":15491},[38,16248],{"className":16249,"style":8852},[154],[38,16251,16253],{"className":16252},[159,160,161,162],[38,16254,16256,16259,16262],{"className":16255},[102,162],[38,16257,601],{"className":16258},[102,103,162],[38,16260,289],{"className":16261},[113,162],[38,16263,302],{"className":16264},[102,162],[38,16266,16267,16270],{"style":15512},[38,16268],{"className":16269,"style":8852},[154],[38,16271,16272],{},[38,16273,15357],{"className":16274},[878,15521,15522],[38,16276,16277,16280],{"style":15525},[38,16278],{"className":16279,"style":8852},[154],[38,16281,16283],{"className":16282},[159,160,161,162],[38,16284,16286],{"className":16285},[102,162],[38,16287,3555],{"className":16288},[102,103,162],[38,16290,408],{"className":16291},[407],[38,16293,16295],{"className":16294},[142],[38,16296,16298],{"className":16297,"style":15547},[146],[38,16299],{},[38,16301],{"className":16302,"style":425},[108],[38,16304,16306,16309],{"className":16305},[102],[38,16307,14806],{"className":16308,"style":3420},[102,103],[38,16310,16312],{"className":16311},[134],[38,16313,16315,16338],{"className":16314},[138,384],[38,16316,16318,16335],{"className":16317},[142],[38,16319,16321],{"className":16320,"style":786},[146],[38,16322,16323,16326],{"style":15141},[38,16324],{"className":16325,"style":155},[154],[38,16327,16329],{"className":16328},[159,160,161,162],[38,16330,16332],{"className":16331},[102,162],[38,16333,15270],{"className":16334},[102,103,162],[38,16336,408],{"className":16337},[407],[38,16339,16341],{"className":16340},[142],[38,16342,16344],{"className":16343,"style":415},[146],[38,16345],{},[38,16347,16349],{"className":16348,"style":1481},[529,1480],[38,16350,330],{"className":16351},[1485,15772],[11,16353,16354],{},"直观来看，均值向量就是这组向量的「中心点」或「平均位置」。在数据分析和机器学习（例如聚类、PCA）中，它常用于表示数据集的整体趋势。均值原型可以看作「在欧氏几何 + 高斯假设下的最优代表点」。如果你的样本嵌入在单位球面上，或服从 vMF 分布，那么欧氏均值就不再是最优的了。",[28,16356,16357],{"id":16357},"几何中位数原型",[11,16359,16360],{},"几何中位数原型（Geometric Median Prototype）指的是：对于一组向量（数据点），找到一个点，使得到所有数据点的欧氏距离之和最小。这个点就是这组向量的几何中位数，并作为代表整个数据集的「原型」。",[11,16362,16363,16364,14757,16392,16420,16421,16615,16616,16645],{},"形式化定义：给定 ",[38,16365,16367,16380],{"className":16366,"translate":42},[41],[38,16368,16370],{"className":16369},[46],[48,16371,16372],{"xmlns":50},[52,16373,16374,16378],{},[55,16375,16376],{},[58,16377,3555],{},[81,16379,3555],{"encoding":83},[38,16381,16383],{"className":16382,"ariaHidden":89},[88],[38,16384,16386,16389],{"className":16385},[93],[38,16387],{"className":16388,"style":4312},[97],[38,16390,3555],{"className":16391},[102,103],[38,16393,16395,16408],{"className":16394,"translate":42},[41],[38,16396,16398],{"className":16397},[46],[48,16399,16400],{"xmlns":50},[52,16401,16402,16406],{},[55,16403,16404],{},[58,16405,74],{},[81,16407,74],{"encoding":83},[38,16409,16411],{"className":16410,"ariaHidden":89},[88],[38,16412,16414,16417],{"className":16413},[93],[38,16415],{"className":16416,"style":4312},[97],[38,16418,74],{"className":16419},[102,103]," 维向量 ",[38,16422,16424,16462],{"className":16423,"translate":42},[41],[38,16425,16427],{"className":16426},[46],[48,16428,16429],{"xmlns":50},[52,16430,16431,16459],{},[55,16432,16433,16439,16441,16447,16449,16451,16453],{},[295,16434,16435,16437],{},[58,16436,14806],{"mathvariant":285},[300,16438,302],{},[62,16440,305],{"separator":89},[295,16442,16443,16445],{},[58,16444,14806],{"mathvariant":285},[300,16446,312],{},[62,16448,305],{"separator":89},[62,16450,317],{},[62,16452,305],{"separator":89},[295,16454,16455,16457],{},[58,16456,14806],{"mathvariant":285},[58,16458,3555],{},[81,16460,16461],{"encoding":83},"\\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_m",[38,16463,16465],{"className":16464,"ariaHidden":89},[88],[38,16466,16468,16471,16511,16514,16517,16557,16560,16563,16566,16569,16572,16575],{"className":16467},[93],[38,16469],{"className":16470,"style":6685},[97],[38,16472,16474,16477],{"className":16473},[102],[38,16475,14806],{"className":16476,"style":6689},[102,351],[38,16478,16480],{"className":16479},[134],[38,16481,16483,16503],{"className":16482},[138,384],[38,16484,16486,16500],{"className":16485},[142],[38,16487,16489],{"className":16488,"style":391},[146],[38,16490,16491,16494],{"style":14861},[38,16492],{"className":16493,"style":155},[154],[38,16495,16497],{"className":16496},[159,160,161,162],[38,16498,302],{"className":16499},[102,162],[38,16501,408],{"className":16502},[407],[38,16504,16506],{"className":16505},[142],[38,16507,16509],{"className":16508,"style":415},[146],[38,16510],{},[38,16512,305],{"className":16513},[421],[38,16515],{"className":16516,"style":425},[108],[38,16518,16520,16523],{"className":16519},[102],[38,16521,14806],{"className":16522,"style":6689},[102,351],[38,16524,16526],{"className":16525},[134],[38,16527,16529,16549],{"className":16528},[138,384],[38,16530,16532,16546],{"className":16531},[142],[38,16533,16535],{"className":16534,"style":391},[146],[38,16536,16537,16540],{"style":14861},[38,16538],{"className":16539,"style":155},[154],[38,16541,16543],{"className":16542},[159,160,161,162],[38,16544,312],{"className":16545},[102,162],[38,16547,408],{"className":16548},[407],[38,16550,16552],{"className":16551},[142],[38,16553,16555],{"className":16554,"style":415},[146],[38,16556],{},[38,16558,305],{"className":16559},[421],[38,16561],{"className":16562,"style":425},[108],[38,16564,317],{"className":16565},[475],[38,16567],{"className":16568,"style":425},[108],[38,16570,305],{"className":16571},[421],[38,16573],{"className":16574,"style":425},[108],[38,16576,16578,16581],{"className":16577},[102],[38,16579,14806],{"className":16580,"style":6689},[102,351],[38,16582,16584],{"className":16583},[134],[38,16585,16587,16607],{"className":16586},[138,384],[38,16588,16590,16604],{"className":16589},[142],[38,16591,16593],{"className":16592,"style":503},[146],[38,16594,16595,16598],{"style":14861},[38,16596],{"className":16597,"style":155},[154],[38,16599,16601],{"className":16600},[159,160,161,162],[38,16602,3555],{"className":16603},[102,103,162],[38,16605,408],{"className":16606},[407],[38,16608,16610],{"className":16609},[142],[38,16611,16613],{"className":16612,"style":415},[146],[38,16614],{},"，几何中位数 ",[38,16617,16619,16633],{"className":16618,"translate":42},[41],[38,16620,16622],{"className":16621},[46],[48,16623,16624],{"xmlns":50},[52,16625,16626,16630],{},[55,16627,16628],{},[58,16629,11],{"mathvariant":285},[81,16631,16632],{"encoding":83},"\\mathbf{p}",[38,16634,16636],{"className":16635,"ariaHidden":89},[88],[38,16637,16639,16642],{"className":16638},[93],[38,16640],{"className":16641,"style":6685},[97],[38,16643,11],{"className":16644},[102,351]," 满足：",[38,16647,16649],{"className":16648,"translate":42},[621],[38,16650,16652,16726],{"className":16651,"translate":42},[41],[38,16653,16655],{"className":16654},[46],[48,16656,16657],{"xmlns":50,"display":630},[52,16658,16659,16723],{},[55,16660,16661,16663,16665,16668,16670,16691,16705,16707,16709,16711,16717],{},[58,16662,11],{"mathvariant":285},[62,16664,289],{},[58,16666,16667],{},"arg",[62,16669,673],{},[664,16671,16672,16679],{},[55,16673,16674,16677],{},[58,16675,16676],{},"min",[62,16678,673],{},[55,16680,16681,16683,16685],{},[58,16682,286],{"mathvariant":285},[62,16684,1420],{},[66,16686,16687,16689],{},[58,16688,71],{"mathvariant":70},[58,16690,74],{},[15353,16692,16693,16695,16703],{},[62,16694,15357],{},[55,16696,16697,16699,16701],{},[58,16698,601],{},[62,16700,289],{},[300,16702,302],{},[58,16704,3555],{},[58,16706,2183],{"mathvariant":333},[58,16708,286],{"mathvariant":285},[62,16710,710],{},[295,16712,16713,16715],{},[58,16714,14806],{"mathvariant":285},[58,16716,601],{},[295,16718,16719,16721],{},[58,16720,2183],{"mathvariant":333},[300,16722,312],{},[81,16724,16725],{"encoding":83},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m \\|\\mathbf{x} - \\mathbf{v}_i\\|_2",[38,16727,16729,16747,16934],{"className":16728,"ariaHidden":89},[88],[38,16730,16732,16735,16738,16741,16744],{"className":16731},[93],[38,16733],{"className":16734,"style":6685},[97],[38,16736,11],{"className":16737},[102,351],[38,16739],{"className":16740,"style":109},[108],[38,16742,289],{"className":16743},[113],[38,16745],{"className":16746,"style":109},[108],[38,16748,16750,16753,16760,16763,16849,16852,16916,16919,16922,16925,16928,16931],{"className":16749},[93],[38,16751],{"className":16752,"style":15410},[97],[38,16754,16756,16757],{"className":16755},[878],"ar",[38,16758,6656],{"style":16759},"margin-right:0.0139em;",[38,16761],{"className":16762,"style":425},[108],[38,16764,16766],{"className":16765},[878,879],[38,16767,16769,16840],{"className":16768},[138,384],[38,16770,16772,16837],{"className":16771},[142],[38,16773,16776,16827],{"className":16774,"style":16775},[146],"height:0.6679em;",[38,16777,16779,16782],{"style":16778},"top:-2.3518em;margin-left:0em;",[38,16780],{"className":16781,"style":760},[154],[38,16783,16785],{"className":16784},[159,160,161,162],[38,16786,16788,16791,16794],{"className":16787},[102,162],[38,16789,286],{"className":16790},[102,351,162],[38,16792,1420],{"className":16793},[113,162],[38,16795,16797,16800],{"className":16796},[102,162],[38,16798,71],{"className":16799},[102,130,162],[38,16801,16803],{"className":16802},[134],[38,16804,16806],{"className":16805},[138],[38,16807,16809],{"className":16808},[142],[38,16810,16813],{"className":16811,"style":16812},[146],"height:0.5935em;",[38,16814,16816,16820],{"style":16815},"top:-2.786em;margin-right:0.0714em;",[38,16817],{"className":16818,"style":16819},[154],"height:2.5em;",[38,16821,16824],{"className":16822},[159,16823,3786,162],"reset-size3",[38,16825,74],{"className":16826},[102,103,162],[38,16828,16829,16832],{"style":913},[38,16830],{"className":16831,"style":760},[154],[38,16833,16834],{},[38,16835,16676],{"className":16836},[878],[38,16838,408],{"className":16839},[407],[38,16841,16843],{"className":16842},[142],[38,16844,16847],{"className":16845,"style":16846},[146],"height:0.7756em;",[38,16848],{},[38,16850],{"className":16851,"style":425},[108],[38,16853,16855],{"className":16854},[878,879],[38,16856,16858,16908],{"className":16857},[138,384],[38,16859,16861,16905],{"className":16860},[142],[38,16862,16864,16884,16894],{"className":16863,"style":15488},[146],[38,16865,16866,16869],{"style":15491},[38,16867],{"className":16868,"style":8852},[154],[38,16870,16872],{"className":16871},[159,160,161,162],[38,16873,16875,16878,16881],{"className":16874},[102,162],[38,16876,601],{"className":16877},[102,103,162],[38,16879,289],{"className":16880},[113,162],[38,16882,302],{"className":16883},[102,162],[38,16885,16886,16889],{"style":15512},[38,16887],{"className":16888,"style":8852},[154],[38,16890,16891],{},[38,16892,15357],{"className":16893},[878,15521,15522],[38,16895,16896,16899],{"style":15525},[38,16897],{"className":16898,"style":8852},[154],[38,16900,16902],{"className":16901},[159,160,161,162],[38,16903,3555],{"className":16904},[102,103,162],[38,16906,408],{"className":16907},[407],[38,16909,16911],{"className":16910},[142],[38,16912,16914],{"className":16913,"style":15547},[146],[38,16915],{},[38,16917],{"className":16918,"style":425},[108],[38,16920,2183],{"className":16921},[102],[38,16923,286],{"className":16924},[102,351],[38,16926],{"className":16927,"style":995},[108],[38,16929,710],{"className":16930},[999],[38,16932],{"className":16933,"style":995},[108],[38,16935,16937,16940,16980],{"className":16936},[93],[38,16938],{"className":16939,"style":1439},[97],[38,16941,16943,16946],{"className":16942},[102],[38,16944,14806],{"className":16945,"style":6689},[102,351],[38,16947,16949],{"className":16948},[134],[38,16950,16952,16972],{"className":16951},[138,384],[38,16953,16955,16969],{"className":16954},[142],[38,16956,16958],{"className":16957,"style":786},[146],[38,16959,16960,16963],{"style":14861},[38,16961],{"className":16962,"style":155},[154],[38,16964,16966],{"className":16965},[159,160,161,162],[38,16967,601],{"className":16968},[102,103,162],[38,16970,408],{"className":16971},[407],[38,16973,16975],{"className":16974},[142],[38,16976,16978],{"className":16977,"style":415},[146],[38,16979],{},[38,16981,16983,16986],{"className":16982},[102],[38,16984,2183],{"className":16985},[102],[38,16987,16989],{"className":16988},[134],[38,16990,16992,17012],{"className":16991},[138,384],[38,16993,16995,17009],{"className":16994},[142],[38,16996,16998],{"className":16997,"style":391},[146],[38,16999,17000,17003],{"style":394},[38,17001],{"className":17002,"style":155},[154],[38,17004,17006],{"className":17005},[159,160,161,162],[38,17007,312],{"className":17008},[102,162],[38,17010,408],{"className":17011},[407],[38,17013,17015],{"className":17014},[142],[38,17016,17018],{"className":17017,"style":415},[146],[38,17019],{},[11,17021,17022,17023,17116,17117,17151],{},"其中 ",[38,17024,17026,17049],{"className":17025,"translate":42},[41],[38,17027,17029],{"className":17028},[46],[48,17030,17031],{"xmlns":50},[52,17032,17033,17046],{},[55,17034,17035,17037,17040],{},[58,17036,2183],{"mathvariant":333},[62,17038,17039],{},"⋅",[295,17041,17042,17044],{},[58,17043,2183],{"mathvariant":333},[300,17045,312],{},[81,17047,17048],{"encoding":83},"\\|\\cdot\\|_2",[38,17050,17052,17070],{"className":17051,"ariaHidden":89},[88],[38,17053,17055,17058,17061,17064,17067],{"className":17054},[93],[38,17056],{"className":17057,"style":1439},[97],[38,17059,2183],{"className":17060},[102],[38,17062],{"className":17063,"style":995},[108],[38,17065,17039],{"className":17066},[999],[38,17068],{"className":17069,"style":995},[108],[38,17071,17073,17076],{"className":17072},[93],[38,17074],{"className":17075,"style":1439},[97],[38,17077,17079,17082],{"className":17078},[102],[38,17080,2183],{"className":17081},[102],[38,17083,17085],{"className":17084},[134],[38,17086,17088,17108],{"className":17087},[138,384],[38,17089,17091,17105],{"className":17090},[142],[38,17092,17094],{"className":17093,"style":391},[146],[38,17095,17096,17099],{"style":394},[38,17097],{"className":17098,"style":155},[154],[38,17100,17102],{"className":17101},[159,160,161,162],[38,17103,312],{"className":17104},[102,162],[38,17106,408],{"className":17107},[407],[38,17109,17111],{"className":17110},[142],[38,17112,17114],{"className":17113,"style":415},[146],[38,17115],{}," 是 ",[38,17118,17120,17136],{"className":17119,"translate":42},[41],[38,17121,17123],{"className":17122},[46],[48,17124,17125],{"xmlns":50},[52,17126,17127,17133],{},[55,17128,17129,17131],{},[58,17130,3104],{},[300,17132,312],{},[81,17134,17135],{"encoding":83},"L2",[38,17137,17139],{"className":17138,"ariaHidden":89},[88],[38,17140,17142,17145,17148],{"className":17141},[93],[38,17143],{"className":17144,"style":7404},[97],[38,17146,3104],{"className":17147},[102,103],[38,17149,312],{"className":17150},[102]," 范数下的欧氏距离。",[11,17153,17154],{},"直观来看，在一维情形下：几何中位数就是通常的中位数（使绝对偏差之和最小）。推广到多维情形：它是空间中使得到所有数据点的「总直线距离」最小的点。可以想象在平面上找一个位置，使得到所有给定点的总旅行距离最小，类似于在几座城市之间找一个使总旅行距离最小的集合点。",[11,17156,17157],{},"几何中位数原型具有很强的稳健性，这是它最重要的特性——即使数据中包含少量异常值，几何中位数仍然能反映「大多数」点的中心。它在旋转和平移下是等变的：如果数据被旋转或平移，几何中位数也会相应变化。它可以位于数据点之间的任意位置（与 K-medoids 不同，后者只能选取实际的数据点）。",[11,17159,17160],{},"由于没有闭式解，通常使用 Weiszfeld 算法（迭代加权平均）：",[38,17162,17164],{"className":17163,"translate":42},[621],[38,17165,17167,17279],{"className":17166,"translate":42},[41],[38,17168,17170],{"className":17169},[46],[48,17171,17172],{"xmlns":50,"display":630},[52,17173,17174,17276],{},[55,17175,17176,17188,17190],{},[295,17177,17178,17180],{},[58,17179,11],{"mathvariant":285},[55,17181,17182,17184,17186],{},[58,17183,3246],{},[62,17185,696],{},[300,17187,302],{},[62,17189,289],{},[636,17191,17192,17236],{},[55,17193,17194,17208],{},[15353,17195,17196,17198,17206],{},[62,17197,15357],{},[55,17199,17200,17202,17204],{},[58,17201,601],{},[62,17203,289],{},[300,17205,302],{},[58,17207,3555],{},[636,17209,17210,17216],{},[295,17211,17212,17214],{},[58,17213,14806],{"mathvariant":285},[58,17215,601],{},[55,17217,17218,17220,17226,17228,17234],{},[58,17219,2183],{"mathvariant":333},[295,17221,17222,17224],{},[58,17223,11],{"mathvariant":285},[58,17225,3246],{},[62,17227,710],{},[295,17229,17230,17232],{},[58,17231,14806],{"mathvariant":285},[58,17233,601],{},[58,17235,2183],{"mathvariant":333},[55,17237,17238,17252],{},[15353,17239,17240,17242,17250],{},[62,17241,15357],{},[55,17243,17244,17246,17248],{},[58,17245,601],{},[62,17247,289],{},[300,17249,302],{},[58,17251,3555],{},[636,17253,17254,17256],{},[300,17255,302],{},[55,17257,17258,17260,17266,17268,17274],{},[58,17259,2183],{"mathvariant":333},[295,17261,17262,17264],{},[58,17263,11],{"mathvariant":285},[58,17265,3246],{},[62,17267,710],{},[295,17269,17270,17272],{},[58,17271,14806],{"mathvariant":285},[58,17273,601],{},[58,17275,2183],{"mathvariant":333},[81,17277,17278],{"encoding":83},"\\mathbf{p}_{t+1} = \\frac{\\sum_{i=1}^m \\frac{\\mathbf{v}_i}{\\|\\mathbf{p}_t - \\mathbf{v}_i\\|}}{\\sum_{i=1}^m \\frac{1}{\\|\\mathbf{p}_t - \\mathbf{v}_i\\|}}",[38,17280,17282,17347],{"className":17281,"ariaHidden":89},[88],[38,17283,17285,17289,17338,17341,17344],{"className":17284},[93],[38,17286],{"className":17287,"style":17288},[97],"height:0.6528em;vertical-align:-0.2083em;",[38,17290,17292,17295],{"className":17291},[102],[38,17293,11],{"className":17294},[102,351],[38,17296,17298],{"className":17297},[134],[38,17299,17301,17330],{"className":17300},[138,384],[38,17302,17304,17327],{"className":17303},[142],[38,17305,17307],{"className":17306,"style":391},[146],[38,17308,17309,17312],{"style":394},[38,17310],{"className":17311,"style":155},[154],[38,17313,17315],{"className":17314},[159,160,161,162],[38,17316,17318,17321,17324],{"className":17317},[102,162],[38,17319,3246],{"className":17320},[102,103,162],[38,17322,696],{"className":17323},[999,162],[38,17325,302],{"className":17326},[102,162],[38,17328,408],{"className":17329},[407],[38,17331,17333],{"className":17332},[142],[38,17334,17336],{"className":17335,"style":3342},[146],[38,17337],{},[38,17339],{"className":17340,"style":109},[108],[38,17342,289],{"className":17343},[113],[38,17345],{"className":17346,"style":109},[108],[38,17348,17350,17354],{"className":17349},[93],[38,17351],{"className":17352,"style":17353},[97],"height:2.9694em;vertical-align:-1.2551em;",[38,17355,17357,17360,17899],{"className":17356},[102],[38,17358],{"className":17359},[371,740],[38,17361,17363],{"className":17362},[636],[38,17364,17366,17890],{"className":17365},[138,384],[38,17367,17369,17887],{"className":17368},[142],[38,17370,17373,17614,17622],{"className":17371,"style":17372},[146],"height:1.7143em;",[38,17374,17376,17379],{"style":17375},"top:-2.2649em;",[38,17377],{"className":17378,"style":760},[154],[38,17380,17382,17448,17451],{"className":17381},[102],[38,17383,17385,17390],{"className":17384},[878],[38,17386,15357],{"className":17387,"style":17389},[878,15521,17388],"small-op","position:relative;top:0em;",[38,17391,17393],{"className":17392},[134],[38,17394,17396,17439],{"className":17395},[138,384],[38,17397,17399,17436],{"className":17398},[142],[38,17400,17403,17424],{"className":17401,"style":17402},[146],"height:0.8043em;",[38,17404,17406,17409],{"style":17405},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[38,17407],{"className":17408,"style":155},[154],[38,17410,17412],{"className":17411},[159,160,161,162],[38,17413,17415,17418,17421],{"className":17414},[102,162],[38,17416,601],{"className":17417},[102,103,162],[38,17419,289],{"className":17420},[113,162],[38,17422,302],{"className":17423},[102,162],[38,17425,17427,17430],{"style":17426},"top:-3.2029em;margin-right:0.05em;",[38,17428],{"className":17429,"style":155},[154],[38,17431,17433],{"className":17432},[159,160,161,162],[38,17434,3555],{"className":17435},[102,103,162],[38,17437,408],{"className":17438},[407],[38,17440,17442],{"className":17441},[142],[38,17443,17446],{"className":17444,"style":17445},[146],"height:0.2997em;",[38,17447],{},[38,17449],{"className":17450,"style":425},[108],[38,17452,17454,17457,17611],{"className":17453},[102],[38,17455],{"className":17456},[371,740],[38,17458,17460],{"className":17459},[636],[38,17461,17463,17602],{"className":17462},[138,384],[38,17464,17466,17599],{"className":17465},[142],[38,17467,17470,17576,17584],{"className":17468,"style":17469},[146],"height:0.8451em;",[38,17471,17473,17476],{"style":17472},"top:-2.655em;",[38,17474],{"className":17475,"style":760},[154],[38,17477,17479],{"className":17478},[159,160,161,162],[38,17480,17482,17485,17528,17531,17573],{"className":17481},[102,162],[38,17483,2183],{"className":17484},[102,162],[38,17486,17488,17491],{"className":17487},[102,162],[38,17489,11],{"className":17490},[102,351,162],[38,17492,17494],{"className":17493},[134],[38,17495,17497,17519],{"className":17496},[138,384],[38,17498,17500,17516],{"className":17499},[142],[38,17501,17504],{"className":17502,"style":17503},[146],"height:0.2963em;",[38,17505,17507,17510],{"style":17506},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[38,17508],{"className":17509,"style":16819},[154],[38,17511,17513],{"className":17512},[159,16823,3786,162],[38,17514,3246],{"className":17515},[102,103,162],[38,17517,408],{"className":17518},[407],[38,17520,17522],{"className":17521},[142],[38,17523,17526],{"className":17524,"style":17525},[146],"height:0.143em;",[38,17527],{},[38,17529,710],{"className":17530},[999,162],[38,17532,17534,17537],{"className":17533},[102,162],[38,17535,14806],{"className":17536,"style":6689},[102,351,162],[38,17538,17540],{"className":17539},[134],[38,17541,17543,17565],{"className":17542},[138,384],[38,17544,17546,17562],{"className":17545},[142],[38,17547,17550],{"className":17548,"style":17549},[146],"height:0.3281em;",[38,17551,17553,17556],{"style":17552},"top:-2.357em;margin-left:-0.016em;margin-right:0.0714em;",[38,17554],{"className":17555,"style":16819},[154],[38,17557,17559],{"className":17558},[159,16823,3786,162],[38,17560,601],{"className":17561},[102,103,162],[38,17563,408],{"className":17564},[407],[38,17566,17568],{"className":17567},[142],[38,17569,17571],{"className":17570,"style":17525},[146],[38,17572],{},[38,17574,2183],{"className":17575},[102,162],[38,17577,17578,17581],{"style":811},[38,17579],{"className":17580,"style":760},[154],[38,17582],{"className":17583,"style":819},[818],[38,17585,17587,17590],{"style":17586},"top:-3.394em;",[38,17588],{"className":17589,"style":760},[154],[38,17591,17593],{"className":17592},[159,160,161,162],[38,17594,17596],{"className":17595},[102,162],[38,17597,302],{"className":17598},[102,162],[38,17600,408],{"className":17601},[407],[38,17603,17605],{"className":17604},[142],[38,17606,17609],{"className":17607,"style":17608},[146],"height:0.52em;",[38,17610],{},[38,17612],{"className":17613},[529,740],[38,17615,17616,17619],{"style":811},[38,17617],{"className":17618,"style":760},[154],[38,17620],{"className":17621,"style":819},[818],[38,17623,17625,17628],{"style":17624},"top:-3.91em;",[38,17626],{"className":17627,"style":760},[154],[38,17629,17631,17691,17694],{"className":17630},[102],[38,17632,17634,17637],{"className":17633},[878],[38,17635,15357],{"className":17636,"style":17389},[878,15521,17388],[38,17638,17640],{"className":17639},[134],[38,17641,17643,17683],{"className":17642},[138,384],[38,17644,17646,17680],{"className":17645},[142],[38,17647,17649,17669],{"className":17648,"style":17402},[146],[38,17650,17651,17654],{"style":17405},[38,17652],{"className":17653,"style":155},[154],[38,17655,17657],{"className":17656},[159,160,161,162],[38,17658,17660,17663,17666],{"className":17659},[102,162],[38,17661,601],{"className":17662},[102,103,162],[38,17664,289],{"className":17665},[113,162],[38,17667,302],{"className":17668},[102,162],[38,17670,17671,17674],{"style":17426},[38,17672],{"className":17673,"style":155},[154],[38,17675,17677],{"className":17676},[159,160,161,162],[38,17678,3555],{"className":17679},[102,103,162],[38,17681,408],{"className":17682},[407],[38,17684,17686],{"className":17685},[142],[38,17687,17689],{"className":17688,"style":17445},[146],[38,17690],{},[38,17692],{"className":17693,"style":425},[108],[38,17695,17697,17700,17884],{"className":17696},[102],[38,17698],{"className":17699},[371,740],[38,17701,17703],{"className":17702},[636],[38,17704,17706,17876],{"className":17705},[138,384],[38,17707,17709,17873],{"className":17708},[142],[38,17710,17713,17813,17821],{"className":17711,"style":17712},[146],"height:0.7212em;",[38,17714,17715,17718],{"style":17472},[38,17716],{"className":17717,"style":760},[154],[38,17719,17721],{"className":17720},[159,160,161,162],[38,17722,17724,17727,17767,17770,17810],{"className":17723},[102,162],[38,17725,2183],{"className":17726},[102,162],[38,17728,17730,17733],{"className":17729},[102,162],[38,17731,11],{"className":17732},[102,351,162],[38,17734,17736],{"className":17735},[134],[38,17737,17739,17759],{"className":17738},[138,384],[38,17740,17742,17756],{"className":17741},[142],[38,17743,17745],{"className":17744,"style":17503},[146],[38,17746,17747,17750],{"style":17506},[38,17748],{"className":17749,"style":16819},[154],[38,17751,17753],{"className":17752},[159,16823,3786,162],[38,17754,3246],{"className":17755},[102,103,162],[38,17757,408],{"className":17758},[407],[38,17760,17762],{"className":17761},[142],[38,17763,17765],{"className":17764,"style":17525},[146],[38,17766],{},[38,17768,710],{"className":17769},[999,162],[38,17771,17773,17776],{"className":17772},[102,162],[38,17774,14806],{"className":17775,"style":6689},[102,351,162],[38,17777,17779],{"className":17778},[134],[38,17780,17782,17802],{"className":17781},[138,384],[38,17783,17785,17799],{"className":17784},[142],[38,17786,17788],{"className":17787,"style":17549},[146],[38,17789,17790,17793],{"style":17552},[38,17791],{"className":17792,"style":16819},[154],[38,17794,17796],{"className":17795},[159,16823,3786,162],[38,17797,601],{"className":17798},[102,103,162],[38,17800,408],{"className":17801},[407],[38,17803,17805],{"className":17804},[142],[38,17806,17808],{"className":17807,"style":17525},[146],[38,17809],{},[38,17811,2183],{"className":17812},[102,162],[38,17814,17815,17818],{"style":811},[38,17816],{"className":17817,"style":760},[154],[38,17819],{"className":17820,"style":819},[818],[38,17822,17824,17827],{"style":17823},"top:-3.4101em;",[38,17825],{"className":17826,"style":760},[154],[38,17828,17830],{"className":17829},[159,160,161,162],[38,17831,17833],{"className":17832},[102,162],[38,17834,17836,17839],{"className":17835},[102,162],[38,17837,14806],{"className":17838,"style":6689},[102,351,162],[38,17840,17842],{"className":17841},[134],[38,17843,17845,17865],{"className":17844},[138,384],[38,17846,17848,17862],{"className":17847},[142],[38,17849,17851],{"className":17850,"style":17549},[146],[38,17852,17853,17856],{"style":17552},[38,17854],{"className":17855,"style":16819},[154],[38,17857,17859],{"className":17858},[159,16823,3786,162],[38,17860,601],{"className":17861},[102,103,162],[38,17863,408],{"className":17864},[407],[38,17866,17868],{"className":17867},[142],[38,17869,17871],{"className":17870,"style":17525},[146],[38,17872],{},[38,17874,408],{"className":17875},[407],[38,17877,17879],{"className":17878},[142],[38,17880,17882],{"className":17881,"style":17608},[146],[38,17883],{},[38,17885],{"className":17886},[529,740],[38,17888,408],{"className":17889},[407],[38,17891,17893],{"className":17892},[142],[38,17894,17897],{"className":17895,"style":17896},[146],"height:1.2551em;",[38,17898],{},[38,17900],{"className":17901},[529,740],[11,17903,17904,17905,17975],{},"初始值可以取均值向量，迭代直至收敛。注意：当 ",[38,17906,17908,17926],{"className":17907,"translate":42},[41],[38,17909,17911],{"className":17910},[46],[48,17912,17913],{"xmlns":50},[52,17914,17915,17923],{},[55,17916,17917],{},[295,17918,17919,17921],{},[58,17920,11],{"mathvariant":285},[58,17922,3246],{},[81,17924,17925],{"encoding":83},"\\mathbf{p}_t",[38,17927,17929],{"className":17928,"ariaHidden":89},[88],[38,17930,17932,17935],{"className":17931},[93],[38,17933],{"className":17934,"style":6685},[97],[38,17936,17938,17941],{"className":17937},[102],[38,17939,11],{"className":17940},[102,351],[38,17942,17944],{"className":17943},[134],[38,17945,17947,17967],{"className":17946},[138,384],[38,17948,17950,17964],{"className":17949},[142],[38,17951,17953],{"className":17952,"style":3379},[146],[38,17954,17955,17958],{"style":394},[38,17956],{"className":17957,"style":155},[154],[38,17959,17961],{"className":17960},[159,160,161,162],[38,17962,3246],{"className":17963},[102,103,162],[38,17965,408],{"className":17966},[407],[38,17968,17970],{"className":17969},[142],[38,17971,17973],{"className":17972,"style":415},[146],[38,17974],{}," 恰好与某个数据点重合时需要进行特殊处理（跳过该点）。",[11,17977,17978],{},"当数据包含异常值或噪声时，它用于表示「典型样本」而非均值。因此，在一些聚类算法中，它被用作簇中心（比 K-means 更稳健）。",[11,17980,17981],{},"在视频背景建模中，多帧的几何中位数可以用作背景（不受前景运动物体影响）。它也用于在多个需求点之间寻找最优服务点（例如医院、仓库选址）。",[11,17983,17984],{},"几何中位数原型是数据点的「空间中心」，使得到所有点的总欧氏距离最小，并且比均值向量对异常值更稳健。",[28,17986,17988,18058],{"id":17987},"lpl_plp-欧氏范数",[38,17989,17991,18009],{"className":17990,"translate":42},[41],[38,17992,17994],{"className":17993},[46],[48,17995,17996],{"xmlns":50},[52,17997,17998,18006],{},[55,17999,18000],{},[295,18001,18002,18004],{},[58,18003,3104],{},[58,18005,11],{},[81,18007,18008],{"encoding":83},"L_p",[38,18010,18012],{"className":18011,"ariaHidden":89},[88],[38,18013,18015,18018],{"className":18014},[93],[38,18016],{"className":18017,"style":13918},[97],[38,18019,18021,18024],{"className":18020},[102],[38,18022,3104],{"className":18023},[102,103],[38,18025,18027],{"className":18026},[134],[38,18028,18030,18050],{"className":18029},[138,384],[38,18031,18033,18047],{"className":18032},[142],[38,18034,18036],{"className":18035,"style":503},[146],[38,18037,18038,18041],{"style":394},[38,18039],{"className":18040,"style":155},[154],[38,18042,18044],{"className":18043},[159,160,161,162],[38,18045,11],{"className":18046},[102,103,162],[38,18048,408],{"className":18049},[407],[38,18051,18053],{"className":18052},[142],[38,18054,18056],{"className":18055,"style":7348},[146],[38,18057],{}," 欧氏范数",[11,18060,18061,18062,18131,18132,18160],{},"欧氏范数是最为人熟知的「向量距离」。不同的范数本质上定义了「距离或大小意味着什么」，不同的定义会直接改变几何结构、优化行为乃至模型的偏好。一般来说，我们讨论的是 ",[38,18063,18065,18082],{"className":18064,"translate":42},[41],[38,18066,18068],{"className":18067},[46],[48,18069,18070],{"xmlns":50},[52,18071,18072,18080],{},[55,18073,18074],{},[295,18075,18076,18078],{},[58,18077,3104],{},[58,18079,11],{},[81,18081,18008],{"encoding":83},[38,18083,18085],{"className":18084,"ariaHidden":89},[88],[38,18086,18088,18091],{"className":18087},[93],[38,18089],{"className":18090,"style":13918},[97],[38,18092,18094,18097],{"className":18093},[102],[38,18095,3104],{"className":18096},[102,103],[38,18098,18100],{"className":18099},[134],[38,18101,18103,18123],{"className":18102},[138,384],[38,18104,18106,18120],{"className":18105},[142],[38,18107,18109],{"className":18108,"style":503},[146],[38,18110,18111,18114],{"style":394},[38,18112],{"className":18113,"style":155},[154],[38,18115,18117],{"className":18116},[159,160,161,162],[38,18118,11],{"className":18119},[102,103,162],[38,18121,408],{"className":18122},[407],[38,18124,18126],{"className":18125},[142],[38,18127,18129],{"className":18128,"style":7348},[146],[38,18130],{}," 范数，其中 ",[38,18133,18135,18148],{"className":18134,"translate":42},[41],[38,18136,18138],{"className":18137},[46],[48,18139,18140],{"xmlns":50},[52,18141,18142,18146],{},[55,18143,18144],{},[58,18145,11],{},[81,18147,11],{"encoding":83},[38,18149,18151],{"className":18150,"ariaHidden":89},[88],[38,18152,18154,18157],{"className":18153},[93],[38,18155],{"className":18156,"style":3505},[97],[38,18158,11],{"className":18159},[102,103]," 是一个参数。",[11,18162,18163,18164,16420,18192,18420],{},"对于 ",[38,18165,18167,18180],{"className":18166,"translate":42},[41],[38,18168,18170],{"className":18169},[46],[48,18171,18172],{"xmlns":50},[52,18173,18174,18178],{},[55,18175,18176],{},[58,18177,74],{},[81,18179,74],{"encoding":83},[38,18181,18183],{"className":18182,"ariaHidden":89},[88],[38,18184,18186,18189],{"className":18185},[93],[38,18187],{"className":18188,"style":4312},[97],[38,18190,74],{"className":18191},[102,103],[38,18193,18195,18245],{"className":18194,"translate":42},[41],[38,18196,18198],{"className":18197},[46],[48,18199,18200],{"xmlns":50},[52,18201,18202,18242],{},[55,18203,18204,18206,18208,18210,18216,18218,18224,18226,18228,18230,18232,18234,18240],{},[58,18205,286],{"mathvariant":285},[62,18207,289],{},[62,18209,293],{"stretchy":292},[295,18211,18212,18214],{},[58,18213,286],{},[300,18215,302],{},[62,18217,305],{"separator":89},[295,18219,18220,18222],{},[58,18221,286],{},[300,18223,312],{},[62,18225,305],{"separator":89},[58,18227,2192],{"mathvariant":333},[58,18229,2192],{"mathvariant":333},[58,18231,2192],{"mathvariant":333},[62,18233,305],{"separator":89},[295,18235,18236,18238],{},[58,18237,286],{},[58,18239,74],{},[62,18241,330],{"stretchy":292},[81,18243,18244],{"encoding":83},"\\mathbf{x} = (x_1, x_2, ..., x_n)",[38,18246,18248,18266],{"className":18247,"ariaHidden":89},[88],[38,18249,18251,18254,18257,18260,18263],{"className":18250},[93],[38,18252],{"className":18253,"style":347},[97],[38,18255,286],{"className":18256},[102,351],[38,18258],{"className":18259,"style":109},[108],[38,18261,289],{"className":18262},[113],[38,18264],{"className":18265,"style":109},[108],[38,18267,18269,18272,18275,18315,18318,18321,18361,18364,18367,18371,18374,18377,18417],{"className":18268},[93],[38,18270],{"className":18271,"style":1439},[97],[38,18273,293],{"className":18274},[371],[38,18276,18278,18281],{"className":18277},[102],[38,18279,286],{"className":18280},[102,103],[38,18282,18284],{"className":18283},[134],[38,18285,18287,18307],{"className":18286},[138,384],[38,18288,18290,18304],{"className":18289},[142],[38,18291,18293],{"className":18292,"style":391},[146],[38,18294,18295,18298],{"style":394},[38,18296],{"className":18297,"style":155},[154],[38,18299,18301],{"className":18300},[159,160,161,162],[38,18302,302],{"className":18303},[102,162],[38,18305,408],{"className":18306},[407],[38,18308,18310],{"className":18309},[142],[38,18311,18313],{"className":18312,"style":415},[146],[38,18314],{},[38,18316,305],{"className":18317},[421],[38,18319],{"className":18320,"style":425},[108],[38,18322,18324,18327],{"className":18323},[102],[38,18325,286],{"className":18326},[102,103],[38,18328,18330],{"className":18329},[134],[38,18331,18333,18353],{"className":18332},[138,384],[38,18334,18336,18350],{"className":18335},[142],[38,18337,18339],{"className":18338,"style":391},[146],[38,18340,18341,18344],{"style":394},[38,18342],{"className":18343,"style":155},[154],[38,18345,18347],{"className":18346},[159,160,161,162],[38,18348,312],{"className":18349},[102,162],[38,18351,408],{"className":18352},[407],[38,18354,18356],{"className":18355},[142],[38,18357,18359],{"className":18358,"style":415},[146],[38,18360],{},[38,18362,305],{"className":18363},[421],[38,18365],{"className":18366,"style":425},[108],[38,18368,18370],{"className":18369},[102],"...",[38,18372,305],{"className":18373},[421],[38,18375],{"className":18376,"style":425},[108],[38,18378,18380,18383],{"className":18379},[102],[38,18381,286],{"className":18382},[102,103],[38,18384,18386],{"className":18385},[134],[38,18387,18389,18409],{"className":18388},[138,384],[38,18390,18392,18406],{"className":18391},[142],[38,18393,18395],{"className":18394,"style":503},[146],[38,18396,18397,18400],{"style":394},[38,18398],{"className":18399,"style":155},[154],[38,18401,18403],{"className":18402},[159,160,161,162],[38,18404,74],{"className":18405},[102,103,162],[38,18407,408],{"className":18408},[407],[38,18410,18412],{"className":18411},[142],[38,18413,18415],{"className":18414,"style":415},[146],[38,18416],{},[38,18418,330],{"className":18419},[529],"，Lp 范数定义为：",[38,18422,18424],{"className":18423,"translate":42},[621],[38,18425,18427,18507],{"className":18426,"translate":42},[41],[38,18428,18430],{"className":18429},[46],[48,18431,18432],{"xmlns":50,"display":630},[52,18433,18434,18504],{},[55,18435,18436,18438,18440,18446,18448,18493,18495,18497,18499,18502],{},[58,18437,2183],{"mathvariant":333},[58,18439,286],{"mathvariant":285},[295,18441,18442,18444],{},[58,18443,2183],{"mathvariant":333},[58,18445,11],{},[62,18447,289],{},[66,18449,18450,18484],{},[55,18451,18452,18454,18468,18470,18476,18482],{},[62,18453,293],{"fence":89},[15353,18455,18456,18458,18466],{},[62,18457,15357],{},[55,18459,18460,18462,18464],{},[58,18461,601],{},[62,18463,289],{},[300,18465,302],{},[58,18467,74],{},[58,18469,7695],{"mathvariant":333},[295,18471,18472,18474],{},[58,18473,286],{},[58,18475,601],{},[66,18477,18478,18480],{},[58,18479,7695],{"mathvariant":333},[58,18481,11],{},[62,18483,330],{"fence":89},[55,18485,18486,18488,18491],{},[300,18487,302],{},[58,18489,18490],{"mathvariant":333},"\u002F",[58,18492,11],{},[62,18494,305],{"separator":89},[108,18496],{"width":4479},[58,18498,11],{},[62,18500,18501],{},"≥",[300,18503,302],{},[81,18505,18506],{"encoding":83},"\\|\\mathbf{x}\\|_p = \\left( \\sum_{i=1}^{n} |x_i|^p \\right)^{1\u002Fp}, \\quad p \\ge 1",[38,18508,18510,18571,18794],{"className":18509,"ariaHidden":89},[88],[38,18511,18513,18516,18519,18522,18562,18565,18568],{"className":18512},[93],[38,18514],{"className":18515,"style":8669},[97],[38,18517,2183],{"className":18518},[102],[38,18520,286],{"className":18521},[102,351],[38,18523,18525,18528],{"className":18524},[102],[38,18526,2183],{"className":18527},[102],[38,18529,18531],{"className":18530},[134],[38,18532,18534,18554],{"className":18533},[138,384],[38,18535,18537,18551],{"className":18536},[142],[38,18538,18540],{"className":18539,"style":503},[146],[38,18541,18542,18545],{"style":394},[38,18543],{"className":18544,"style":155},[154],[38,18546,18548],{"className":18547},[159,160,161,162],[38,18549,11],{"className":18550},[102,103,162],[38,18552,408],{"className":18553},[407],[38,18555,18557],{"className":18556},[142],[38,18558,18560],{"className":18559,"style":7348},[146],[38,18561],{},[38,18563],{"className":18564,"style":109},[108],[38,18566,289],{"className":18567},[113],[38,18569],{"className":18570,"style":109},[108],[38,18572,18574,18578,18770,18773,18776,18779,18782,18785,18788,18791],{"className":18573},[93],[38,18575],{"className":18576,"style":18577},[97],"height:3.3056em;vertical-align:-1.2777em;",[38,18579,18581,18738],{"className":18580},[475],[38,18582,18584,18590,18657,18660,18663,18703,18732],{"className":18583},[475],[38,18585,18587],{"className":18586,"style":1481},[371,1480],[38,18588,293],{"className":18589},[1485,15772],[38,18591,18593],{"className":18592},[878,879],[38,18594,18596,18649],{"className":18595},[138,384],[38,18597,18599,18646],{"className":18598},[142],[38,18600,18602,18622,18632],{"className":18601,"style":15488},[146],[38,18603,18604,18607],{"style":15491},[38,18605],{"className":18606,"style":8852},[154],[38,18608,18610],{"className":18609},[159,160,161,162],[38,18611,18613,18616,18619],{"className":18612},[102,162],[38,18614,601],{"className":18615},[102,103,162],[38,18617,289],{"className":18618},[113,162],[38,18620,302],{"className":18621},[102,162],[38,18623,18624,18627],{"style":15512},[38,18625],{"className":18626,"style":8852},[154],[38,18628,18629],{},[38,18630,15357],{"className":18631},[878,15521,15522],[38,18633,18634,18637],{"style":15525},[38,18635],{"className":18636,"style":8852},[154],[38,18638,18640],{"className":18639},[159,160,161,162],[38,18641,18643],{"className":18642},[102,162],[38,18644,74],{"className":18645},[102,103,162],[38,18647,408],{"className":18648},[407],[38,18650,18652],{"className":18651},[142],[38,18653,18655],{"className":18654,"style":15547},[146],[38,18656],{},[38,18658],{"className":18659,"style":425},[108],[38,18661,7695],{"className":18662},[102],[38,18664,18666,18669],{"className":18665},[102],[38,18667,286],{"className":18668},[102,103],[38,18670,18672],{"className":18671},[134],[38,18673,18675,18695],{"className":18674},[138,384],[38,18676,18678,18692],{"className":18677},[142],[38,18679,18681],{"className":18680,"style":786},[146],[38,18682,18683,18686],{"style":394},[38,18684],{"className":18685,"style":155},[154],[38,18687,18689],{"className":18688},[159,160,161,162],[38,18690,601],{"className":18691},[102,103,162],[38,18693,408],{"className":18694},[407],[38,18696,18698],{"className":18697},[142],[38,18699,18701],{"className":18700,"style":415},[146],[38,18702],{},[38,18704,18706,18709],{"className":18705},[102],[38,18707,7695],{"className":18708},[102],[38,18710,18712],{"className":18711},[134],[38,18713,18715],{"className":18714},[138],[38,18716,18718],{"className":18717},[142],[38,18719,18721],{"className":18720,"style":1871},[146],[38,18722,18723,18726],{"style":1892},[38,18724],{"className":18725,"style":155},[154],[38,18727,18729],{"className":18728},[159,160,161,162],[38,18730,11],{"className":18731},[102,103,162],[38,18733,18735],{"className":18734,"style":1481},[529,1480],[38,18736,330],{"className":18737},[1485,15772],[38,18739,18741],{"className":18740},[134],[38,18742,18744],{"className":18743},[138],[38,18745,18747],{"className":18746},[142],[38,18748,18751],{"className":18749,"style":18750},[146],"height:2.0279em;",[38,18752,18754,18757],{"style":18753},"top:-4.2029em;margin-right:0.05em;",[38,18755],{"className":18756,"style":155},[154],[38,18758,18760],{"className":18759},[159,160,161,162],[38,18761,18763,18767],{"className":18762},[102,162],[38,18764,18766],{"className":18765},[102,162],"1\u002F",[38,18768,11],{"className":18769},[102,103,162],[38,18771],{"className":18772,"style":425},[108],[38,18774,305],{"className":18775},[421],[38,18777],{"className":18778,"style":10740},[108],[38,18780],{"className":18781,"style":425},[108],[38,18783,11],{"className":18784},[102,103],[38,18786],{"className":18787,"style":109},[108],[38,18789,18501],{"className":18790},[113],[38,18792],{"className":18793,"style":109},[108],[38,18795,18797,18800],{"className":18796},[93],[38,18798],{"className":18799,"style":2429},[97],[38,18801,302],{"className":18802},[102],[11,18804,18805,18833,18834,18885,18886,18956],{},[38,18806,18808,18821],{"className":18807,"translate":42},[41],[38,18809,18811],{"className":18810},[46],[48,18812,18813],{"xmlns":50},[52,18814,18815,18819],{},[55,18816,18817],{},[58,18818,11],{},[81,18820,11],{"encoding":83},[38,18822,18824],{"className":18823,"ariaHidden":89},[88],[38,18825,18827,18830],{"className":18826},[93],[38,18828],{"className":18829,"style":3505},[97],[38,18831,11],{"className":18832},[102,103]," 取不同值会得到不同范数下的距离。当 ",[38,18835,18837,18855],{"className":18836,"translate":42},[41],[38,18838,18840],{"className":18839},[46],[48,18841,18842],{"xmlns":50},[52,18843,18844,18852],{},[55,18845,18846,18848,18850],{},[58,18847,11],{},[62,18849,289],{},[300,18851,302],{},[81,18853,18854],{"encoding":83},"p=1",[38,18856,18858,18876],{"className":18857,"ariaHidden":89},[88],[38,18859,18861,18864,18867,18870,18873],{"className":18860},[93],[38,18862],{"className":18863,"style":3505},[97],[38,18865,11],{"className":18866},[102,103],[38,18868],{"className":18869,"style":109},[108],[38,18871,289],{"className":18872},[113],[38,18874],{"className":18875,"style":109},[108],[38,18877,18879,18882],{"className":18878},[93],[38,18880],{"className":18881,"style":2429},[97],[38,18883,302],{"className":18884},[102]," 时，得到 ",[38,18887,18889,18907],{"className":18888,"translate":42},[41],[38,18890,18892],{"className":18891},[46],[48,18893,18894],{"xmlns":50},[52,18895,18896,18904],{},[55,18897,18898],{},[295,18899,18900,18902],{},[58,18901,3104],{},[300,18903,302],{},[81,18905,18906],{"encoding":83},"L_1",[38,18908,18910],{"className":18909,"ariaHidden":89},[88],[38,18911,18913,18916],{"className":18912},[93],[38,18914],{"className":18915,"style":13610},[97],[38,18917,18919,18922],{"className":18918},[102],[38,18920,3104],{"className":18921},[102,103],[38,18923,18925],{"className":18924},[134],[38,18926,18928,18948],{"className":18927},[138,384],[38,18929,18931,18945],{"className":18930},[142],[38,18932,18934],{"className":18933,"style":391},[146],[38,18935,18936,18939],{"style":394},[38,18937],{"className":18938,"style":155},[154],[38,18940,18942],{"className":18941},[159,160,161,162],[38,18943,302],{"className":18944},[102,162],[38,18946,408],{"className":18947},[407],[38,18949,18951],{"className":18950},[142],[38,18952,18954],{"className":18953,"style":415},[146],[38,18955],{}," 范数下的距离：",[38,18958,18960],{"className":18959,"translate":42},[621],[38,18961,18963,19055],{"className":18962,"translate":42},[41],[38,18964,18966],{"className":18965},[46],[48,18967,18968],{"xmlns":50,"display":630},[52,18969,18970,19052],{},[55,18971,18972,18974,18976,18982,18984,18998,19000,19006,19008,19010,19012,19018,19020,19022,19024,19030,19032,19034,19036,19038,19040,19042,19044,19050],{},[58,18973,2183],{"mathvariant":333},[58,18975,286],{"mathvariant":285},[295,18977,18978,18980],{},[58,18979,2183],{"mathvariant":333},[300,18981,302],{},[62,18983,289],{},[15353,18985,18986,18988,18996],{},[62,18987,15357],{},[55,18989,18990,18992,18994],{},[58,18991,601],{},[62,18993,289],{},[300,18995,302],{},[58,18997,74],{},[58,18999,7695],{"mathvariant":333},[295,19001,19002,19004],{},[58,19003,286],{},[58,19005,601],{},[58,19007,7695],{"mathvariant":333},[62,19009,289],{},[58,19011,7695],{"mathvariant":333},[295,19013,19014,19016],{},[58,19015,286],{},[300,19017,302],{},[58,19019,7695],{"mathvariant":333},[62,19021,696],{},[58,19023,7695],{"mathvariant":333},[295,19025,19026,19028],{},[58,19027,286],{},[300,19029,312],{},[58,19031,7695],{"mathvariant":333},[62,19033,696],{},[58,19035,2192],{"mathvariant":333},[58,19037,2192],{"mathvariant":333},[58,19039,2192],{"mathvariant":333},[62,19041,696],{},[58,19043,7695],{"mathvariant":333},[295,19045,19046,19048],{},[58,19047,286],{},[58,19049,74],{},[58,19051,7695],{"mathvariant":333},[81,19053,19054],{"encoding":83},"\\|\\mathbf{x}\\|_1 = \\sum_{i=1}^{n} |x_i| = |x_1| + |x_2| + ... + |x_n|",[38,19056,19058,19119,19250,19311,19372,19390],{"className":19057,"ariaHidden":89},[88],[38,19059,19061,19064,19067,19070,19110,19113,19116],{"className":19060},[93],[38,19062],{"className":19063,"style":1439},[97],[38,19065,2183],{"className":19066},[102],[38,19068,286],{"className":19069},[102,351],[38,19071,19073,19076],{"className":19072},[102],[38,19074,2183],{"className":19075},[102],[38,19077,19079],{"className":19078},[134],[38,19080,19082,19102],{"className":19081},[138,384],[38,19083,19085,19099],{"className":19084},[142],[38,19086,19088],{"className":19087,"style":391},[146],[38,19089,19090,19093],{"style":394},[38,19091],{"className":19092,"style":155},[154],[38,19094,19096],{"className":19095},[159,160,161,162],[38,19097,302],{"className":19098},[102,162],[38,19100,408],{"className":19101},[407],[38,19103,19105],{"className":19104},[142],[38,19106,19108],{"className":19107,"style":415},[146],[38,19109],{},[38,19111],{"className":19112,"style":109},[108],[38,19114,289],{"className":19115},[113],[38,19117],{"className":19118,"style":109},[108],[38,19120,19122,19125,19192,19195,19198,19238,19241,19244,19247],{"className":19121},[93],[38,19123],{"className":19124,"style":15410},[97],[38,19126,19128],{"className":19127},[878,879],[38,19129,19131,19184],{"className":19130},[138,384],[38,19132,19134,19181],{"className":19133},[142],[38,19135,19137,19157,19167],{"className":19136,"style":15488},[146],[38,19138,19139,19142],{"style":15491},[38,19140],{"className":19141,"style":8852},[154],[38,19143,19145],{"className":19144},[159,160,161,162],[38,19146,19148,19151,19154],{"className":19147},[102,162],[38,19149,601],{"className":19150},[102,103,162],[38,19152,289],{"className":19153},[113,162],[38,19155,302],{"className":19156},[102,162],[38,19158,19159,19162],{"style":15512},[38,19160],{"className":19161,"style":8852},[154],[38,19163,19164],{},[38,19165,15357],{"className":19166},[878,15521,15522],[38,19168,19169,19172],{"style":15525},[38,19170],{"className":19171,"style":8852},[154],[38,19173,19175],{"className":19174},[159,160,161,162],[38,19176,19178],{"className":19177},[102,162],[38,19179,74],{"className":19180},[102,103,162],[38,19182,408],{"className":19183},[407],[38,19185,19187],{"className":19186},[142],[38,19188,19190],{"className":19189,"style":15547},[146],[38,19191],{},[38,19193],{"className":19194,"style":425},[108],[38,19196,7695],{"className":19197},[102],[38,19199,19201,19204],{"className":19200},[102],[38,19202,286],{"className":19203},[102,103],[38,19205,19207],{"className":19206},[134],[38,19208,19210,19230],{"className":19209},[138,384],[38,19211,19213,19227],{"className":19212},[142],[38,19214,19216],{"className":19215,"style":786},[146],[38,19217,19218,19221],{"style":394},[38,19219],{"className":19220,"style":155},[154],[38,19222,19224],{"className":19223},[159,160,161,162],[38,19225,601],{"className":19226},[102,103,162],[38,19228,408],{"className":19229},[407],[38,19231,19233],{"className":19232},[142],[38,19234,19236],{"className":19235,"style":415},[146],[38,19237],{},[38,19239,7695],{"className":19240},[102],[38,19242],{"className":19243,"style":109},[108],[38,19245,289],{"className":19246},[113],[38,19248],{"className":19249,"style":109},[108],[38,19251,19253,19256,19259,19299,19302,19305,19308],{"className":19252},[93],[38,19254],{"className":19255,"style":1439},[97],[38,19257,7695],{"className":19258},[102],[38,19260,19262,19265],{"className":19261},[102],[38,19263,286],{"className":19264},[102,103],[38,19266,19268],{"className":19267},[134],[38,19269,19271,19291],{"className":19270},[138,384],[38,19272,19274,19288],{"className":19273},[142],[38,19275,19277],{"className":19276,"style":391},[146],[38,19278,19279,19282],{"style":394},[38,19280],{"className":19281,"style":155},[154],[38,19283,19285],{"className":19284},[159,160,161,162],[38,19286,302],{"className":19287},[102,162],[38,19289,408],{"className":19290},[407],[38,19292,19294],{"className":19293},[142],[38,19295,19297],{"className":19296,"style":415},[146],[38,19298],{},[38,19300,7695],{"className":19301},[102],[38,19303],{"className":19304,"style":995},[108],[38,19306,696],{"className":19307},[999],[38,19309],{"className":19310,"style":995},[108],[38,19312,19314,19317,19320,19360,19363,19366,19369],{"className":19313},[93],[38,19315],{"className":19316,"style":1439},[97],[38,19318,7695],{"className":19319},[102],[38,19321,19323,19326],{"className":19322},[102],[38,19324,286],{"className":19325},[102,103],[38,19327,19329],{"className":19328},[134],[38,19330,19332,19352],{"className":19331},[138,384],[38,19333,19335,19349],{"className":19334},[142],[38,19336,19338],{"className":19337,"style":391},[146],[38,19339,19340,19343],{"style":394},[38,19341],{"className":19342,"style":155},[154],[38,19344,19346],{"className":19345},[159,160,161,162],[38,19347,312],{"className":19348},[102,162],[38,19350,408],{"className":19351},[407],[38,19353,19355],{"className":19354},[142],[38,19356,19358],{"className":19357,"st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距离就是曼哈顿距离（只能沿坐标轴方向移动）。例如，",[38,19516,19518,19547],{"className":19517,"translate":42},[41],[38,19519,19521],{"className":19520},[46],[48,19522,19523],{"xmlns":50},[52,19524,19525,19544],{},[55,19526,19527,19529,19531,19533,19535,19537,19539,19542],{},[58,19528,286],{"mathvariant":285},[62,19530,289],{},[62,19532,293],{"stretchy":292},[300,19534,13114],{},[62,19536,305],{"separator":89},[62,19538,710],{},[300,19540,19541],{},"4",[62,19543,330],{"stretchy":292},[81,19545,19546],{"encoding":83},"\\mathbf{x} = (3, -4)",[38,19548,19550,19568],{"className":19549,"ariaHidden":89},[88],[38,19551,19553,19556,19559,19562,19565],{"className":19552},[93],[38,19554],{"className":19555,"style":347},[97],[38,19557,286],{"className":19558},[102,351],[38,19560],{"className":19561,"style":109},[108],[38,19563,289],{"className":19564},[113],[38,19566],{"className":19567,"style":109},[108],[38,19569,19571,19574,19577,19580,19583,19586,19589,19592],{"className":19570},[93],[38,19572],{"className":19573,"style":1439},[97],[38,19575,293],{"className":19576},[371],[38,19578,13114],{"className":19579},[102],[38,19581,305],{"className":19582},[421],[38,19584],{"className":19585,"style":425},[108],[38,19587,710],{"className":19588},[102],[38,19590,19541],{"className":19591},[102],[38,19593,330],{"className":19594},[529]," 的 ",[38,19597,19599,19616],{"className":19598,"translate":42},[41],[38,19600,19602],{"className":19601},[46],[48,19603,19604],{"xmlns":50},[52,19605,19606,19614],{},[55,19607,19608],{},[295,19609,19610,19612],{},[58,19611,3104],{},[300,19613,302],{},[81,19615,18906],{"encoding":83},[38,19617,19619],{"className":19618,"ariaHidden":89},[88],[38,19620,19622,19625],{"className":19621},[93],[38,19623],{"className":19624,"style":13610},[97],[38,19626,19628,19631],{"className":19627},[102],[38,19629,3104],{"className":19630},[102,103],[38,19632,19634],{"className":19633},[134],[38,19635,19637,19657],{"className":19636},[138,384],[38,19638,19640,19654],{"className":19639},[142],[38,19641,19643],{"className":19642,"style":391},[146],[38,19644,19645,19648],{"style":394},[38,19646],{"className":19647,"style":155},[154],[38,19649,19651],{"className":19650},[159,160,161,162],[38,19652,302],{"className":19653},[102,162],[38,19655,408],{"className":19656},[407],[38,19658,19660],{"className":19659},[142],[38,19661,19663],{"className":19662,"style":415},[146],[38,19664],{}," 范数为 ",[38,19667,19669,19722],{"className":19668,"translate":42},[41],[38,19670,19672],{"className":19671},[46],[48,19673,19674],{"xmlns":50},[52,19675,19676,19719],{},[55,19677,19678,19680,19682,19688,19690,19692,19694,19696,19698,19700,19702,19704,19706,19708,19710,19712,19714,19716],{},[58,19679,2183],{"mathvariant":333},[58,19681,286],{"mathvariant":285},[295,19683,19684,19686],{},[58,19685,2183],{"mathvariant":333},[300,19687,302],{},[62,19689,289],{},[58,19691,7695],{"mathvariant":333},[300,19693,13114],{},[58,19695,7695],{"mathvariant":333},[62,19697,696],{},[58,19699,7695],{"mathvariant":333},[62,19701,710],{},[300,19703,19541],{},[58,19705,7695],{"mathvariant":333},[62,19707,289],{},[300,19709,13114],{},[62,19711,696],{},[300,19713,19541],{},[62,19715,289],{},[300,19717,19718],{},"7",[81,19720,19721],{"encoding":83},"\\|\\mathbf{x}\\|_1 = |3| + |-4| = 3 + 4 = 7",[38,19723,19725,19786,19805,19823,19842,19861,19879],{"className":19724,"ariaHidden":89},[88],[38,19726,19728,19731,19734,19737,19777,19780,19783],{"className":19727},[93],[38,19729],{"className":19730,"style":1439},[97],[38,19732,2183],{"className":19733},[102],[38,19735,286],{"className":19736},[102,351],[38,19738,19740,19743],{"className":19739},[102],[38,19741,2183],{"className":19742},[102],[38,19744,19746],{"className":19745},[134],[38,19747,19749,19769],{"className":19748},[138,384],[38,19750,19752,19766],{"className":19751},[142],[38,19753,19755],{"className":19754,"style":391},[146],[38,19756,19757,19760],{"style":394},[38,19758],{"className":19759,"style":155},[154],[38,19761,19763],{"className":19762},[159,160,161,162],[38,19764,302],{"className":19765},[102,162],[38,19767,408],{"className":19768},[407],[38,19770,19772],{"className":19771},[142],[38,19773,19775],{"className":19774,"style":415},[146],[38,19776],{},[38,19778],{"className":19779,"style":109},[108],[38,19781,289],{"className":19782},[113],[38,19784],{"className":19785,"style":109},[108],[38,19787,19789,19792,19796,19799,19802],{"className":19788},[93],[38,19790],{"className":19791,"style":1439},[97],[38,19793,19795],{"className":19794},[102],"∣3∣",[38,19797],{"className":19798,"style":995},[108],[38,19800,696],{"className":19801},[999],[38,19803],{"className":19804,"style":995},[108],[38,19806,19808,19811,19814,19817,19820],{"className":19807},[93],[38,19809],{"className":19810,"style":1439},[97],[38,19812,7695],{"className":19813},[102],[38,19815],{"className":19816,"style":995},[108],[38,19818,710],{"className":19819},[999],[38,19821],{"className":19822,"style":995},[108],[38,19824,19826,19829,19833,19836,19839],{"className":19825},[93],[38,19827],{"className":19828,"style":1439},[97],[38,19830,19832],{"className":19831},[102],"4∣",[38,19834],{"className":19835,"style":109},[108],[38,19837,289],{"className":19838},[113],[38,19840],{"className":19841,"style":109},[108],[38,19843,19845,19849,19852,19855,19858],{"className":19844},[93],[38,19846],{"className":19847,"style":19848},[97],"height:0.7278em;vertical-align:-0.0833em;",[38,19850,13114],{"className":19851},[102],[38,19853],{"className":19854,"style":995},[108],[38,19856,696],{"className":19857},[999],[38,19859],{"className":19860,"style":995},[108],[38,19862,19864,19867,19870,19873,19876],{"className":19863},[93],[38,19865],{"className":19866,"style":2429},[97],[38,19868,19541],{"className":19869},[102],[38,19871],{"className":19872,"style":109},[108],[38,19874,289],{"className":19875},[113],[38,19877],{"className":19878,"style":109},[108],[38,19880,19882,19885],{"className":19881},[93],[38,19883],{"className":19884,"style":2429},[97],[38,19886,19718],{"className":19887},[102],[11,19889,19890,19891,19942,19943,20013],{},"当 ",[38,19892,19894,19912],{"className":19893,"translate":42},[41],[38,19895,19897],{"className":19896},[46],[48,19898,19899],{"xmlns":50},[52,19900,19901,19909],{},[55,19902,19903,19905,19907],{},[58,19904,11],{},[62,19906,289],{},[300,19908,312],{},[81,19910,19911],{"encoding":83},"p=2",[38,19913,19915,19933],{"className":19914,"ariaHidden":89},[88],[38,19916,19918,19921,19924,19927,19930],{"className":19917},[93],[38,19919],{"className":19920,"style":3505},[97],[38,19922,11],{"className":19923},[102,103],[38,19925],{"className":19926,"style":109},[108],[38,19928,289],{"className":19929},[113],[38,19931],{"className":19932,"style":109},[108],[38,19934,19936,19939],{"className":19935},[93],[38,19937],{"className":19938,"style":2429},[97],[38,19940,312],{"className":19941},[102]," 时，",[38,19944,19946,19964],{"className":19945,"translate":42},[41],[38,19947,19949],{"className":19948},[46],[48,19950,19951],{"xmlns":50},[52,19952,19953,19961],{},[55,19954,19955],{},[295,19956,19957,19959],{},[58,19958,3104],{},[300,19960,312],{},[81,19962,19963],{"encoding":83},"L_2",[38,19965,19967],{"className":19966,"ariaHidden":89},[88],[38,19968,19970,19973],{"className":19969},[93],[38,19971],{"className":19972,"style":13610},[97],[38,19974,19976,19979],{"className":19975},[102],[38,19977,3104],{"className":19978},[102,103],[38,19980,19982],{"className":19981},[134],[38,19983,19985,20005],{"className":19984},[138,384],[38,19986,19988,20002],{"className":19987},[142],[38,19989,19991],{"className":19990,"style":391},[146],[38,19992,19993,19996],{"style":394},[38,19994],{"className":19995,"style":155},[154],[38,19997,19999],{"className":19998},[159,160,161,162],[38,20000,312],{"className":20001},[102,162],[38,20003,408],{"className":20004},[407],[38,20006,20008],{"className":20007},[142],[38,20009,20011],{"className":20010,"style":415},[146],[38,20012],{}," 范数下的距离定义为：",[38,20015,20017],{"className":20016,"translate":42},[621],[38,20018,20020,20113],{"className":20019,"translate":42},[41],[38,20021,20023],{"className":20022},[46],[48,20024,20025],{"xmlns":50,"display":630},[52,20026,20027,20110],{},[55,20028,20029,20031,20033,20039,20041,20068,20070],{},[58,20030,2183],{"mathvariant":333},[58,20032,286],{"mathvariant":285},[295,20034,20035,20037],{},[58,20036,2183],{"mathvariant":333},[300,20038,312],{},[62,20040,289],{},[20042,20043,20044],"msqrt",{},[55,20045,20046,20060],{},[15353,20047,20048,20050,20058],{},[62,20049,15357],{},[55,20051,20052,20054,20056],{},[58,20053,601],{},[62,20055,289],{},[300,20057,302],{},[58,20059,74],{},[8499,20061,20062,20064,20066],{},[58,20063,286],{},[58,20065,601],{},[300,20067,312],{},[62,20069,289],{},[20042,20071,20072],{},[55,20073,20074,20082,20084,20092,20094,20096,20098,20100,20102],{},[8499,20075,20076,20078,20080],{},[58,20077,286],{},[300,20079,302],{},[300,20081,312],{},[62,20083,696],{},[8499,20085,20086,20088,20090],{},[58,20087,286],{},[300,20089,312],{},[300,20091,312],{},[62,20093,696],{},[58,20095,2192],{"mathvariant":333},[58,20097,2192],{"mathvariant":333},[58,20099,2192],{"mathvariant":333},[62,20101,696],{},[8499,20103,20104,20106,20108],{},[58,20105,286],{},[58,20107,74],{},[300,20109,312],{},[81,20111,20112],{"encoding":83},"\\|\\mathbf{x}\\|_2 = \\sqrt{\\sum_{i=1}^{n} x_i^2} = \\sqrt{x_1^2 + x_2^2 + ... + x_n^2}",[38,20114,20116,20177,20374],{"className":20115,"ariaHidden":89},[88],[38,20117,20119,20122,20125,20128,20168,20171,20174],{"className":20118},[93],[38,20120],{"className":20121,"style":1439},[97],[38,20123,2183],{"className":20124},[102],[38,20126,286],{"className":20127},[102,351],[38,20129,20131,20134],{"className":20130},[102],[38,20132,2183],{"className":20133},[102],[38,20135,20137],{"className":20136},[134],[38,20138,20140,20160],{"className":20139},[138,384],[38,20141,20143,20157],{"className":20142},[142],[38,20144,20146],{"className":20145,"style":391},[146],[38,20147,20148,20151],{"style":394},[38,20149],{"className":20150,"style":155},[154],[38,20152,20154],{"className":20153},[159,160,161,162],[38,20155,312],{"className":20156},[102,162],[38,20158,408],{"className":20159},[407],[38,20161,20163],{"className":20162},[142],[38,20164,20166],{"className":20165,"style":415},[146],[38,20167],{},[38,20169],{"className":20170,"style":109},[108],[38,20172,289],{"className":20173},[113],[38,20175],{"className":20176,"style":109},[108],[38,20178,20180,20184,20365,20368,20371],{"className":20179},[93],[38,20181],{"className":20182,"style":20183},[97],"height:3.1568em;vertical-align:-1.2777em;",[38,20185,20188],{"className":20186},[102,20187],"sqrt",[38,20189,20191,20357],{"className":20190},[138,384],[38,20192,20194,20354],{"className":20193},[142],[38,20195,20198,20334],{"className":20196,"style":20197},[146],"height:1.8791em;",[38,20199,20203,20207],{"className":20200,"style":20202},[20201],"svg-align","top:-5.1168em;",[38,20204],{"className":20205,"style":20206},[154],"height:5.1168em;",[38,20208,20211,20278,20281],{"className":20209,"style":20210},[102],"padding-left:1.056em;",[38,20212,20214],{"className":20213},[878,879],[38,20215,20217,20270],{"className":20216},[138,384],[38,20218,20220,20267],{"className":20219},[142],[38,20221,20223,20243,20253],{"className":20222,"style":15488},[146],[38,20224,20225,20228],{"style":15491},[38,20226],{"className":20227,"style":8852},[154],[38,20229,20231],{"className":20230},[159,160,161,162],[38,20232,20234,20237,20240],{"className":20233},[102,162],[38,20235,601],{"className":20236},[102,103,162],[38,20238,289],{"className":20239},[113,162],[38,20241,302],{"className":20242},[102,162],[38,20244,20245,20248],{"style":15512},[38,20246],{"className":20247,"style":8852},[154],[38,20249,20250],{},[38,20251,15357],{"className":20252},[878,15521,15522],[38,20254,20255,20258],{"style":15525},[38,20256],{"className":20257,"style":8852},[154],[38,20259,20261],{"className":20260},[159,160,161,162],[38,20262,20264],{"className":20263},[102,162],[38,20265,74],{"className":20266},[102,103,162],[38,20268,408],{"className":20269},[407],[38,20271,20273],{"className":20272},[142],[38,20274,20276],{"className":20275,"style":15547},[146],[38,20277],{},[38,20279],{"className":20280,"style":425},[108],[38,20282,20284,20287],{"className":20283},[102],[38,20285,286],{"className":20286},[102,103],[38,20288,20290],{"className":20289},[134],[38,20291,20293,20325],{"className":20292},[138,384],[38,20294,20296,20322],{"className":20295},[142],[38,20297,20299,20311],{"className":20298,"style":8931},[146],[38,20300,20302,20305],{"style":20301},"top:-2.4231em;margin-left:0em;margin-right:0.05em;",[38,20303],{"className":20304,"style":155},[154],[38,20306,20308],{"className":20307},[159,160,161,162],[38,20309,601],{"className":20310},[102,103,162],[38,20312,20313,20316],{"style":8946},[38,20314],{"className":20315,"style":155},[154],[38,20317,20319],{"className":20318},[159,160,161,162],[38,20320,312],{"className":20321},[102,162],[38,20323,408],{"className":20324},[407],[38,20326,20328],{"className":20327},[142],[38,20329,20332],{"className":20330,"style":20331},[146],"height:0.2769em;",[38,20333],{},[38,20335,20337,20340],{"style":20336},"top:-3.8391em;",[38,20338],{"className":20339,"style":20206},[154],[38,20341,20345],{"className":20342,"style":20344},[20343],"hide-tail","min-width:0.742em;height:3.1968em;",[4842,20346,20351],{"xmlns":4844,"width":20347,"height":20348,"viewBox":20349,"preserveAspectRatio":20350},"400em","3.1968em","0 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80H400000v40H742z",[38,20355,408],{"className":20356},[407],[38,20358,20360],{"className":20359},[142],[38,20361,20363],{"className":20362,"style":15547},[146],[38,20364],{},[38,20366],{"className":20367,"style":109},[108],[38,20369,289],{"className":20370},[113],[38,20372],{"className":20373,"style":109},[108],[38,20375,20377,20381],{"className":20376},[93],[38,20378],{"className":20379,"style":20380},[97],"height:1.84em;vertical-align:-0.5413em;",[38,20382,20384],{"className":20383},[102,20187],[38,20385,20387,20609],{"className":20386},[138,384],[38,20388,20390,20606],{"className":20389},[142],[38,20391,20394,20589],{"className":20392,"style":20393},[146],"height:1.2987em;",[38,20395,20398,20402],{"className":20396,"style":20397},[20201],"top:-3.8em;",[38,20399],{"className":20400,"style":20401},[154],"height:3.8em;",[38,20403,20406,20457,20460,20463,20466,20517,20520,20523,20526,20529,20532,20535,20538],{"className":20404,"style":20405},[102],"padding-left:1em;",[38,20407,20409,20412],{"className":20408},[102],[38,20410,286],{"className":20411},[102,103],[38,20413,20415],{"className":20414},[134],[38,20416,20418,20449],{"className":20417},[138,384],[38,20419,20421,20446],{"className":20420},[142],[38,20422,20424,20435],{"className":20423,"style":8931},[146],[38,20425,20426,20429],{"style":8934},[38,20427],{"className":20428,"style":155},[154],[38,20430,20432],{"className":20431},[159,160,161,162],[38,20433,302],{"className":20434},[102,162],[38,20436,20437,20440],{"style":8946},[38,20438],{"className":20439,"style":155},[154],[38,20441,20443],{"className":20442},[159,160,161,162],[38,20444,312],{"className":20445},[102,162],[38,20447,408],{"className":20448},[407],[38,20450,20452],{"className":20451},[142],[38,20453,20455],{"className":20454,"style":8965},[146],[38,20456],{},[38,20458],{"className":20459,"style":995},[108],[38,20461,696],{"className":20462},[999],[38,20464],{"className":20465,"style":995},[108],[38,20467,20469,20472],{"className":20468},[102],[38,20470,286],{"className":20471},[102,103],[38,20473,20475],{"className":20474},[134],[38,20476,20478,20509],{"className":20477},[138,384],[38,20479,20481,20506],{"className":20480},[142],[38,20482,20484,20495],{"className":20483,"style":8931},[146],[38,20485,20486,20489],{"style":8934},[38,20487],{"className":20488,"style":155},[154],[38,20490,20492],{"className":20491},[159,160,161,162],[38,20493,312],{"className":20494},[102,162],[38,20496,20497,20500],{"style":8946},[38,20498],{"className":20499,"style":155},[154],[38,20501,20503],{"className":20502},[159,160,161,162],[38,20504,312],{"className":20505},[102,162],[38,20507,408],{"className":20508},[407],[38,20510,20512],{"className":20511},[142],[38,20513,20515],{"className":20514,"style":8965},[146],[38,20516],{},[38,20518],{"className":20519,"style":995},[108],[38,20521,696],{"className":20522},[999],[38,20524],{"className":20525,"style":995},[108],[38,20527,18370],{"className":20528},[102],[38,20530],{"className":20531,"style":995},[108],[38,20533,696],{"className":20534},[999],[38,20536],{"className":20537,"style":995},[108],[38,20539,20541,20544],{"className":20540},[102],[38,20542,286],{"className":20543},[102,103],[38,20545,20547],{"className":20546},[134],[38,20548,20550,20581],{"className":20549},[138,384],[38,20551,20553,20578],{"className":20552},[142],[38,20554,20556,20567],{"className":20555,"style":9582},[146],[38,20557,20558,20561],{"style":9585},[38,20559],{"className":20560,"style":155},[154],[38,20562,20564],{"className":20563},[159,160,161,162],[38,20565,74],{"className":20566},[102,103,162],[38,20568,20569,20572],{"style":9597},[38,20570],{"className":20571,"style":155},[154],[38,20573,20575],{"className":20574},[159,160,161,162],[38,20576,312],{"className":20577},[102,162],[38,20579,408],{"className":20580},[407],[38,20582,20584],{"className":20583},[142],[38,20585,20587],{"className":20586,"style":9616},[146],[38,20588],{},[38,20590,20592,20595],{"style":20591},"top:-3.2587em;",[38,20593],{"className":20594,"style":20401},[154],[38,20596,20599],{"className":20597,"style":20598},[20343],"min-width:1.02em;height:1.88em;",[4842,20600,20603],{"xmlns":4844,"width":20347,"height":20601,"viewBox":20602,"preserveAspectRatio":20350},"1.88em","0 0 400000 1944",[4849,20604],{"d":20605},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[38,20607,408],{"className":20608},[407],[38,20610,20612],{"className":20611},[142],[38,20613,20616],{"className":20614,"style":20615},[146],"height:0.5413em;",[38,20617],{},[11,20619,20620,20621,19595,20699,19665,20768,21139,21140,21242],{},"直观来看，它就是几何中从原点到该点的直线距离。例如，",[38,20622,20624,20651],{"className":20623,"translate":42},[41],[38,20625,20627],{"className":20626},[46],[48,20628,20629],{"xmlns":50},[52,20630,20631,20649],{},[55,20632,20633,20635,20637,20639,20641,20643,20645,20647],{},[58,20634,286],{"mathvariant":285},[62,20636,289],{},[62,20638,293],{"stretchy":292},[300,20640,13114],{},[62,20642,305],{"separator":89},[62,20644,710],{},[300,20646,19541],{},[62,20648,330],{"stretchy":292},[81,20650,19546],{"encoding":83},[38,20652,20654,20672],{"className":20653,"ariaHidden":89},[88],[38,20655,20657,20660,20663,20666,20669],{"className":20656},[93],[38,20658],{"className":20659,"style":347},[97],[38,20661,286],{"className":20662},[102,351],[38,20664],{"className":20665,"style":109},[108],[38,20667,289],{"className":20668},[113],[38,20670],{"className":20671,"style":109},[108],[38,20673,20675,20678,20681,20684,20687,20690,20693,20696],{"className":20674},[93],[38,20676],{"className":20677,"style":1439},[97],[38,20679,293],{"className":20680},[371],[38,20682,13114],{"className":20683},[102],[38,20685,305],{"className":20686},[421],[38,20688],{"className":20689,"style":425},[108],[38,20691,710],{"className":20692},[102],[38,20694,19541],{"className":20695},[102],[38,20697,330],{"className":20698},[529],[38,20700,20702,20719],{"className":20701,"translate":42},[41],[38,20703,20705],{"className":20704},[46],[48,20706,20707],{"xmlns":50},[52,20708,20709,20717],{},[55,20710,20711],{},[295,20712,20713,20715],{},[58,20714,3104],{},[300,20716,312],{},[81,20718,19963],{"encoding":83},[38,20720,20722],{"className":20721,"ariaHidden":89},[88],[38,20723,20725,20728],{"className":20724},[93],[38,20726],{"className":20727,"style":13610},[97],[38,20729,20731,20734],{"className":20730},[102],[38,20732,3104],{"className":20733},[102,103],[38,20735,20737],{"className":20736},[134],[38,20738,20740,20760],{"className":20739},[138,384],[38,20741,20743,20757],{"className":20742},[142],[38,20744,20746],{"className":20745,"style":391},[146],[38,20747,20748,20751],{"style":394},[38,20749],{"className":20750,"style":155},[154],[38,20752,20754],{"className":20753},[159,160,161,162],[38,20755,312],{"className":20756},[102,162],[38,20758,408],{"className":20759},[407],[38,20761,20763],{"className":20762},[142],[38,20764,20766],{"className":20765,"style":415},[146],[38,20767],{},[38,20769,20771,20838],{"className":20770,"translate":42},[41],[38,20772,20774],{"className":20773},[46],[48,20775,20776],{"xmlns":50},[52,20777,20778,20835],{},[55,20779,20780,20782,20784,20790,20792,20816,20818,20830,20832],{},[58,20781,2183],{"mathvariant":333},[58,20783,286],{"mathvariant":285},[295,20785,20786,20788],{},[58,20787,2183],{"mathvariant":333},[300,20789,312],{},[62,20791,289],{},[20042,20793,20794],{},[55,20795,20796,20802,20804,20806,20808,20810],{},[66,20797,20798,20800],{},[300,20799,13114],{},[300,20801,312],{},[62,20803,696],{},[62,20805,293],{"stretchy":292},[62,20807,710],{},[300,20809,19541],{},[66,20811,20812,20814],{},[62,20813,330],{"stretchy":292},[300,20815,312],{},[62,20817,289],{},[20042,20819,20820],{},[55,20821,20822,20825,20827],{},[300,20823,20824],{},"9",[62,20826,696],{},[300,20828,20829],{},"16",[62,20831,289],{},[300,20833,20834],{},"5",[81,20836,20837],{"encoding":83},"\\|\\mathbf{x}\\|_2 = \\sqrt{3^2 + (-4)^2} = \\sqrt{9+16} = 5",[38,20839,20841,20902,21047,21130],{"className":20840,"ariaHidden":89},[88],[38,20842,20844,20847,20850,20853,20893,20896,20899],{"className":20843},[93],[38,20845],{"className":20846,"style":1439},[97],[38,20848,2183],{"className":20849},[102],[38,20851,286],{"className":20852},[102,351],[38,20854,20856,20859],{"className":20855},[102],[38,20857,2183],{"className":20858},[102],[38,20860,20862],{"className":20861},[134],[38,20863,20865,20885],{"className":20864},[138,384],[38,20866,20868,20882],{"className":20867},[142],[38,20869,20871],{"className":20870,"style":391},[146],[38,20872,20873,20876],{"style":394},[38,20874],{"className":20875,"style":155},[154],[38,20877,20879],{"className":20878},[159,160,161,162],[38,20880,312],{"className":20881},[102,162],[38,20883,408],{"className":20884},[407],[38,20886,20888],{"className":20887},[142],[38,20889,20891],{"className":20890,"style":415},[146],[38,20892],{},[38,20894],{"className":20895,"style":109},[108],[38,20897,289],{"className":20898},[113],[38,20900],{"className":20901,"style":109},[108],[38,20903,20905,20909,21038,21041,21044],{"className":20904},[93],[38,20906],{"className":20907,"style":20908},[97],"height:1.24em;vertical-align:-0.305em;",[38,20910,20912],{"className":20911},[102,20187],[38,20913,20915,21029],{"className":20914},[138,384],[38,20916,20918,21026],{"className":20917},[142],[38,20919,20922,21009],{"className":20920,"style":20921},[146],"height:0.935em;",[38,20923,20926,20930],{"className":20924,"style":20925},[20201],"top:-3.2em;",[38,20927],{"className":20928,"style":20929},[154],"height:3.2em;",[38,20931,20933,20962,20965,20968,20971,20974,20977,20980],{"className":20932,"style":20405},[102],[38,20934,20936,20939],{"className":20935},[102],[38,20937,13114],{"className":20938},[102],[38,20940,20942],{"className":20941},[134],[38,20943,20945],{"className":20944},[138],[38,20946,20948],{"className":20947},[142],[38,20949,20951],{"className":20950,"style":9582},[146],[38,20952,20953,20956],{"style":9597},[38,20954],{"className":20955,"style":155},[154],[38,20957,20959],{"className":20958},[159,160,161,162],[38,20960,312],{"className":20961},[102,162],[38,20963],{"className":20964,"style":995},[108],[38,20966,696],{"className":20967},[999],[38,20969],{"className":20970,"style":995},[108],[38,20972,293],{"className":20973},[371],[38,20975,710],{"className":20976},[102],[38,20978,19541],{"className":20979},[102],[38,20981,20983,20986],{"className":20982},[529],[38,20984,330],{"className":20985},[529],[38,20987,20989],{"className":20988},[134],[38,20990,20992],{"className":20991},[138],[38,20993,20995],{"className":20994},[142],[38,20996,20998],{"className":20997,"style":9582},[146],[38,20999,21000,21003],{"style":9597},[38,21001],{"className":21002,"style":155},[154],[38,21004,21006],{"className":21005},[159,160,161,162],[38,21007,312],{"className":21008},[102,162],[38,21010,21012,21015],{"style":21011},"top:-2.895em;",[38,21013],{"className":21014,"style":20929},[154],[38,21016,21019],{"className":21017,"style":21018},[20343],"min-width:1.02em;height:1.28em;",[4842,21020,21023],{"xmlns":4844,"width":20347,"height":21021,"viewBox":21022,"preserveAspectRatio":20350},"1.28em","0 0 400000 1296",[4849,21024],{"d":21025},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[38,21027,408],{"className":21028},[407],[38,21030,21032],{"className":21031},[142],[38,21033,21036],{"className":21034,"style":21035},[146],"height:0.305em;",[38,21037],{},[38,21039],{"className":21040,"style":109},[108],[38,21042,289],{"className":21043},[113],[38,21045],{"className":21046,"style":109},[108],[38,21048,21050,21054,21121,21124,21127],{"className":21049},[93],[38,21051],{"className":21052,"style":21053},[97],"height:1.04em;vertical-align:-0.1744em;",[38,21055,21057],{"className":21056},[102,20187],[38,21058,21060,21112],{"className":21059},[138,384],[38,21061,21063,21109],{"className":21062},[142],[38,21064,21067,21092],{"className":21065,"style":21066},[146],"height:0.8656em;",[38,21068,21070,21073],{"className":21069,"style":913},[20201],[38,21071],{"className":21072,"style":760},[154],[38,21074,21077,21080,21083,21086,21089],{"className":21075,"style":21076},[102],"padding-left:0.833em;",[38,21078,20824],{"className":21079},[102],[38,21081],{"className":21082,"style":995},[108],[38,21084,696],{"className":21085},[999],[38,21087],{"className":21088,"style":995},[108],[38,21090,20829],{"className":21091},[102],[38,21093,21095,21098],{"style":21094},"top:-2.8256em;",[38,21096],{"className":21097,"style":760},[154],[38,21099,21102],{"className":21100,"style":21101},[20343],"min-width:0.853em;height:1.08em;",[4842,21103,21106],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},"1.08em","0 0 400000 1080",[4849,21107],{"d":21108},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[38,21110,408],{"className":21111},[407],[38,21113,21115],{"className":21114},[142],[38,21116,21119],{"className":21117,"style":21118},[146],"height:0.1744em;",[38,21120],{},[38,21122],{"className":21123,"style":109},[108],[38,21125,289],{"className":21126},[113],[38,21128],{"className":21129,"style":109},[108],[38,21131,21133,21136],{"className":21132},[93],[38,21134],{"className":21135,"style":2429},[97],[38,21137,20834],{"className":21138},[102],"。几何上，当 ",[38,21141,21143,21169],{"className":21142,"translate":42},[41],[38,21144,21146],{"className":21145},[46],[48,21147,21148],{"xmlns":50},[52,21149,21150,21166],{},[55,21151,21152,21154,21156,21162,21164],{},[58,21153,2183],{"mathvariant":333},[58,21155,286],{"mathvariant":285},[295,21157,21158,21160],{},[58,21159,2183],{"mathvariant":333},[300,21161,312],{},[62,21163,289],{},[300,21165,302],{},[81,21167,21168],{"encoding":83},"\\|\\mathbf{x}\\|_2 = 1",[38,21170,21172,21233],{"className":21171,"ariaHidden":89},[88],[38,21173,21175,21178,21181,21184,21224,21227,21230],{"className":21174},[93],[38,21176],{"className":21177,"style":1439},[97],[38,21179,2183],{"className":21180},[102],[38,21182,286],{"className":21183},[102,351],[38,21185,21187,21190],{"className":21186},[102],[38,21188,2183],{"className":21189},[102],[38,21191,21193],{"className":21192},[134],[38,21194,21196,21216],{"className":21195},[138,384],[38,21197,21199,21213],{"className":21198},[142],[38,21200,21202],{"className":21201,"style":391},[146],[38,21203,21204,21207],{"style":394},[38,21205],{"className":21206,"style":155},[154],[38,21208,21210],{"className":21209},[159,160,161,162],[38,21211,312],{"className":21212},[102,162],[38,21214,408],{"className":21215},[407],[38,21217,21219],{"className":21218},[142],[38,21220,21222],{"className":21221,"style":415},[146],[38,21223],{},[38,21225],{"className":21226,"style":109},[108],[38,21228,289],{"className":21229},[113],[38,21231],{"className":21232,"style":109},[108],[38,21234,21236,21239],{"className":21235},[93],[38,21237],{"className":21238,"style":2429},[97],[38,21240,302],{"className":21241},[102]," 时，向量落在单位圆上。",[11,21244,21245,21246,21315,21316,21385,21386,21455,21456,21484,21485,21531,21532,21584],{},"总而言之，",[38,21247,21249,21266],{"className":21248,"translate":42},[41],[38,21250,21252],{"className":21251},[46],[48,21253,21254],{"xmlns":50},[52,21255,21256,21264],{},[55,21257,21258],{},[295,21259,21260,21262],{},[58,21261,3104],{},[300,21263,302],{},[81,21265,18906],{"encoding":83},[38,21267,21269],{"className":21268,"ariaHidden":89},[88],[38,21270,21272,21275],{"className":21271},[93],[38,21273],{"className":21274,"style":13610},[97],[38,21276,21278,21281],{"className":21277},[102],[38,21279,3104],{"className":21280},[102,103],[38,21282,21284],{"className":21283},[134],[38,21285,21287,21307],{"className":21286},[138,384],[38,21288,21290,21304],{"className":21289},[142],[38,21291,21293],{"className":21292,"style":391},[146],[38,21294,21295,21298],{"style":394},[38,21296],{"className":21297,"style":155},[154],[38,21299,21301],{"className":21300},[159,160,161,162],[38,21302,302],{"className":21303},[102,162],[38,21305,408],{"className":21306},[407],[38,21308,21310],{"className":21309},[142],[38,21311,21313],{"className":21312,"style":415},[146],[38,21314],{}," 范数下的距离是各坐标绝对值之和（曼哈顿距离），会产生稀疏解。",[38,21317,21319,21336],{"className":21318,"translate":42},[41],[38,21320,21322],{"className":21321},[46],[48,21323,21324],{"xmlns":50},[52,21325,21326,21334],{},[55,21327,21328],{},[295,21329,21330,21332],{},[58,21331,3104],{},[300,21333,312],{},[81,21335,19963],{"encoding":83},[38,21337,21339],{"className":21338,"ariaHidden":89},[88],[38,21340,21342,21345],{"className":21341},[93],[38,21343],{"className":21344,"style":13610},[97],[38,21346,21348,21351],{"className":21347},[102],[38,21349,3104],{"className":21350},[102,103],[38,21352,21354],{"className":21353},[134],[38,21355,21357,21377],{"className":21356},[138,384],[38,21358,21360,21374],{"className":21359},[142],[38,21361,21363],{"className":21362,"style":391},[146],[38,21364,21365,21368],{"style":394},[38,21366],{"className":21367,"style":155},[154],[38,21369,21371],{"className":21370},[159,160,161,162],[38,21372,312],{"className":21373},[102,162],[38,21375,408],{"className":21376},[407],[38,21378,21380],{"className":21379},[142],[38,21381,21383],{"className":21382,"style":415},[146],[38,21384],{}," 范数是平方和的平方根（欧氏距离），光滑且易于优化。一般的 ",[38,21387,21389,21406],{"className":21388,"translate":42},[41],[38,21390,21392],{"className":21391},[46],[48,21393,21394],{"xmlns":50},[52,21395,21396,21404],{},[55,21397,21398],{},[295,21399,21400,21402],{},[58,21401,3104],{},[58,21403,11],{},[81,21405,18008],{"encoding":83},[38,21407,21409],{"className":21408,"ariaHidden":89},[88],[38,21410,21412,21415],{"className":21411},[93],[38,21413],{"className":21414,"style":13918},[97],[38,21416,21418,21421],{"className":21417},[102],[38,21419,3104],{"className":21420},[102,103],[38,21422,21424],{"className":21423},[134],[38,21425,21427,21447],{"className":21426},[138,384],[38,21428,21430,21444],{"className":21429},[142],[38,21431,21433],{"className":21432,"style":503},[146],[38,21434,21435,21438],{"style":394},[38,21436],{"className":21437,"style":155},[154],[38,21439,21441],{"className":21440},[159,160,161,162],[38,21442,11],{"className":21443},[102,103,162],[38,21445,408],{"className":21446},[407],[38,21448,21450],{"className":21449},[142],[38,21451,21453],{"className":21452,"style":7348},[146],[38,21454],{}," 公式是各分量 ",[38,21457,21459,21472],{"className":21458,"translate":42},[41],[38,21460,21462],{"className":21461},[46],[48,21463,21464],{"xmlns":50},[52,21465,21466,21470],{},[55,21467,21468],{},[58,21469,11],{},[81,21471,11],{"encoding":83},[38,21473,21475],{"className":21474,"ariaHidden":89},[88],[38,21476,21478,21481],{"className":21477},[93],[38,21479],{"className":21480,"style":3505},[97],[38,21482,11],{"className":21483},[102,103]," 次幂之和的 ",[38,21486,21488,21510],{"className":21487,"translate":42},[41],[38,21489,21491],{"className":21490},[46],[48,21492,21493],{"xmlns":50},[52,21494,21495,21507],{},[55,21496,21497,21499,21501,21503,21505],{},[62,21498,293],{"stretchy":292},[300,21500,302],{},[58,21502,18490],{"mathvariant":333},[58,21504,11],{},[62,21506,330],{"stretchy":292},[81,21508,21509],{"encoding":83},"(1\u002Fp)",[38,21511,21513],{"className":21512,"ariaHidden":89},[88],[38,21514,21516,21519,21522,21525,21528],{"className":21515},[93],[38,21517],{"className":21518,"style":1439},[97],[38,21520,293],{"className":21521},[371],[38,21523,18766],{"className":21524},[102],[38,21526,11],{"className":21527},[102,103],[38,21529,330],{"className":21530},[529]," 次方，当 ",[38,21533,21535,21554],{"className":21534,"translate":42},[41],[38,21536,21538],{"className":21537},[46],[48,21539,21540],{"xmlns":50},[52,21541,21542,21551],{},[55,21543,21544,21546,21548],{},[58,21545,11],{},[62,21547,77],{},[58,21549,21550],{"mathvariant":333},"∞",[81,21552,21553],{"encoding":83},"p \\to \\infty",[38,21555,21557,21575],{"className":21556,"ariaHidden":89},[88],[38,21558,21560,21563,21566,21569,21572],{"className":21559},[93],[38,21561],{"className":21562,"style":3505},[97],[38,21564,11],{"className":21565},[102,103],[38,21567],{"className":21568,"style":109},[108],[38,21570,77],{"className":21571},[113],[38,21573],{"className":21574,"style":109},[108],[38,21576,21578,21581],{"className":21577},[93],[38,21579],{"className":21580,"style":4312},[97],[38,21582,21550],{"className":21583},[102]," 时趋近于最大值。",[28,21586,21587],{"id":21587},"逐坐标中位数原型",[11,21589,21590],{},"逐坐标中位数原型（Coordinate-wise Median Prototype）指的是：对于多维数据集，分别在每个维度（坐标）上计算中位数，然后将这些中位数组合成新向量，作为整个数据集的「原型」或中心代表。",[11,21592,21593,21594,14757,21622,16420,21650,21843],{},"假设有 ",[38,21595,21597,21610],{"className":21596,"translate":42},[41],[38,21598,21600],{"className":21599},[46],[48,21601,21602],{"xmlns":50},[52,21603,21604,21608],{},[55,21605,21606],{},[58,21607,3555],{},[81,21609,3555],{"encoding":83},[38,21611,21613],{"className":21612,"ariaHidden":89},[88],[38,21614,21616,21619],{"className":21615},[93],[38,21617],{"className":21618,"style":4312},[97],[38,21620,3555],{"className":21621},[102,103],[38,21623,21625,21638],{"className":21624,"translate":42},[41],[38,21626,21628],{"className":21627},[46],[48,21629,21630],{"xmlns":50},[52,21631,21632,21636],{},[55,21633,21634],{},[58,21635,74],{},[81,21637,74],{"encoding":83},[38,21639,21641],{"className":21640,"ariaHidden":89},[88],[38,21642,21644,21647],{"className":21643},[93],[38,21645],{"className":21646,"style":4312},[97],[38,21648,74],{"className":21649},[102,103],[38,21651,21653,21690],{"className":21652,"translate":42},[41],[38,21654,21656],{"className":21655},[46],[48,21657,21658],{"xmlns":50},[52,21659,21660,21688],{},[55,21661,21662,21668,21670,21676,21678,21680,21682],{},[295,21663,21664,21666],{},[58,21665,14806],{"mathvariant":285},[300,21667,302],{},[62,21669,305],{"separator":89},[295,21671,21672,21674],{},[58,21673,14806],{"mathvariant":285},[300,21675,312],{},[62,21677,305],{"separator":89},[62,21679,317],{},[62,21681,305],{"separator":89},[295,21683,21684,21686],{},[58,21685,14806],{"mathvariant":285},[58,21687,3555],{},[81,21689,16461],{"encoding":83},[38,21691,21693],{"className":21692,"ariaHidden":89},[88],[38,21694,21696,21699,21739,21742,21745,21785,21788,21791,21794,21797,21800,21803],{"className":21695},[93],[38,21697],{"className":21698,"style":6685},[97],[38,21700,21702,21705],{"className":21701},[102],[38,21703,14806],{"className":21704,"style":6689},[102,351],[38,21706,21708],{"className":21707},[134],[38,21709,21711,21731],{"className":21710},[138,384],[38,21712,21714,21728],{"className":21713},[142],[38,21715,21717],{"className":21716,"style":391},[146],[38,21718,21719,21722],{"style":14861},[38,21720],{"className":21721,"style":155},[154],[38,21723,21725],{"className":21724},[159,160,161,162],[38,21726,302],{"className":21727},[102,162],[38,21729,408],{"className":21730},[407],[38,21732,21734],{"className":21733},[142],[38,21735,21737],{"className":21736,"style":415},[146],[38,21738],{},[38,21740,305],{"className":21741},[421],[38,21743],{"className":21744,"style":425},[108],[38,21746,21748,21751],{"className":21747},[102],[38,21749,14806],{"className":21750,"style":6689},[102,351],[38,21752,21754],{"className":21753},[134],[38,21755,21757,21777],{"className":21756},[138,384],[38,21758,21760,21774],{"className":21759},[142],[38,21761,21763],{"className":21762,"style":391},[146],[38,21764,21765,21768],{"style":14861},[38,21766],{"className":21767,"style":155},[154],[38,21769,21771],{"className":21770},[159,160,161,162],[38,21772,312],{"className":21773},[102,162],[38,21775,408],{"className":21776},[407],[38,21778,21780],{"className":21779},[142],[38,21781,21783],{"className":21782,"style":415},[146],[38,21784],{},[38,21786,305],{"className":21787},[421],[38,21789],{"className":21790,"style":425},[108],[38,21792,317],{"className":21793},[475],[38,21795],{"className":21796,"style":425},[108],[38,21798,305],{"className":21799},[421],[38,21801],{"className":21802,"style":425},[108],[38,21804,21806,21809],{"className":21805},[102],[38,21807,14806],{"className":21808,"style":6689},[102,351],[38,21810,21812],{"className":21811},[134],[38,21813,21815,21835],{"className":21814},[138,384],[38,21816,21818,21832],{"className":21817},[142],[38,21819,21821],{"className":21820,"style":503},[146],[38,21822,21823,21826],{"style":14861},[38,21824],{"className":21825,"style":155},[154],[38,21827,21829],{"className":21828},[159,160,161,162],[38,21830,3555],{"className":21831},[102,103,162],[38,21833,408],{"className":21834},[407],[38,21836,21838],{"className":21837},[142],[38,21839,21841],{"className":21840,"style":415},[146],[38,21842],{},"，其中",[38,21845,21847],{"className":21846,"translate":42},[621],[38,21848,21850,21911],{"className":21849,"translate":42},[41],[38,21851,21853],{"className":21852},[46],[48,21854,21855],{"xmlns":50,"display":630},[52,21856,21857,21909],{},[55,21858,21859,21865,21867,21869,21879,21881,21891,21893,21895,21897,21907],{},[295,21860,21861,21863],{},[58,21862,14806],{"mathvariant":285},[58,21864,601],{},[62,21866,289],{},[62,21868,293],{"stretchy":292},[295,21870,21871,21873],{},[58,21872,14806],{},[55,21874,21875,21877],{},[58,21876,601],{},[300,21878,302],{},[62,21880,305],{"separator":89},[295,21882,21883,21885],{},[58,21884,14806],{},[55,21886,21887,21889],{},[58,21888,601],{},[300,21890,312],{},[62,21892,305],{"separator":89},[62,21894,317],{},[62,21896,305],{"separator":89},[295,21898,21899,21901],{},[58,21900,14806],{},[55,21902,21903,21905],{},[58,21904,601],{},[58,21906,74],{},[62,21908,330],{"stretchy":292},[81,21910,15053],{"encoding":83},[38,21912,21914,21969],{"className":21913,"ariaHidden":89},[88],[38,21915,21917,21920,21960,21963,21966],{"className":21916},[93],[38,21918],{"className":21919,"style":1148},[97],[38,21921,21923,21926],{"className":21922},[102],[38,21924,14806],{"className":21925,"style":6689},[102,351],[38,21927,21929],{"className":21928},[134],[38,21930,21932,21952],{"className":21931},[138,384],[38,21933,21935,21949],{"className":21934},[142],[38,21936,21938],{"className":21937,"style":786},[146],[38,21939,21940,21943],{"style":14861},[38,21941],{"className":21942,"style":155},[154],[38,21944,21946],{"className":21945},[159,160,161,162],[38,21947,601],{"className":21948},[102,103,162],[38,21950,408],{"className":21951},[407],[38,21953,21955],{"className":21954},[142],[38,21956,21958],{"className":21957,"style":415},[146],[38,21959],{},[38,21961],{"className":21962,"style":109},[108],[38,21964,289],{"className":21965},[113],[38,21967],{"className":21968,"style":109},[108],[38,21970,21972,21975,21978,22024,22027,22030,22076,22079,22082,22085,22088,22091,22094,22137],{"className":21971},[93],[38,21973],{"className":21974,"style":1439},[97],[38,21976,293],{"className":21977},[371],[38,21979,21981,21984],{"className":21980},[102],[38,21982,14806],{"className":21983,"style":3420},[102,103],[38,21985,21987],{"className":21986},[134],[38,21988,21990,22016],{"className":21989},[138,384],[38,21991,21993,22013],{"className":21992},[142],[38,21994,21996],{"className":21995,"style":786},[146],[38,21997,21998,22001],{"style":15141},[38,21999],{"className":22000,"style":155},[154],[38,22002,22004],{"className":22003},[159,160,161,162],[38,22005,22007,22010],{"className":22006},[102,162],[38,22008,601],{"className":22009},[102,103,162],[38,22011,302],{"className":22012},[102,162],[38,22014,408],{"className":22015},[407],[38,22017,22019],{"className":22018},[142],[38,22020,22022],{"className":22021,"style":415},[146],[38,22023],{},[38,22025,305],{"className":22026},[421],[38,22028],{"className":22029,"style":425},[108],[38,22031,22033,22036],{"className":22032},[102],[38,22034,14806],{"className":22035,"style":3420},[102,103],[38,22037,22039],{"className":22038},[134],[38,22040,22042,22068],{"className":22041},[138,384],[38,22043,22045,22065],{"className":22044},[142],[38,22046,22048],{"className":22047,"style":786},[146],[38,22049,22050,22053],{"style":15141},[38,22051],{"className":22052,"style":155},[154],[38,22054,22056],{"className":22055},[159,160,161,162],[38,22057,22059,22062],{"className":22058},[102,162],[38,22060,601],{"className":22061},[102,103,162],[38,22063,312],{"className":22064},[102,162],[38,22066,408],{"className":22067},[407],[38,22069,22071],{"className":22070},[142],[38,22072,22074],{"className":22073,"style":415},[146],[38,22075],{},[38,22077,305],{"className":22078},[421],[38,22080],{"className":22081,"style":425},[108],[38,22083,317],{"className":22084},[475],[38,22086],{"className":22087,"style":425},[108],[38,22089,305],{"className":22090},[421],[38,22092],{"className":22093,"style":425},[108],[38,22095,22097,22100],{"className":22096},[102],[38,22098,14806],{"className":22099,"style":3420},[102,103],[38,22101,22103],{"className":22102},[134],[38,22104,22106,22129],{"className":22105},[138,384],[38,22107,22109,22126],{"className":22108},[142],[38,22110,22112],{"className":22111,"style":786},[146],[38,22113,22114,22117],{"style":15141},[38,22115],{"className":22116,"style":155},[154],[38,22118,22120],{"className":22119},[159,160,161,162],[38,22121,22123],{"className":22122},[102,162],[38,22124,15270],{"className":22125},[102,103,162],[38,22127,408],{"className":22128},[407],[38,22130,22132],{"className":22131},[142],[38,22133,22135],{"className":22134,"style":415},[146],[38,22136],{},[38,22138,330],{"className":22139},[529],[11,22141,22142,22143,22369],{},"逐坐标中位数原型 ",[38,22144,22146,22192],{"className":22145,"translate":42},[41],[38,22147,22149],{"className":22148},[46],[48,22150,22151],{"xmlns":50},[52,22152,22153,22189],{},[55,22154,22155,22157,22159,22161,22167,22169,22175,22177,22179,22181,22187],{},[58,22156,11],{"mathvariant":285},[62,22158,289],{},[62,22160,293],{"stretchy":292},[295,22162,22163,22165],{},[58,22164,11],{},[300,22166,302],{},[62,22168,305],{"separator":89},[295,22170,22171,22173],{},[58,22172,11],{},[300,22174,312],{},[62,22176,305],{"separator":89},[62,22178,317],{},[62,22180,305],{"separator":89},[295,22182,22183,22185],{},[58,22184,11],{},[58,22186,74],{},[62,22188,330],{"stretchy":292},[81,22190,22191],{"encoding":83},"\\mathbf{p} = (p_1, p_2, \\dots, p_n)",[38,22193,22195,22213],{"className":22194,"ariaHidden":89},[88],[38,22196,22198,22201,22204,22207,22210],{"className":22197},[93],[38,22199],{"className":22200,"style":6685},[97],[38,22202,11],{"className":22203},[102,351],[38,22205],{"className":22206,"style":109},[108],[38,22208,289],{"className":22209},[113],[38,22211],{"className":22212,"style":109},[108],[38,22214,22216,22219,22222,22262,22265,22268,22308,22311,22314,22317,22320,22323,22326,22366],{"className":22215},[93],[38,22217],{"className":22218,"style":1439},[97],[38,22220,293],{"className":22221},[371],[38,22223,22225,22228],{"className":22224},[102],[38,22226,11],{"className":22227},[102,103],[38,22229,22231],{"className":22230},[134],[38,22232,22234,22254],{"className":22233},[138,384],[38,22235,22237,22251],{"className":22236},[142],[38,22238,22240],{"className":22239,"style":391},[146],[38,22241,22242,22245],{"style":394},[38,22243],{"className":22244,"style":155},[154],[38,22246,22248],{"className":22247},[159,160,161,162],[38,22249,302],{"className":22250},[102,162],[38,22252,408],{"className":22253},[407],[38,22255,22257],{"className":22256},[142],[38,22258,22260],{"className":22259,"style":415},[146],[38,22261],{},[38,22263,305],{"className":22264},[421],[38,22266],{"className":22267,"style":425},[108],[38,22269,22271,22274],{"className":22270},[102],[38,22272,11],{"className":22273},[102,103],[38,22275,22277],{"className":22276},[134],[38,22278,22280,22300],{"className":22279},[138,384],[38,22281,22283,22297],{"className":22282},[142],[38,22284,22286],{"className":22285,"style":391},[146],[38,22287,22288,22291],{"style":394},[38,22289],{"className":22290,"style":155},[154],[38,22292,22294],{"className":22293},[159,160,161,162],[38,22295,312],{"className":22296},[102,162],[38,22298,408],{"className":22299},[407],[38,22301,22303],{"className":22302},[142],[38,22304,22306],{"className":22305,"style":415},[146],[38,22307],{},[38,22309,305],{"className":22310},[421],[38,22312],{"className":22313,"style":425},[108],[38,22315,317],{"className":22316},[475],[38,22318],{"className":22319,"style":425},[108],[38,22321,305],{"className":22322},[421],[38,22324],{"className":22325,"style":425},[108],[38,22327,22329,22332],{"className":22328},[102],[38,22330,11],{"className":22331},[102,103],[38,22333,22335],{"className":22334},[134],[38,22336,22338,22358],{"className":22337},[138,384],[38,22339,22341,22355],{"className":22340},[142],[38,22342,22344],{"className":22343,"style":503},[146],[38,22345,22346,22349],{"style":394},[38,22347],{"className":22348,"style":155},[154],[38,22350,22352],{"className":22351},[159,160,161,162],[38,22353,74],{"className":22354},[102,103,162],[38,22356,408],{"className":22357},[407],[38,22359,22361],{"className":22360},[142],[38,22362,22364],{"className":22363,"style":415},[146],[38,22365],{},[38,22367,330],{"className":22368},[529]," 定义为：",[38,22371,22373],{"className":22372,"translate":42},[621],[38,22374,22376,22463],{"className":22375,"translate":42},[41],[38,22377,22379],{"className":22378},[46],[48,22380,22381],{"xmlns":50,"display":630},[52,22382,22383,22460],{},[55,22384,22385,22391,22393,22396,22398,22408,22410,22420,22422,22424,22426,22436,22438,22440,22442,22444,22446,22448,22450,22452,22454,22456,22458],{},[295,22386,22387,22389],{},[58,22388,11],{},[58,22390,10308],{},[62,22392,289],{},[3265,22394,22395],{},"中位数",[62,22397,293],{"stretchy":292},[295,22399,22400,22402],{},[58,22401,14806],{},[55,22403,22404,22406],{},[300,22405,302],{},[58,22407,10308],{},[62,22409,305],{"separator":89},[295,22411,22412,22414],{},[58,22413,14806],{},[55,22415,22416,22418],{},[300,22417,312],{},[58,22419,10308],{},[62,22421,305],{"separator":89},[62,22423,317],{},[62,22425,305],{"separator":89},[295,22427,22428,22430],{},[58,22429,14806],{},[55,22431,22432,22434],{},[58,22433,3555],{},[58,22435,10308],{},[62,22437,330],{"stretchy":292},[62,22439,305],{"separator":89},[108,22441],{"width":4479},[58,22443,10308],{},[62,22445,289],{},[300,22447,302],{},[62,22449,305],{"separator":89},[300,22451,312],{},[62,22453,305],{"separator":89},[62,22455,317],{},[62,22457,305],{"separator":89},[58,22459,74],{},[81,22461,22462],{"encoding":83},"p_j = \\text{中位数}(v_{1j}, v_{2j}, \\dots, v_{mj}), \\quad j = 1, 2, \\dots, n",[38,22464,22466,22522,22723],{"className":22465,"ariaHidden":89},[88],[38,22467,22469,22473,22513,22516,22519],{"className":22468},[93],[38,22470],{"className":22471,"style":22472},[97],"height:0.7167em;vertical-align:-0.2861em;",[38,22474,22476,22479],{"className":22475},[102],[38,22477,11],{"className":22478},[102,103],[38,22480,22482],{"className":22481},[134],[38,22483,22485,22505],{"className":22484},[138,384],[38,22486,22488,22502],{"className":22487},[142],[38,22489,22491],{"className":22490,"style":786},[146],[38,22492,22493,22496],{"style":394},[38,22494],{"className":22495,"style":155},[154],[38,22497,22499],{"className":22498},[159,160,161,162],[38,22500,10308],{"className":22501,"style":10470},[102,103,162],[38,22503,408],{"className":22504},[407],[38,22506,22508],{"className":22507},[142],[38,22509,22511],{"className":22510,"style":7348},[146],[38,22512],{},[38,22514],{"className":22515,"style":109},[108],[38,22517,289],{"className":22518},[113],[38,22520],{"className":22521,"style":109},[108],[38,22523,22525,22528,22536,22539,22585,22588,22591,22637,22640,22643,22646,22649,22652,22655,22699,22702,22705,22708,22711,22714,22717,22720],{"className":22524},[93],[38,22526],{"className":22527,"style":8669},[97],[38,22529,22532],{"className":22530},[102,22531],"text",[38,22533,22395],{"className":22534},[102,22535],"cjk_fallback",[38,22537,293],{"className":22538},[371],[38,22540,22542,22545],{"className":22541},[102],[38,22543,14806],{"className":22544,"style":3420},[102,103],[38,22546,22548],{"className":22547},[134],[38,22549,22551,22577],{"className":22550},[138,384],[38,22552,22554,22574],{"className":22553},[142],[38,22555,22557],{"className":22556,"style":786},[146],[38,22558,22559,22562],{"style":15141},[38,22560],{"className":22561,"style":155},[154],[38,22563,22565],{"className":22564},[159,160,161,162],[38,22566,22568,22571],{"className":22567},[102,162],[38,22569,302],{"className":22570},[102,162],[38,22572,10308],{"className":22573,"style":10470},[102,103,162],[38,22575,408],{"className":22576},[407],[38,22578,22580],{"className":22579},[142],[38,22581,22583],{"className":22582,"style":7348},[146],[38,22584],{},[38,22586,305],{"className":22587},[421],[38,22589],{"className":22590,"style":425},[108],[38,22592,22594,22597],{"className":22593},[102],[38,22595,14806],{"className":22596,"style":3420},[102,103],[38,22598,22600],{"className":22599},[134],[38,22601,22603,22629],{"className":22602},[138,384],[38,22604,22606,22626],{"className":22605},[142],[38,22607,22609],{"className":22608,"style":786},[146],[38,22610,22611,22614],{"style":15141},[38,22612],{"className":22613,"style":155},[154],[38,22615,22617],{"className":22616},[159,160,161,162],[38,22618,22620,22623],{"className":22619},[102,162],[38,22621,312],{"className":22622},[102,162],[38,22624,10308],{"className":22625,"style":10470},[102,103,162],[38,22627,408],{"className":22628},[407],[38,22630,22632],{"className":22631},[142],[38,22633,22635],{"className":22634,"style":7348},[146],[38,22636],{},[38,22638,305],{"className":22639},[421],[38,22641],{"className":22642,"style":425},[108],[38,22644,317],{"className":22645},[475],[38,22647],{"className":22648,"style":425},[108],[38,22650,305],{"className":22651},[421],[38,22653],{"className":22654,"style":425},[108],[38,22656,22658,22661],{"className":22657},[102],[38,22659,14806],{"className":22660,"style":3420},[102,103],[38,22662,22664],{"className":22663},[134],[38,22665,22667,22691],{"className":22666},[138,384],[38,22668,22670,22688],{"className":22669},[142],[38,22671,22673],{"className":22672,"style":786},[146],[38,22674,22675,22678],{"style":15141},[38,22676],{"className":22677,"style":155},[154],[38,22679,22681],{"className":22680},[159,160,161,162],[38,22682,22684],{"className":22683},[102,162],[38,22685,22687],{"className":22686,"style":10470},[102,103,162],"mj",[38,22689,408],{"className":22690},[407],[38,22692,22694],{"className":22693},[142],[38,22695,22697],{"className":22696,"style":7348},[146],[38,22698],{},[38,22700,330],{"className":22701},[529],[38,22703,305],{"className":22704},[421],[38,22706],{"className":22707,"style":10740},[108],[38,22709],{"className":22710,"style":425},[108],[38,22712,10308],{"className":22713,"style":10470},[102,103],[38,22715],{"className":22716,"style":109},[108],[38,22718,289],{"className":22719},[113],[38,22721],{"className":22722,"style":109},[108],[38,22724,22726,22730,22733,22736,22739,22742,22745,22748,22751,22754,22757,22760],{"className":22725},[93],[38,22727],{"className":22728,"style":22729},[97],"height:0.8389em;vertical-align:-0.1944em;",[38,22731,302],{"className":22732},[102],[38,22734,305],{"className":22735},[421],[38,22737],{"className":22738,"style":425},[108],[38,22740,312],{"className":22741},[102],[38,22743,305],{"className":22744},[421],[38,22746],{"className":22747,"style":425},[108],[38,22749,317],{"className":22750},[475],[38,22752],{"className":22753,"style":425},[108],[38,22755,305],{"className":22756},[421],[38,22758],{"className":22759,"style":425},[108],[38,22761,74],{"className":22762},[102,103],[11,22764,22765],{},"也就是说，每个分量独立地取所有数据点在该维度上的中位数。",[11,22767,22768],{},"如果旋转 45 度，逐坐标中位数的结果会发生变化（不具有旋转等变性），而几何中位数则会相应旋转。",[11,22770,22771,22772,22838,22839,22883],{},"在计算上，它只需对每个维度排序 ",[38,22773,22775,22802],{"className":22774,"translate":42},[41],[38,22776,22778],{"className":22777},[46],[48,22779,22780],{"xmlns":50},[52,22781,22782,22799],{},[55,22783,22784,22786,22788,22790,22793,22795,22797],{},[58,22785,13105],{},[62,22787,293],{"stretchy":292},[58,22789,3555],{},[58,22791,22792],{},"log",[62,22794,673],{},[58,22796,3555],{},[62,22798,330],{"stretchy":292},[81,22800,22801],{"encoding":83},"O(m \\log m)",[38,22803,22805],{"className":22804,"ariaHidden":89},[88],[38,22806,22808,22811,22814,22817,22820,22823,22829,22832,22835],{"className":22807},[93],[38,22809],{"className":22810,"style":1439},[97],[38,22812,13105],{"className":22813,"style":2500},[102,103],[38,22815,293],{"className":22816},[371],[38,22818,3555],{"className":22819},[102,103],[38,22821],{"className":22822,"style":425},[108],[38,22824,22826,22827],{"className":22825},[878],"lo",[38,22828,6656],{"style":16759},[38,22830],{"className":22831,"style":425},[108],[38,22833,3555],{"className":22834},[102,103],[38,22836,330],{"className":22837},[529]," 或使用选择算法 ",[38,22840,22842,22862],{"className":22841,"translate":42},[41],[38,22843,22845],{"className":22844},[46],[48,22846,22847],{"xmlns":50},[52,22848,22849,22859],{},[55,22850,22851,22853,22855,22857],{},[58,22852,13105],{},[62,22854,293],{"stretchy":292},[58,22856,3555],{},[62,22858,330],{"stretchy":292},[81,22860,22861],{"encoding":83},"O(m)",[38,22863,22865],{"className":22864,"ariaHidden":89},[88],[38,22866,22868,22871,22874,22877,22880],{"className":22867},[93],[38,22869],{"className":22870,"style":1439},[97],[38,22872,13105],{"className":22873,"style":2500},[102,103],[38,22875,293],{"className":22876},[371],[38,22878,3555],{"className":22879},[102,103],[38,22881,330],{"className":22882},[529],"，并且最多可以抵抗 50% 的任意极端异常值。",[11,22885,22886],{},"然而，坐标轴的选择会影响结果。如果数据存在相关性（例如倾斜的椭圆），逐坐标中位数可能会完全偏离真实的中心。它必然落在每个维度的取值范围之内，因此位于数据沿坐标轴的包围盒内，但不一定位于数据点的凸包内（因为每个维度都是独立选取的）。在机器学习中，它缺乏全局优化目标，不同于中位数或几何中位数那样最小化某种距离之和。它不适合用于目标函数。",[11,22888,22889],{},"当数据存在异常值且各维度相对独立时，它对异常值具有稳健性。",[11,22891,22892,22893,22944],{},"例如，在三维空间中，我们有一个 ",[38,22894,22896,22914],{"className":22895,"translate":42},[41],[38,22897,22899],{"className":22898},[46],[48,22900,22901],{"xmlns":50},[52,22902,22903,22911],{},[55,22904,22905,22907,22909],{},[300,22906,19541],{},[62,22908,4402],{},[300,22910,13114],{},[81,22912,22913],{"encoding":83},"4 \\times 3",[38,22915,22917,22935],{"className":22916,"ariaHidden":89},[88],[38,22918,22920,22923,22926,22929,22932],{"className":22919},[93],[38,22921],{"className":22922,"style":19848},[97],[38,22924,19541],{"className":22925},[102],[38,22927],{"className":22928,"style":995},[108],[38,22930,4402],{"className":22931},[999],[38,22933],{"className":22934,"style":995},[108],[38,22936,22938,22941],{"className":22937},[93],[38,22939],{"className":22940,"style":2429},[97],[38,22942,13114],{"className":22943},[102]," 的向量矩阵：",[38,22946,22948],{"className":22947,"translate":42},[621],[38,22949,22951,23054],{"className":22950,"translate":42},[41],[38,22952,22954],{"className":22953},[46],[48,22955,22956],{"xmlns":50,"display":630},[52,22957,22958,23051],{},[55,22959,22960,22962,23049],{},[62,22961,293],{"fence":89},[4475,22963,22965,22991,23019],{"rowspacing":4477,"columnalign":22964,"columnspacing":4479},"center center center center",[4481,22966,22967,22973,22979,22985],{},[4484,22968,22969],{},[2631,22970,22971],{"scriptlevel":683,"displaystyle":292},[300,22972,20834],{},[4484,22974,22975],{},[2631,22976,22977],{"scriptlevel":683,"displaystyle":292},[300,22978,13114],{},[4484,22980,22981],{},[2631,22982,22983],{"scriptlevel":683,"displaystyle":292},[300,22984,19718],{},[4484,22986,22987],{},[2631,22988,22989],{"scriptlevel":683,"displaystyle":292},[300,22990,19541],{},[4481,22992,22993,23000,23006,23013],{},[4484,22994,22995],{},[2631,22996,22997],{"scriptlevel":683,"displaystyle":292},[300,22998,22999],{},"8",[4484,23001,23002],{},[2631,23003,23004],{"scriptlevel":683,"displaystyle":292},[300,23005,20834],{},[4484,23007,23008],{},[2631,23009,23010],{"scriptlevel":683,"displaystyle":292},[300,23011,23012],{},"6",[4484,23014,23015],{},[2631,23016,23017],{"scriptlevel":683,"displaystyle":292},[300,23018,19718],{},[4481,23020,23021,23028,23035,23042],{},[4484,23022,23023],{},[2631,23024,23025],{"scriptlevel":683,"displaystyle":292},[300,23026,23027],{},"12",[4484,23029,23030],{},[2631,23031,23032],{"scriptlevel":683,"displaystyle":292},[300,23033,23034],{},"15",[4484,23036,23037],{},[2631,23038,23039],{"scriptlevel":683,"displaystyle":292},[300,23040,23041],{},"10",[4484,23043,23044],{},[2631,23045,23046],{"scriptlevel":683,"displaystyle":292},[300,23047,23048],{},"13",[62,23050,330],{"fence":89},[81,23052,23053],{"encoding":83},"\\begin{pmatrix}\n5 & 3 & 7 & 4 \\\\\n8 & 5 & 6 & 7 \\\\\n12 & 15 & 10 & 13\n\\end{pmatrix}",[38,23055,23057],{"className":23056,"ariaHidden":89},[88],[38,23058,23060,23064],{"className":23059},[93],[38,23061],{"className":23062,"style":23063},[97],"height:3.6em;vertical-align:-1.55em;",[38,23065,23067,23112,23363],{"className":23066},[475],[38,23068,23070],{"className":23069},[371],[38,23071,23073],{"className":23072},[1485,4820],[38,23074,23076,23103],{"className":23075},[138,384],[38,23077,23079,23100],{"className":23078},[142],[38,23080,23083],{"className":23081,"style":23082},[146],"height:2.05em;",[38,23084,23086,23090],{"style":23085},"top:-4.05em;",[38,23087],{"className":23088,"style":23089},[154],"height:5.6em;",[38,23091,23093],{"style":23092},"width:0.875em;height:3.6em;",[4842,23094,23097],{"xmlns":4844,"width":4845,"height":23095,"viewBox":23096},"3.6em","0 0 875 3600",[4849,23098],{"d":23099},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z",[38,23101,408],{"className":23102},[407],[38,23104,23106],{"className":23105},[142],[38,23107,23110],{"className":23108,"style":23109},[146],"height:1.55em;",[38,23111],{},[38,23113,23115],{"className":23114},[102],[38,23116,23118,23177,23180,23183,23239,23242,23245,23301,23304,23307],{"className":23117},[4475],[38,23119,23121],{"className":23120},[4873],[38,23122,23124,23169],{"className":23123},[138,384],[38,23125,23127,23166],{"className":23126},[142],[38,23128,23130,23142,23154],{"className":23129,"style":23082},[146],[38,23131,23133,23136],{"style":23132},"top:-4.21em;",[38,23134],{"className":23135,"style":760},[154],[38,23137,23139],{"className":23138},[102],[38,23140,20834],{"className":23141},[102],[38,23143,23145,23148],{"style":23144},"top:-3.01em;",[38,23146],{"className":23147,"style":760},[154],[38,23149,23151],{"className":23150},[102],[38,23152,22999],{"className":23153},[102],[38,23155,23157,23160],{"style":23156},"top:-1.81em;",[38,23158],{"className":23159,"style":760},[154],[38,23161,23163],{"className":23162},[102],[38,23164,23027],{"className":23165},[102],[38,23167,408],{"className":23168},[407],[38,23170,23172],{"className":23171},[142],[38,23173,23175],{"className":23174,"style":23109},[146],[38,23176],{},[38,23178],{"className":23179,"style":5223},[5222],[38,23181],{"className":23182,"style":5223},[5222],[38,23184,23186],{"className":23185},[4873],[38,23187,23189,23231],{"className":23188},[138,384],[38,23190,23192,23228],{"className":23191},[142],[38,23193,23195,23206,23217],{"className":23194,"style":23082},[146],[38,23196,23197,23200],{"style":23132},[38,23198],{"className":23199,"style":760},[154],[38,23201,23203],{"className":23202},[102],[38,23204,13114],{"className":23205},[102],[38,23207,23208,23211],{"style":23144},[38,23209],{"className":23210,"style":760},[154],[38,23212,23214],{"className":23213},[102],[38,23215,20834],{"className":23216},[102],[38,23218,23219,23222],{"style":23156},[38,23220],{"className":23221,"style":760},[154],[38,23223,23225],{"className":23224},[102],[38,23226,23034],{"className":23227},[102],[38,23229,408],{"className":23230},[407],[38,23232,23234],{"className":23233},[142],[38,23235,23237],{"className":23236,"style":23109},[146],[38,23238],{},[38,23240],{"className":23241,"style":5223},[5222],[38,23243],{"className":23244,"style":5223},[5222],[38,23246,23248],{"className":23247},[4873],[38,23249,23251,23293],{"className":23250},[138,384],[38,23252,23254,23290],{"className":23253},[142],[38,23255,23257,23268,23279],{"className":23256,"style":23082},[146],[38,23258,23259,23262],{"style":23132},[38,23260],{"className":23261,"style":760},[154],[38,23263,23265],{"className":23264},[102],[38,23266,19718],{"className":23267},[102],[38,23269,23270,23273],{"style":23144},[38,23271],{"className":23272,"style":760},[154],[38,23274,23276],{"className":23275},[102],[38,23277,23012],{"className":23278},[102],[38,23280,23281,23284],{"style":23156},[38,23282],{"className":23283,"style":760},[154],[38,23285,23287],{"className":23286},[102],[38,23288,23041],{"className":23289},[102],[38,23291,408],{"className":23292},[407],[38,23294,23296],{"className":23295},[142],[38,23297,23299],{"className":23298,"style":23109},[146],[38,23300],{},[38,23302],{"className":23303,"style":5223},[5222],[38,23305],{"className":23306,"style":5223},[5222],[38,23308,23310],{"className":23309},[4873],[38,23311,23313,23355],{"className":23312},[138,384],[38,23314,23316,23352],{"className":23315},[142],[38,23317,23319,23330,23341],{"className":23318,"style":23082},[146],[38,23320,23321,23324],{"style":23132},[38,23322],{"className":23323,"style":760},[154],[38,23325,23327],{"className":23326},[102],[38,23328,19541],{"className":23329},[102],[38,23331,23332,23335],{"style":23144},[38,23333],{"className":23334,"style":760},[154],[38,23336,23338],{"className":23337},[102],[38,23339,19718],{"className":23340},[102],[38,23342,23343,23346],{"style":23156},[38,23344],{"className":23345,"style":760},[154],[38,23347,23349],{"className":23348},[102],[38,23350,23048],{"className":23351},[102],[38,23353,408],{"className":23354},[407],[38,23356,23358],{"className":23357},[142],[38,23359,23361],{"className":23360,"style":23109},[146],[38,23362],{},[38,23364,23366],{"className":23365},[529],[38,23367,23369],{"className":23368},[1485,4820],[38,23370,23372,23393],{"className":23371},[138,384],[38,23373,23375,23390],{"className":23374},[142],[38,23376,23378],{"className":23377,"style":23082},[146],[38,23379,23380,23383],{"style":23085},[38,23381],{"className":23382,"style":23089},[154],[38,23384,23385],{"style":23092},[4842,23386,23387],{"xmlns":4844,"width":4845,"height":23095,"viewBox":23096},[4849,23388],{"d":23389},"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[38,23391,408],{"className":23392},[407],[38,23394,23396],{"className":23395},[142],[38,23397,23399],{"className":23398,"style":23109},[146],[38,23400],{},[11,23402,23403,23404,23433,23434,23463,23464,1964],{},"第 1 维的中位数是 ",[38,23405,23407,23421],{"className":23406,"translate":42},[41],[38,23408,23410],{"className":23409},[46],[48,23411,23412],{"xmlns":50},[52,23413,23414,23419],{},[55,23415,23416],{},[300,23417,23418],{},"4.5",[81,23420,23418],{"encoding":83},[38,23422,23424],{"className":23423,"ariaHidden":89},[88],[38,23425,23427,23430],{"className":23426},[93],[38,23428],{"className":23429,"style":2429},[97],[38,23431,23418],{"className":23432},[102],"，第 2 维是 ",[38,23435,23437,23451],{"className":23436,"translate":42},[41],[38,23438,23440],{"className":23439},[46],[48,23441,23442],{"xmlns":50},[52,23443,23444,23449],{},[55,23445,23446],{},[300,23447,23448],{},"6.5",[81,23450,23448],{"encoding":83},[38,23452,23454],{"className":23453,"ariaHidden":89},[88],[38,23455,23457,23460],{"className":23456},[93],[38,23458],{"className":23459,"style":2429},[97],[38,23461,23448],{"className":23462},[102],"，第 3 维是 ",[38,23465,23467,23481],{"className":23466,"translate":42},[41],[38,23468,23470],{"className":23469},[46],[48,23471,23472],{"xmlns":50},[52,23473,23474,23479],{},[55,23475,23476],{},[300,23477,23478],{},"12.5",[81,23480,23478],{"encoding":83},[38,23482,23484],{"className":23483,"ariaHidden":89},[88],[38,23485,23487,23490],{"className":23486},[93],[38,23488],{"className":23489,"style":2429},[97],[38,23491,23478],{"className":23492},[102],[11,23494,23495,23496,1964],{},"它的逐坐标中位数原型为 ",[38,23497,23499,23530],{"className":23498,"translate":42},[41],[38,23500,23502],{"className":23501},[46],[48,23503,23504],{"xmlns":50},[52,23505,23506,23527],{},[55,23507,23508,23510,23512,23514,23516,23518,23520],{},[62,23509,293],{"stretchy":292},[300,23511,23418],{},[62,23513,305],{"separator":89},[300,23515,23448],{},[62,23517,305],{"separator":89},[300,23519,23478],{},[66,23521,23522,23524],{},[62,23523,330],{"stretchy":292},[58,23525,23526],{},"T",[81,23528,23529],{"encoding":83},"(4.5, 6.5, 12.5)^T",[38,23531,23533],{"className":23532,"ariaHidden":89},[88],[38,23534,23536,23540,23543,23546,23549,23552,23555,23558,23561,23564],{"className":23535},[93],[38,23537],{"className":23538,"style":23539},[97],"height:1.0913em;vertical-align:-0.25em;",[38,23541,293],{"className":23542},[371],[38,23544,23418],{"className":23545},[102],[38,23547,305],{"className":23548},[421],[38,23550],{"className":23551,"style":425},[108],[38,23553,23448],{"className":23554},[102],[38,23556,305],{"className":23557},[421],[38,23559],{"className":23560,"style":425},[108],[38,23562,23478],{"className":23563},[102],[38,23565,23567,23570],{"className":23566},[529],[38,23568,330],{"className":23569},[529],[38,23571,23573],{"className":23572},[134],[38,23574,23576],{"className":23575},[138],[38,23577,23579],{"className":23578},[142],[38,23580,23583],{"className":23581,"style":23582},[146],"height:0.8413em;",[38,23584,23585,23588],{"style":150},[38,23586],{"className":23587,"style":155},[154],[38,23589,23591],{"className":23590},[159,160,161,162],[38,23592,23526],{"className":23593,"style":23594},[102,103,162],"margin-right:0.1389em;",[28,23596,23597],{"id":23597},"推广到一般范数",[11,23599,23600],{},"在定义了欧氏范数之后，让我们重新审视均值原型、几何中位数原型和逐坐标中位数原型的定义。显然，我们可以用欧氏范数来定义它们。",[11,23602,23603,23604,23673],{},"均值原型是使得到所有点的欧氏距离平方和（",[38,23605,23607,23624],{"className":23606,"translate":42},[41],[38,23608,23610],{"className":23609},[46],[48,23611,23612],{"xmlns":50},[52,23613,23614,23622],{},[55,23615,23616],{},[295,23617,23618,23620],{},[58,23619,3104],{},[300,23621,312],{},[81,23623,19963],{"encoding":83},[38,23625,23627],{"className":23626,"ariaHidden":89},[88],[38,23628,23630,23633],{"className":23629},[93],[38,23631],{"className":23632,"style":13610},[97],[38,23634,23636,23639],{"className":23635},[102],[38,23637,3104],{"className":23638},[102,103],[38,23640,23642],{"className":23641},[134],[38,23643,23645,23665],{"className":23644},[138,384],[38,23646,23648,23662],{"className":23647},[142],[38,23649,23651],{"className":23650,"style":391},[146],[38,23652,23653,23656],{"style":394},[38,23654],{"className":23655,"style":155},[154],[38,23657,23659],{"className":23658},[159,160,161,162],[38,23660,312],{"className":23661},[102,162],[38,23663,408],{"className":23664},[407],[38,23666,23668],{"className":23667},[142],[38,23669,23671],{"className":23670,"style":415},[146],[38,23672],{}," 距离范数）最小的点：",[38,23675,23677],{"className":23676,"translate":42},[621],[38,23678,23680,23754],{"className":23679,"translate":42},[41],[38,23681,23683],{"className":23682},[46],[48,23684,23685],{"xmlns":50,"display":630},[52,23686,23687,23751],{},[55,23688,23689,23691,23693,23695,23697,23717,23731,23733,23735,23737,23743],{},[58,23690,15303],{"mathvariant":285},[62,23692,289],{},[58,23694,16667],{},[62,23696,673],{},[664,23698,23699,23705],{},[55,23700,23701,23703],{},[58,23702,16676],{},[62,23704,673],{},[55,23706,23707,23709,23711],{},[58,23708,286],{"mathvariant":285},[62,23710,1420],{},[66,23712,23713,23715],{},[58,23714,71],{"mathvariant":70},[58,23716,74],{},[15353,23718,23719,23721,23729],{},[62,23720,15357],{},[55,23722,23723,23725,23727],{},[58,23724,601],{},[62,23726,289],{},[300,23728,302],{},[58,23730,3555],{},[58,23732,2183],{"mathvariant":333},[58,23734,286],{"mathvariant":285},[62,23736,710],{},[295,23738,23739,23741],{},[58,23740,14806],{"mathvariant":285},[58,23742,601],{},[8499,23744,23745,23747,23749],{},[58,23746,2183],{"mathvariant":333},[300,23748,312],{},[300,23750,312],{},[81,23752,23753],{"encoding":83},"\\mathbf{\\mu} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m \\|\\mathbf{x} - \\mathbf{v}_i\\|_2^2",[38,23755,23757,23775,23953],{"className":23756,"ariaHidden":89},[88],[38,23758,23760,23763,23766,23769,23772],{"className":23759},[93],[38,23761],{"className":23762,"style":3505},[97],[38,23764,15303],{"className":23765},[102,103],[38,23767],{"className":23768,"style":109},[108],[38,23770,289],{"className":23771},[113],[38,23773],{"className":23774,"style":109},[108],[38,23776,23778,23781,23786,23789,23868,23871,23935,23938,23941,23944,23947,23950],{"className":23777},[93],[38,23779],{"className":23780,"style":15410},[97],[38,23782,16756,23784],{"className":23783},[878],[38,23785,6656],{"style":16759},[38,23787],{"className":23788,"style":425},[108],[38,23790,23792],{"className":23791},[878,879],[38,23793,23795,23860],{"className":23794},[138,384],[38,23796,23798,23857],{"className":23797},[142],[38,23799,23801,23847],{"className":23800,"style":16775},[146],[38,23802,23803,23806],{"style":16778},[38,23804],{"className":23805,"style":760},[154],[38,23807,23809],{"className":23808},[159,160,161,162],[38,23810,23812,23815,23818],{"className":23811},[102,162],[38,23813,286],{"className":23814},[102,351,162],[38,23816,1420],{"className":23817},[113,162],[38,23819,23821,23824],{"className":23820},[102,162],[38,23822,71],{"className":23823},[102,130,162],[38,23825,23827],{"className":23826},[134],[38,23828,23830],{"className":23829},[138],[38,23831,23833],{"className":23832},[142],[38,23834,23836],{"className":23835,"style":16812},[146],[38,23837,23838,23841],{"style":16815},[38,23839],{"className":23840,"style":16819},[154],[38,23842,23844],{"className":23843},[159,16823,3786,162],[38,23845,74],{"className":23846},[102,103,162],[38,23848,23849,23852],{"style":913},[38,23850],{"className":23851,"style":760},[154],[38,23853,23854],{},[38,23855,16676],{"className":23856},[878],[38,23858,408],{"className":23859},[407],[38,23861,23863],{"className":23862},[142],[38,23864,23866],{"className":23865,"style":16846},[146],[38,23867],{},[38,23869],{"className":23870,"style":425},[108],[38,23872,23874],{"className":23873},[878,879],[38,23875,23877,23927],{"className":23876},[138,384],[38,23878,23880,23924],{"className":23879},[142],[38,23881,23883,23903,23913],{"className":23882,"style":15488},[146],[38,23884,23885,23888],{"style":15491},[38,23886],{"className":23887,"style":8852},[154],[38,23889,23891],{"className":23890},[159,160,161,162],[38,23892,23894,23897,23900],{"className":23893},[102,162],[38,23895,601],{"className":23896},[102,103,162],[38,23898,289],{"className":23899},[113,162],[38,23901,302],{"className":23902},[102,162],[38,23904,23905,23908],{"style":15512},[38,23906],{"className":23907,"style":8852},[154],[38,23909,23910],{},[38,23911,15357],{"className":23912},[878,15521,15522],[38,23914,23915,23918],{"style":15525},[38,23916],{"className":23917,"style":8852},[154],[38,23919,23921],{"className":23920},[159,160,161,162],[38,23922,3555],{"className":23923},[102,103,162],[38,23925,408],{"className":23926},[407],[38,23928,23930],{"className":23929},[142],[38,23931,23933],{"className":23932,"style":15547},[146],[38,23934],{},[38,23936],{"className":23937,"style":425},[108],[38,23939,2183],{"className":23940},[102],[38,23942,286],{"className":23943},[102,351],[38,23945],{"className":23946,"style":995},[108],[38,23948,710],{"className":23949},[999],[38,23951],{"className":23952,"style":995},[108],[38,23954,23956,23959,23999],{"className":23955},[93],[38,23957],{"className":23958,"style":11468},[97],[38,23960,23962,23965],{"className":23961},[102],[38,23963,14806],{"className":23964,"style":6689},[102,351],[38,23966,23968],{"className":23967},[134],[38,23969,23971,23991],{"className":23970},[138,384],[38,23972,23974,23988],{"className":23973},[142],[38,23975,23977],{"className":23976,"style":786},[146],[38,23978,23979,23982],{"style":14861},[38,23980],{"className":23981,"style":155},[154],[38,23983,23985],{"className":23984},[159,160,161,162],[38,23986,601],{"className":23987},[102,103,162],[38,23989,408],{"className":23990},[407],[38,23992,23994],{"className":23993},[142],[38,23995,23997],{"className":23996,"style":415},[146],[38,23998],{},[38,24000,24002,24005],{"className":24001},[102],[38,24003,2183],{"className":24004},[102],[38,24006,24008],{"className":24007},[134],[38,24009,24011,24042],{"className":24010},[138,384],[38,24012,24014,24039],{"className":24013},[142],[38,24015,24017,24028],{"className":24016,"style":11499},[146],[38,24018,24019,24022],{"style":9585},[38,24020],{"className":24021,"style":155},[154],[38,24023,24025],{"className":24024},[159,160,161,162],[38,24026,312],{"className":24027},[102,162],[38,24029,24030,24033],{"style":1892},[38,24031],{"className":24032,"style":155},[154],[38,24034,24036],{"className":24035},[159,160,161,162],[38,24037,312],{"className":24038},[102,162],[38,24040,408],{"className":24041},[407],[38,24043,24045],{"className":24044},[142],[38,24046,24048],{"className":24047,"style":9616},[146],[38,24049],{},[11,24051,24052,24053,23673],{},"几何中位数原型是使得到所有点的欧氏距离之和（",[38,24054,24056,24073],{"className":24055,"translate":42},[41],[38,24057,24059],{"className":24058},[46],[48,24060,24061],{"xmlns":50},[52,24062,24063,24071],{},[55,24064,24065],{},[295,24066,24067,24069],{},[58,24068,3104],{},[300,24070,312],{},[81,24072,19963],{"encoding":83},[38,24074,24076],{"className":24075,"ariaHidden":89},[88],[38,24077,24079,24082],{"className":24078},[93],[38,24080],{"className":24081,"style":13610},[97],[38,24083,24085,24088],{"className":24084},[102],[38,24086,3104],{"className":24087},[102,103],[38,24089,24091],{"className":24090},[134],[38,24092,24094,24114],{"className":24093},[138,384],[38,24095,24097,24111],{"className":24096},[142],[38,24098,24100],{"className":24099,"style":391},[146],[38,24101,24102,24105],{"style":394},[38,24103],{"className":24104,"style":155},[154],[38,24106,24108],{"className":24107},[159,160,161,162],[38,24109,312],{"className":24110},[102,162],[38,24112,408],{"className":24113},[407],[38,24115,24117],{"className":24116},[142],[38,24118,24120],{"className":24119,"style":415},[146],[38,24121],{},[38,24123,24125],{"className":24124,"translate":42},[621],[38,24126,24128,24204],{"className":24127,"translate":42},[41],[38,24129,24131],{"className":24130},[46],[48,24132,24133],{"xmlns":50,"display":630},[52,24134,24135,24201],{},[55,24136,24137,24139,24141,24143,24145,24165,24179,24181,24183,24185,24187,24193,24195],{},[58,24138,11],{"mathvariant":285},[62,24140,289],{},[58,24142,16667],{},[62,24144,673],{},[664,24146,24147,24153],{},[55,24148,24149,24151],{},[58,24150,16676],{},[62,24152,673],{},[55,24154,24155,24157,24159],{},[58,24156,286],{"mathvariant":285},[62,24158,1420],{},[66,24160,24161,24163],{},[58,24162,71],{"mathvariant":70},[58,24164,74],{},[15353,24166,24167,24169,24177],{},[62,24168,15357],{},[55,24170,24171,24173,24175],{},[58,24172,601],{},[62,24174,289],{},[300,24176,302],{},[58,24178,3555],{},[58,24180,7695],{"mathvariant":333},[58,24182,7695],{"mathvariant":333},[58,24184,286],{"mathvariant":285},[62,24186,710],{},[295,24188,24189,24191],{},[58,24190,14806],{"mathvariant":285},[58,24192,601],{},[58,24194,7695],{"mathvariant":333},[295,24196,24197,24199],{},[58,24198,7695],{"mathvariant":333},[300,24200,312],{},[81,24202,24203],{"encoding":83},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m ||\\mathbf{x} - \\mathbf{v}_i||_2",[38,24205,24207,24225,24404],{"className":24206,"ariaHidden":89},[88],[38,24208,24210,24213,24216,24219,24222],{"className":24209},[93],[38,24211],{"className":24212,"style":6685},[97],[38,24214,11],{"className":24215},[102,351],[38,24217],{"className":24218,"style":109},[108],[38,24220,289],{"className":24221},[113],[38,24223],{"className":24224,"style":109},[108],[38,24226,24228,24231,24236,24239,24318,24321,24385,24388,24392,24395,24398,24401],{"className":24227},[93],[38,24229],{"className":24230,"style":15410},[97],[38,24232,16756,24234],{"className":24233},[878],[38,24235,6656],{"style":16759},[38,24237],{"className":24238,"style":425},[108],[38,24240,24242],{"className":24241},[878,879],[38,24243,24245,24310],{"className":24244},[138,384],[38,24246,24248,24307],{"className":24247},[142],[38,24249,24251,24297],{"className":24250,"style":16775},[146],[38,24252,24253,24256],{"style":16778},[38,24254],{"className":24255,"style":760},[154],[38,24257,24259],{"className":24258},[159,160,161,162],[38,24260,24262,24265,24268],{"className":24261},[102,162],[38,24263,286],{"className":24264},[102,351,162],[38,24266,1420],{"className":24267},[113,162],[38,24269,24271,24274],{"className":24270},[102,162],[38,24272,71],{"className":24273},[102,130,162],[38,24275,24277],{"className":24276},[134],[38,24278,24280],{"className":24279},[138],[38,24281,24283],{"className":24282},[142],[38,24284,24286],{"className":24285,"style":16812},[146],[38,24287,24288,24291],{"style":16815},[38,24289],{"className":24290,"style":16819},[154],[38,24292,24294],{"className":24293},[159,16823,3786,162],[38,24295,74],{"className":24296},[102,103,162],[38,24298,24299,24302],{"style":913},[38,24300],{"className":24301,"style":760},[154],[38,24303,24304],{},[38,24305,16676],{"className":24306},[878],[38,24308,408],{"className":24309},[407],[38,24311,24313],{"className":24312},[142],[38,24314,24316],{"className":24315,"style":16846},[146],[38,24317],{},[38,24319],{"className":24320,"style":425},[108],[38,24322,24324],{"className":24323},[878,879],[38,24325,24327,24377],{"className":24326},[138,384],[38,24328,24330,24374],{"className":24329},[142],[38,24331,24333,24353,24363],{"className":24332,"style":15488},[146],[38,24334,24335,24338],{"style":15491},[38,24336],{"className":24337,"style":8852},[154],[38,24339,24341],{"className":24340},[159,160,161,162],[38,24342,24344,24347,24350],{"className":24343},[102,162],[38,24345,601],{"className":24346},[102,103,162],[38,24348,289],{"className":24349},[113,162],[38,24351,302],{"className":24352},[102,162],[38,24354,24355,24358],{"style":15512},[38,24356],{"className":24357,"style":8852},[154],[38,24359,24360],{},[38,24361,15357],{"className":24362},[878,15521,15522],[38,24364,24365,24368],{"style":15525},[38,24366],{"className":24367,"style":8852},[154],[38,24369,24371],{"className":24370},[159,160,161,162],[38,24372,3555],{"className":24373},[102,103,162],[38,24375,408],{"className":24376},[407],[38,24378,24380],{"className":24379},[142],[38,24381,24383],{"className":24382,"style":15547},[146],[38,24384],{},[38,24386],{"className":24387,"style":425},[108],[38,24389,24391],{"className":24390},[102],"∣∣",[38,24393,286],{"className":24394},[102,351],[38,24396],{"className":24397,"style":995},[108],[38,24399,710],{"className":24400},[999],[38,24402],{"className":24403,"style":995},[108],[38,24405,24407,24410,24450,24453],{"className":24406},[93],[38,24408],{"className":24409,"style":1439},[97],[38,24411,24413,24416],{"className":24412},[102],[38,24414,14806],{"className":24415,"style":6689},[102,351],[38,24417,24419],{"className":24418},[134],[38,24420,24422,24442],{"className":24421},[138,384],[38,24423,24425,24439],{"className":24424},[142],[38,24426,24428],{"className":24427,"style":786},[146],[38,24429,24430,24433],{"style":14861},[38,24431],{"className":24432,"style":155},[154],[38,24434,24436],{"className":24435},[159,160,161,162],[38,24437,601],{"className":24438},[102,103,162],[38,24440,408],{"className":24441},[407],[38,24443,24445],{"className":24444},[142],[38,24446,24448],{"className":24447,"style":415},[146],[38,24449],{},[38,24451,7695],{"className":24452},[102],[38,24454,24456,24459],{"className":24455},[102],[38,24457,7695],{"className":24458},[102],[38,24460,24462],{"className":24461},[134],[38,24463,24465,24485],{"className":24464},[138,384],[38,24466,24468,24482],{"className":24467},[142],[38,24469,24471],{"className":24470,"style":391},[146],[38,24472,24473,24476],{"style":394},[38,24474],{"className":24475,"style":155},[154],[38,24477,24479],{"className":24478},[159,160,161,162],[38,24480,312],{"className":24481},[102,162],[38,24483,408],{"className":24484},[407],[38,24486,24488],{"className":24487},[142],[38,24489,24491],{"className":24490,"style":415},[146],[38,24492],{},[11,24494,24495,24496,23673],{},"逐坐标中位数原型是使得到所有点的曼哈顿距离之和（",[38,24497,24499,24516],{"className":24498,"translate":42},[41],[38,24500,24502],{"className":24501},[46],[48,24503,24504],{"xmlns":50},[52,24505,24506,24514],{},[55,24507,24508],{},[295,24509,24510,24512],{},[58,24511,3104],{},[300,24513,302],{},[81,24515,18906],{"encoding":83},[38,24517,24519],{"className":24518,"ariaHidden":89},[88],[38,24520,24522,24525],{"className":24521},[93],[38,24523],{"className":24524,"style":13610},[97],[38,24526,24528,24531],{"className":24527},[102],[38,24529,3104],{"className":24530},[102,103],[38,24532,24534],{"className":24533},[134],[38,24535,24537,24557],{"className":24536},[138,384],[38,24538,24540,24554],{"className":24539},[142],[38,24541,24543],{"className":24542,"style":391},[146],[38,24544,24545,24548],{"style":394},[38,24546],{"className":24547,"style":155},[154],[38,24549,24551],{"className":24550},[159,160,161,162],[38,24552,302],{"className":24553},[102,162],[38,24555,408],{"className":24556},[407],[38,24558,24560],{"className":24559},[142],[38,24561,24563],{"className":24562,"style":415},[146],[38,24564],{},[38,24566,24568],{"className":24567,"translate":42},[621],[38,24569,24571,24647],{"className":24570,"translate":42},[41],[38,24572,24574],{"className":24573},[46],[48,24575,24576],{"xmlns":50,"display":630},[52,24577,24578,24644],{},[55,24579,24580,24582,24584,24586,24588,24608,24622,24624,24626,24628,24630,24636,24638],{},[58,24581,11],{"mathvariant":285},[62,24583,289],{},[58,24585,16667],{},[62,24587,673],{},[664,24589,24590,24596],{},[55,24591,24592,24594],{},[58,24593,16676],{},[62,24595,673],{},[55,24597,24598,24600,24602],{},[58,24599,286],{"mathvariant":285},[62,24601,1420],{},[66,24603,24604,24606],{},[58,24605,71],{"mathvariant":70},[58,24607,74],{},[15353,24609,24610,24612,24620],{},[62,24611,15357],{},[55,24613,24614,24616,24618],{},[58,24615,601],{},[62,24617,289],{},[300,24619,302],{},[58,24621,3555],{},[58,24623,7695],{"mathvariant":333},[58,24625,7695],{"mathvariant":333},[58,24627,286],{"mathvariant":285},[62,24629,710],{},[295,24631,24632,24634],{},[58,24633,14806],{"mathvariant":285},[58,24635,601],{},[58,24637,7695],{"mathvariant":333},[295,24639,24640,24642],{},[58,24641,7695],{"mathvariant":333},[300,24643,302],{},[81,24645,24646],{"encoding":83},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m ||\\mathbf{x} - \\mathbf{v}_i||_1",[38,24648,24650,24668,24846],{"className":24649,"ariaHidden":89},[88],[38,24651,24653,24656,24659,24662,24665],{"className":24652},[93],[38,24654],{"className":24655,"style":6685},[97],[38,24657,11],{"className":24658},[102,351],[38,24660],{"className":24661,"style":109},[108],[38,24663,289],{"className":24664},[113],[38,24666],{"className":24667,"style":109},[108],[38,24669,24671,24674,24679,24682,24761,24764,24828,24831,24834,24837,24840,24843],{"className":24670},[93],[38,24672],{"className":24673,"style":15410},[97],[38,24675,16756,24677],{"className":24676},[878],[38,24678,6656],{"style":16759},[38,24680],{"className":24681,"style":425},[108],[38,24683,24685],{"className":24684},[878,879],[38,24686,24688,24753],{"className":24687},[138,384],[38,24689,24691,24750],{"className":24690},[142],[38,24692,24694,24740],{"className":24693,"style":16775},[146],[38,24695,24696,24699],{"style":16778},[38,24697],{"className":24698,"style":760},[154],[38,24700,24702],{"className":24701},[159,160,161,162],[38,24703,24705,24708,24711],{"className":24704},[102,162],[38,24706,286],{"className":24707},[102,351,162],[38,24709,1420],{"className":24710},[113,162],[38,24712,24714,24717],{"className":24713},[102,162],[38,24715,71],{"className":24716},[102,130,162],[38,24718,24720],{"className":24719},[134],[38,24721,24723],{"className":24722},[138],[38,24724,24726],{"className":24725},[142],[38,24727,24729],{"className":24728,"style":16812},[146],[38,24730,24731,24734],{"style":16815},[38,24732],{"className":24733,"style":16819},[154],[38,24735,24737],{"className":24736},[159,16823,3786,162],[38,24738,74],{"className":24739},[102,103,162],[38,24741,24742,24745],{"style":913},[38,24743],{"className":24744,"style":760},[154],[38,24746,24747],{},[38,24748,16676],{"className":24749},[878],[38,24751,408],{"className":24752},[407],[38,24754,24756],{"className":24755},[142],[38,24757,24759],{"className":24758,"style":16846},[146],[38,24760],{},[38,24762],{"className":24763,"style":425},[108],[38,24765,24767],{"className":24766},[878,879],[38,24768,24770,24820],{"className":24769},[138,384],[38,24771,24773,24817],{"className":24772},[142],[38,24774,24776,24796,24806],{"className":24775,"style":15488},[146],[38,24777,24778,24781],{"style":15491},[38,24779],{"className":24780,"style":8852},[154],[38,24782,24784],{"className":24783},[159,160,161,162],[38,24785,24787,24790,24793],{"className":24786},[102,162],[38,24788,601],{"className":24789},[102,103,162],[38,24791,289],{"className":24792},[113,162],[38,24794,302],{"className":24795},[102,162],[38,24797,24798,24801],{"style":15512},[38,24799],{"className":24800,"style":8852},[154],[38,24802,24803],{},[38,24804,15357],{"className":24805},[878,15521,15522],[38,24807,24808,24811],{"style":15525},[38,24809],{"className":24810,"style":8852},[154],[38,24812,24814],{"className":24813},[159,160,161,162],[38,24815,3555],{"className":24816},[102,103,162],[38,24818,408],{"className":24819},[407],[38,24821,24823],{"className":24822},[142],[38,24824,24826],{"className":24825,"style":15547},[146],[38,24827],{},[38,24829],{"className":24830,"style":425},[108],[38,24832,24391],{"className":24833},[102],[38,24835,286],{"className":24836},[102,351],[38,24838],{"className":24839,"style":995},[108],[38,24841,710],{"className":24842},[999],[38,24844],{"className":24845,"style":995},[108],[38,24847,24849,24852,24892,24895],{"className":24848},[93],[38,24850],{"className":24851,"style":1439},[97],[38,24853,24855,24858],{"className":24854},[102],[38,24856,14806],{"className":24857,"style":6689},[102,351],[38,24859,24861],{"className":24860},[134],[38,24862,24864,24884],{"className":24863},[138,384],[38,24865,24867,24881],{"className":24866},[142],[38,24868,24870],{"className":24869,"style":786},[146],[38,24871,24872,24875],{"style":14861},[38,24873],{"className":24874,"style":155},[154],[38,24876,24878],{"className":24877},[159,160,161,162],[38,24879,601],{"className":24880},[102,103,162],[38,24882,408],{"className":24883},[407],[38,24885,24887],{"className":24886},[142],[38,24888,24890],{"className":24889,"style":415},[146],[38,24891],{},[38,24893,7695],{"className":24894},[102],[38,24896,24898,24901],{"className":24897},[102],[38,24899,7695],{"className":24900},[102],[38,24902,24904],{"className":24903},[134],[38,24905,24907,24927],{"className":24906},[138,384],[38,24908,24910,24924],{"className":24909},[142],[38,24911,24913],{"className":24912,"style":391},[146],[38,24914,24915,24918],{"style":394},[38,24916],{"className":24917,"style":155},[154],[38,24919,24921],{"className":24920},[159,160,161,162],[38,24922,302],{"className":24923},[102,162],[38,24925,408],{"className":24926},[407],[38,24928,24930],{"className":24929},[142],[38,24931,24933],{"className":24932,"style":415},[146],[38,24934],{},[11,24936,24937],{},"至此，我们可以统一上述所有概念：",[38,24939,24941],{"className":24940,"translate":42},[621],[38,24942,24944,25019],{"className":24943,"translate":42},[41],[38,24945,24947],{"className":24946},[46],[48,24948,24949],{"xmlns":50,"display":630},[52,24950,24951,25016],{},[55,24952,24953,24955,24957,24959,24961,24981,24995,24997,24999,25001,25007],{},[58,24954,11],{"mathvariant":285},[62,24956,289],{},[58,24958,16667],{},[62,24960,673],{},[664,24962,24963,24969],{},[55,24964,24965,24967],{},[58,24966,16676],{},[62,24968,673],{},[55,24970,24971,24973,24975],{},[58,24972,286],{"mathvariant":285},[62,24974,1420],{},[66,24976,24977,24979],{},[58,24978,71],{"mathvariant":70},[58,24980,74],{},[15353,24982,24983,24985,24993],{},[62,24984,15357],{},[55,24986,24987,24989,24991],{},[58,24988,601],{},[62,24990,289],{},[300,24992,302],{},[58,24994,3555],{},[58,24996,2183],{"mathvariant":333},[58,24998,286],{"mathvariant":285},[62,25000,710],{},[295,25002,25003,25005],{},[58,25004,14806],{"mathvariant":285},[58,25006,601],{},[8499,25008,25009,25011,25013],{},[58,25010,2183],{"mathvariant":333},[58,25012,11],{},[58,25014,25015],{},"q",[81,25017,25018],{"encoding":83},"\\mathbf{p} = \\arg\\min_{\\mathbf{x} \\in \\mathbb{R}^n} \\sum_{i=1}^m \\|\\mathbf{x} - \\mathbf{v}_i\\|_p^q",[38,25020,25022,25040,25218],{"className":25021,"ariaHidden":89},[88],[38,25023,25025,25028,25031,25034,25037],{"className":25024},[93],[38,25026],{"className":25027,"style":6685},[97],[38,25029,11],{"className":25030},[102,351],[38,25032],{"className":25033,"style":109},[108],[38,25035,289],{"className":25036},[113],[38,25038],{"className":25039,"style":109},[108],[38,25041,25043,25046,25051,25054,25133,25136,25200,25203,25206,25209,25212,25215],{"className":25042},[93],[38,25044],{"className":25045,"style":15410},[97],[38,25047,16756,25049],{"className":25048},[878],[38,25050,6656],{"style":16759},[38,25052],{"className":25053,"style":425},[108],[38,25055,25057],{"className":25056},[878,879],[38,25058,25060,25125],{"className":25059},[138,384],[38,25061,25063,25122],{"className":25062},[142],[38,25064,25066,25112],{"className":25065,"style":16775},[146],[38,25067,25068,25071],{"style":16778},[38,25069],{"className":25070,"style":760},[154],[38,25072,25074],{"className":25073},[159,160,161,162],[38,25075,25077,25080,25083],{"className":25076},[102,162],[38,25078,286],{"className":25079},[102,351,162],[38,25081,1420],{"className":25082},[113,162],[38,25084,25086,25089],{"className":25085},[102,162],[38,25087,71],{"className":25088},[102,130,162],[38,25090,25092],{"className":25091},[134],[38,25093,25095],{"className":25094},[138],[38,25096,25098],{"className":25097},[142],[38,25099,25101],{"className":25100,"style":16812},[146],[38,25102,25103,25106],{"style":16815},[38,25104],{"className":25105,"style":16819},[154],[38,25107,25109],{"className":25108},[159,16823,3786,162],[38,25110,74],{"className":25111},[102,103,162],[38,25113,25114,25117],{"style":913},[38,25115],{"className":25116,"style":760},[154],[38,25118,25119],{},[38,25120,16676],{"className":25121},[878],[38,25123,408],{"className":25124},[407],[38,25126,25128],{"className":25127},[142],[38,25129,25131],{"className":25130,"style":16846},[146],[38,25132],{},[38,25134],{"className":25135,"style":425},[108],[38,25137,25139],{"className":25138},[878,879],[38,25140,25142,25192],{"className":25141},[138,384],[38,25143,25145,25189],{"className":25144},[142],[38,25146,25148,25168,25178],{"className":25147,"style":15488},[146],[38,25149,25150,25153],{"style":15491},[38,25151],{"className":25152,"style":8852},[154],[38,25154,25156],{"className":25155},[159,160,161,162],[38,25157,25159,25162,25165],{"className":25158},[102,162],[38,25160,601],{"className":25161},[102,103,162],[38,25163,289],{"className":25164},[113,162],[38,25166,302],{"className":25167},[102,162],[38,25169,25170,25173],{"style":15512},[38,25171],{"className":25172,"style":8852},[154],[38,25174,25175],{},[38,25176,15357],{"className":25177},[878,15521,15522],[38,25179,25180,25183],{"style":15525},[38,25181],{"className":25182,"style":8852},[154],[38,25184,25186],{"className":25185},[159,160,161,162],[38,25187,3555],{"className":25188},[102,103,162],[38,25190,408],{"className":25191},[407],[38,25193,25195],{"className":25194},[142],[38,25196,25198],{"className":25197,"style":15547},[146],[38,25199],{},[38,25201],{"className":25202,"style":425},[108],[38,25204,2183],{"className":25205},[102],[38,25207,286],{"className":25208},[102,351],[38,25210],{"className":25211,"style":995},[108],[38,25213,710],{"className":25214},[999],[38,25216],{"className":25217,"style":995},[108],[38,25219,25221,25225,25265],{"className":25220},[93],[38,25222],{"className":25223,"style":25224},[97],"height:1.1331em;vertical-align:-0.3831em;",[38,25226,25228,25231],{"className":25227},[102],[38,25229,14806],{"className":25230,"style":6689},[102,351],[38,25232,25234],{"className":25233},[134],[38,25235,25237,25257],{"className":25236},[138,384],[38,25238,25240,25254],{"className":25239},[142],[38,25241,25243],{"className":25242,"style":786},[146],[38,25244,25245,25248],{"style":14861},[38,25246],{"className":25247,"style":155},[154],[38,25249,25251],{"className":25250},[159,160,161,162],[38,25252,601],{"className":25253},[102,103,162],[38,25255,408],{"className":25256},[407],[38,25258,25260],{"className":25259},[142],[38,25261,25263],{"className":25262,"style":415},[146],[38,25264],{},[38,25266,25268,25271],{"className":25267},[102],[38,25269,2183],{"className":25270},[102],[38,25272,25274],{"className":25273},[134],[38,25275,25277,25308],{"className":25276},[138,384],[38,25278,25280,25305],{"className":25279},[142],[38,25281,25283,25294],{"className":25282,"style":1871},[146],[38,25284,25285,25288],{"style":9585},[38,25286],{"className":25287,"style":155},[154],[38,25289,25291],{"className":25290},[159,160,161,162],[38,25292,11],{"className":25293},[102,103,162],[38,25295,25296,25299],{"style":1892},[38,25297],{"className":25298,"style":155},[154],[38,25300,25302],{"className":25301},[159,160,161,162],[38,25303,25015],{"className":25304,"style":3420},[102,103,162],[38,25306,408],{"className":25307},[407],[38,25309,25311],{"className":25310},[142],[38,25312,25315],{"className":25313,"style":25314},[146],"height:0.3831em;",[38,25316],{},[11,25318,17022,25319,25347,25348,25376,25377,25405,25406,25434,25435,25463],{},[38,25320,25322,25335],{"className":25321,"translate":42},[41],[38,25323,25325],{"className":25324},[46],[48,25326,25327],{"xmlns":50},[52,25328,25329,25333],{},[55,25330,25331],{},[58,25332,11],{},[81,25334,11],{"encoding":83},[38,25336,25338],{"className":25337,"ariaHidden":89},[88],[38,25339,25341,25344],{"className":25340},[93],[38,25342],{"className":25343,"style":3505},[97],[38,25345,11],{"className":25346},[102,103]," 是距离范数，",[38,25349,25351,25364],{"className":25350,"translate":42},[41],[38,25352,25354],{"className":25353},[46],[48,25355,25356],{"xmlns":50},[52,25357,25358,25362],{},[55,25359,25360],{},[58,25361,25015],{},[81,25363,25015],{"encoding":83},[38,25365,25367],{"className":25366,"ariaHidden":89},[88],[38,25368,25370,25373],{"className":25369},[93],[38,25371],{"className":25372,"style":3505},[97],[38,25374,25015],{"className":25375,"style":3420},[102,103]," 是该范数下距离所取的幂次。",[38,25378,25380,25393],{"className":25379,"translate":42},[41],[38,25381,25383],{"className":25382},[46],[48,25384,25385],{"xmlns":50},[52,25386,25387,25391],{},[55,25388,25389],{},[58,25390,11],{},[81,25392,11],{"encoding":83},[38,25394,25396],{"className":25395,"ariaHidden":89},[88],[38,25397,25399,25402],{"className":25398},[93],[38,25400],{"className":25401,"style":3505},[97],[38,25403,11],{"className":25404},[102,103]," 决定了如何度量距离，而 ",[38,25407,25409,25422],{"className":25408,"translate":42},[41],[38,25410,25412],{"className":25411},[46],[48,25413,25414],{"xmlns":50},[52,25415,25416,25420],{},[55,25417,25418],{},[58,25419,25015],{},[81,25421,25015],{"encoding":83},[38,25423,25425],{"className":25424,"ariaHidden":89},[88],[38,25426,25428,25431],{"className":25427},[93],[38,25429],{"className":25430,"style":3505},[97],[38,25432,25015],{"className":25433,"style":3420},[102,103]," 决定了对异常值的惩罚程度：",[38,25436,25438,25451],{"className":25437,"translate":42},[41],[38,25439,25441],{"className":25440},[46],[48,25442,25443],{"xmlns":50},[52,25444,25445,25449],{},[55,25446,25447],{},[58,25448,25015],{},[81,25450,25015],{"encoding":83},[38,25452,25454],{"className":25453,"ariaHidden":89},[88],[38,25455,25457,25460],{"className":25456},[93],[38,25458],{"className":25459,"style":3505},[97],[38,25461,25015],{"className":25462,"style":3420},[102,103]," 越大，对异常值越敏感。",[11,25465,25466],{},"这一公式也描述了广义 Weber 选址问题（Generalized Weber Location Problem）的核心目标函数。",{"title":14638,"searchDepth":14639,"depth":14639,"links":25468},[25469,25470,25471,25473,25474],{"id":14722,"depth":14639,"text":14722},{"id":16357,"depth":14639,"text":16357},{"id":17987,"depth":14639,"text":25472},"LpL_pLp​ 欧氏范数",{"id":21587,"depth":14639,"text":21587},{"id":23597,"depth":14639,"text":23597},"当处理的数据不是简单向量（例如复杂矩阵、概率分布、曲面上的点），无法直接求和再除以 NNN 时，Fréchet 均值就变得必不可少。","ChongQing",{},"\u002Fblog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype",{"title":14681,"description":25475},"blog\u002F2026\u002F2026-04-25-the-frechet-mean-prototype","Fréchet 均值原型是度量空间中使到所有数据点距离平方和最小的中心点。它将求平均的运算从平坦向量空间推广到任意弯曲空间或复杂对象。",[14675,14676],"2026-04-25T00:00:00+08:00","4LNv_FY6RMSjfHRN9m3HtVtcl6piAT90rqonedlqAYw",{"id":25486,"title":25487,"body":25488,"description":25492,"draft":14664,"enableComment":14665,"extension":14666,"image":14638,"important":14664,"location":14669,"meta":38484,"navigation":14665,"ogImage":14669,"path":38485,"seo":38486,"stem":38487,"summary":38488,"tags":38489,"time":38490,"__hash__":38491},"blog\u002Fblog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space.md","零曲率与正曲率空间中的维数灾难",{"type":8,"value":25489,"toc":38472},[25490,25493,25496,25499,25502,25565,25799,25936,26325,27265,27431,28639,28773,28834,29197,29200,29654,29657,30072,30132,30401,30433,30436,30499,30654,30685,30923,31081,31227,31472,31475,31644,31763,31766,32057,32060,32248,32293,32538,32541,32985,32988,33093,33187,33190,33668,33671,33815,33818,33949,33951,34344,34347,34350,34425,34529,34532,34696,34699,34763,34765,34905,34908,35046,35049,35052,35084,35280,35283,35286,35289,35431,35670,35771,36018,36021,36412,36415,36471,36856,36859,36862,36865,36868,36871,37002,37005,37188,37190,37417,37420,37425,37428,37680,37683,37761,37922,37924,38102,38104,38245,38348,38351,38354,38357,38368,38371],[11,25491,25492],{},"维数灾难并非某种特定的几何结构，而是由高维空间的内在性质所引发的一类现象。特别是，测度集中与距离集中等效应会导致度量区分能力退化，使许多基于距离的方法失效。",[11,25494,25495],{},"本文从数学与统计的角度，介绍不同曲率空间中的维数灾难效应。",[28,25497,25498],{"id":25498},"欧氏空间",[11,25500,25501],{},"要引入维数灾难，我们需要从空间中的样本分布说起。",[11,25503,25504,25505,25564],{},"先从一维情形开始：考虑数值范围 ",[38,25506,25508,25534],{"className":25507,"translate":42},[41],[38,25509,25511],{"className":25510},[46],[48,25512,25513],{"xmlns":50},[52,25514,25515,25531],{},[55,25516,25517,25520,25522,25524,25526,25528],{},[62,25518,25519],{"stretchy":292},"[",[62,25521,710],{},[300,25523,302],{},[62,25525,305],{"separator":89},[300,25527,302],{},[62,25529,25530],{"stretchy":292},"]",[81,25532,25533],{"encoding":83},"[-1,1]",[38,25535,25537],{"className":25536,"ariaHidden":89},[88],[38,25538,25540,25543,25546,25549,25552,25555,25558,25561],{"className":25539},[93],[38,25541],{"className":25542,"style":1439},[97],[38,25544,25519],{"className":25545},[371],[38,25547,710],{"className":25548},[102],[38,25550,302],{"className":25551},[102],[38,25553,305],{"className":25554},[421],[38,25556],{"className":25557,"style":425},[108],[38,25559,302],{"className":25560},[102],[38,25562,25530],{"className":25563},[529]," 内的一段线段，并假设所有样本在这条线段上随机且均匀地分布。同时，该线段的总测度为 1 个单位长度。",[11,25566,25567,25568,25626,25627,25683,25684,25746,25747,1964],{},"现在考虑包含 90% 测度的内部区域，即区间 ",[38,25569,25571,25596],{"className":25570,"translate":42},[41],[38,25572,25574],{"className":25573},[46],[48,25575,25576],{"xmlns":50},[52,25577,25578,25593],{},[55,25579,25580,25582,25584,25587,25589,25591],{},[62,25581,25519],{"stretchy":292},[62,25583,710],{},[300,25585,25586],{},"0.9",[62,25588,305],{"separator":89},[300,25590,25586],{},[62,25592,25530],{"stretchy":292},[81,25594,25595],{"encoding":83},"[-0.9, 0.9]",[38,25597,25599],{"className":25598,"ariaHidden":89},[88],[38,25600,25602,25605,25608,25611,25614,25617,25620,25623],{"className":25601},[93],[38,25603],{"className":25604,"style":1439},[97],[38,25606,25519],{"className":25607},[371],[38,25609,710],{"className":25610},[102],[38,25612,25586],{"className":25613},[102],[38,25615,305],{"className":25616},[421],[38,25618],{"className":25619,"style":425},[108],[38,25621,25586],{"className":25622},[102],[38,25624,25530],{"className":25625},[529],"，并将剩余的 10% 视为具有一定厚度的薄外壳。显然，90% 的样本随机分布在 ",[38,25628,25630,25653],{"className":25629,"translate":42},[41],[38,25631,25633],{"className":25632},[46],[48,25634,25635],{"xmlns":50},[52,25636,25637,25651],{},[55,25638,25639,25641,25643,25645,25647,25649],{},[62,25640,25519],{"stretchy":292},[62,25642,710],{},[300,25644,25586],{},[62,25646,305],{"separator":89},[300,25648,25586],{},[62,25650,25530],{"stretchy":292},[81,25652,25595],{"encoding":83},[38,25654,25656],{"className":25655,"ariaHidden":89},[88],[38,25657,25659,25662,25665,25668,25671,25674,25677,25680],{"className":25658},[93],[38,25660],{"className":25661,"style":1439},[97],[38,25663,25519],{"className":25664},[371],[38,25666,710],{"className":25667},[102],[38,25669,25586],{"className":25670},[102],[38,25672,305],{"className":25673},[421],[38,25675],{"className":25676,"style":425},[108],[38,25678,25586],{"className":25679},[102],[38,25681,25530],{"className":25682},[529]," 内，而其余 10% 位于外壳区域，即 ",[38,25685,25687,25713],{"className":25686,"translate":42},[41],[38,25688,25690],{"className":25689},[46],[48,25691,25692],{"xmlns":50},[52,25693,25694,25710],{},[55,25695,25696,25698,25700,25702,25704,25706,25708],{},[62,25697,25519],{"stretchy":292},[62,25699,710],{},[300,25701,302],{},[62,25703,305],{"separator":89},[62,25705,710],{},[300,25707,25586],{},[62,25709,330],{"stretchy":292},[81,25711,25712],{"encoding":83},"[-1, -0.9)",[38,25714,25716],{"className":25715,"ariaHidden":89},[88],[38,25717,25719,25722,25725,25728,25731,25734,25737,25740,25743],{"className":25718},[93],[38,25720],{"className":25721,"style":1439},[97],[38,25723,25519],{"className":25724},[371],[38,25726,710],{"className":25727},[102],[38,25729,302],{"className":25730},[102],[38,25732,305],{"className":25733},[421],[38,25735],{"className":25736,"style":425},[108],[38,25738,710],{"className":25739},[102],[38,25741,25586],{"className":25742},[102],[38,25744,330],{"className":25745},[529]," 和 ",[38,25748,25750,25772],{"className":25749,"translate":42},[41],[38,25751,25753],{"className":25752},[46],[48,25754,25755],{"xmlns":50},[52,25756,25757,25769],{},[55,25758,25759,25761,25763,25765,25767],{},[62,25760,293],{"stretchy":292},[300,25762,25586],{},[62,25764,305],{"separator":89},[300,25766,302],{},[62,25768,25530],{"stretchy":292},[81,25770,25771],{"encoding":83},"(0.9, 1]",[38,25773,25775],{"className":25774,"ariaHidden":89},[88],[38,25776,25778,25781,25784,25787,25790,25793,25796],{"className":25777},[93],[38,25779],{"className":25780,"style":1439},[97],[38,25782,293],{"className":25783},[371],[38,25785,25586],{"className":25786},[102],[38,25788,305],{"className":25789},[421],[38,25791],{"className":25792,"style":425},[108],[38,25794,302],{"className":25795},[102],[38,25797,25530],{"className":25798},[529],[11,25800,25801,25802,1964],{},"显然，我们可以推出一维情形下分布在薄外壳上的样本比例为 ",[38,25803,25805,25841],{"className":25804,"translate":42},[41],[38,25806,25808],{"className":25807},[46],[48,25809,25810],{"xmlns":50},[52,25811,25812,25838],{},[55,25813,25814,25831,25833,25835],{},[8499,25815,25816,25819,25822],{},[58,25817,25818],{},"P",[3265,25820,25821],{},"外壳",[55,25823,25824,25827,25829],{},[58,25825,25826],{},"d",[62,25828,289],{},[300,25830,302],{},[62,25832,289],{},[300,25834,23041],{},[58,25836,25837],{"mathvariant":333},"%",[81,25839,25840],{"encoding":83},"P_\\text{外壳}^{d=1} = 10\\%",[38,25842,25844,25925],{"className":25843,"ariaHidden":89},[88],[38,25845,25847,25851,25916,25919,25922],{"className":25846},[93],[38,25848],{"className":25849,"style":25850},[97],"height:1.1244em;vertical-align:-0.2753em;",[38,25852,25854,25857],{"className":25853},[102],[38,25855,25818],{"className":25856,"style":23594},[102,103],[38,25858,25860],{"className":25859},[134],[38,25861,25863,25907],{"className":25862},[138,384],[38,25864,25866,25904],{"className":25865},[142],[38,25867,25869,25884],{"className":25868,"style":545},[146],[38,25870,25872,25875],{"style":25871},"top:-2.4247em;margin-left:-0.1389em;margin-right:0.05em;",[38,25873],{"className":25874,"style":155},[154],[38,25876,25878],{"className":25877},[159,160,161,162],[38,25879,25881],{"className":25880},[102,22531,162],[38,25882,25821],{"className":25883},[102,22535,162],[38,25885,25886,25889],{"style":150},[38,25887],{"className":25888,"style":155},[154],[38,25890,25892],{"className":25891},[159,160,161,162],[38,25893,25895,25898,25901],{"className":25894},[102,162],[38,25896,25826],{"className":25897},[102,103,162],[38,25899,289],{"className":25900},[113,162],[38,25902,302],{"className":25903},[102,162],[38,25905,408],{"className":25906},[407],[38,25908,25910],{"className":25909},[142],[38,25911,25914],{"className":25912,"style":25913},[146],"height:0.2753em;",[38,25915],{},[38,25917],{"className":25918,"style":109},[108],[38,25920,289],{"className":25921},[113],[38,25923],{"className":25924,"style":109},[108],[38,25926,25928,25932],{"className":25927},[93],[38,25929],{"className":25930,"style":25931},[97],"height:0.8056em;vertical-align:-0.0556em;",[38,25933,25935],{"className":25934},[102],"10%",[11,25937,25938,25939,26029,26030,26166,26167,26324],{},"现在，让我们把类似的情形推广到二维平面中的圆。众所周知，圆的面积可以用公式 ",[38,25940,25942,25969],{"className":25941,"translate":42},[41],[38,25943,25945],{"className":25944},[46],[48,25946,25947],{"xmlns":50},[52,25948,25949,25966],{},[55,25950,25951,25954,25956,25959],{},[58,25952,25953],{},"S",[62,25955,289],{},[58,25957,25958],{},"π",[66,25960,25961,25964],{},[58,25962,25963],{},"r",[300,25965,312],{},[81,25967,25968],{"encoding":83},"S = \\pi r^2",[38,25970,25972,25991],{"className":25971,"ariaHidden":89},[88],[38,25973,25975,25978,25982,25985,25988],{"className":25974},[93],[38,25976],{"className":25977,"style":7404},[97],[38,25979,25953],{"className":25980,"style":25981},[102,103],"margin-right:0.0576em;",[38,25983],{"className":25984,"style":109},[108],[38,25986,289],{"className":25987},[113],[38,25989],{"className":25990,"style":109},[108],[38,25992,25994,25997,26000],{"className":25993},[93],[38,25995],{"className":25996,"style":9002},[97],[38,25998,25958],{"className":25999,"style":3420},[102,103],[38,26001,26003,26006],{"className":26002},[102],[38,26004,25963],{"className":26005,"style":2500},[102,103],[38,26007,26009],{"className":26008},[134],[38,26010,26012],{"className":26011},[138],[38,26013,26015],{"className":26014},[142],[38,26016,26018],{"className":26017,"style":9002},[146],[38,26019,26020,26023],{"style":150},[38,26021],{"className":26022,"style":155},[154],[38,26024,26026],{"className":26025},[159,160,161,162],[38,26027,312],{"className":26028},[102,162]," 计算。同样地，我们考虑包含 90% 测度的内部区域，即 ",[38,26031,26033,26064],{"className":26032,"translate":42},[41],[38,26034,26036],{"className":26035},[46],[48,26037,26038],{"xmlns":50},[52,26039,26040,26061],{},[55,26041,26042,26048,26050,26057,26059],{},[66,26043,26044,26046],{},[58,26045,286],{},[300,26047,312],{},[62,26049,696],{},[66,26051,26052,26055],{},[58,26053,26054],{},"y",[300,26056,312],{},[62,26058,2879],{},[300,26060,25586],{},[81,26062,26063],{"encoding":83},"x^2 + y^2 \\leq 0.9",[38,26065,26067,26112,26157],{"className":26066,"ariaHidden":89},[88],[38,26068,26070,26074,26103,26106,26109],{"className":26069},[93],[38,26071],{"className":26072,"style":26073},[97],"height:0.8974em;vertical-align:-0.0833em;",[38,26075,26077,26080],{"className":26076},[102],[38,26078,286],{"className":26079},[102,103],[38,26081,26083],{"className":26082},[134],[38,26084,26086],{"className":26085},[138],[38,26087,26089],{"className":26088},[142],[38,26090,26092],{"className":26091,"style":9002},[146],[38,26093,26094,26097],{"style":150},[38,26095],{"className":26096,"style":155},[154],[38,26098,26100],{"className":26099},[159,160,161,162],[38,26101,312],{"className":26102},[102,162],[38,26104],{"className":26105,"style":995},[108],[38,26107,696],{"className":26108},[999],[38,26110],{"className":26111,"style":995},[108],[38,26113,26115,26119,26148,26151,26154],{"className":26114},[93],[38,26116],{"className":26117,"style":26118},[97],"height:1.0085em;vertical-align:-0.1944em;",[38,26120,26122,26125],{"className":26121},[102],[38,26123,26054],{"className":26124,"style":3420},[102,103],[38,26126,26128],{"className":26127},[134],[38,26129,26131],{"className":26130},[138],[38,26132,26134],{"className":26133},[142],[38,26135,26137],{"className":26136,"style":9002},[146],[38,26138,26139,26142],{"style":150},[38,26140],{"className":26141,"style":155},[154],[38,26143,26145],{"className":26144},[159,160,161,162],[38,26146,312],{"className":26147},[102,162],[38,26149],{"className":26150,"style":109},[108],[38,26152,2879],{"className":26153},[113],[38,26155],{"className":26156,"style":109},[108],[38,26158,26160,26163],{"className":26159},[93],[38,26161],{"className":26162,"style":2429},[97],[38,26164,25586],{"className":26165},[102],"，以及剩余的有限外壳 ",[38,26168,26170,26205],{"className":26169,"translate":42},[41],[38,26171,26173],{"className":26172},[46],[48,26174,26175],{"xmlns":50},[52,26176,26177,26202],{},[55,26178,26179,26181,26184,26190,26192,26198,26200],{},[300,26180,25586],{},[62,26182,26183],{},"\u003C",[66,26185,26186,26188],{},[58,26187,286],{},[300,26189,312],{},[62,26191,696],{},[66,26193,26194,26196],{},[58,26195,26054],{},[300,26197,312],{},[62,26199,2879],{},[300,26201,302],{},[81,26203,26204],{"encoding":83},"0.9 \u003C x^2 + y^2 \\leq 1",[38,26206,26208,26227,26271,26315],{"className":26207,"ariaHidden":89},[88],[38,26209,26211,26215,26218,26221,26224],{"className":26210},[93],[38,26212],{"className":26213,"style":26214},[97],"height:0.6835em;vertical-align:-0.0391em;",[38,26216,25586],{"className":26217},[102],[38,26219],{"className":26220,"style":109},[108],[38,26222,26183],{"className":26223},[113],[38,26225],{"className":26226,"style":109},[108],[38,26228,26230,26233,26262,26265,26268],{"className":26229},[93],[38,26231],{"className":26232,"style":26073},[97],[38,26234,26236,26239],{"className":26235},[102],[38,26237,286],{"className":26238},[102,103],[38,26240,26242],{"className":26241},[134],[38,26243,26245],{"className":26244},[138],[38,26246,26248],{"className":26247},[142],[38,26249,26251],{"className":26250,"style":9002},[146],[38,26252,26253,26256],{"style":150},[38,26254],{"className":26255,"style":155},[154],[38,26257,26259],{"className":26258},[159,160,161,162],[38,26260,312],{"className":26261},[102,162],[38,26263],{"className":26264,"style":995},[108],[38,26266,696],{"className":26267},[999],[38,26269],{"className":26270,"style":995},[108],[38,26272,26274,26277,26306,26309,26312],{"className":26273},[93],[38,26275],{"className":26276,"style":26118},[97],[38,26278,26280,26283],{"className":26279},[102],[38,26281,26054],{"className":26282,"style":3420},[102,103],[38,26284,26286],{"className":26285},[134],[38,26287,26289],{"className":26288},[138],[38,26290,26292],{"className":26291},[142],[38,26293,26295],{"className":26294,"style":9002},[146],[38,26296,26297,26300],{"style":150},[38,26298],{"className":26299,"style":155},[154],[38,26301,26303],{"className":26302},[159,160,161,162],[38,26304,312],{"className":26305},[102,162],[38,26307],{"className":26308,"style":109},[108],[38,26310,2879],{"className":26311},[113],[38,26313],{"className":26314,"style":109},[108],[38,26316,26318,26321],{"className":26317},[93],[38,26319],{"className":26320,"style":2429},[97],[38,26322,302],{"className":26323},[102],"。接下来，我们需要计算分布在这两个区域中的样本比例。",[38,26326,26328],{"className":26327,"translate":42},[621],[38,26329,26331,26506],{"className":26330,"translate":42},[41],[38,26332,26334],{"className":26333},[46],[48,26335,26336],{"xmlns":50,"display":630},[52,26337,26338,26503],{},[4475,26339,26342,26380,26446],{"rowspacing":26340,"columnalign":26341,"columnspacing":14638},"0.25em","right",[4481,26343,26344],{},[4484,26345,26346],{},[2631,26347,26348],{"scriptlevel":683,"displaystyle":89},[55,26349,26350,26357,26359,26361,26363,26365,26367,26373,26375,26378],{},[295,26351,26352,26354],{},[58,26353,25953],{},[3265,26355,26356],{},"内部",[62,26358,289],{},[58,26360,25958],{},[62,26362,17039],{},[62,26364,293],{"stretchy":292},[300,26366,25586],{},[66,26368,26369,26371],{},[62,26370,330],{"stretchy":292},[300,26372,312],{},[62,26374,289],{},[300,26376,26377],{},"0.81",[58,26379,25958],{},[4481,26381,26382],{},[4484,26383,26384],{},[2631,26385,26386],{"scriptlevel":683,"displaystyle":89},[55,26387,26388,26394,26396,26403,26405,26411,26413,26415,26417,26423,26425,26427,26429,26431,26433,26439,26441,26444],{},[295,26389,26390,26392],{},[58,26391,25953],{},[3265,26393,25821],{},[62,26395,289],{},[295,26397,26398,26400],{},[58,26399,25953],{},[3265,26401,26402],{},"单位",[62,26404,710],{},[295,26406,26407,26409],{},[58,26408,25953],{},[3265,26410,26356],{},[62,26412,289],{},[58,26414,25958],{},[62,26416,17039],{},[66,26418,26419,26421],{},[300,26420,302],{},[300,26422,312],{},[62,26424,710],{},[58,26426,25958],{},[62,26428,17039],{},[62,26430,293],{"stretchy":292},[300,26432,25586],{},[66,26434,26435,26437],{},[62,26436,330],{"stretchy":292},[300,26438,312],{},[62,26440,289],{},[300,26442,26443],{},"0.19",[58,26445,25958],{},[4481,26447,26448],{},[4484,26449,26450],{},[2631,26451,26452],{"scriptlevel":683,"displaystyle":89},[55,26453,26454,26468,26470,26484,26486,26496,26498,26501],{},[8499,26455,26456,26458,26460],{},[58,26457,25818],{},[3265,26459,25821],{},[55,26461,26462,26464,26466],{},[58,26463,25826],{},[62,26465,289],{},[300,26467,312],{},[62,26469,289],{},[636,26471,26472,26478],{},[295,26473,26474,26476],{},[58,26475,25953],{},[3265,26477,25821],{},[295,26479,26480,26482],{},[58,26481,25953],{},[3265,26483,26402],{},[62,26485,289],{},[636,26487,26488,26494],{},[55,26489,26490,26492],{},[300,26491,26443],{},[58,26493,25958],{},[58,26495,25958],{},[62,26497,289],{},[300,26499,26500],{},"19",[58,26502,25837],{"mathvariant":333},[81,26504,26505],{"encoding":83},"\\begin{aligned}\nS_\\text{内部} = \\pi \\cdot (0.9)^2 = 0.81 \\pi \\\\\nS_\\text{外壳} = S_\\text{单位} - S_\\text{内部} = \\pi \\cdot 1^2 - \\pi \\cdot (0.9)^{2} = 0.19\\pi \\\\\nP^{d=2}_\\text{外壳} = \\frac{S_\\text{外壳}}{S_\\text{单位}} = \\frac{0.19 \\pi}{\\pi} = 19\\%  \n\\end{aligned}",[38,26507,26509],{"className":26508,"ariaHidden":89},[88],[38,26510,26512,26516],{"className":26511},[93],[38,26513],{"className":26514,"style":26515},[97],"height:5.2445em;vertical-align:-2.3723em;",[38,26517,26519],{"className":26518},[102],[38,26520,26522],{"className":26521},[4475],[38,26523,26526],{"className":26524},[26525],"col-align-r",[38,26527,26529,27256],{"className":26528},[138,384],[38,26530,26532,27253],{"className":26531},[142],[38,26533,26536,26661,26941],{"className":26534,"style":26535},[146],"height:2.8723em;",[38,26537,26539,26543],{"style":26538},"top:-5.3685em;",[38,26540],{"className":26541,"style":26542},[154],"height:3.3603em;",[38,26544,26546,26590,26593,26596,26599,26602,26605,26608,26611,26614,26617,26646,26649,26652,26655,26658],{"className":26545},[102],[38,26547,26549,26552],{"className":26548},[102],[38,26550,25953],{"className":26551,"style":25981},[102,103],[38,26553,26555],{"className":26554},[134],[38,26556,26558,26582],{"className":26557},[138,384],[38,26559,26561,26579],{"className":26560},[142],[38,26562,26564],{"className":26563,"style":7469},[146],[38,26565,26567,26570],{"style":26566},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[38,26568],{"className":26569,"style":155},[154],[38,26571,26573],{"className":26572},[159,160,161,162],[38,26574,26576],{"className":26575},[102,22531,162],[38,26577,26356],{"className":26578},[102,22535,162],[38,26580,408],{"className":26581},[407],[38,26583,26585],{"className":26584},[142],[38,26586,26588],{"className":26587,"style":415},[146],[38,26589],{},[38,26591],{"className":26592,"style":109},[108],[38,26594,289],{"className":26595},[113],[38,26597],{"className":26598,"style":109},[108],[38,26600,25958],{"className":26601,"style":3420},[102,103],[38,26603],{"className":26604,"style":995},[108],[38,26606,17039],{"className":26607},[999],[38,26609],{"className":26610,"style":995},[108],[38,26612,293],{"className":26613},[371],[38,26615,25586],{"className":26616},[102],[38,26618,26620,26623],{"className":26619},[529],[38,26621,330],{"className":26622},[529],[38,26624,26626],{"className":26625},[134],[38,26627,26629],{"className":26628},[138],[38,26630,26632],{"className":26631},[142],[38,26633,26635],{"className":26634,"style":11499},[146],[38,26636,26637,26640],{"style":1892},[38,26638],{"className":26639,"style":155},[154],[38,26641,26643],{"className":26642},[159,160,161,162],[38,26644,312],{"className":26645},[102,162],[38,26647],{"className":26648,"style":109},[108],[38,26650,289],{"className":26651},[113],[38,26653],{"className":26654,"style":109},[108],[38,26656,26377],{"className":26657},[102],[38,26659,25958],{"className":26660,"style":3420},[102,103],[38,26662,26664,26667],{"style":26663},"top:-3.8444em;",[38,26665],{"className":26666,"style":26542},[154],[38,26668,26670,26713,26716,26719,26722,26765,26768,26771,26774,26817,26820,26823,26826,26829,26832,26835,26838,26867,26870,26873,26876,26879,26882,26885,26888,26891,26894,26926,26929,26932,26935,26938],{"className":26669},[102],[38,26671,26673,26676],{"className":26672},[102],[38,26674,25953],{"className":26675,"style":25981},[102,103],[38,26677,26679],{"className":26678},[134],[38,26680,26682,26705],{"className":26681},[138,384],[38,26683,26685,26702],{"className":26684},[142],[38,26686,26688],{"className":26687,"style":7469},[146],[38,26689,26690,26693],{"style":26566},[38,26691],{"className":26692,"style":155},[154],[38,26694,26696],{"className":26695},[159,160,161,162],[38,26697,26699],{"className":26698},[102,22531,162],[38,26700,25821],{"className":26701},[102,22535,162],[38,26703,408],{"className":26704},[407],[38,26706,26708],{"className":26707},[142],[38,26709,26711],{"className":26710,"style":415},[146],[38,26712],{},[38,26714],{"className":26715,"style":109},[108],[38,26717,289],{"className":26718},[113],[38,26720],{"className":26721,"style":109},[108],[38,26723,26725,26728],{"className":26724},[102],[38,26726,25953],{"className":26727,"style":25981},[102,103],[38,26729,26731],{"className":26730},[134],[38,26732,26734,26757],{"className":26733},[138,384],[38,26735,26737,26754],{"className":26736},[142],[38,26738,26740],{"className":26739,"style":7469},[146],[38,26741,26742,26745],{"style":26566},[38,26743],{"className":26744,"style":155},[154],[38,26746,26748],{"className":26747},[159,160,161,162],[38,26749,26751],{"className":26750},[102,22531,162],[38,26752,26402],{"className":26753},[102,22535,162],[38,26755,408],{"className":26756},[407],[38,26758,26760],{"className":26759},[142],[38,26761,26763],{"className":26762,"style":415},[146],[38,26764],{},[38,26766],{"className":26767,"style":995},[108],[38,26769,710],{"className":26770},[999],[38,26772],{"className":26773,"style":995},[108],[38,26775,26777,26780],{"className":26776},[102],[38,26778,25953],{"className":26779,"style":25981},[102,103],[38,26781,26783],{"className":26782},[134],[38,26784,26786,26809],{"className":26785},[138,384],[38,26787,26789,26806],{"className":26788}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= \\frac{4}{3} \\pi \\cdot (0.9)^3 = \\frac{4}{3} \\pi \\times 0.729  \\\\\nV_\\text{外壳} = V_\\text{单位} - V_\\text{内部} = \\frac{4}{3} \\pi \\cdot 1^3 - \\frac{4}{3} \\pi \\cdot (0.9)^{3} = \\frac{4}{3} \\pi \\times 0.271 \\\\\nP^{d=3}_\\text{外壳} = \\frac{V_\\text{外壳}}{V_\\text{单位}} = 27.1\\%  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",[38,28643,28645,28661],{"className":28644,"translate":42},[41],[38,28646,28648],{"className":28647},[46],[48,28649,28650],{"xmlns":50},[52,28651,28652,28658],{},[55,28653,28654,28656],{},[300,28655,23041],{},[58,28657,25837],{"mathvariant":333},[81,28659,28660],{"encoding":83},"10\\%",[38,28662,28664],{"className":28663,"ariaHidden":89},[88],[38,28665,28667,28670],{"className":28666},[93],[38,28668],{"className":28669,"style":25931},[97],[38,28671,25935],{"className":28672},[102]," 增至 ",[38,28675,28677,28693],{"className":28676,"translate":42},[41],[38,28678,28680],{"className":28679},[46],[48,28681,28682],{"xmlns":50},[52,28683,28684,28690],{},[55,28685,28686,28688],{},[300,28687,27624],{},[58,28689,25837],{"mathvariant":333},[81,28691,28692],{"encoding":83},"27.1\\%",[38,28694,28696],{"className":28695,"ariaHidden":89},[88],[38,28697,28699,28702],{"className":28698},[93],[38,28700],{"className":28701,"style":25931},[97],[38,28703,28626],{"className":28704},[102],"；而分布在内腔中的样本比例持续减小，从 ",[38,28707,28709,28726],{"className":28708,"translate":42},[41],[38,28710,28712],{"className":28711},[46],[48,28713,28714],{"xmlns":50},[52,28715,28716,28723],{},[55,28717,28718,28721],{},[300,28719,28720],{},"90",[58,28722,25837],{"mathvariant":333},[81,28724,28725],{"encoding":83},"90\\%",[38,28727,28729],{"className":28728,"ariaHidden":89},[88],[38,28730,28732,28735],{"className":28731},[93],[38,28733],{"className":28734,"style":25931},[97],[38,28736,28738],{"className":28737},[102],"90%"," 降至 ",[38,28741,28743,28760],{"className":28742,"translate":42},[41],[38,28744,28746],{"className":28745},[46],[48,28747,28748],{"xmlns":50},[52,28749,28750,28757],{},[55,28751,28752,28755],{},[300,28753,28754],{},"72.9",[58,28756,25837],{"mathvariant":333},[81,28758,28759],{"encoding":83},"72.9\\%",[38,28761,28763],{"className":28762,"ariaHidden":89},[88],[38,28764,28766,28769],{"className":28765},[93],[38,28767],{"className":28768,"style":25931},[97],[38,28770,28772],{"className":28771},[102],"72.9%",[11,28774,28775,28776,28804,28805,28833],{},"为了严格验证这一结论，我们需要一般的数学推导。对于 ",[38,28777,28779,28792],{"className":28778,"translate":42},[41],[38,28780,28782],{"className":28781},[46],[48,28783,28784],{"xmlns":50},[52,28785,28786,28790],{},[55,28787,28788],{},[58,28789,25826],{},[81,28791,25826],{"encoding":83},[38,28793,28795],{"className":28794,"ariaHidden":89},[88],[38,28796,28798,28801],{"className":28797},[93],[38,28799],{"className":28800,"style":889},[97],[38,28802,25826],{"className":28803},[102,103]," 维空间，半径为 ",[38,28806,28808,28821],{"className":28807,"translate":42},[41],[38,28809,28811],{"className":28810},[46],[48,28812,28813],{"xmlns":50},[52,28814,28815,28819],{},[55,28816,28817],{},[58,28818,25963],{},[81,28820,25963],{"encoding":83},[38,28822,28824],{"className":28823,"ariaHidden":89},[88],[38,28825,28827,28830],{"className":28826},[93],[38,28828],{"className":28829,"style":4312},[97],[38,28831,25963],{"className":28832,"style":2500},[102,103]," 的超球体体积为：",[38,28835,28837],{"className":28836,"translate":42},[621],[38,28838,28840,28907],{"className":28839,"translate":42},[41],[38,28841,28843],{"className":28842},[46],[48,28844,28845],{"xmlns":50,"display":630},[52,28846,28847,28904],{},[55,28848,28849,28855,28857,28859,28861,28863,28898],{},[66,28850,28851,28853],{},[58,28852,27282],{},[58,28854,25826],{},[62,28856,293],{"stretchy":292},[58,28858,25963],{},[62,28860,330],{"stretchy":292},[62,28862,289],{},[636,28864,28865,28877],{},[66,28866,28867,28869],{},[58,28868,25958],{},[55,28870,28871,28873,28875],{},[58,28872,25826],{},[58,28874,18490],{"mathvariant":333},[300,28876,312],{},[55,28878,28879,28882],{},[58,28880,28881],{"mathvariant":333},"Γ",[55,28883,28884,28886,28892,28894,28896],{},[62,28885,293],{"fence":89},[636,28887,28888,28890],{},[58,28889,25826],{},[300,28891,312],{},[62,28893,696],{},[300,28895,302],{},[62,28897,330],{"fence":89},[66,28899,28900,28902],{},[58,28901,25963],{},[58,28903,25826],{},[81,28905,28906],{"encoding":83}," V^d(r) = \\frac{\\pi^{d\u002F2}}{\\Gamma\\left(\\frac{d}{2}+1\\right)} r^d",[38,28908,28910,28963],{"className":28909,"ariaHidden":89},[88],[38,28911,28913,28916,28945,28948,28951,28954,28957,28960],{"className":28912},[93],[38,28914],{"className":28915,"style":2277},[97],[38,28917,28919,28922],{"className":28918},[102],[38,28920,27282],{"className":28921,"style":995},[102,103],[38,28923,28925],{"className":28924},[134],[38,28926,28928],{"className":28927},[138],[38,28929,28931],{"className":28930},[142],[38,28932,28934],{"className":28933,"style":2308},[146],[38,28935,28936,28939],{"style":1892},[38,28937],{"className":28938,"style":155},[154],[38,28940,28942],{"className":28941},[159,160,161,162],[38,28943,25826],{"className":28944},[102,103,162],[38,28946,293],{"className":28947},[371],[38,28949,25963],{"className":28950,"style":2500},[102,103],[38,28952,330],{"className":28953},[529],[38,28955],{"className":28956,"style":109},[108],[38,28958,289],{"className":28959},[113],[38,28961],{"className":28962,"style":109},[108],[38,28964,28966,28970,29168],{"className":28965},[93],[38,28967],{"className":28968,"style":28969},[97],"height:2.6851em;vertical-align:-1.1201em;",[38,28971,28973,28976,29165],{"className":28972},[102],[38,28974],{"className":28975},[371,740],[38,28977,28979],{"className":28978},[636],[38,28980,28982,29156],{"className":28981},[138,384],[38,28983,28985,29153],{"className":28984},[142],[38,28986,28989,29100,29108],{"className":28987,"style":28988},[146],"height:1.565em;",[38,28990,28992,28995],{"style":28991},"top:-2.2299em;",[38,28993],{"className":28994,"style":760},[154],[38,28996,28998,29001,29004],{"className":28997},[102],[38,28999,28881],{"className":29000},[102],[38,29002],{"className":29003,"style":425},[108],[38,29005,29007,29013,29082,29085,29088,29091,29094],{"className":29006},[475],[38,29008,29010],{"className":29009,"style":1481},[371,1480],[38,29011,293],{"className":29012},[1485,3786],[38,29014,29016,29019,29079],{"className":29015},[102],[38,29017],{"className":29018},[371,740],[38,29020,29022],{"className":29021},[636],[38,29023,29025,29071],{"className":29024},[138,384],[38,29026,29028,29068],{"className":29027},[142],[38,29029,29032,29046,29054],{"className":29030,"style":29031},[146],"height:0.8801em;",[38,29033,29034,29037],{"style":17472},[38,29035],{"className":29036,"style":760},[154],[38,29038,29040],{"className":29039},[159,160,161,162],[38,29041,29043],{"className":29042},[102,162],[38,29044,312],{"className":29045},[102,162],[38,29047,29048,29051],{"style":811},[38,29049],{"className":29050,"style":760},[154],[38,29052],{"className":29053,"style":819},[818],[38,29055,29056,29059],{"style":17586},[38,29057],{"className":29058,"style":760},[154],[38,29060,29062],{"className":29061},[159,160,161,162],[38,29063,29065],{"className":29064},[102,162],[38,29066,25826],{"className":29067},[102,103,162],[38,29069,408],{"className":29070},[407],[38,29072,29074],{"className":29073},[142],[38,29075,29077],{"className":29076,"style":27393},[146],[38,29078],{},[38,29080],{"className":29081},[529,740],[38,29083],{"className":29084,"style":995},[108],[38,29086,696],{"className":29087},[999],[38,29089],{"className":29090,"style":995},[108],[38,29092,302],{"className":29093},[102],[38,29095,29097],{"className":29096,"style":1481},[529,1480],[38,29098,330],{"className":29099},[1485,3786],[38,29101,29102,29105],{"style":811},[38,29103],{"className":29104,"style":760},[154],[38,29106],{"className":29107,"style":819},[818],[38,29109,29110,29113],{"style":822},[38,29111],{"className":29112,"style":760},[154],[38,29114,29116],{"className":29115},[102],[38,29117,29119,29122],{"className":29118},[102],[38,29120,25958],{"className":29121,"style":3420},[102,103],[38,29123,29125],{"className":29124},[134],[38,29126,29128],{"className":29127},[138],[38,29129,29131],{"className":29130},[142],[38,29132,29135],{"className":29133,"style":29134},[146],"height:0.888em;",[38,29136,29137,29140],{"style":150},[38,29138],{"className":29139,"style":155},[154],[38,29141,29143],{"className":29142},[159,160,161,162],[38,29144,29146,29149],{"className":29145},[102,162],[38,29147,25826],{"className":29148},[102,103,162],[38,29150,29152],{"className":29151},[102,162],"\u002F2",[38,29154,408],{"className":29155},[407],[38,29157,29159],{"className":29158},[142],[38,29160,29163],{"className":29161,"style":29162},[146],"height:1.1201em;",[38,29164],{},[38,29166],{"className":29167},[529,740],[38,29169,29171,29174],{"className":29170},[102],[38,29172,25963],{"className":29173,"style":2500},[102,103],[38,29175,29177],{"className":29176},[134],[38,29178,29180],{"className":29179},[138],[38,29181,29183],{"className":29182},[142],[38,29184,29186],{"className":29185,"style":2308},[146],[38,29187,29188,29191],{"style":1892},[38,29189],{"className":29190,"style":155},[154],[38,29192,29194],{"className":29193},[159,160,161,162],[38,29195,25826],{"className":29196},[102,103,162],[11,29198,29199],{},"其中：",[7370,29201,29202,29249,29543],{},[7373,29203,29204,29248],{},[38,29205,29207,29227],{"className":29206,"translate":42},[41],[38,29208,29210],{"className":29209},[46],[48,29211,29212],{"xmlns":50},[52,29213,29214,29224],{},[55,29215,29216,29218,29220,29222],{},[58,29217,28881],{"mathvariant":333},[62,29219,293],{"stretchy":292},[62,29221,17039],{},[62,29223,330],{"stretchy":292},[81,29225,29226],{"encoding":83},"\\Gamma(\\cdot)",[38,29228,29230],{"className":29229,"ariaHidden":89},[88],[38,29231,29233,29236,29239,29242,29245],{"className":29232},[93],[38,29234],{"className":29235,"style":1439},[97],[38,29237,28881],{"className":29238},[102],[38,29240,293],{"className":29241},[371],[38,29243,17039],{"className":29244},[102],[38,29246,330],{"className":29247},[529]," 是伽马函数；",[7373,29250,19890,29251,29279,29280,29542],{},[38,29252,29254,29267],{"className":29253,"translate":42},[41],[38,29255,29257],{"className":29256},[46],[48,29258,29259],{"xmlns":50},[52,29260,29261,29265],{},[55,29262,29263],{},[58,29264,25826],{},[81,29266,25826],{"encoding":83},[38,29268,29270],{"className":29269,"ariaHidden":89},[88],[38,29271,29273,29276],{"className":29272},[93],[38,29274],{"className":29275,"style":889},[97],[38,29277,25826],{"className":29278},[102,103]," 为正整数且为偶数时，",[38,29281,29283,29330],{"className":29282,"translate":42},[41],[38,29284,29286],{"className":29285},[46],[48,29287,29288],{"xmlns":50},[52,29289,29290,29327],{},[55,29291,29292,29294,29310,29312,29324],{},[58,29293,28881],{"mathvariant":333},[55,29295,29296,29298,29304,29306,29308],{},[62,29297,293],{"fence":89},[636,29299,29300,29302],{},[58,29301,25826],{},[300,29303,312],{},[62,29305,696],{},[300,29307,302],{},[62,29309,330],{"fence":89},[62,29311,289],{},[55,29313,29314,29316,29322],{},[62,29315,293],{"fence":89},[636,29317,29318,29320],{},[58,29319,25826],{},[300,29321,312],{},[62,29323,330],{"fence":89},[62,29325,29326],{"stretchy":292},"!",[81,29328,29329],{"encoding":83},"\\Gamma\\left(\\frac{d}{2}+1\\right) = \\left(\\frac{d}{2}\\right)!",[38,29331,29333,29450],{"className":29332,"ariaHidden":89},[88],[38,29334,29336,29340,29343,29346,29441,29444,29447],{"className":29335},[93],[38,29337],{"className":29338,"style":29339},[97],"height:1.2301em;vertical-align:-0.35em;",[38,29341,28881],{"className":29342},[102],[38,29344],{"className":29345,"style":425},[108],[38,29347,29349,29355,29423,29426,29429,29432,29435],{"className":29348},[475],[38,29350,29352],{"className":29351,"style":1481},[371,1480],[38,29353,293],{"className":29354},[1485,3786],[38,29356,29358,29361,29420],{"className":29357},[102],[38,29359],{"className":29360},[371,740],[38,29362,29364],{"className":29363},[636],[38,29365,29367,29412],{"className":29366},[138,384],[38,29368,29370,29409],{"className":29369},[142],[38,29371,29373,29387,29395],{"className":29372,"style":29031},[146],[38,29374,29375,29378],{"style":17472},[38,29376],{"className":29377,"style":760},[154],[38,29379,29381],{"className":29380},[159,160,161,162],[38,29382,29384],{"className":29383},[102,162],[38,29385,312],{"className":29386},[102,162],[38,29388,29389,29392],{"style":811},[38,29390],{"className":29391,"style":760},[154],[38,29393],{"className":29394,"style":819},[818],[38,29396,29397,29400],{"style":17586},[38,29398],{"className":29399,"style":760},[154],[38,29401,29403],{"className":29402},[159,160,161,162],[38,29404,29406],{"className":29405},[102,162],[38,29407,25826],{"className":29408},[102,103,162],[38,29410,408],{"className":29411},[407],[38,29413,29415],{"className":29414},[142],[38,29416,29418],{"className":29417,"style":27393},[146],[38,29419],{},[38,29421],{"className":29422},[529,740],[38,29424],{"className":29425,"style":995},[108],[38,29427,696],{"className":29428},[999],[38,29430],{"className":29431,"style":995},[108],[38,29433,302],{"className":29434},[102],[38,29436,29438],{"className":29437,"style":1481},[529,1480],[38,29439,330],{"className":29440},[1485,3786],[38,29442],{"className":29443,"style":109},[108],[38,29445,289],{"className":29446},[113],[38,29448],{"className":29449,"style":109},[108],[38,29451,29453,29456,29539],{"className":29452},[93],[38,29454],{"className":29455,"style":29339},[97],[38,29457,29459,29465,29533],{"className":29458},[475],[38,29460,29462],{"className":29461,"style":1481},[371,1480],[38,29463,293],{"className":29464},[1485,3786],[38,29466,29468,29471,29530],{"className":29467},[102],[38,29469],{"className":29470},[371,740],[38,29472,29474],{"className":29473},[636],[38,29475,29477,29522],{"className":29476},[138,384],[38,29478,29480,29519],{"className":29479},[142],[38,29481,29483,29497,29505],{"className":29482,"style":29031},[146],[38,29484,29485,29488],{"style":17472},[38,29486],{"className":29487,"style":760},[154],[38,29489,29491],{"className":29490},[159,160,161,162],[38,29492,29494],{"className":29493},[102,162],[38,29495,312],{"className":29496},[102,162],[38,29498,29499,29502],{"style":811},[38,29500],{"className":29501,"style":760},[154],[38,29503],{"className":29504,"style":819},[818],[38,29506,29507,29510],{"style":17586},[38,29508],{"className":29509,"style":760},[154],[38,29511,29513],{"className":29512},[159,160,161,162],[38,29514,29516],{"className":29515},[102,162],[38,29517,25826],{"className":29518},[102,103,162],[38,29520,408],{"className":29521},[407],[38,29523,29525],{"className":29524},[142],[38,29526,29528],{"className":29527,"style":27393},[146],[38,29529],{},[38,29531],{"className":29532},[529,740],[38,29534,29536],{"className":29535,"style":1481},[529,1480],[38,29537,330],{"className":29538},[1485,3786],[38,29540,29326],{"className":29541},[529],"；",[7373,29544,11836,29545,29573,29574,29653],{},[38,29546,29548,29561],{"className":29547,"translate":42},[41],[38,29549,29551],{"className":29550},[46],[48,29552,29553],{"xmlns":50},[52,29554,29555,29559],{},[55,29556,29557],{},[58,29558,25826],{},[81,29560,25826],{"encoding":83},[38,29562,29564],{"className":29563,"ariaHidden":89},[88],[38,29565,29567,29570],{"className":29566},[93],[38,29568],{"className":29569,"style":889},[97],[38,29571,25826],{"className":29572},[102,103]," 为奇数，伽马函数将给出含 ",[38,29575,29577,29593],{"className":29576,"translate":42},[41],[38,29578,29580],{"className":29579},[46],[48,29581,29582],{"xmlns":50},[52,29583,29584,29590],{},[55,29585,29586],{},[20042,29587,29588],{},[58,29589,25958],{},[81,29591,29592],{"encoding":83},"\\sqrt{\\pi}",[38,29594,29596],{"className":29595,"ariaHidden":89},[88],[38,29597,29599,29603],{"className":29598},[93],[38,29600],{"className":29601,"style":29602},[97],"height:1.04em;vertical-align:-0.2397em;",[38,29604,29606],{"className":29605},[102,20187],[38,29607,29609,29644],{"className":29608},[138,384],[38,29610,29612,29641],{"className":29611},[142],[38,29613,29616,29628],{"className":29614,"style":29615},[146],"height:0.8003em;",[38,29617,29619,29622],{"className":29618,"style":913},[20201],[38,29620],{"className":29621,"style":760},[154],[38,29623,29625],{"className":29624,"style":21076},[102],[38,29626,25958],{"className":29627,"style":3420},[102,103],[38,29629,29631,29634],{"style":29630},"top:-2.7603em;",[38,29632],{"className":29633,"style":760},[154],[38,29635,29637],{"className":29636,"style":21101},[20343],[4842,29638,29639],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,29640],{"d":21108},[38,29642,408],{"className":29643},[407],[38,29645,29647],{"className":29646},[142],[38,29648,29651],{"className":29649,"style":29650},[146],"height:0.2397em;",[38,29652],{}," 的表达式。",[11,29655,29656],{},"我们有：",[38,29658,29660],{"className":29659,"translate":42},[621],[38,29661,29663,29723],{"className":29662,"translate":42},[41],[38,29664,29666],{"className":29665},[46],[48,29667,29668],{"xmlns":50,"display":630},[52,29669,29670,29720],{},[55,29671,29672,29680,29682,29700,29702,29708,29710,29712,29714],{},[8499,29673,29674,29676,29678],{},[58,29675,25818],{},[3265,29677,25821],{},[58,29679,25826],{},[62,29681,289],{},[636,29683,29684,29692],{},[8499,29685,29686,29688,29690],{},[58,29687,27282],{},[3265,29689,25821],{},[58,29691,25826],{},[8499,29693,29694,29696,29698],{},[58,29695,27282],{},[3265,29697,26402],{},[58,29699,25826],{},[62,29701,289],{},[66,29703,29704,29706],{},[300,29705,302],{},[58,29707,25826],{},[62,29709,710],{},[62,29711,293],{"stretchy":292},[300,29713,25586],{},[66,29715,29716,29718],{},[62,29717,330],{"stretchy":292},[58,29719,25826],{},[81,29721,29722],{"encoding":83},"P^{d}_\\text{外壳} = \\frac{V^d_\\text{外壳}}{V^d_\\text{单位}} = \n1^d - (0.9)^d",[38,29724,29726,29799,29986,30031],{"className":29725,"ariaHidden":89},[88],[38,29727,29729,29733,29790,29793,29796],{"className":29728},[93],[38,29730],{"className":29731,"style":29732},[97],"height:1.1461em;vertical-align:-0.247em;",[38,29734,29736,29739],{"className":29735},[102],[38,29737,25818],{"className":29738,"style":23594},[102,103],[38,29740,29742],{"className":29741},[134],[38,29743,29745,29782],{"className":29744},[138,384],[38,29746,29748,29779],{"className":29747},[142],[38,29749,29751,29765],{"className":29750,"style":2308},[146],[38,29752,29753,29756],{"style":26970},[38,29754],{"className":29755,"style":155},[154],[38,29757,29759],{"className":29758},[159,160,161,162],[38,29760,29762],{"className":29761},[102,22531,162],[38,29763,25821],{"className":29764},[102,22535,162],[38,29766,29767,29770],{"style":1892},[38,29768],{"className":29769,"style":155},[154],[38,29771,29773],{"className":29772},[159,160,161,162],[38,29774,29776],{"className":29775},[102,162],[38,29777,25826],{"className":29778},[102,103,162],[38,29780,408],{"className":29781},[407],[38,29783,29785],{"className":29784},[142],[38,29786,29788],{"className":29787,"style":9616},[146],[38,29789],{},[38,29791],{"className":29792,"style":109},[108],[38,29794,289],{"className":29795},[113],[38,29797],{"className":29798,"style":109},[108],[38,29800,29802,29806,29977,29980,29983],{"className":29801},[93],[38,29803],{"className":29804,"style":29805},[97],"height:2.5405em;vertical-align:-1.0144em;",[38,29807,29809,29812,29974],{"className":29808},[102],[38,29810],{"className":29811},[371,740],[38,29813,29815],{"className":29814},[636],[38,29816,29818,29965],{"className":29817},[138,384],[38,29819,29821,29962],{"className":29820},[142],[38,29822,29825,29891,29899],{"className":29823,"style":29824},[146],"height:1.5261em;",[38,29826,29828,29831],{"style":29827},"top:-2.2791em;",[38,29829],{"className":29830,"style":760},[154],[38,29832,29834],{"className":29833},[102],[38,29835,29837,29840],{"className":29836},[102],[38,29838,27282],{"className":29839,"style":995},[102,103],[38,29841,29843],{"className":29842},[134],[38,29844,29846,29882],{"className":29845},[138,384],[38,29847,29849,29879],{"className":29848},[142],[38,29850,29853,29868],{"className":29851,"style":29852},[146],"height:0.8309em;",[38,29854,29856,29859],{"style":29855},"top:-2.4065em;margin-left:-0.2222em;margin-right:0.05em;",[38,29857],{"className":29858,"style":155},[154],[38,29860,29862],{"className":29861},[159,160,161,162],[38,29863,29865],{"className":29864},[102,22531,162],[38,29866,26402],{"className":29867},[102,22535,162],[38,29869,29870,29873],{"style":8946},[38,29871],{"className":29872,"style":155},[154],[38,29874,29876],{"className":29875},[159,160,161,162],[38,29877,25826],{"className":29878},[102,103,162],[38,29880,408],{"className":29881},[407],[38,29883,29885],{"className":29884},[142],[38,29886,29889],{"className":29887,"style":29888},[146],"height:0.2935em;",[38,29890],{},[38,29892,29893,29896],{"style":811},[38,29894],{"className":29895,"style":760},[154],[38,29897],{"className":29898,"style":819},[818],[38,29900,29901,29904],{"style":822},[38,29902],{"className":29903,"style":760},[154],[38,29905,29907],{"className":29906},[102],[38,29908,29910,29913],{"className":29909},[102],[38,29911,27282],{"className":29912,"style":995},[102,103],[38,29914,29916],{"className":29915},[134],[38,29917,29919,29954],{"className":29918},[138,384],[38,29920,29922,29951],{"className":29921},[142],[38,29923,29925,29940],{"className":29924,"style":545},[146],[38,29926,29928,29931],{"style":29927},"top:-2.4247em;margin-left:-0.2222em;margin-right:0.05em;",[38,29929],{"className":29930,"style":155},[154],[38,29932,29934],{"className":29933},[159,160,161,162],[38,29935,29937],{"className":29936},[102,22531,162],[38,29938,25821],{"className":29939},[102,22535,162],[38,29941,29942,29945],{"style":150},[38,29943],{"className":29944,"style":155},[154],[38,29946,29948],{"className":29947},[159,160,161,162],[38,29949,25826],{"className":29950},[102,103,162],[38,29952,408],{"className":29953},[407],[38,29955,29957],{"className":29956},[142],[38,29958,29960],{"className":29959,"style":25913},[146],[38,29961],{},[38,29963,408],{"className":29964},[407],[38,29966,29968],{"className":29967},[142],[38,29969,29972],{"className":29970,"style":29971},[146],"height:1.0144em;",[38,29973],{},[38,29975],{"className":29976},[529,740],[38,29978],{"className":29979,"style":109},[108],[38,29981,289],{"className":29982},[113],[38,29984],{"className":29985,"style":109},[108],[38,29987,29989,29993,30022,30025,30028],{"className":29988},[93],[38,29990],{"className":29991,"style":29992},[97],"height:0.9824em;vertical-align:-0.0833em;",[38,29994,29996,29999],{"className":29995},[102],[38,29997,302],{"className":29998},[102],[38,30000,30002],{"className":30001},[134],[38,30003,30005],{"className":30004},[138],[38,30006,30008],{"className":30007},[142],[38,30009,30011],{"className":30010,"style":2308},[146],[38,30012,30013,30016],{"style":1892},[38,30014],{"className":30015,"style":155},[154],[38,30017,30019],{"className":30018},[159,160,161,162],[38,30020,25826],{"className":30021},[102,103,162],[38,30023],{"className":30024,"style":995},[108],[38,30026,710],{"className":30027},[999],[38,30029],{"className":30030,"style":995},[108],[38,30032,30034,30037,30040,30043],{"className":30033},[93],[38,30035],{"className":30036,"style":2277},[97],[38,30038,293],{"className":30039},[371],[38,30041,25586],{"className":30042},[102],[38,30044,30046,30049],{"className":30045},[529],[38,30047,330],{"className":30048},[529],[38,30050,30052],{"className":30051},[134],[38,30053,30055],{"className":30054},[138],[38,30056,30058],{"className":30057},[142],[38,30059,30061],{"className":30060,"style":2308},[146],[38,30062,30063,30066],{"style":1892},[38,30064],{"className":30065,"style":155},[154],[38,30067,30069],{"className":30068},[159,160,161,162],[38,30070,25826],{"className":30071},[102,103,162],[11,30073,30074,30075,30131],{},"对于高维情形 ",[38,30076,30078,30098],{"className":30077,"translate":42},[41],[38,30079,30081],{"className":30080},[46],[48,30082,30083],{"xmlns":50},[52,30084,30085,30095],{},[55,30086,30087,30089,30091,30093],{},[58,30088,25826],{},[62,30090,77],{},[62,30092,696],{},[58,30094,21550],{"mathvariant":333},[81,30096,30097],{"encoding":83},"d \\rightarrow +\\infty",[38,30099,30101,30119],{"className":30100,"ariaHidden":89},[88],[38,30102,30104,30107,30110,30113,30116],{"className":30103},[93],[38,30105],{"className":30106,"style":889},[97],[38,30108,25826],{"className":30109},[102,103],[38,30111],{"className":30112,"style":109},[108],[38,30114,77],{"className":30115},[113],[38,30117],{"className":30118,"style":109},[108],[38,30120,30122,30125,30128],{"className":30121},[93],[38,30123],{"className":30124,"style":4417},[97],[38,30126,696],{"className":30127},[102],[38,30129,21550],{"className":30130},[102],"，我们有以下极限：",[38,30133,30135],{"className":30134,"translate":42},[621],[38,30136,30138,30198],{"className":30137,"translate":42},[41],[38,30139,30141],{"className":30140},[46],[48,30142,30143],{"xmlns":50,"display":630},[52,30144,30145,30195],{},[55,30146,30147,30165,30171,30173,30175,30177,30183,30185,30187,30189,30191,30193],{},[664,30148,30149,30155],{},[55,30150,30151,30153],{},[58,30152,670],{},[62,30154,673],{},[55,30156,30157,30159,30161,30163],{},[58,30158,25826],{},[62,30160,77],{},[62,30162,696],{},[58,30164,21550],{"mathvariant":333},[66,30166,30167,30169],{},[300,30168,302],{},[58,30170,25826],{},[62,30172,710],{},[62,30174,293],{"stretchy":292},[300,30176,25586],{},[66,30178,30179,30181],{},[62,30180,330],{"stretchy":292},[58,30182,25826],{},[62,30184,289],{},[300,30186,302],{},[62,30188,710],{},[300,30190,683],{},[62,30192,289],{},[300,30194,302],{},[81,30196,30197],{"encoding":83},"\\lim_{d \\rightarrow +\\infty} 1^d - (0.9)^d = 1 - 0 = 1",[38,30199,30201,30306,30356,30374,30392],{"className":30200,"ariaHidden":89},[88],[38,30202,30204,30208,30265,30268,30297,30300,30303],{"className":30203},[93],[38,30205],{"className":30206,"style":30207},[97],"height:1.7095em;vertical-align:-0.8104em;",[38,30209,30211],{"className":30210},[878,879],[38,30212,30214,30256],{"className":30213},[138,384],[38,30215,30217,30253],{"className":30216},[142],[38,30218,30220,30243],{"className":30219,"style":889},[146],[38,30221,30222,30225],{"style":892},[38,30223],{"className":30224,"style":760},[154],[38,30226,30228],{"className":30227},[159,160,161,162],[38,30229,30231,30234,30237,30240],{"className":30230},[102,162],[38,30232,25826],{"className":30233},[102,103,162],[38,30235,77],{"className":30236},[113,162],[38,30238,696],{"className":30239},[102,162],[38,30241,21550],{"className":30242},[102,162],[38,30244,30245,30248],{"style":913},[38,30246],{"className":30247,"style":760},[154],[38,30249,30250],{},[38,30251,670],{"className":30252},[878],[38,30254,408],{"className":30255},[407],[38,30257,30259],{"className":30258},[142],[38,30260,30263],{"className":30261,"style":30262},[146],"height:0.8104em;",[38,30264],{},[38,30266],{"className":30267,"style":425},[108],[38,30269,30271,30274],{"className":30270},[102],[38,30272,302],{"className":30273},[102],[38,30275,30277],{"className":30276},[134],[38,30278,30280],{"className":30279},[138],[38,30281,30283],{"className":30282},[142],[38,30284,30286],{"className":30285,"style":2308},[146],[38,30287,30288,30291],{"style":1892},[38,30289],{"className":30290,"style":155},[154],[38,30292,30294],{"className":30293},[159,160,161,162],[38,30295,25826],{"className":30296},[102,103,162],[38,30298],{"className":30299,"style":995},[108],[38,30301,710],{"className":30302},[999],[38,30304],{"className":30305,"style":995},[108],[38,30307,30309,30312,30315,30318,30347,30350,30353],{"className":30308},[93],[38,30310],{"className":30311,"style":2277},[97],[38,30313,293],{"className":30314},[371],[38,30316,25586],{"className":30317},[102],[38,30319,30321,30324],{"className":30320},[529],[38,30322,330],{"className":30323},[529],[38,30325,30327],{"className":30326},[134],[38,30328,30330],{"className":30329},[138],[38,30331,30333],{"className":30332},[142],[38,30334,30336],{"className":30335,"style":2308},[146],[38,30337,30338,30341],{"style":1892},[38,30339],{"className":30340,"style":155},[154],[38,30342,30344],{"className":30343},[159,160,161,162],[38,30345,25826],{"className":30346},[102,103,162],[38,30348],{"className":30349,"style":109},[108],[38,30351,289],{"className":30352},[113],[38,30354],{"className":30355,"style":109},[108],[38,30357,30359,30362,30365,30368,30371],{"className":30358},[93],[38,30360],{"className":30361,"style":19848},[97],[38,30363,302],{"className":30364},[102],[38,30366],{"className":30367,"style":995},[108],[38,30369,710],{"className":30370},[999],[38,30372],{"className":30373,"style":995},[108],[38,30375,30377,30380,30383,30386,30389],{"className":30376},[93],[38,30378],{"className":30379,"style":2429},[97],[38,30381,683],{"className":30382},[102],[38,30384],{"className":30385,"style":109},[108],[38,30387,289],{"className":30388},[113],[38,30390],{"className":30391,"style":109},[108],[38,30393,30395,30398],{"className":30394},[93],[38,30396],{"className":30397,"style":2429},[97],[38,30399,302],{"className":30400},[102],[11,30402,30403,30404,30432],{},"也就是说，随着维数 ",[38,30405,30407,30420],{"className":30406,"translate":42},[41],[38,30408,30410],{"className":30409},[46],[48,30411,30412],{"xmlns":50},[52,30413,30414,30418],{},[55,30415,30416],{},[58,30417,25826],{},[81,30419,25826],{"encoding":83},[38,30421,30423],{"className":30422,"ariaHidden":89},[88],[38,30424,30426,30429],{"className":30425},[93],[38,30427],{"className":30428,"style":889},[97],[38,30430,25826],{"className":30431},[102,103]," 增大，随机均匀样本趋向于分布在外壳上。这一现象被称为薄壳效应（Thin-Shell Effect）。",[31,30434,30435],{"id":30435},"欧氏距离",[11,30437,30438,30439,30498],{},"考虑高维空间 ",[38,30440,30442,30460],{"className":30441,"translate":42},[41],[38,30443,30445],{"className":30444},[46],[48,30446,30447],{"xmlns":50},[52,30448,30449,30457],{},[55,30450,30451],{},[66,30452,30453,30455],{},[58,30454,71],{"mathvariant":70},[58,30456,25826],{},[81,30458,30459],{"encoding":83},"\\mathbb{R}^d",[38,30461,30463],{"className":30462,"ariaHidden":89},[88],[38,30464,30466,30469],{"className":30465},[93],[38,30467],{"className":30468,"style":545},[97],[38,30470,30472,30475],{"className":30471},[102],[38,30473,71],{"className":30474},[102,130],[38,30476,30478],{"className":30477},[134],[38,30479,30481],{"className":30480},[138],[38,30482,30484],{"className":30483},[142],[38,30485,30487],{"className":30486,"style":545},[146],[38,30488,30489,30492],{"style":150},[38,30490],{"className":30491,"style":155},[154],[38,30493,30495],{"className":30494},[159,160,161,162],[38,30496,25826],{"className":30497},[102,103,162]," 中的随机向量：",[38,30500,30502],{"className":30501,"translate":42},[621],[38,30503,30505,30547],{"className":30504,"translate":42},[41],[38,30506,30508],{"className":30507},[46],[48,30509,30510],{"xmlns":50,"display":630},[52,30511,30512,30544],{},[55,30513,30514,30516,30518,30520,30523,30526,30529,30531,30533,30535,30542],{},[58,30515,286],{},[62,30517,305],{"separator":89},[58,30519,26054],{},[62,30521,30522],{},"∼",[3265,30524,30525],{},"i.i.d. ",[58,30527,14701],{"mathvariant":30528},"script",[62,30530,293],{"stretchy":292},[300,30532,683],{},[62,30534,305],{"separator":89},[295,30536,30537,30540],{},[58,30538,30539],{},"I",[58,30541,25826],{},[62,30543,330],{"stretchy":292},[81,30545,30546],{"encoding":83},"x, y \\sim \\text{i.i.d.} \\ \\mathcal{N}(0, I_d)",[38,30548,30550,30577],{"className":30549,"ariaHidden":89},[88],[38,30551,30553,30556,30559,30562,30565,30568,30571,30574],{"className":30552},[93],[38,30554],{"className":30555,"style":3505},[97],[38,30557,286],{"className":30558},[102,103],[38,30560,305],{"className":30561},[421],[38,30563],{"className":30564,"style":425},[108],[38,30566,26054],{"className":30567,"style":3420},[102,103],[38,30569],{"className":30570,"style":109},[108],[38,30572,30522],{"className":30573},[113],[38,30575],{"className":30576,"style":109},[108],[38,30578,30580,30583,30590,30593,30598,30601,30604,30607,30610,30651],{"className":30579},[93],[38,30581],{"className":30582,"style":1439},[97],[38,30584,30586],{"className":30585},[102,22531],[38,30587,30589],{"className":30588},[102],"i.i.d.",[38,30591,15653],{"className":30592},[108],[38,30594,14701],{"className":30595,"style":30597},[102,30596],"mathcal","margin-right:0.1474em;",[38,30599,293],{"className":30600},[371],[38,30602,683],{"className":30603},[102],[38,30605,305],{"className":30606},[421],[38,30608],{"className":30609,"style":425},[108],[38,30611,30613,30616],{"className":30612},[102],[38,30614,30539],{"className":30615,"style":7756},[102,103],[38,30617,30619],{"className":30618},[134],[38,30620,30622,30643],{"className":30621},[138,384],[38,30623,30625,30640],{"className":30624},[142],[38,30626,30628],{"className":30627,"style":4759},[146],[38,30629,30631,30634],{"style":30630},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[38,30632],{"className":30633,"style":155},[154],[38,30635,30637],{"className":30636},[159,160,161,162],[38,30638,25826],{"className":30639},[102,103,162],[38,30641,408],{"className":30642},[407],[38,30644,30646],{"className":30645},[142],[38,30647,30649],{"className":30648,"style":415},[146],[38,30650],{},[38,30652,330],{"className":30653},[529],[11,30655,13448,30656,30684],{},[38,30657,30659,30672],{"className":30658,"translate":42},[41],[38,30660,30662],{"className":30661},[46],[48,30663,30664],{"xmlns":50},[52,30665,30666,30670],{},[55,30667,30668],{},[58,30669,286],{},[81,30671,286],{"encoding":83},[38,30673,30675],{"className":30674,"ariaHidden":89},[88],[38,30676,30678,30681],{"className":30677},[93],[38,30679],{"className":30680,"style":4312},[97],[38,30682,286],{"className":30683},[102,103]," 的模长为：",[38,30686,30688],{"className":30687,"translate":42},[621],[38,30689,30691,30741],{"className":30690,"translate":42},[41],[38,30692,30694],{"className":30693},[46],[48,30695,30696],{"xmlns":50,"display":630},[52,30697,30698,30738],{},[55,30699,30700,30702,30704,30706,30708,30714,30716,30730],{},[58,30701,7695],{"mathvariant":333},[58,30703,7695],{"mathvariant":333},[58,30705,286],{},[58,30707,7695],{"mathvariant":333},[66,30709,30710,30712],{},[58,30711,7695],{"mathvariant":333},[300,30713,312],{},[62,30715,289],{},[15353,30717,30718,30720,30728],{},[62,30719,15357],{},[55,30721,30722,30724,30726],{},[58,30723,601],{},[62,30725,289],{},[300,30727,302],{},[58,30729,25826],{},[8499,30731,30732,30734,30736],{},[58,30733,286],{},[58,30735,601],{},[300,30737,312],{},[81,30739,30740],{"encoding":83},"||x||^2 = \\sum_{i=1}^d x_i^2",[38,30742,30744,30797],{"className":30743,"ariaHidden":89},[88],[38,30745,30747,30750,30753,30756,30759,30788,30791,30794],{"className":30746},[93],[38,30748],{"className":30749,"style":11468},[97],[38,30751,24391],{"className":30752},[102],[38,30754,286],{"className":30755},[102,103],[38,30757,7695],{"className":30758},[102],[38,30760,30762,30765],{"className":30761},[102],[38,30763,7695],{"className":30764},[102],[38,30766,30768],{"className":30767},[134],[38,30769,30771],{"className":30770},[138],[38,30772,30774],{"className":30773},[142],[38,30775,30777],{"className":30776,"style":11499},[146],[38,30778,30779,30782],{"style":1892},[38,30780],{"className":30781,"style":155},[154],[38,30783,30785],{"className":30784},[159,160,161,162],[38,30786,312],{"className":30787},[102,162],[38,30789],{"className":30790,"style":109},[108],[38,30792,289],{"className":30793},[113],[38,30795],{"className":30796,"style":109},[108],[38,30798,30800,30804,30869,30872],{"className":30799},[93],[38,30801],{"className":30802,"style":30803},[97],"height:3.1138em;vertical-align:-1.2777em;",[38,30805,30807],{"className":30806},[878,879],[38,30808,30810,30861],{"className":30809},[138,384],[38,30811,30813,30858],{"className":30812},[142],[38,30814,30817,30837,30847],{"className":30815,"style":30816},[146],"height:1.8361em;",[38,30818,30819,30822],{"style":15491},[38,30820],{"className":30821,"style":8852},[154],[38,30823,30825],{"className":30824},[159,160,161,162],[38,30826,30828,30831,30834],{"className":30827},[102,162],[38,30829,601],{"className":30830},[102,103,162],[38,30832,289],{"className":30833},[113,162],[38,30835,302],{"className":30836},[102,162],[38,30838,30839,30842],{"style":15512},[38,30840],{"className":30841,"style":8852},[154],[38,30843,30844],{},[38,30845,15357],{"className":30846},[878,15521,15522],[38,30848,30849,30852],{"style":15525},[38,30850],{"className":30851,"style":8852},[154],[38,30853,30855],{"className":30854},[159,160,161,162],[38,30856,25826],{"className":30857},[102,103,162],[38,30859,408],{"className":30860},[407],[38,30862,30864],{"className":30863},[142],[38,30865,30867],{"className":30866,"style":15547},[146],[38,30868],{},[38,30870],{"className":30871,"style":425},[108],[38,30873,30875,30878],{"className":30874},[102],[38,30876,286],{"className":30877},[102,103],[38,30879,30881],{"className":30880},[134],[38,30882,30884,30915],{"className":30883},[138,384],[38,30885,30887,30912],{"className":30886},[142],[38,30888,30890,30901],{"className":30889,"style":11499},[146],[38,30891,30892,30895],{"style":9585},[38,30893],{"className":30894,"style":155},[154],[38,30896,30898],{"className":30897},[159,160,161,162],[38,30899,601],{"className":30900},[102,103,162],[38,30902,30903,30906],{"style":1892},[38,30904],{"className":30905,"style":155},[154],[38,30907,30909],{"className":30908},[159,160,161,162],[38,30910,312],{"className":30911},[102,162],[38,30913,408],{"className":30914},[407],[38,30916,30918],{"className":30917},[142],[38,30919,30921],{"className":30920,"style":9616},[146],[38,30922],{},[11,30924,30925,30926,31080],{},"由于 ",[38,30927,30929,30964],{"className":30928,"translate":42},[41],[38,30930,30932],{"className":30931},[46],[48,30933,30934],{"xmlns":50},[52,30935,30936,30961],{},[55,30937,30938,30946,30948,30955,30957,30959],{},[8499,30939,30940,30942,30944],{},[58,30941,286],{},[58,30943,601],{},[300,30945,312],{},[62,30947,30522],{},[66,30949,30950,30953],{},[58,30951,30952],{},"χ",[300,30954,312],{},[62,30956,293],{"stretchy":292},[300,30958,302],{},[62,30960,330],{"stretchy":292},[81,30962,30963],{"encoding":83},"x_i^2 \\sim \\chi^2(1)",[38,30965,30967,31036],{"className":30966,"ariaHidden":89},[88],[38,30968,30970,30974,31027,31030,31033],{"className":30969},[93],[38,30971],{"className":30972,"style":30973},[97],"height:1.0728em;vertical-align:-0.2587em;",[38,30975,30977,30980],{"className":30976},[102],[38,30978,286],{"className":30979},[102,103],[38,30981,30983],{"className":30982},[134],[38,30984,30986,31018],{"className":30985},[138,384],[38,30987,30989,31015],{"className":30988},[142],[38,30990,30992,31004],{"className":30991,"style":9002},[146],[38,30993,30995,30998],{"style":30994},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[38,30996],{"className":30997,"style":155},[154],[38,30999,31001],{"className":31000},[159,160,161,162],[38,31002,601],{"className":31003},[102,103,162],[38,31005,31006,31009],{"style":150},[38,31007],{"className":31008,"style":155},[154],[38,31010,31012],{"className":31011},[159,160,161,162],[38,31013,312],{"className":31014},[102,162],[38,31016,408],{"className":31017},[407],[38,31019,31021],{"className":31020},[142],[38,31022,31025],{"className":31023,"style":31024},[146],"height:0.2587em;",[38,31026],{},[38,31028],{"className":31029,"style":109},[108],[38,31031,30522],{"className":31032},[113],[38,31034],{"className":31035,"style":109},[108],[38,31037,31039,31042,31071,31074,31077],{"className":31038},[93],[38,31040],{"className":31041,"style":13129},[97],[38,31043,31045,31048],{"className":31044},[102],[38,31046,30952],{"className":31047},[102,103],[38,31049,31051],{"className":31050},[134],[38,31052,31054],{"className":31053},[138],[38,31055,31057],{"className":31056},[142],[38,31058,31060],{"className":31059,"style":9002},[146],[38,31061,31062,31065],{"style":150},[38,31063],{"className":31064,"style":155},[154],[38,31066,31068],{"className":31067},[159,160,161,162],[38,31069,312],{"className":31070},[102,162],[38,31072,293],{"className":31073},[371],[38,31075,302],{"className":31076},[102],[38,31078,330],{"className":31079},[529],"，我们有：",[38,31082,31084],{"className":31083,"translate":42},[621],[38,31085,31087,31127],{"className":31086,"translate":42},[41],[38,31088,31090],{"className":31089},[46],[48,31091,31092],{"xmlns":50,"display":630},[52,31093,31094,31124],{},[55,31095,31096,31098,31100,31102,31104,31110,31112,31118,31120,31122],{},[58,31097,7695],{"mathvariant":333},[58,31099,7695],{"mathvariant":333},[58,31101,286],{},[58,31103,7695],{"mathvariant":333},[66,31105,31106,31108],{},[58,31107,7695],{"mathvariant":333},[300,31109,312],{},[62,31111,30522],{},[66,31113,31114,31116],{},[58,31115,30952],{},[300,31117,312],{},[62,31119,293],{"stretchy":292},[58,31121,25826],{},[62,31123,330],{"stretchy":292},[81,31125,31126],{"encoding":83},"||x||^2 \\sim \\chi^2(d)",[38,31128,31130,31183],{"className":31129,"ariaHidden":89},[88],[38,31131,31133,31136,31139,31142,31145,31174,31177,31180],{"className":31132},[93],[38,31134],{"className":31135,"style":11468},[97],[38,31137,24391],{"className":31138},[102],[38,31140,286],{"className":31141},[102,103],[38,31143,7695],{"className":31144},[102],[38,31146,31148,31151],{"className":31147},[102],[38,31149,7695],{"className":31150},[102],[38,31152,31154],{"className":31153},[134],[38,31155,31157],{"className":31156},[138],[38,31158,31160],{"className":31159},[142],[38,31161,31163],{"className":31162,"style":11499},[146],[38,31164,31165,31168],{"style":1892},[38,31166],{"className":31167,"style":155},[154],[38,31169,31171],{"className":31170},[159,160,161,162],[38,31172,312],{"className":31173},[102,162],[38,31175],{"className":31176,"style":109},[108],[38,31178,30522],{"className":31179},[113],[38,31181],{"className":31182,"style":109},[108],[38,31184,31186,31189,31218,31221,31224],{"className":31185},[93],[38,31187],{"className":31188,"style":11468},[97],[38,31190,31192,31195],{"className":31191},[102],[38,31193,30952],{"className":31194},[102,103],[38,31196,31198],{"className":31197},[134],[38,31199,31201],{"className":31200},[138],[38,31202,31204],{"className":31203},[142],[38,31205,31207],{"className":31206,"style":11499},[146],[38,31208,31209,31212],{"style":1892},[38,31210],{"className":31211,"style":155},[154],[38,31213,31215],{"className":31214},[159,160,161,162],[38,31216,312],{"className":31217},[102,162],[38,31219,293],{"className":31220},[371],[38,31222,25826],{"className":31223},[102,103],[38,31225,330],{"className":31226},[529],[11,31228,31229,31230,31304,31305,31333,31334,31391,31392,31420,31421,1964],{},"这里，",[38,31231,31233,31257],{"className":31232,"translate":42},[41],[38,31234,31236],{"className":31235},[46],[48,31237,31238],{"xmlns":50},[52,31239,31240,31254],{},[55,31241,31242,31248,31250,31252],{},[66,31243,31244,31246],{},[58,31245,30952],{},[300,31247,312],{},[62,31249,293],{"stretchy":292},[58,31251,6809],{},[62,31253,330],{"stretchy":292},[81,31255,31256],{"encoding":83},"\\chi^2(k)",[38,31258,31260],{"className":31259,"ariaHidden":89},[88],[38,31261,31263,31266,31295,31298,31301],{"className":31262},[93],[38,31264],{"className":31265,"style":13129},[97],[38,31267,31269,31272],{"className":31268},[102],[38,31270,30952],{"className":31271},[102,103],[38,31273,31275],{"className":31274},[134],[38,31276,31278],{"className":31277},[138],[38,31279,31281],{"className":31280},[142],[38,31282,31284],{"className":31283,"style":9002},[146],[38,31285,31286,31289],{"style":150},[38,31287],{"className":31288,"style":155},[154],[38,31290,31292],{"className":31291},[159,160,161,162],[38,31293,312],{"className":31294},[102,162],[38,31296,293],{"className":31297},[371],[38,31299,6809],{"className":31300,"style":6913},[102,103],[38,31302,330],{"className":31303},[529]," 是卡方分布（Chi-Squared Distribution），定义为 ",[38,31306,31308,31321],{"className":31307,"translate":42},[41],[38,31309,31311],{"className":31310},[46],[48,31312,31313],{"xmlns":50},[52,31314,31315,31319],{},[55,31316,31317],{},[58,31318,6809],{},[81,31320,6809],{"encoding":83},[38,31322,31324],{"className":31323,"ariaHidden":89},[88],[38,31325,31327,31330],{"className":31326},[93],[38,31328],{"className":31329,"style":889},[97],[38,31331,6809],{"className":31332,"style":6913},[102,103]," 个相互独立、服从标准正态分布 ",[38,31335,31337,31361],{"className":31336,"translate":42},[41],[38,31338,31340],{"className":31339},[46],[48,31341,31342],{"xmlns":50},[52,31343,31344,31358],{},[55,31345,31346,31348,31350,31352,31354,31356],{},[58,31347,14701],{"mathvariant":30528},[62,31349,293],{"stretchy":292},[300,31351,683],{},[62,31353,305],{"separator":89},[300,31355,302],{},[62,31357,330],{"stretchy":292},[81,31359,31360],{"encoding":83},"\\mathcal{N}(0,1)",[38,31362,31364],{"className":31363,"ariaHidden":89},[88],[38,31365,31367,31370,31373,31376,31379,31382,31385,31388],{"className":31366},[93],[38,31368],{"className":31369,"style":1439},[97],[38,31371,14701],{"className":31372,"style":30597},[102,30596],[38,31374,293],{"className":31375},[371],[38,31377,683],{"className":31378},[102],[38,31380,305],{"className":31381},[421],[38,31383],{"className":31384,"style":425},[108],[38,31386,302],{"className":31387},[102],[38,31389,330],{"className":31390},[529]," 的随机变量的平方和。在本例中，由于模长的平方是 ",[38,31393,31395,31408],{"className":31394,"translate":42},[41],[38,31396,31398],{"className":31397},[46],[48,31399,31400],{"xmlns":50},[52,31401,31402,31406],{},[55,31403,31404],{},[58,31405,25826],{},[81,31407,25826],{"encoding":83},[38,31409,31411],{"className":31410,"ariaHidden":89},[88],[38,31412,31414,31417],{"className":31413},[93],[38,31415],{"className":31416,"style":889},[97],[38,31418,25826],{"className":31419},[102,103]," 个这样的变量之和，所以自由度为 ",[38,31422,31424,31442],{"className":31423,"translate":42},[41],[38,31425,31427],{"className":31426},[46],[48,31428,31429],{"xmlns":50},[52,31430,31431,31439],{},[55,31432,31433,31435,31437],{},[58,31434,6809],{},[62,31436,289],{},[58,31438,25826],{},[81,31440,31441],{"encoding":83},"k=d",[38,31443,31445,31463],{"className":31444,"ariaHidden":89},[88],[38,31446,31448,31451,31454,31457,31460],{"className":31447},[93],[38,31449],{"className":31450,"style":889},[97],[38,31452,6809],{"className":31453,"style":6913},[102,103],[38,31455],{"className":31456,"style":109},[108],[38,31458,289],{"className":31459},[113],[38,31461],{"className":31462,"style":109},[108],[38,31464,31466,31469],{"className":31465},[93],[38,31467],{"className":31468,"style":889},[97],[38,31470,25826],{"className":31471},[102,103],[11,31473,31474],{},"根据大数定律：",[38,31476,31478],{"className":31477,"translate":42},[621],[38,31479,31481,31517],{"className":31480,"translate":42},[41],[38,31482,31484],{"className":31483},[46],[48,31485,31486],{"xmlns":50,"display":630},[52,31487,31488,31514],{},[55,31489,31490,31496,31498,31500,31502,31504,31510,31512],{},[636,31491,31492,31494],{},[300,31493,302],{},[58,31495,25826],{},[58,31497,7695],{"mathvariant":333},[58,31499,7695],{"mathvariant":333},[58,31501,286],{},[58,31503,7695],{"mathvariant":333},[66,31505,31506,31508],{},[58,31507,7695],{"mathvariant":333},[300,31509,312],{},[62,31511,77],{},[300,31513,302],{},[81,31515,31516],{"encoding":83},"\\frac{1}{d}||x||^2 \\to 1",[38,31518,31520,31635],{"className":31519,"ariaHidden":89},[88],[38,31521,31523,31526,31588,31591,31594,31597,31626,31629,31632],{"className":31522},[93],[38,31524],{"className":31525,"style":11305},[97],[38,31527,31529,31532,31585],{"className":31528},[102],[38,31530],{"className":31531},[371,740],[38,31533,31535],{"className":31534},[636],[38,31536,31538,31577],{"className":31537},[138,384],[38,31539,31541,31574],{"className":31540},[142],[38,31542,31544,31555,31563],{"className":31543,"style":11324},[146],[38,31545,31546,31549],{"style":756},[38,31547],{"className":31548,"style":760},[154],[38,31550,31552],{"className":31551},[102],[38,31553,25826],{"className":31554},[102,103],[38,31556,31557,31560],{"style":811},[38,31558],{"className":31559,"style":760},[154],[38,31561],{"className":31562,"style":819},[818],[38,31564,31565,31568],{"style":822},[38,31566],{"className":31567,"style":760},[154],[38,31569,31571],{"className":31570},[102],[38,31572,302],{"className":31573},[102],[38,31575,408],{"className":31576},[407],[38,31578,31580],{"className":31579},[142],[38,31581,31583],{"className":31582,"style":1079},[146],[38,31584],{},[38,31586],{"className":31587},[529,740],[38,31589,24391],{"className":31590},[102],[38,31592,286],{"className":31593},[102,103],[38,31595,7695],{"className":31596},[102],[38,31598,31600,31603],{"className":31599},[102],[38,31601,7695],{"className":31602},[102],[38,31604,31606],{"className":31605},[134],[38,31607,31609],{"className":31608},[138],[38,31610,31612],{"className":31611},[142],[38,31613,31615],{"className":31614,"style":11499},[146],[38,31616,31617,31620],{"style":1892},[38,31618],{"className":31619,"style":155},[154],[38,31621,31623],{"className":31622},[159,160,161,162],[38,31624,312],{"className":31625},[102,162],[38,31627],{"className":31628,"style":109},[108],[38,31630,77],{"className":31631},[113],[38,31633],{"className":31634,"style":109},[108],[38,31636,31638,31641],{"className":31637},[93],[38,31639],{"className":31640,"style":2429},[97],[38,31642,302],{"className":31643},[102],[11,31645,31646,31647,1964],{},"也就是说，向量的模长近似为 ",[38,31648,31650,31679],{"className":31649,"translate":42},[41],[38,31651,31653],{"className":31652},[46],[48,31654,31655],{"xmlns":50},[52,31656,31657,31676],{},[55,31658,31659,31661,31663,31665,31667,31669,31672],{},[58,31660,7695],{"mathvariant":333},[58,31662,7695],{"mathvariant":333},[58,31664,286],{},[58,31666,7695],{"mathvariant":333},[58,31668,7695],{"mathvariant":333},[62,31670,31671],{},"≈",[20042,31673,31674],{},[58,31675,25826],{},[81,31677,31678],{"encoding":83},"||x|| \\approx \\sqrt{d}",[38,31680,31682,31706],{"className":31681,"ariaHidden":89},[88],[38,31683,31685,31688,31691,31694,31697,31700,31703],{"className":31684},[93],[38,31686],{"className":31687,"style":1439},[97],[38,31689,24391],{"className":31690},[102],[38,31692,286],{"className":31693},[102,103],[38,31695,24391],{"className":31696},[102],[38,31698],{"className":31699,"style":109},[108],[38,31701,31671],{"className":31702},[113],[38,31704],{"className":31705,"style":109},[108],[38,31707,31709,31713],{"className":31708},[93],[38,31710],{"className":31711,"style":31712},[97],"height:1.04em;vertical-align:-0.1078em;",[38,31714,31716],{"className":31715},[102,20187],[38,31717,31719,31754],{"className":31718},[138,384],[38,31720,31722,31751],{"className":31721},[142],[38,31723,31726,31738],{"className":31724,"style":31725},[146],"height:0.9322em;",[38,31727,31729,31732],{"className":31728,"style":913},[20201],[38,31730],{"className":31731,"style":760},[154],[38,31733,31735],{"className":31734,"style":21076},[102],[38,31736,25826],{"className":31737},[102,103],[38,31739,31741,31744],{"style":31740},"top:-2.8922em;",[38,31742],{"className":31743,"style":760},[154],[38,31745,31747],{"className":31746,"style":21101},[20343],[4842,31748,31749],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,31750],{"d":21108},[38,31752,408],{"className":31753},[407],[38,31755,31757],{"className":31756},[142],[38,31758,31761],{"className":31759,"style":31760},[146],"height:0.1078em;",[38,31762],{},[11,31764,31765],{},"欧氏距离为：",[38,31767,31769],{"className":31768,"translate":42},[621],[38,31770,31772,31850],{"className":31771,"translate":42},[41],[38,31773,31775],{"className":31774},[46],[48,31776,31777],{"xmlns":50,"display":630},[52,31778,31779,31847],{},[55,31780,31781,31783,31785,31787,31789,31791,31793,31799,31801,31803,31805,31807,31809,31815,31817,31819,31821,31823,31825,31831,31833,31835,31838,31840,31842,31844],{},[58,31782,7695],{"mathvariant":333},[58,31784,7695],{"mathvariant":333},[58,31786,286],{},[62,31788,710],{},[58,31790,26054],{},[58,31792,7695],{"mathvariant":333},[66,31794,31795,31797],{},[58,31796,7695],{"mathvariant":333},[300,31798,312],{},[62,31800,289],{},[58,31802,7695],{"mathvariant":333},[58,31804,7695],{"mathvariant":333},[58,31806,286],{},[58,31808,7695],{"mathvariant":333},[66,31810,31811,31813],{},[58,31812,7695],{"mathvariant":333},[300,31814,312],{},[62,31816,696],{},[58,31818,7695],{"mathvariant":333},[58,31820,7695],{"mathvariant":333},[58,31822,26054],{},[58,31824,7695],{"mathvariant":333},[66,31826,31827,31829],{},[58,31828,7695],{"mathvariant":333},[300,31830,312],{},[62,31832,710],{},[300,31834,312],{},[62,31836,31837],{"stretchy":292},"⟨",[58,31839,286],{},[62,31841,305],{"separator":89},[58,31843,26054],{},[62,31845,31846],{"stretchy":292},"⟩",[81,31848,31849],{"encoding":83},"||x - y||^2 = ||x||^2 + ||y||^2 - 2 \\langle x, y \\rangle",[38,31851,31853,31874,31924,31977,32030],{"className":31852,"ariaHidden":89},[88],[38,31854,31856,31859,31862,31865,31868,31871],{"className":31855},[93],[38,31857],{"className":31858,"style":1439},[97],[38,31860,24391],{"className":31861},[102],[38,31863,286],{"className":31864},[102,103],[38,31866],{"className":31867,"style":995},[108],[38,31869,710],{"className":31870},[999],[38,31872],{"className":31873,"style":995},[108],[38,31875,31877,31880,31883,31886,31915,31918,31921],{"className":31876},[93],[38,31878],{"className":31879,"style":11468},[97],[38,31881,26054],{"className":31882,"style":3420},[102,103],[38,31884,7695],{"className":31885},[102],[38,31887,31889,31892],{"className":31888},[102],[38,31890,7695],{"className":31891},[102],[38,31893,31895],{"className":31894},[134],[38,31896,31898],{"className":31897},[138],[38,31899,31901],{"className":31900},[142],[38,31902,31904],{"className":31903,"style":11499},[146],[38,31905,31906,31909],{"style":1892},[38,31907],{"className":31908,"style":155},[154],[38,31910,31912],{"className":31911},[159,160,161,162],[38,31913,312],{"className":31914},[102,162],[38,31916],{"className":31917,"style":109},[108],[38,31919,289],{"className":31920},[113],[38,31922],{"className":31923,"style":109},[108],[38,31925,31927,31930,31933,31936,31939,31968,31971,31974],{"className":31926},[93],[38,31928],{"className":31929,"style":11468},[97],[38,31931,24391],{"className":31932},[102],[38,31934,286],{"className":31935},[102,103],[38,31937,7695],{"className":31938},[102],[38,31940,31942,31945],{"className":31941},[102],[38,31943,7695],{"className":31944},[102],[38,31946,31948],{"className":31947},[134],[38,31949,31951],{"className":31950},[138],[38,31952,31954],{"className":31953},[142],[38,31955,31957],{"className":31956,"style":11499},[146],[38,31958,31959,31962],{"style":1892},[38,31960],{"className":31961,"style":155},[154],[38,31963,31965],{"className":31964},[159,160,161,162],[38,31966,312],{"className":31967},[102,162],[38,31969],{"className":31970,"style":995},[108],[38,31972,696],{"className":31973},[999],[38,31975],{"className":31976,"style":995},[108],[38,31978,31980,31983,31986,31989,31992,32021,32024,32027],{"className":31979},[93],[38,31981],{"className":31982,"style":11468},[97],[38,31984,24391],{"className":31985},[102],[38,31987,26054],{"className":31988,"style":3420},[102,103],[38,31990,7695],{"className":31991},[102],[38,31993,31995,31998],{"className":31994},[102],[38,31996,7695],{"className":31997},[102],[38,31999,32001],{"className":32000},[134],[38,32002,32004],{"className":32003},[138],[38,32005,32007],{"className":32006},[142],[38,32008,32010],{"className":32009,"style":11499},[146],[38,32011,32012,32015],{"style":1892},[38,32013],{"className":32014,"style":155},[154],[38,32016,32018],{"className":32017},[159,160,161,162],[38,32019,312],{"className":32020},[102,162],[38,32022],{"className":32023,"style":995},[108],[38,32025,710],{"className":32026},[999],[38,32028],{"className":32029,"style":995},[108],[38,32031,32033,32036,32039,32042,32045,32048,32051,32054],{"className":32032},[93],[38,32034],{"className":32035,"style":1439},[97],[38,32037,312],{"className":32038},[102],[38,32040,31837],{"className":32041},[371],[38,32043,286],{"className":32044},[102,103],[38,32046,305],{"className":32047},[421],[38,32049],{"className":32050,"style":425},[108],[38,32052,26054],{"className":32053,"style":3420},[102,103],[38,32055,31846],{"className":32056},[529],[11,32058,32059],{},"我们已知：",[38,32061,32063],{"className":32062,"translate":42},[621],[38,32064,32066,32118],{"className":32065,"translate":42},[41],[38,32067,32069],{"className":32068},[46],[48,32070,32071],{"xmlns":50,"display":630},[52,32072,32073,32115],{},[55,32074,32075,32077,32079,32081,32083,32089,32091,32093,32095,32097,32099,32101,32103,32105,32111,32113],{},[58,32076,7695],{"mathvariant":333},[58,32078,7695],{"mathvariant":333},[58,32080,286],{},[58,32082,7695],{"mathvariant":333},[66,32084,32085,32087],{},[58,32086,7695],{"mathvariant":333},[300,32088,312],{},[62,32090,31671],{},[58,32092,25826],{},[62,32094,305],{"separator":89},[108,32096],{"width":4479},[58,32098,7695],{"mathvariant":333},[58,32100,7695],{"mathvariant":333},[58,32102,26054],{},[58,32104,7695],{"mathvariant":333},[66,32106,32107,32109],{},[58,32108,7695],{"mathvariant":333},[300,32110,312],{},[62,32112,31671],{},[58,32114,25826],{},[81,32116,32117],{"encoding":83},"||x||^2 \\approx d, \\quad ||y||^2 \\approx d",[38,32119,32121,32174,32239],{"className":32120,"ariaHidden":89},[88],[38,32122,32124,32127,32130,32133,32136,32165,32168,32171],{"className":32123},[93],[38,32125],{"className":32126,"style":11468},[97],[38,32128,24391],{"className":32129},[102],[38,32131,286],{"className":32132},[102,103],[38,32134,7695],{"className":32135},[102],[38,32137,32139,32142],{"className":32138},[102],[38,32140,7695],{"className":32141},[102],[38,32143,32145],{"className":32144},[134],[38,32146,32148],{"className":32147},[138],[38,32149,32151],{"className":32150},[142],[38,32152,32154],{"className":32153,"style":11499},[146],[38,32155,32156,32159],{"style":1892},[38,32157],{"className":32158,"style":155},[154],[38,32160,32162],{"className":32161},[159,160,161,162],[38,32163,312],{"className":32164},[102,162],[38,32166],{"className":32167,"style":109},[108],[38,32169,31671],{"className":32170},[113],[38,32172],{"className":32173,"style":109},[108],[38,32175,32177,32180,32183,32186,32189,32192,32195,32198,32201,32230,32233,32236],{"className":32176},[93],[38,32178],{"className":32179,"style":11468},[97],[38,32181,25826],{"className":32182},[102,103],[38,32184,305],{"className":32185},[421],[38,32187],{"className":32188,"style":10740},[108],[38,32190],{"className":32191,"style":425},[108],[38,32193,24391],{"className":32194},[102],[38,32196,26054],{"className":32197,"style":3420},[102,103],[38,32199,7695],{"className":32200},[102],[38,32202,32204,32207],{"className":32203},[102],[38,32205,7695],{"className":32206},[102],[38,32208,32210],{"className":32209},[134],[38,32211,32213],{"className":32212},[138],[38,32214,32216],{"className":32215},[142],[38,32217,32219],{"className":32218,"style":11499},[146],[38,32220,32221,32224],{"style":1892},[38,32222],{"className":32223,"style":155},[154],[38,32225,32227],{"className":32226},[159,160,161,162],[38,32228,312],{"className":32229},[102,162],[38,32231],{"className":32232,"style":109},[108],[38,32234,31671],{"className":32235},[113],[38,32237],{"className":32238,"style":109},[108],[38,32240,32242,32245],{"className":32241},[93],[38,32243],{"className":32244,"style":889},[97],[38,32246,25826],{"className":32247},[102,103],[11,32249,13448,32250,32292],{},[38,32251,32253,32271],{"className":32252,"translate":42},[41],[38,32254,32256],{"className":32255},[46],[48,32257,32258],{"xmlns":50},[52,32259,32260,32268],{},[55,32261,32262,32264,32266],{},[58,32263,286],{},[62,32265,305],{"separator":89},[58,32267,26054],{},[81,32269,32270],{"encoding":83},"x, y",[38,32272,32274],{"className":32273,"ariaHidden":89},[88],[38,32275,32277,32280,32283,32286,32289],{"className":32276},[93],[38,32278],{"className":32279,"style":3505},[97],[38,32281,286],{"className":32282},[102,103],[38,32284,305],{"className":32285},[421],[38,32287],{"className":32288,"style":425},[108],[38,32290,26054],{"className":32291,"style":3420},[102,103]," 的内积为：",[38,32294,32296],{"className":32295,"translate":42},[621],[38,32297,32299,32349],{"className":32298,"translate":42},[41],[38,32300,32302],{"className":32301},[46],[48,32303,32304],{"xmlns":50,"display":630},[52,32305,32306,32346],{},[55,32307,32308,32310,32312,32314,32316,32318,32320,32334,32340],{},[62,32309,31837],{"stretchy":292},[58,32311,286],{},[62,32313,305],{"separator":89},[58,32315,26054],{},[62,32317,31846],{"stretchy":292},[62,32319,289],{},[15353,32321,32322,32324,32332],{},[62,32323,15357],{},[55,32325,32326,32328,32330],{},[58,32327,601],{},[62,32329,289],{},[300,32331,302],{},[58,32333,25826],{},[295,32335,32336,32338],{},[58,32337,286],{},[58,32339,601],{},[295,32341,32342,32344],{},[58,32343,26054],{},[58,32345,601],{},[81,32347,32348],{"encoding":83},"\\langle x, y \\rangle = \\sum_{i=1}^d x_i y_i",[38,32350,32352,32385],{"className":32351,"ariaHidden":89},[88],[38,32353,32355,32358,32361,32364,32367,32370,32373,32376,32379,32382],{"className":32354},[93],[38,32356],{"className":32357,"style":1439},[97],[38,32359,31837],{"className":32360},[371],[38,32362,286],{"className":32363},[102,103],[38,32365,305],{"className":32366},[421],[38,32368],{"className":32369,"style":425},[108],[38,32371,26054],{"className":32372,"style":3420},[102,103],[38,32374,31846],{"className":32375},[529],[38,32377],{"className":32378,"style":109},[108],[38,32380,289],{"className":32381},[113],[38,32383],{"className":32384,"style":109},[108],[38,32386,32388,32391,32455,32458,32498],{"className":32387},[93],[38,32389],{"className":32390,"style":30803},[97],[38,32392,32394],{"className":32393},[878,879],[38,32395,32397,32447],{"className":32396},[138,384],[38,32398,32400,32444],{"className":32399},[142],[38,32401,32403,32423,32433],{"className":32402,"style":30816},[146],[38,32404,32405,32408],{"style":15491},[38,32406],{"className":32407,"style":8852},[154],[38,32409,32411],{"className":32410},[159,160,161,162],[38,32412,32414,32417,32420],{"className":32413},[102,162],[38,32415,601],{"className":32416},[102,103,162],[38,32418,289],{"className":32419},[113,162],[38,32421,302],{"className":32422},[102,162],[38,32424,32425,32428],{"style":15512},[38,32426],{"className":32427,"style":8852},[154],[38,32429,32430],{},[38,32431,15357],{"className":32432},[878,15521,15522],[38,32434,32435,32438],{"style":15525},[38,32436],{"className":32437,"style":8852},[154],[38,32439,32441],{"className":32440},[159,160,161,162],[38,32442,25826],{"className":32443},[102,103,162],[38,32445,408],{"className":32446},[407],[38,32448,32450],{"className":32449},[142],[38,32451,32453],{"className":32452,"style":15547},[146],[38,32454],{},[38,32456],{"className":32457,"style":425},[108],[38,32459,32461,32464],{"className":32460},[102],[38,32462,286],{"className":32463},[102,103],[38,32465,32467],{"className":32466},[134],[38,32468,32470,32490],{"className":32469},[138,384],[38,32471,32473,32487],{"className":32472},[142],[38,32474,32476],{"className":32475,"style":786},[146],[38,32477,32478,32481],{"style":394},[38,32479],{"className":32480,"style":155},[154],[38,32482,32484],{"className":32483},[159,160,161,162],[38,32485,601],{"className":32486},[102,103,162],[38,32488,408],{"className":32489},[407],[38,32491,32493],{"className":32492},[142],[38,32494,32496],{"className":32495,"style":415},[146],[38,32497],{},[38,32499,32501,32504],{"className":32500},[102],[38,32502,26054],{"className":32503,"style":3420},[102,103],[38,32505,32507],{"className":32506},[134],[38,32508,32510,32530],{"className":32509},[138,384],[38,32511,32513,32527],{"className":32512},[142],[38,32514,32516],{"className":32515,"style":786},[146],[38,32517,32518,32521],{"style":15141},[38,32519],{"className":32520,"style":155},[154],[38,32522,32524],{"className":32523},[159,160,161,162],[38,32525,601],{"className":32526},[102,103,162],[38,32528,408],{"className":32529},[407],[38,32531,32533],{"className":32532},[142],[38,32534,32536],{"className":32535,"style":415},[146],[38,32537],{},[11,32539,32540],{},"由于：",[7370,32542,32543,32670,32826],{},[7373,32544,32545,32669],{},[38,32546,32548,32574],{"className":32547,"translate":42},[41],[38,32549,32551],{"className":32550},[46],[48,32552,32553],{"xmlns":50},[52,32554,32555,32571],{},[55,32556,32557,32563,32565],{},[295,32558,32559,32561],{},[58,32560,286],{},[58,32562,601],{},[62,32564,305],{"separator":89},[295,32566,32567,32569],{},[58,32568,26054],{},[58,32570,601],{},[81,32572,32573],{"encoding":83},"x_i, y_i",[38,32575,32577],{"className":32576,"ariaHidden":89},[88],[38,32578,32580,32583,32623,32626,32629],{"className":32579},[93],[38,32581],{"className":32582,"style":3505},[97],[38,32584,32586,32589],{"className":32585},[102],[38,32587,286],{"className":32588},[102,103],[38,32590,32592],{"className":32591},[134],[38,32593,32595,32615],{"className":32594},[138,384],[38,32596,32598,32612],{"className":32597},[142],[38,32599,32601],{"className":32600,"style":786},[146],[38,32602,32603,32606],{"style":394},[38,32604],{"className":32605,"style":155},[154],[38,32607,32609],{"className":32608},[159,160,161,162],[38,32610,601],{"className":32611},[102,103,162],[38,32613,408],{"className":32614},[407],[38,32616,32618],{"className":32617},[142],[38,32619,32621],{"className":32620,"style":415},[146],[38,32622],{},[38,32624,305],{"className":32625},[421],[38,32627],{"className":32628,"style":425},[108],[38,32630,32632,32635],{"className":32631},[102],[38,32633,26054],{"className":32634,"style":3420},[102,103],[38,32636,32638],{"className":32637},[134],[38,32639,32641,32661],{"className":32640},[138,384],[38,32642,32644,32658],{"className":32643},[142],[38,32645,32647],{"className":32646,"style":786},[146],[38,32648,32649,32652],{"style":15141},[38,32650],{"className":32651,"style":155},[154],[38,32653,32655],{"className":32654},[159,160,161,162],[38,32656,601],{"className":32657},[102,103,162],[38,32659,408],{"className":32660},[407],[38,32662,32664],{"className":32663},[142],[38,32665,32667],{"className":32666,"style":415},[146],[38,32668],{}," 相互独立",[7373,32671,32672],{},[38,32673,32675,32710],{"className":32674,"translate":42},[41],[38,32676,32678],{"className":32677},[46],[48,32679,32680],{"xmlns":50},[52,32681,32682,32707],{},[55,32683,32684,32687,32689,32695,32701,32703,32705],{},[58,32685,32686],{},"E",[62,32688,25519],{"stretchy":292},[295,32690,32691,32693],{},[58,32692,286],{},[58,32694,601],{},[295,32696,32697,32699],{},[58,32698,26054],{},[58,32700,601],{},[62,32702,25530],{"stretchy":292},[62,32704,289],{},[300,32706,683],{},[81,32708,32709],{"encoding":83},"E[x_i y_i] = 0",[38,32711,32713,32817],{"className":32712,"ariaHidden":89},[88],[38,32714,32716,32719,32722,32725,32765,32805,32808,32811,32814],{"className":32715},[93],[38,32717],{"className":32718,"style":1439},[97],[38,32720,32686],{"className":32721,"style":25981},[102,103],[38,32723,25519],{"className":32724},[371],[38,32726,32728,32731],{"className":32727},[102],[38,32729,286],{"className":32730},[102,103],[38,32732,32734],{"className":32733},[134],[38,32735,32737,32757],{"className":32736},[138,384],[38,32738,32740,32754],{"className":32739},[142],[38,32741,32743],{"className":32742,"style":786},[146],[38,32744,32745,32748],{"style":394},[38,32746],{"className":32747,"style":155},[154],[38,32749,32751],{"className":32750},[159,160,161,162],[38,32752,601],{"className":32753},[102,103,162],[38,32755,408],{"className":32756},[407],[38,32758,32760],{"className":32759},[142],[38,32761,32763],{"className":32762,"style":415},[146],[38,32764],{},[38,32766,32768,32771],{"className":32767},[102],[38,32769,26054],{"className":32770,"style":3420},[102,103],[38,32772,32774],{"className":32773},[134],[38,32775,32777,32797],{"className":32776},[138,384],[38,32778,32780,32794],{"className":32779},[142],[38,32781,32783],{"className":32782,"style":786},[146],[38,32784,32785,32788],{"style":15141},[38,32786],{"className":32787,"style":155},[154],[38,32789,32791],{"className":32790},[159,160,161,162],[38,32792,601],{"className":32793},[102,103,162],[38,32795,408],{"className":32796},[407],[38,32798,32800],{"className":32799},[142],[38,32801,32803],{"className":32802,"style":415},[146],[38,32804],{},[38,32806,25530],{"className":32807},[529],[38,32809],{"className":32810,"style":109},[108],[38,32812,289],{"className":32813},[113],[38,32815],{"className":32816,"style":109},[108],[38,32818,32820,32823],{"className":32819},[93],[38,32821],{"className":32822,"style":2429},[97],[38,32824,683],{"className":32825},[102],[7373,32827,32828],{},[38,32829,32831,32866],{"className":32830,"translate":42},[41],[38,32832,32834],{"className":32833},[46],[48,32835,32836],{"xmlns":50},[52,32837,32838,32863],{},[55,32839,32840,32843,32845,32851,32857,32859,32861],{},[3265,32841,32842],{},"Var",[62,32844,293],{"stretchy":292},[295,32846,32847,32849],{},[58,32848,286],{},[58,32850,601],{},[295,32852,32853,32855],{},[58,32854,26054],{},[58,32856,601],{},[62,32858,330],{"stretchy":292},[62,32860,289],{},[300,32862,302],{},[81,32864,32865],{"encoding":83},"\\text{Var}(x_i y_i) = 1",[38,32867,32869,32976],{"className":32868,"ariaHidden":89},[88],[38,32870,32872,32875,32881,32884,32924,32964,32967,32970,32973],{"className":32871},[93],[38,32873],{"className":32874,"style":1439},[97],[38,32876,32878],{"className":32877},[102,22531],[38,32879,32842],{"className":32880},[102],[38,32882,293],{"className":32883},[371],[38,32885,32887,32890],{"className":32886},[102],[38,32888,286],{"className":32889},[102,103],[38,32891,32893],{"className":32892},[134],[38,32894,32896,32916],{"className":32895},[138,384],[38,32897,32899,32913],{"className":32898},[142],[38,32900,32902],{"className":32901,"style":786},[146],[38,32903,32904,32907],{"style":394},[38,32905],{"className":32906,"style":155},[154],[38,32908,32910],{"className":32909},[159,160,161,162],[38,32911,601],{"className":32912},[102,103,162],[38,32914,408],{"className":32915},[407],[38,32917,32919],{"className":32918},[142],[38,32920,32922],{"className":32921,"style":415},[146],[38,32923],{},[38,32925,32927,32930],{"className":32926},[102],[38,32928,26054],{"className":32929,"style":3420},[102,103],[38,32931,32933],{"className":32932},[134],[38,32934,32936,32956],{"className":32935},[138,384],[38,32937,32939,32953],{"className":32938},[142],[38,32940,32942],{"className":32941,"style":786},[146],[38,32943,32944,32947],{"style":15141},[38,32945],{"className":32946,"style":155},[154],[38,32948,32950],{"className":32949},[159,160,161,162],[38,32951,601],{"className":32952},[102,103,162],[38,32954,408],{"className":32955},[407],[38,32957,32959],{"className":32958},[142],[38,32960,32962],{"className":32961,"style":415},[146],[38,32963],{},[38,32965,330],{"className":32966},[529],[38,32968],{"className":32969,"style":109},[108],[38,32971,289],{"className":32972},[113],[38,32974],{"className":32975,"style":109},[108],[38,32977,32979,32982],{"className":32978},[93],[38,32980],{"className":32981,"style":2429},[97],[38,32983,302],{"className":32984},[102],[11,32986,32987],{},"因此：",[38,32989,32991],{"className":32990,"translate":42},[621],[38,32992,32994,33030],{"className":32993,"translate":42},[41],[38,32995,32997],{"className":32996},[46],[48,32998,32999],{"xmlns":50,"display":630},[52,33000,33001,33027],{},[55,33002,33003,33005,33007,33009,33011,33013,33015,33017,33019,33021,33023,33025],{},[62,33004,31837],{"stretchy":292},[58,33006,286],{},[62,33008,305],{"separator":89},[58,33010,26054],{},[62,33012,31846],{"stretchy":292},[62,33014,30522],{},[58,33016,14701],{"mathvariant":30528},[62,33018,293],{"stretchy":292},[300,33020,683],{},[62,33022,305],{"separator":89},[58,33024,25826],{},[62,33026,330],{"stretchy":292},[81,33028,33029],{"encoding":83},"\\langle x, y \\rangle \\sim \\mathcal{N}(0, d)",[38,33031,33033,33066],{"className":33032,"ariaHidden":89},[88],[38,33034,33036,33039,33042,33045,33048,33051,33054,33057,33060,33063],{"className":33035},[93],[38,33037],{"className":33038,"style":1439},[97],[38,33040,31837],{"className":33041},[371],[38,33043,286],{"className":33044},[102,103],[38,33046,305],{"className":33047},[421],[38,33049],{"className":33050,"style":425},[108],[38,33052,26054],{"className":33053,"style":3420},[102,103],[38,33055,31846],{"className":33056},[529],[38,33058],{"className":33059,"style":109},[108],[38,33061,30522],{"className":33062},[113],[38,33064],{"className":33065,"style":109},[108],[38,33067,33069,33072,33075,33078,33081,33084,33087,33090],{"className":33068},[93],[38,33070],{"className":33071,"style":1439},[97],[38,33073,14701],{"className":33074,"style":30597},[102,30596],[38,33076,293],{"className":33077},[371],[38,33079,683],{"className":33080},[102],[38,33082,305],{"className":33083},[421],[38,33085],{"className":33086,"style":425},[108],[38,33088,25826],{"className":33089},[102,103],[38,33091,330],{"className":33092},[529],[11,33094,33095,33096,1964],{},"其量级为 ",[38,33097,33099,33121],{"className":33098,"translate":42},[41],[38,33100,33102],{"className":33101},[46],[48,33103,33104],{"xmlns":50},[52,33105,33106,33118],{},[55,33107,33108,33110,33112,33116],{},[58,33109,13105],{},[62,33111,293],{"stretchy":292},[20042,33113,33114],{},[58,33115,25826],{},[62,33117,330],{"stretchy":292},[81,33119,33120],{"encoding":83},"O(\\sqrt{d})",[38,33122,33124],{"className":33123,"ariaHidden":89},[88],[38,33125,33127,33131,33134,33137,33184],{"className":33126},[93],[38,33128],{"className":33129,"style":33130},[97],"height:1.1822em;vertical-align:-0.25em;",[38,33132,13105],{"className":33133,"style":2500},[102,103],[38,33135,293],{"className":33136},[371],[38,33138,33140],{"className":33139},[102,20187],[38,33141,33143,33176],{"className":33142},[138,384],[38,33144,33146,33173],{"className":33145},[142],[38,33147,33149,33161],{"className":33148,"style":31725},[146],[38,33150,33152,33155],{"className":33151,"style":913},[20201],[38,33153],{"className":33154,"style":760},[154],[38,33156,33158],{"className":33157,"style":21076},[102],[38,33159,25826],{"className":33160},[102,103],[38,33162,33163,33166],{"style":31740},[38,33164],{"className":33165,"style":760},[154],[38,33167,33169],{"className":33168,"style":21101},[20343],[4842,33170,33171],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,33172],{"d":21108},[38,33174,408],{"className":33175},[407],[38,33177,33179],{"className":33178},[142],[38,33180,33182],{"className":33181,"style":31760},[146],[38,33183],{},[38,33185,330],{"className":33186},[529],[11,33188,33189],{},"代回距离方程，可得：",[38,33191,33193],{"className":33192,"translate":42},[621],[38,33194,33196,33295],{"className":33195,"translate":42},[41],[38,33197,33199],{"className":33198},[46],[48,33200,33201],{"xmlns":50,"display":630},[52,33202,33203,33292],{},[55,33204,33205,33207,33209,33211,33213,33215,33217,33223,33225,33227,33229,33231,33233,33235,33237,33239,33241,33245,33247,33249,33251,33253,33255,33257,33259,33263,33265,33268,33270,33272,33274,33276,33278,33280,33286,33288,33290],{},[58,33206,7695],{"mathvariant":333},[58,33208,7695],{"mathvariant":333},[58,33210,286],{},[62,33212,710],{},[58,33214,26054],{},[58,33216,7695],{"mathvariant":333},[66,33218,33219,33221],{},[58,33220,7695],{"mathvariant":333},[300,33222,312],{},[62,33224,289],{},[58,33226,25826],{},[62,33228,696],{},[58,33230,25826],{},[62,33232,710],{},[300,33234,312],{},[62,33236,17039],{},[58,33238,13105],{},[62,33240,293],{"stretchy":292},[20042,33242,33243],{},[58,33244,25826],{},[62,33246,330],{"stretchy":292},[62,33248,289],{},[300,33250,312],{},[58,33252,25826],{},[62,33254,696],{},[58,33256,13105],{},[62,33258,293],{"stretchy":292},[20042,33260,33261],{},[58,33262,25826],{},[62,33264,330],{"stretchy":292},[108,33266],{"linebreak":33267},"newline",[58,33269,7695],{"mathvariant":333},[58,33271,7695],{"mathvariant":333},[58,33273,286],{},[62,33275,710],{},[58,33277,26054],{},[58,33279,7695],{"mathvariant":333},[66,33281,33282,33284],{},[58,33283,7695],{"mathvariant":333},[300,33285,312],{},[62,33287,31671],{},[300,33289,312],{},[58,33291,25826],{},[81,33293,33294],{"encoding":83},"||x - y||^2\n= d + d - 2 \\cdot O(\\sqrt{d})\n= 2d + O(\\sqrt{d}) \\\\\n||x - y||^2 \\approx 2d",[38,33296,33298,33319,33369,33388,33406,33424,33499,33520,33582,33585,33606,33656],{"className":33297,"ariaHidden":89},[88],[38,33299,33301,33304,33307,33310,33313,33316],{"className":33300},[93],[38,33302],{"className":33303,"style":1439},[97],[38,33305,24391],{"className":33306},[102],[38,33308,286],{"className":33309},[102,103],[38,33311],{"className":33312,"style":995},[108],[38,33314,710],{"className":33315},[999],[38,33317],{"className":33318,"style":995},[108],[38,33320,33322,33325,33328,33331,33360,33363,33366],{"className":33321},[93],[38,33323],{"className":33324,"style":11468},[97],[38,33326,26054],{"className":33327,"style":3420},[102,103],[38,33329,7695],{"className":33330},[102],[38,33332,33334,33337],{"className":33333},[102],[38,33335,7695],{"className":33336},[102],[38,33338,33340],{"className":33339},[134],[38,33341,33343],{"className":33342},[138],[38,33344,33346],{"className":33345},[142],[38,33347,33349],{"className":33348,"style":11499},[146],[38,33350,33351,33354],{"style":1892},[38,33352],{"className":33353,"style":155},[154],[38,33355,33357],{"className":33356},[159,160,161,162],[38,33358,312],{"className":33359},[102,162],[38,33361],{"className":33362,"style":109},[108],[38,33364,289],{"className":33365},[113],[38,33367],{"className":33368,"style":109},[108],[38,33370,33372,33376,33379,33382,33385],{"className":33371},[93],[38,33373],{"className":33374,"style":33375},[97],"height:0.7778em;vertical-align:-0.0833em;",[38,33377,25826],{"className":33378},[102,103],[38,33380],{"className":33381,"style":995},[108],[38,33383,696],{"className":33384},[999],[38,33386],{"className":33387,"style":995},[108],[38,33389,33391,33394,33397,33400,33403],{"className":33390},[93],[38,33392],{"className":33393,"style":33375},[97],[38,33395,25826],{"className":33396},[102,103],[38,33398],{"className":33399,"style":995},[108],[38,33401,710],{"className":33402},[999],[38,33404],{"className":33405,"style":995},[108],[38,33407,33409,33412,33415,33418,33421],{"className":33408},[93],[38,33410],{"className":33411,"style":2429},[97],[38,33413,312],{"className":33414},[102],[38,33416],{"className":33417,"style":995},[108],[38,33419,17039],{"className":33420},[999],[38,33422],{"className":33423,"style":995},[108],[38,33425,33427,33431,33434,33437,33487,33490,33493,33496],{"className":33426},[93],[38,33428],{"className":33429,"style":33430},[97],"height:1.2311em;vertical-align:-0.25em;",[38,33432,13105],{"className":33433,"style":2500},[102,103],[38,33435,293],{"className":33436},[371],[38,33438,33440],{"className":33439},[102,20187],[38,33441,33443,33478],{"className":33442},[138,384],[38,33444,33446,33475],{"className":33445},[142],[38,33447,33450,33462],{"className":33448,"style":33449},[146],"height:0.9811em;",[38,33451,33453,33456],{"className":33452,"style":913},[20201],[38,33454],{"className":33455,"style":760},[154],[38,33457,33459],{"className":33458,"style":21076},[102],[38,33460,25826],{"className":33461},[102,103],[38,33463,33465,33468],{"style":33464},"top:-2.9411em;",[38,33466],{"className":33467,"style":760},[154],[38,33469,33471],{"className":33470,"style":21101},[20343],[4842,33472,33473],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,33474],{"d":21108},[38,33476,408],{"className":33477},[407],[38,33479,33481],{"className":33480},[142],[38,33482,33485],{"className":33483,"style":33484},[146],"height:0.0589em;",[38,33486],{},[38,33488,330],{"className":33489},[529],[38,33491],{"className":33492,"style":109},[108],[38,33494,289],{"className":33495},[113],[38,33497],{"className":33498,"style":109},[108],[38,33500,33502,33505,33508,33511,33514,33517],{"className":33501},[93],[38,33503],{"className":33504,"style":33375},[97],[38,33506,312],{"className":33507},[102],[38,33509,25826],{"className":33510},[102,103],[38,33512],{"className":33513,"style":995},[108],[38,33515,696],{"className":33516},[999],[38,33518],{"className":33519,"style":995},[108],[38,33521,33523,33526,33529,33532,33579],{"className":33522},[93],[38,33524],{"className":33525,"style":33430},[97],[38,33527,13105],{"className":33528,"style":2500},[102,103],[38,33530,293],{"className":33531},[371],[38,33533,33535],{"className":33534},[102,20187],[38,33536,33538,33571],{"className":33537},[138,384],[38,33539,33541,33568],{"className":33540},[142],[38,33542,33544,33556],{"className":33543,"style":33449},[146],[38,33545,33547,33550],{"className":33546,"style":913},[20201],[38,33548],{"className":33549,"style":760},[154],[38,33551,33553],{"className":33552,"style":21076},[102],[38,33554,25826],{"className":33555},[102,103],[38,33557,33558,33561],{"style":33464},[38,33559],{"className":33560,"style":760},[154],[38,33562,33564],{"className":33563,"style":21101},[20343],[4842,33565,33566],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,33567],{"d":21108},[38,33569,408],{"className":33570},[407],[38,33572,33574],{"className":33573},[142],[38,33575,33577],{"className":33576,"style":33484},[146],[38,33578],{},[38,33580,330],{"className":33581},[529],[38,33583],{"className":33584},[108,33267],[38,33586,33588,33591,33594,33597,33600,33603],{"className":33587},[93],[38,33589],{"className":33590,"style":1439},[97],[38,33592,24391],{"className":33593},[102],[38,33595,286],{"className":33596},[102,103],[38,33598],{"className":33599,"style":995},[108],[38,33601,710],{"className":33602},[999],[38,33604],{"className":33605,"style":995},[108],[38,33607,33609,33612,33615,33618,33647,33650,33653],{"className":33608},[93],[38,33610],{"className":33611,"style":11468},[97],[38,33613,26054],{"className":33614,"style":3420},[102,103],[38,33616,7695],{"className":33617},[102],[38,33619,33621,33624],{"className":33620},[102],[38,33622,7695],{"className":33623},[102],[38,33625,33627],{"className":33626},[134],[38,33628,33630],{"className":33629},[138],[38,33631,33633],{"className":33632},[142],[38,33634,33636],{"className":33635,"style":11499},[146],[38,33637,33638,33641],{"style":1892},[38,33639],{"className":33640,"style":155},[154],[38,33642,33644],{"className":33643},[159,160,161,162],[38,33645,312],{"className":33646},[102,162],[38,33648],{"className":33649,"style":109},[108],[38,33651,31671],{"className":33652},[113],[38,33654],{"className":33655,"style":109},[108],[38,33657,33659,33662,33665],{"className":33658},[93],[38,33660],{"className":33661,"style":889},[97],[38,33663,312],{"className":33664},[102],[38,33666,25826],{"className":33667},[102,103],[11,33669,33670],{},"于是：",[38,33672,33674],{"className":33673,"translate":42},[621],[38,33675,33677,33713],{"className":33676,"translate":42},[41],[38,33678,33680],{"className":33679},[46],[48,33681,33682],{"xmlns":50,"display":630},[52,33683,33684,33710],{},[55,33685,33686,33688,33690,33692,33694,33696,33698,33700,33702],{},[58,33687,7695],{"mathvariant":333},[58,33689,7695],{"mathvariant":333},[58,33691,286],{},[62,33693,710],{},[58,33695,26054],{},[58,33697,7695],{"mathvariant":333},[58,33699,7695],{"mathvariant":333},[62,33701,31671],{},[20042,33703,33704],{},[55,33705,33706,33708],{},[300,33707,312],{},[58,33709,25826],{},[81,33711,33712],{"encoding":83},"||x - y|| \\approx \\sqrt{2d}",[38,33714,33716,33737,33758],{"className":33715,"ariaHidden":89},[88],[38,33717,33719,33722,33725,33728,33731,33734],{"className":33718},[93],[38,33720],{"className":33721,"style":1439},[97],[38,33723,24391],{"className":33724},[102],[38,33726,286],{"className":33727},[102,103],[38,33729],{"className":33730,"style":995},[108],[38,33732,710],{"className":33733},[999],[38,33735],{"className":33736,"style":995},[108],[38,33738,33740,33743,33746,33749,33752,33755],{"className":33739},[93],[38,33741],{"className":33742,"style":1439},[97],[38,33744,26054],{"className":33745,"style":3420},[102,103],[38,33747,24391],{"className":33748},[102],[38,33750],{"className":33751,"style":109},[108],[38,33753,31671],{"className":33754},[113],[38,33756],{"className":33757,"style":109},[108],[38,33759,33761,33765],{"className":33760},[93],[38,33762],{"className":33763,"style":33764},[97],"height:1.04em;vertical-align:-0.0589em;",[38,33766,33768],{"className":33767},[102,20187],[38,33769,33771,33807],{"className":33770},[138,384],[38,33772,33774,33804],{"className":33773},[142],[38,33775,33777,33792],{"className":33776,"style":33449},[146],[38,33778,33780,33783],{"className":33779,"style":913},[20201],[38,33781],{"className":33782,"style":760},[154],[38,33784,33786,33789],{"className":33785,"style":21076},[102],[38,33787,312],{"className":33788},[102],[38,33790,25826],{"className":33791},[102,103],[38,33793,33794,33797],{"style":33464},[38,33795],{"className":33796,"style":760},[154],[38,33798,33800],{"className":33799,"style":21101},[20343],[4842,33801,33802],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,33803],{"d":21108},[38,33805,408],{"className":33806},[407],[38,33808,33810],{"className":33809},[142],[38,33811,33813],{"className":33812,"style":33484},[146],[38,33814],{},[11,33816,33817],{},"再看相对波动：",[7370,33819,33820,33857],{},[7373,33821,33822,33823],{},"均值：",[38,33824,33826,33842],{"className":33825,"translate":42},[41],[38,33827,33829],{"className":33828},[46],[48,33830,33831],{"xmlns":50},[52,33832,33833,33839],{},[55,33834,33835,33837],{},[300,33836,312],{},[58,33838,25826],{},[81,33840,33841],{"encoding":83},"2d",[38,33843,33845],{"className":33844,"ariaHidden":89},[88],[38,33846,33848,33851,33854],{"className":33847},[93],[38,33849],{"className":33850,"style":889},[97],[38,33852,312],{"className":33853},[102],[38,33855,25826],{"className":33856},[102,103],[7373,33858,33859,33860],{},"标准差：",[38,33861,33863,33884],{"className":33862,"translate":42},[41],[38,33864,33866],{"className":33865},[46],[48,33867,33868],{"xmlns":50},[52,33869,33870,33882],{},[55,33871,33872,33874,33876,33880],{},[58,33873,13105],{},[62,33875,293],{"stretchy":292},[20042,33877,33878],{},[58,33879,25826],{},[62,33881,330],{"stretchy":292},[81,33883,33120],{"encoding":83},[38,33885,33887],{"className":33886,"ariaHidden":89},[88],[38,33888,33890,33893,33896,33899,33946],{"className":33889},[93],[38,33891],{"className":33892,"style":33130},[97],[38,33894,13105],{"className":33895,"style":2500},[102,103],[38,33897,293],{"className":33898},[371],[38,33900,33902],{"className":33901},[102,20187],[38,33903,33905,33938],{"className":33904},[138,384],[38,33906,33908,33935],{"className":33907},[142],[38,33909,33911,33923],{"className":33910,"style":31725},[146],[38,33912,33914,33917],{"className":33913,"style":913},[20201],[38,33915],{"className":33916,"style":760},[154],[38,33918,33920],{"className":33919,"style":21076},[102],[38,33921,25826],{"className":33922},[102,103],[38,33924,33925,33928],{"style":31740},[38,33926],{"className":33927,"style":760},[154],[38,33929,33931],{"className":33930,"style":21101},[20343],[4842,33932,33933],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,33934],{"d":21108},[38,33936,408],{"className":33937},[407],[38,33939,33941],{"className":33940},[142],[38,33942,33944],{"className":33943,"style":31760},[146],[38,33945],{},[38,33947,330],{"className":33948},[529],[11,33950,32987],{},[38,33952,33954],{"className":33953,"translate":42},[621],[38,33955,33957,34001],{"className":33956,"translate":42},[41],[38,33958,33960],{"className":33959},[46],[48,33961,33962],{"xmlns":50,"display":630},[52,33963,33964,33998],{},[55,33965,33966,33974,33976,33984,33986,33994,33996],{},[636,33967,33968,33971],{},[3265,33969,33970],{},"标准差",[3265,33972,33973],{},"均值",[62,33975,30522],{},[636,33977,33978,33982],{},[20042,33979,33980],{},[58,33981,25826],{},[58,33983,25826],{},[62,33985,289],{},[636,33987,33988,33990],{},[300,33989,302],{},[20042,33991,33992],{},[58,33993,25826],{},[62,33995,77],{},[300,33997,683],{},[81,33999,34000],{"encoding":83},"\\frac{\\text{标准差}}{\\text{均值}} \\sim \\frac{\\sqrt{d}}{d} = \\frac{1}{\\sqrt{d}} \\to 0",[38,34002,34004,34088,34211,34335],{"className":34003,"ariaHidden":89},[88],[38,34005,34007,34011,34079,34082,34085],{"className":34006},[93],[38,34008],{"className":34009,"style":34010},[97],"height:2.0463em;vertical-align:-0.686em;",[38,34012,34014,34017,34076],{"className":34013},[102],[38,34015],{"className":34016},[371,740],[38,34018,34020],{"className":34019},[636],[38,34021,34023,34068],{"className":34022},[138,384],[38,34024,34026,34065],{"className":34025},[142],[38,34027,34029,34043,34051],{"className":34028,"style":27041},[146],[38,34030,34031,34034],{"style":756},[38,34032],{"className":34033,"style":760},[154],[38,34035,34037],{"className":34036},[102],[38,34038,34040],{"className":34039},[102,22531],[38,34041,33973],{"className":34042},[102,22535],[38,34044,34045,34048],{"style":811},[38,34046],{"className":34047,"style":760},[154],[38,34049],{"className":34050,"style":819},[818],[38,34052,34053,34056],{"style":822},[38,34054],{"className":34055,"style":760},[154],[38,34057,34059],{"className":34058},[102],[38,34060,34062],{"className":34061},[102,22531],[38,34063,33970],{"className":34064},[102,22535],[38,34066,408],{"className":34067},[407],[38,34069,34071],{"className":34070},[142],[38,34072,34074],{"className":34073,"style":1079},[146],[38,34075],{},[38,34077],{"className":34078},[529,740],[38,34080],{"className":34081,"style":109},[108],[38,34083,30522],{"className":34084},[113],[38,34086],{"className":34087,"style":109},[108],[38,34089,34091,34095,34202,34205,34208],{"className":34090},[93],[38,34092],{"className":34093,"style":34094},[97],"height:2.2952em;vertical-align:-0.686em;",[38,34096,34098,34101,34199],{"className":34097},[102],[38,34099],{"className":34100},[371,740],[38,34102,34104],{"className":34103},[636],[38,34105,34107,34191],{"className":34106},[138,384],[38,34108,34110,34188],{"className":34109},[142],[38,34111,34114,34125,34133],{"className":34112,"style":34113},[146],"height:1.6092em;",[38,34115,34116,34119],{"style":756},[38,34117],{"className":34118,"style":760},[154],[38,34120,34122],{"className":34121},[102],[38,34123,25826],{"className":34124},[102,103],[38,34126,34127,34130],{"style":811},[38,34128],{"className":34129,"style":760},[154],[38,34131],{"className":34132,"style":819},[818],[38,34134,34135,34138],{"style":822},[38,34136],{"className":34137,"style":760},[154],[38,34139,34141],{"className":34140},[102],[38,34142,34144],{"className":34143},[102,20187],[38,34145,34147,34180],{"className":34146},[138,384],[38,34148,34150,34177],{"className":34149},[142],[38,34151,34153,34165],{"className":34152,"style":31725},[146],[38,34154,34156,34159],{"className":34155,"style":913},[20201],[38,34157],{"className":34158,"style":760},[154],[38,34160,34162],{"className":34161,"style":21076},[102],[38,34163,25826],{"className":34164},[102,103],[38,34166,34167,34170],{"style":31740},[38,34168],{"className":34169,"style":760},[154],[38,34171,34173],{"className":34172,"style":21101},[20343],[4842,34174,34175],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,34176],{"d":21108},[38,34178,408],{"className":34179},[407],[38,34181,34183],{"className":34182},[142],[38,34184,34186],{"className":34185,"style":31760},[146],[38,34187],{},[38,34189,408],{"className":34190},[407],[38,34192,34194],{"className":34193},[142],[38,34195,34197],{"className":34196,"style":1079},[146],[38,34198],{},[38,34200],{"className":34201},[529,740],[38,34203],{"className":34204,"style":109},[108],[38,34206,289],{"className":34207},[113],[38,34209],{"className":34210,"style":109},[108],[38,34212,34214,34218,34326,34329,34332],{"className":34213},[93],[38,34215],{"className":34216,"style":34217},[97],"height:2.2514em;vertical-align:-0.93em;",[38,34219,34221,34224,34323],{"className":34220},[102],[38,34222],{"className":34223},[371,740],[38,34225,34227],{"className":34226},[636],[38,34228,34230,34314],{"className":34229},[138,384],[38,34231,34233,34311],{"className":34232},[142],[38,34234,34236,34292,34300],{"className":34235,"style":11324},[146],[38,34237,34239,34242],{"style":34238},"top:-2.1778em;",[38,34240],{"className":34241,"style":760},[154],[38,34243,34245],{"className":34244},[102],[38,34246,34248],{"className":34247},[102,20187],[38,34249,34251,34284],{"className":34250},[138,384],[38,34252,34254,34281],{"className":34253},[142],[38,34255,34257,34269],{"className":34256,"style":31725},[146],[38,34258,34260,34263],{"className":34259,"style":913},[20201],[38,34261],{"className":34262,"style":760},[154],[38,34264,34266],{"className":34265,"style":21076},[102],[38,34267,25826],{"className":34268},[102,103],[38,34270,34271,34274],{"style":31740},[38,34272],{"className":34273,"style":760},[154],[38,34275,34277],{"className":34276,"style":21101},[20343],[4842,34278,34279],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,34280],{"d":21108},[38,34282,408],{"className":34283},[407],[38,34285,34287],{"className":34286},[142],[38,34288,34290],{"className":34289,"style":31760},[146],[38,34291],{},[38,34293,34294,34297],{"style":811},[38,34295],{"className":34296,"style":760},[154],[38,34298],{"className":34299,"style":819},[818],[38,34301,34302,34305],{"style":822},[38,34303],{"className":34304,"style":760},[154],[38,34306,34308],{"className":34307},[102],[38,34309,302],{"className":34310},[102],[38,34312,408],{"className":34313},[407],[38,34315,34317],{"className":34316},[142],[38,34318,34321],{"className":34319,"style":34320},[146],"height:0.93em;",[38,34322],{},[38,34324],{"className":34325},[529,740],[38,34327],{"className":34328,"style":109},[108],[38,34330,77],{"className":34331},[113],[38,34333],{"className":34334,"style":109},[108],[38,34336,34338,34341],{"className":34337},[93],[38,34339],{"className":34340,"style":2429},[97],[38,34342,683],{"className":34343},[102],[11,34345,34346],{},"结论：距离的相对波动趋于 0，也就是说，所有点之间的距离几乎相同。",[31,34348,34349],{"id":34349},"角度距离",[11,34351,34352,34353,34424],{},"如果将向量在 ",[38,34354,34356,34375],{"className":34355,"translate":42},[41],[38,34357,34359],{"className":34358},[46],[48,34360,34361],{"xmlns":50},[52,34362,34363,34372],{},[55,34364,34365],{},[295,34366,34367,34370],{},[58,34368,34369],{"mathvariant":333},"ℓ",[300,34371,312],{},[81,34373,34374],{"encoding":83},"\\ell_2",[38,34376,34378],{"className":34377,"ariaHidden":89},[88],[38,34379,34381,34384],{"className":34380},[93],[38,34382],{"className":34383,"style":3360},[97],[38,34385,34387,34390],{"className":34386},[102],[38,34388,34369],{"className":34389},[102],[38,34391,34393],{"className":34392},[134],[38,34394,34396,34416],{"className":34395},[138,384],[38,34397,34399,34413],{"className":34398},[142],[38,34400,34402],{"className":34401,"style":391},[146],[38,34403,34404,34407],{"style":394},[38,34405],{"className":34406,"style":155},[154],[38,34408,34410],{"className":34409},[159,160,161,162],[38,34411,312],{"className":34412},[102,162],[38,34414,408],{"className":34415},[407],[38,34417,34419],{"className":34418},[142],[38,34420,34422],{"className":34421,"style":415},[146],[38,34423],{}," 归一化后投影到单位超球面上：",[38,34426,34428],{"className":34427,"translate":42},[621],[38,34429,34431,34469],{"className":34430,"translate":42},[41],[38,34432,34434],{"className":34433},[46],[48,34435,34436],{"xmlns":50,"display":630},[52,34437,34438,34466],{},[55,34439,34440,34442,34444,34446,34448,34450,34452,34454,34456,34458,34460,34462,34464],{},[58,34441,7695],{"mathvariant":333},[58,34443,7695],{"mathvariant":333},[58,34445,286],{},[58,34447,7695],{"mathvariant":333},[58,34449,7695],{"mathvariant":333},[62,34451,289],{},[58,34453,7695],{"mathvariant":333},[58,34455,7695],{"mathvariant":333},[58,34457,26054],{},[58,34459,7695],{"mathvariant":333},[58,34461,7695],{"mathvariant":333},[62,34463,289],{},[300,34465,302],{},[81,34467,34468],{"encoding":83},"||x|| = ||y|| = 1",[38,34470,34472,34496,34520],{"className":34471,"ariaHidden":89},[88],[38,34473,34475,34478,34481,34484,34487,34490,34493],{"className":34474},[93],[38,34476],{"className":34477,"style":1439},[97],[38,34479,24391],{"className":34480},[102],[38,34482,286],{"className":34483},[102,103],[38,34485,24391],{"className":34486},[102],[38,34488],{"className":34489,"style":109},[108],[38,34491,289],{"className":34492},[113],[38,34494],{"className":34495,"style":109},[108],[38,34497,34499,34502,34505,34508,34511,34514,34517],{"className":34498},[93],[38,34500],{"className":34501,"style":1439},[97],[38,34503,24391],{"className":34504},[102],[38,34506,26054],{"className":34507,"style":3420},[102,103],[38,34509,24391],{"className":34510},[102],[38,34512],{"className":34513,"style":109},[108],[38,34515,289],{"className":34516},[113],[38,34518],{"className":34519,"style":109},[108],[38,34521,34523,34526],{"className":34522},[93],[38,34524],{"className":34525,"style":2429},[97],[38,34527,302],{"className":34528},[102],[11,34530,34531],{},"那么：",[38,34533,34535],{"className":34534,"translate":42},[621],[38,34536,34538,34583],{"className":34537,"translate":42},[41],[38,34539,34541],{"className":34540},[46],[48,34542,34543],{"xmlns":50,"display":630},[52,34544,34545,34580],{},[55,34546,34547,34549,34551,34553,34555,34557,34559,34565,34567,34569,34571,34573,34576,34578],{},[58,34548,7695],{"mathvariant":333},[58,34550,7695],{"mathvariant":333},[58,34552,286],{},[62,34554,710],{},[58,34556,26054],{},[58,34558,7695],{"mathvariant":333},[66,34560,34561,34563],{},[58,34562,7695],{"mathvariant":333},[300,34564,312],{},[62,34566,289],{},[300,34568,312],{},[62,34570,710],{},[300,34572,312],{},[58,34574,34575],{},"cos",[62,34577,673],{},[58,34579,3109],{},[81,34581,34582],{"encoding":83},"||x - y||^2 = 2 - 2\\cos\\theta",[38,34584,34586,34607,34657,34675],{"className":34585,"ariaHidden":89},[88],[38,34587,34589,34592,34595,34598,34601,34604],{"className":34588},[93],[38,34590],{"className":34591,"style":1439},[97],[38,34593,24391],{"className":34594},[102],[38,34596,286],{"className":34597},[102,103],[38,34599],{"className":34600,"style":995},[108],[38,34602,710],{"className":34603},[999],[38,34605],{"className":34606,"style":995},[108],[38,34608,34610,34613,34616,34619,34648,34651,34654],{"className":34609},[93],[38,34611],{"className":34612,"style":11468},[97],[38,34614,26054],{"className":34615,"style":3420},[102,103],[38,34617,7695],{"className":34618},[102],[38,34620,34622,34625],{"className":34621},[102],[38,34623,7695],{"className":34624},[102],[38,34626,34628],{"className":34627},[134],[38,34629,34631],{"className":34630},[138],[38,34632,34634],{"className":34633},[142],[38,34635,34637],{"className":34636,"style":11499},[146],[38,34638,34639,34642],{"style":1892},[38,34640],{"className":34641,"style":155},[154],[38,34643,34645],{"className":34644},[159,160,161,162],[38,34646,312],{"className":34647},[102,162],[38,34649],{"className":34650,"style":109},[108],[38,34652,289],{"className":34653},[113],[38,34655],{"className":34656,"style":109},[108],[38,34658,34660,34663,34666,34669,34672],{"className":34659},[93],[38,34661],{"className":34662,"style":19848},[97],[38,34664,312],{"className":34665},[102],[38,34667],{"className":34668,"style":995},[108],[38,34670,710],{"className":34671},[999],[38,34673],{"className":34674,"style":995},[108],[38,34676,34678,34681,34684,34687,34690,34693],{"className":34677},[93],[38,34679],{"className":34680,"style":889},[97],[38,34682,312],{"className":34683},[102],[38,34685],{"className":34686,"style":425},[108],[38,34688,34575],{"className":34689},[878],[38,34691],{"className":34692,"style":425},[108],[38,34694,3109],{"className":34695,"style":2500},[102,103],[11,34697,34698],{},"在高维情形下：",[38,34700,34702],{"className":34701,"translate":42},[621],[38,34703,34705,34727],{"className":34704,"translate":42},[41],[38,34706,34708],{"className":34707},[46],[48,34709,34710],{"xmlns":50,"display":630},[52,34711,34712,34724],{},[55,34713,34714,34716,34718,34720,34722],{},[58,34715,34575],{},[62,34717,673],{},[58,34719,3109],{},[62,34721,31671],{},[300,34723,683],{},[81,34725,34726],{"encoding":83},"\\cos\\theta \\approx 0",[38,34728,34730,34754],{"className":34729,"ariaHidden":89},[88],[38,34731,34733,34736,34739,34742,34745,34748,34751],{"className":34732},[93],[38,34734],{"className":34735,"style":889},[97],[38,34737,34575],{"className":34738},[878],[38,34740],{"className":34741,"style":425},[108],[38,34743,3109],{"className":34744,"style":2500},[102,103],[38,34746],{"className":34747,"style":109},[108],[38,34749,31671],{"className":34750},[113],[38,34752],{"className":34753,"style":109},[108],[38,34755,34757,34760],{"className":34756},[93],[38,34758],{"className":34759,"style":2429},[97],[38,34761,683],{"className":34762},[102],[11,34764,32987],{},[38,34766,34768],{"className":34767,"translate":42},[621],[38,34769,34771,34803],{"className":34770,"translate":42},[41],[38,34772,34774],{"className":34773},[46],[48,34775,34776],{"xmlns":50,"display":630},[52,34777,34778,34800],{},[55,34779,34780,34782,34784,34786,34788,34790,34792,34794,34796],{},[58,34781,7695],{"mathvariant":333},[58,34783,7695],{"mathvariant":333},[58,34785,286],{},[62,34787,710],{},[58,34789,26054],{},[58,34791,7695],{"mathvariant":333},[58,34793,7695],{"mathvariant":333},[62,34795,31671],{},[20042,34797,34798],{},[300,34799,312],{},[81,34801,34802],{"encoding":83},"||x - y|| \\approx \\sqrt{2}",[38,34804,34806,34827,34848],{"className":34805,"ariaHidden":89},[88],[38,34807,34809,34812,34815,34818,34821,34824],{"className":34808},[93],[38,34810],{"className":34811,"style":1439},[97],[38,34813,24391],{"className":34814},[102],[38,34816,286],{"className":34817},[102,103],[38,34819],{"className":34820,"style":995},[108],[38,34822,710],{"className":34823},[999],[38,34825],{"className":34826,"style":995},[108],[38,34828,34830,34833,34836,34839,34842,34845],{"className":34829},[93],[38,34831],{"className":34832,"style":1439},[97],[38,34834,26054],{"className":34835,"style":3420},[102,103],[38,34837,24391],{"className":34838},[102],[38,34840],{"className":34841,"style":109},[108],[38,34843,31671],{"className":34844},[113],[38,34846],{"className":34847,"style":109},[108],[38,34849,34851,34855],{"className":34850},[93],[38,34852],{"className":34853,"style":34854},[97],"height:1.04em;vertical-align:-0.0839em;",[38,34856,34858],{"className":34857},[102,20187],[38,34859,34861,34896],{"className":34860},[138,384],[38,34862,34864,34893],{"className":34863},[142],[38,34865,34868,34880],{"className":34866,"style":34867},[146],"height:0.9561em;",[38,34869,34871,34874],{"className":34870,"style":913},[20201],[38,34872],{"className":34873,"style":760},[154],[38,34875,34877],{"className":34876,"style":21076},[102],[38,34878,312],{"className":34879},[102],[38,34881,34883,34886],{"style":34882},"top:-2.9161em;",[38,34884],{"className":34885,"style":760},[154],[38,34887,34889],{"className":34888,"style":21101},[20343],[4842,34890,34891],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,34892],{"d":21108},[38,34894,408],{"className":34895},[407],[38,34897,34899],{"className":34898},[142],[38,34900,34903],{"className":34901,"style":34902},[146],"height:0.0839em;",[38,34904],{},[11,34906,34907],{},"所有向量之间的角度距离几乎等距。",[11,34909,34910,34911,35000,35001,35045],{},"在高维空间中，根据大数定律，随机向量的范数集中在一个常数附近。此外，独立向量之间的内积只以 ",[38,34912,34914,34935],{"className":34913,"translate":42},[41],[38,34915,34917],{"className":34916},[46],[48,34918,34919],{"xmlns":50},[52,34920,34921,34933],{},[55,34922,34923,34925,34927,34931],{},[58,34924,13105],{},[62,34926,293],{"stretchy":292},[20042,34928,34929],{},[58,34930,25826],{},[62,34932,330],{"stretchy":292},[81,34934,33120],{"encoding":83},[38,34936,34938],{"className":34937,"ariaHidden":89},[88],[38,34939,34941,34944,34947,34950,34997],{"className":34940},[93],[38,34942],{"className":34943,"style":33130},[97],[38,34945,13105],{"className":34946,"style":2500},[102,103],[38,34948,293],{"className":34949},[371],[38,34951,34953],{"className":34952},[102,20187],[38,34954,34956,34989],{"className":34955},[138,384],[38,34957,34959,34986],{"className":34958},[142],[38,34960,34962,34974],{"className":34961,"style":31725},[146],[38,34963,34965,34968],{"className":34964,"style":913},[20201],[38,34966],{"className":34967,"style":760},[154],[38,34969,34971],{"className":34970,"style":21076},[102],[38,34972,25826],{"className":34973},[102,103],[38,34975,34976,34979],{"style":31740},[38,34977],{"className":34978,"style":760},[154],[38,34980,34982],{"className":34981,"style":21101},[20343],[4842,34983,34984],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,34985],{"d":21108},[38,34987,408],{"className":34988},[407],[38,34990,34992],{"className":34991},[142],[38,34993,34995],{"className":34994,"style":31760},[146],[38,34996],{},[38,34998,330],{"className":34999},[529]," 的量级增长，与范数平方的 ",[38,35002,35004,35024],{"className":35003,"translate":42},[41],[38,35005,35007],{"className":35006},[46],[48,35008,35009],{"xmlns":50},[52,35010,35011,35021],{},[55,35012,35013,35015,35017,35019],{},[58,35014,13105],{},[62,35016,293],{"stretchy":292},[58,35018,25826],{},[62,35020,330],{"stretchy":292},[81,35022,35023],{"encoding":83},"O(d)",[38,35025,35027],{"className":35026,"ariaHidden":89},[88],[38,35028,35030,35033,35036,35039,35042],{"className":35029},[93],[38,35031],{"className":35032,"style":1439},[97],[38,35034,13105],{"className":35035,"style":2500},[102,103],[38,35037,293],{"className":35038},[371],[38,35040,25826],{"className":35041},[102,103],[38,35043,330],{"className":35044},[529]," 量级相比可以忽略不计。其结果是，两两距离集中在一个常数附近，从而出现几乎所有点都近似等距的现象。",[28,35047,35048],{"id":35048},"正曲率黎曼空间",[11,35050,35051],{},"在欧氏空间中，维数灾难源于体积的多项式增长以及坐标的独立结构。然而，在正曲率黎曼流形中，几何本身从根本上重塑了体积分布与距离行为。",[11,35053,35054,35055,35083],{},"一个典型的例子是 ",[38,35056,35058,35071],{"className":35057,"translate":42},[41],[38,35059,35061],{"className":35060},[46],[48,35062,35063],{"xmlns":50},[52,35064,35065,35069],{},[55,35066,35067],{},[58,35068,25826],{},[81,35070,25826],{"encoding":83},[38,35072,35074],{"className":35073,"ariaHidden":89},[88],[38,35075,35077,35080],{"className":35076},[93],[38,35078],{"className":35079,"style":889},[97],[38,35081,25826],{"className":35082},[102,103]," 维单位球面：",[38,35085,35087],{"className":35086,"translate":42},[621],[38,35088,35090,35144],{"className":35089,"translate":42},[41],[38,35091,35093],{"className":35092},[46],[48,35094,35095],{"xmlns":50,"display":630},[52,35096,35097,35141],{},[55,35098,35099,35111,35113],{},[66,35100,35101,35103],{},[58,35102,25953],{"mathvariant":70},[55,35104,35105,35107,35109],{},[58,35106,25826],{},[62,35108,710],{},[300,35110,302],{},[62,35112,289],{},[55,35114,35115,35117,35119,35125,35127,35129,35131,35133,35135,35137,35139],{},[58,35116,286],{},[62,35118,1420],{},[66,35120,35121,35123],{},[58,35122,71],{"mathvariant":70},[58,35124,25826],{},[62,35126,64],{},[58,35128,7695],{"mathvariant":333},[58,35130,7695],{"mathvariant":333},[58,35132,286],{},[58,35134,7695],{"mathvariant":333},[58,35136,7695],{"mathvariant":333},[62,35138,289],{},[300,35140,302],{},[81,35142,35143],{"encoding":83},"\\mathbb{S}^{d-1} = { x \\in \\mathbb{R}^d : ||x|| = 1 }",[38,35145,35147,35200],{"className":35146,"ariaHidden":89},[88],[38,35148,35150,35153,35191,35194,35197],{"className":35149},[93],[38,35151],{"className":35152,"style":2308},[97],[38,35154,35156,35159],{"className":35155},[102],[38,35157,25953],{"className":35158},[102,130],[38,35160,35162],{"className":35161},[134],[38,35163,35165],{"className":35164},[138],[38,35166,35168],{"className":35167},[142],[38,35169,35171],{"className":35170,"style":2308},[146],[38,35172,35173,35176],{"style":1892},[38,35174],{"className":35175,"style":155},[154],[38,35177,35179],{"className":35178},[159,160,161,162],[38,35180,35182,35185,35188],{"className":35181},[102,162],[38,35183,25826],{"className":35184},[102,103,162],[38,35186,710],{"className":35187},[999,162],[38,35189,302],{"className":35190},[102,162],[38,35192],{"className":35193,"style":109},[108],[38,35195,289],{"className":35196},[113],[38,35198],{"className":35199,"style":109},[108],[38,35201,35203,35206],{"className":35202},[93],[38,35204],{"className":35205,"style":2277},[97],[38,35207,35209,35212,35215,35218,35221,35250,35253,35256,35259,35262,35265,35268,35271,35274,35277],{"className":35208},[102],[38,35210,286],{"className":35211},[102,103],[38,35213],{"className":35214,"style":109},[108],[38,35216,1420],{"className":35217},[113],[38,35219],{"className":35220,"style":109},[108],[38,35222,35224,35227],{"className":35223},[102],[38,35225,71],{"className":35226},[102,130],[38,35228,35230],{"className":35229},[134],[38,35231,35233],{"className":35232},[138],[38,35234,35236],{"className":35235},[142],[38,35237,35239],{"className":35238,"style":2308},[146],[38,35240,35241,35244],{"style":1892},[38,35242],{"className":35243,"style":155},[154],[38,35245,35247],{"className":35246},[159,160,161,162],[38,35248,25826],{"className":35249},[102,103,162],[38,35251],{"className":35252,"style":109},[108],[38,35254,64],{"className":35255},[113],[38,35257],{"className":35258,"style":109},[108],[38,35260,24391],{"className":35261},[102],[38,35263,286],{"className":35264},[102,103],[38,35266,24391],{"className":35267},[102],[38,35269],{"className":35270,"style":109},[108],[38,35272,289],{"className":35273},[113],[38,35275],{"className":35276,"style":109},[108],[38,35278,302],{"className":35279},[102],[11,35281,35282],{},"它是一个具有常正截面曲率的空间。",[31,35284,35285],{"id":35285},"球面上的测度集中",[11,35287,35288],{},"与欧氏空间中体积沿径向扩散不同，在球面上所有点都位于固定半径的流形上。因此，「径向壳层」被测地带来取代。",[11,35290,35291,35292,35430],{},"设固定一点 ",[38,35293,35295,35327],{"className":35294,"translate":42},[41],[38,35296,35298],{"className":35297},[46],[48,35299,35300],{"xmlns":50},[52,35301,35302,35324],{},[55,35303,35304,35310,35312],{},[295,35305,35306,35308],{},[58,35307,286],{},[300,35309,683],{},[62,35311,1420],{},[66,35313,35314,35316],{},[58,35315,25953],{"mathvariant":70},[55,35317,35318,35320,35322],{},[58,35319,25826],{},[62,35321,710],{},[300,35323,302],{},[81,35325,35326],{"encoding":83},"x_0 \\in \\mathbb{S}^{d-1}",[38,35328,35330,35386],{"className":35329,"ariaHidden":89},[88],[38,35331,35333,35337,35377,35380,35383],{"className":35332},[93],[38,35334],{"className":35335,"style":35336},[97],"height:0.6891em;vertical-align:-0.15em;",[38,35338,35340,35343],{"className":35339},[102],[38,35341,286],{"className":35342},[102,103],[38,35344,35346],{"className":35345},[134],[38,35347,35349,35369],{"className":35348},[138,384],[38,35350,35352,35366],{"className":35351},[142],[38,35353,35355],{"className":35354,"style":391},[146],[38,35356,35357,35360],{"style":394},[38,35358],{"className":35359,"style":155},[154],[38,35361,35363],{"className":35362},[159,160,161,162],[38,35364,683],{"className":35365},[102,162],[38,35367,408],{"className":35368},[407],[38,35370,35372],{"className":35371},[142],[38,35373,35375],{"className":35374,"style":415},[146],[38,35376],{},[38,35378],{"className":35379,"style":109},[108],[38,35381,1420],{"className":35382},[113],[38,35384],{"className":35385,"style":109},[108],[38,35387,35389,35392],{"className":35388},[93],[38,35390],{"className":35391,"style":545},[97],[38,35393,35395,35398],{"className":35394},[102],[38,35396,25953],{"className":35397},[102,130],[38,35399,35401],{"className":35400},[134],[38,35402,35404],{"className":35403},[138],[38,35405,35407],{"className":35406},[142],[38,35408,35410],{"className":35409,"style":545},[146],[38,35411,35412,35415],{"style":150},[38,35413],{"className":35414,"style":155},[154],[38,35416,35418],{"className":35417},[159,160,161,162],[38,35419,35421,35424,35427],{"className":35420},[102,162],[38,35422,25826],{"className":35423},[102,103,162],[38,35425,710],{"className":35426},[999,162],[38,35428,302],{"className":35429},[102,162],"，并定义测地球：",[38,35432,35434],{"className":35433,"translate":42},[621],[38,35435,35437,35501],{"className":35436,"translate":42},[41],[38,35438,35440],{"className":35439},[46],[48,35441,35442],{"xmlns":50,"display":630},[52,35443,35444,35498],{},[55,35445,35446,35449,35451,35453,35455,35457],{},[58,35447,35448],{},"B",[62,35450,293],{"stretchy":292},[58,35452,3109],{},[62,35454,330],{"stretchy":292},[62,35456,289],{},[55,35458,35459,35461,35463,35475,35477,35480,35482,35484,35486,35492,35494,35496],{},[58,35460,286],{},[62,35462,1420],{},[66,35464,35465,35467],{},[58,35466,25953],{"mathvariant":70},[55,35468,35469,35471,35473],{},[58,35470,25826],{},[62,35472,710],{},[300,35474,302],{},[62,35476,64],{},[58,35478,35479],{"mathvariant":333},"∠",[62,35481,293],{"stretchy":292},[58,35483,286],{},[62,35485,305],{"separator":89},[295,35487,35488,35490],{},[58,35489,286],{},[300,35491,683],{},[62,35493,330],{"stretchy":292},[62,35495,2879],{},[58,35497,3109],{},[81,35499,35500],{"encoding":83},"B(\\theta) = { x \\in \\mathbb{S}^{d-1} : \\angle(x, x_0) \\leq \\theta }",[38,35502,35504,35532],{"className":35503,"ariaHidden":89},[88],[38,35505,35507,35510,35514,35517,35520,35523,35526,35529],{"className":35506},[93],[38,35508],{"className":35509,"style":1439},[97],[38,35511,35448],{"className":35512,"style":35513},[102,103],"margin-right:0.0502em;",[38,35515,293],{"className":35516},[371],[38,35518,3109],{"className":35519,"style":2500},[102,103],[38,35521,330],{"className":35522},[529],[38,35524],{"className":35525,"style":109},[108],[38,35527,289],{"className":35528},[113],[38,35530],{"className":35531,"style":109},[108],[38,35533,35535,35538],{"className":35534},[93],[38,35536],{"className":35537,"style":2277},[97],[38,35539,35541,35544,35547,35550,35553,35591,35594,35597,35600,35603,35606,35609,35612,35615,35655,35658,35661,35664,35667],{"className":35540},[102],[38,35542,286],{"className":35543},[102,103],[38,35545],{"className":35546,"style":109},[108],[38,35548,1420],{"className":35549},[113],[38,35551],{"className":35552,"style":109},[108],[38,35554,35556,35559],{"className":35555},[102],[38,35557,25953],{"className":35558},[102,130],[38,35560,35562],{"className":35561},[134],[38,35563,35565],{"className":35564},[138],[38,35566,35568],{"className":35567},[142],[38,35569,35571],{"className":35570,"style":2308},[146],[38,35572,35573,35576],{"style":1892},[38,35574],{"className":35575,"style":155},[154],[38,35577,35579],{"className":35578},[159,160,161,162],[38,35580,35582,35585,35588],{"className":35581},[102,162],[38,35583,25826],{"className":35584},[102,103,162],[38,35586,710],{"className":35587},[999,162],[38,35589,302],{"className":35590},[102,162],[38,35592],{"className":35593,"style":109},[108],[38,35595,64],{"className":35596},[113],[38,35598],{"className":35599,"style":109},[108],[38,35601,35479],{"className":35602},[102],[38,35604,293],{"className":35605},[371],[38,35607,286],{"className":35608},[102,103],[38,35610,305],{"className":35611},[421],[38,35613],{"className":35614,"style":425},[108],[38,35616,35618,35621],{"className":35617},[102],[38,35619,286],{"className":35620},[102,103],[38,35622,35624],{"className":35623},[134],[38,35625,35627,35647],{"className":35626},[138,384],[38,35628,35630,35644],{"className":35629},[142],[38,35631,35633],{"className":35632,"style":391},[146],[38,35634,35635,35638],{"style":394},[38,35636],{"className":35637,"style":155},[154],[38,35639,35641],{"className":35640},[159,160,161,162],[38,35642,683],{"className":35643},[102,162],[38,35645,408],{"className":35646},[407],[38,35648,35650],{"className":35649},[142],[38,35651,35653],{"className":35652,"style":415},[146],[38,35654],{},[38,35656,330],{"className":35657},[529],[38,35659],{"className":35660,"style":109},[108],[38,35662,2879],{"className":35663},[113],[38,35665],{"className":35666,"style":109},[108],[38,35668,3109],{"className":35669,"style":2500},[102,103],[11,35671,35672,35673,35770],{},"该区域的测度依赖于 ",[38,35674,35676,35711],{"className":35675,"translate":42},[41],[38,35677,35679],{"className":35678},[46],[48,35680,35681],{"xmlns":50},[52,35682,35683,35708],{},[55,35684,35685,35702,35704,35706],{},[66,35686,35687,35694],{},[55,35688,35689,35692],{},[58,35690,35691],{},"sin",[62,35693,673],{},[55,35695,35696,35698,35700],{},[58,35697,25826],{},[62,35699,710],{},[300,35701,312],{},[62,35703,293],{"stretchy":292},[58,35705,3109],{},[62,35707,330],{"stretchy":292},[81,35709,35710],{"encoding":83},"\\sin^{d-2}(\\theta)",[38,35712,35714],{"className":35713,"ariaHidden":89},[88],[38,35715,35717,35721,35761,35764,35767],{"className":35716},[93],[38,35718],{"className":35719,"style":35720},[97],"height:1.1569em;vertical-align:-0.25em;",[38,35722,35724,35727],{"className":35723},[878],[38,35725,35691],{"className":35726},[878],[38,35728,35730],{"className":35729},[134],[38,35731,35733],{"className":35732},[138],[38,35734,35736],{"className":35735},[142],[38,35737,35740],{"className":35738,"style":35739},[146],"height:0.9069em;",[38,35741,35743,35746],{"style":35742},"top:-3.1208em;margin-right:0.05em;",[38,35744],{"className":35745,"style":155},[154],[38,35747,35749],{"className":35748},[159,160,161,162],[38,35750,35752,35755,35758],{"className":35751},[102,162],[38,35753,25826],{"className":35754},[102,103,162],[38,35756,710],{"className":35757},[999,162],[38,35759,312],{"className":35760},[102,162],[38,35762,293],{"className":35763},[371],[38,35765,3109],{"className":35766,"style":2500},[102,103],[38,35768,330],{"className":35769},[529],"：",[38,35772,35774],{"className":35773,"translate":42},[621],[38,35775,35777,35851],{"className":35776,"translate":42},[41],[38,35778,35780],{"className":35779},[46],[48,35781,35782],{"xmlns":50,"display":630},[52,35783,35784,35848],{},[55,35785,35786,35795,35797,35799,35801,35803,35805,35807,35810,35819,35835,35837,35840,35842,35844,35846],{},[55,35787,35788,35790,35792],{},[58,35789,27282],{"mathvariant":333},[58,35791,2178],{"mathvariant":333},[58,35793,35794],{"mathvariant":333},"l",[62,35796,293],{"stretchy":292},[58,35798,35448],{},[62,35800,293],{"stretchy":292},[58,35802,3109],{},[62,35804,330],{"stretchy":292},[62,35806,330],{"stretchy":292},[62,35808,35809],{},"∝",[8499,35811,35812,35815,35817],{},[62,35813,35814],{},"∫",[300,35816,683],{},[58,35818,3109],{},[66,35820,35821,35827],{},[55,35822,35823,35825],{},[58,35824,35691],{},[62,35826,673],{},[55,35828,35829,35831,35833],{},[58,35830,25826],{},[62,35832,710],{},[300,35834,312],{},[62,35836,293],{"stretchy":292},[58,35838,35839],{},"ϕ",[62,35841,330],{"stretchy":292},[62,35843,305],{"separator":89},[58,35845,25826],{},[58,35847,35839],{},[81,35849,35850],{"encoding":83},"\\mathrm{Vol}(B(\\theta)) \\propto \\int_0^\\theta \\sin^{d-2}(\\phi), d\\phi",[38,35852,35854,35893],{"className":35853,"ariaHidden":89},[88],[38,35855,35857,35860,35868,35871,35874,35877,35880,35884,35887,35890],{"className":35856},[93],[38,35858],{"className":35859,"style":1439},[97],[38,35861,35863],{"className":35862},[102],[38,35864,35867],{"className":35865},[102,35866],"mathrm","Vol",[38,35869,293],{"className":35870},[371],[38,35872,35448],{"className":35873,"style":35513},[102,103],[38,35875,293],{"className":35876},[371],[38,35878,3109],{"className":35879,"style":2500},[102,103],[38,35881,35883],{"className":35882},[529],"))",[38,35885],{"className":35886,"style":109},[108],[38,35888,35809],{"className":35889},[113],[38,35891],{"className":35892,"style":109},[108],[38,35894,35896,35900,35956,35959,35997,36000,36003,36006,36009,36012,36015],{"className":35895},[93],[38,35897],{"className":35898,"style":35899},[97],"height:2.511em;vertical-align:-0.9119em;",[38,35901,35903,35907],{"className":35902},[878],[38,35904,35814],{"className":35905,"style":35906},[878,15521,15522],"margin-right:0.4445em;position:relative;top:-0.0011em;",[38,35908,35910],{"className":35909},[134],[38,35911,35913,35947],{"className":35912},[138,384],[38,35914,35916,35944],{"className":35915},[142],[38,35917,35920,35932],{"className":35918,"style":35919},[146],"height:1.599em;",[38,35921,35923,35926],{"style":35922},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[38,35924],{"className":35925,"style":155},[154],[38,35927,35929],{"className":35928},[159,160,161,162],[38,35930,683],{"className":35931},[102,162],[38,35933,35935,35938],{"style":35934},"top:-3.8129em;margin-right:0.05em;",[38,35936],{"className":35937,"style":155},[154],[38,35939,35941],{"className":35940},[159,160,161,162],[38,35942,3109],{"className":35943,"style":2500},[102,103,162],[38,35945,408],{"className":35946},[407],[38,35948,35950],{"className":35949},[142],[38,35951,35954],{"className":35952,"style":35953},[146],"height:0.9119em;",[38,35955],{},[38,35957],{"className":35958,"style":425},[108],[38,35960,35962,35965],{"className":35961},[878],[38,35963,35691],{"className":35964},[878],[38,35966,35968],{"className":35967},[134],[38,35969,35971],{"className":35970},[138],[38,35972,35974],{"className":35973},[142],[38,35975,35977],{"className":35976,"style":35739},[146],[38,35978,35979,35982],{"style":35742},[38,35980],{"className":35981,"style":155},[154],[38,35983,35985],{"className":35984},[159,160,161,162],[38,35986,35988,35991,35994],{"className":35987},[102,162],[38,35989,25826],{"className":35990},[102,103,162],[38,35992,710],{"className":35993},[999,162],[38,35995,312],{"className":35996},[102,162],[38,35998,293],{"className":35999},[371],[38,36001,35839],{"className":36002},[102,103],[38,36004,330],{"className":36005},[529],[38,36007,305],{"className":36008},[421],[38,36010],{"className":36011,"style":425},[108],[38,36013,25826],{"className":36014},[102,103],[38,36016,35839],{"className":36017},[102,103],[11,36019,36020],{},"现在观察关键现象：",[7370,36022,36023,36147],{},[7373,36024,19890,36025,36053,36054,36146],{},[38,36026,36028,36041],{"className":36027,"translate":42},[41],[38,36029,36031],{"className":36030},[46],[48,36032,36033],{"xmlns":50},[52,36034,36035,36039],{},[55,36036,36037],{},[58,36038,25826],{},[81,36040,25826],{"encoding":83},[38,36042,36044],{"className":36043,"ariaHidden":89},[88],[38,36045,36047,36050],{"className":36046},[93],[38,36048],{"className":36049,"style":889},[97],[38,36051,25826],{"className":36052},[102,103]," 较小时，",[38,36055,36057,36090],{"className":36056,"translate":42},[41],[38,36058,36060],{"className":36059},[46],[48,36061,36062],{"xmlns":50},[52,36063,36064,36088],{},[55,36065,36066,36082,36084,36086],{},[66,36067,36068,36074],{},[55,36069,36070,36072],{},[58,36071,35691],{},[62,36073,673],{},[55,36075,36076,36078,36080],{},[58,36077,25826],{},[62,36079,710],{},[300,36081,312],{},[62,36083,293],{"stretchy":292},[58,36085,3109],{},[62,36087,330],{"stretchy":292},[81,36089,35710],{"encoding":83},[38,36091,36093],{"className":36092,"ariaHidden":89},[88],[38,36094,36096,36099,36137,36140,36143],{"className":36095},[93],[38,36097],{"className":36098,"style":35720},[97],[38,36100,36102,36105],{"className":36101},[878],[38,36103,35691],{"className":36104},[878],[38,36106,36108],{"className":36107},[134],[38,36109,36111],{"className":36110},[138],[38,36112,36114],{"className":36113},[142],[38,36115,36117],{"className":36116,"style":35739},[146],[38,36118,36119,36122],{"style":35742},[38,36120],{"className":36121,"style":155},[154],[38,36123,36125],{"className":36124},[159,160,161,162],[38,36126,36128,36131,36134],{"className":36127},[102,162],[38,36129,25826],{"className":36130},[102,103,162],[38,36132,710],{"className":36133},[999,162],[38,36135,312],{"className":36136},[102,162],[38,36138,293],{"className":36139},[371],[38,36141,3109],{"className":36142,"style":2500},[102,103],[38,36144,330],{"className":36145},[529]," 相对平坦",[7373,36148,19890,36149,19942,36200,36292,36293],{},[38,36150,36152,36170],{"className":36151,"translate":42},[41],[38,36153,36155],{"className":36154},[46],[48,36156,36157],{"xmlns":50},[52,36158,36159,36167],{},[55,36160,36161,36163,36165],{},[58,36162,25826],{},[62,36164,77],{},[58,36166,21550],{"mathvariant":333},[81,36168,36169],{"encoding":83},"d \\to \\infty",[38,36171,36173,36191],{"className":36172,"ariaHidden":89},[88],[38,36174,36176,36179,36182,36185,36188],{"className":36175},[93],[38,36177],{"className":36178,"style":889},[97],[38,36180,25826],{"className":36181},[102,103],[38,36183],{"className":36184,"style":109},[108],[38,36186,77],{"className":36187},[113],[38,36189],{"className":36190,"style":109},[108],[38,36192,36194,36197],{"className":36193},[93],[38,36195],{"className":36196,"style":4312},[97],[38,36198,21550],{"className":36199},[102],[38,36201,36203,36236],{"className":36202,"translate":42},[41],[38,36204,36206],{"className":36205},[46],[48,36207,36208],{"xmlns":50},[52,36209,36210,36234],{},[55,36211,36212,36228,36230,36232],{},[66,36213,36214,36220],{},[55,36215,36216,36218],{},[58,36217,35691],{},[62,36219,673],{},[55,36221,36222,36224,36226],{},[58,36223,25826],{},[62,36225,710],{},[300,36227,312],{},[62,36229,293],{"stretchy":292},[58,36231,3109],{},[62,36233,330],{"stretchy":292},[81,36235,35710],{"encoding":83},[38,36237,36239],{"className":36238,"ariaHidden":89},[88],[38,36240,36242,36245,36283,36286,36289],{"className":36241},[93],[38,36243],{"className":36244,"style":35720},[97],[38,36246,36248,36251],{"className":36247},[878],[38,36249,35691],{"className":36250},[878],[38,36252,36254],{"className":36253},[134],[38,36255,36257],{"className":36256},[138],[38,36258,36260],{"className":36259},[142],[38,36261,36263],{"className":36262,"style":35739},[146],[38,36264,36265,36268],{"style":35742},[38,36266],{"className":36267,"style":155},[154],[38,36269,36271],{"className":36270},[159,160,161,162],[38,36272,36274,36277,36280],{"className":36273},[102,162],[38,36275,25826],{"className":36276},[102,103,162],[38,36278,710],{"className":36279},[999,162],[38,36281,312],{"className":36282},[102,162],[38,36284,293],{"className":36285},[371],[38,36287,3109],{"className":36288,"style":2500},[102,103],[38,36290,330],{"className":36291},[529]," 在以下位置出现尖锐的峰：\n",[38,36294,36296],{"className":36295,"translate":42},[621],[38,36297,36299,36321],{"className":36298,"translate":42},[41],[38,36300,36302],{"className":36301},[46],[48,36303,36304],{"xmlns":50,"display":630},[52,36305,36306,36318],{},[55,36307,36308,36310,36312],{},[58,36309,3109],{},[62,36311,289],{},[636,36313,36314,36316],{},[58,36315,25958],{},[300,36317,312],{},[81,36319,36320],{"encoding":83},"\\theta = \\frac{\\pi}{2}",[38,36322,36324,36342],{"className":36323,"ariaHidden":89},[88],[38,36325,36327,36330,36333,36336,36339],{"className":36326},[93],[38,36328],{"className":36329,"style":889},[97],[38,36331,3109],{"className":36332,"style":2500},[102,103],[38,36334],{"className":36335,"style":109},[108],[38,36337,289],{"className":36338},[113],[38,36340],{"className":36341,"style":109},[108],[38,36343,36345,36349],{"className":36344},[93],[38,36346],{"className":36347,"style":36348},[97],"height:1.7936em;vertical-align:-0.686em;",[38,36350,36352,36355,36409],{"className":36351},[102],[38,36353],{"className":36354},[371,740],[38,36356,36358],{"className":36357},[636],[38,36359,36361,36401],{"className":36360},[138,384],[38,36362,36364,36398],{"className":36363},[142],[38,36365,36368,36379,36387],{"className":36366,"style":36367},[146],"height:1.1076em;",[38,36369,36370,36373],{"style":756},[38,36371],{"className":36372,"style":760},[154],[38,36374,36376],{"className":36375},[102],[38,36377,312],{"className":36378},[102],[38,36380,36381,36384],{"style":811},[38,36382],{"className":36383,"style":760},[154],[38,36385],{"className":36386,"style":819},[818],[38,36388,36389,36392],{"style":822},[38,36390],{"className":36391,"style":760},[154],[38,36393,36395],{"className":36394},[102],[38,36396,25958],{"className":36397,"style":3420},[102,103],[38,36399,408],{"className":36400},[407],[38,36402,36404],{"className":36403},[142],[38,36405,36407],{"className":36406,"style":1079},[146],[38,36408],{},[38,36410],{"className":36411},[529,740],[11,36413,36414],{},"这导致：大部分质量集中在与任意固定方向正交的赤道附近。",[11,36416,36417,36418,35770],{},"更正式地说，对任意固定的 ",[38,36419,36421,36441],{"className":36420,"translate":42},[41],[38,36422,36424],{"className":36423},[46],[48,36425,36426],{"xmlns":50},[52,36427,36428,36438],{},[55,36429,36430,36433,36436],{},[58,36431,36432],{},"ϵ",[62,36434,36435],{},">",[300,36437,683],{},[81,36439,36440],{"encoding":83},"\\epsilon > 0",[38,36442,36444,36462],{"className":36443,"ariaHidden":89},[88],[38,36445,36447,36450,36453,36456,36459],{"className":36446},[93],[38,36448],{"className":36449,"style":12573},[97],[38,36451,36432],{"className":36452},[102,103],[38,36454],{"className":36455,"style":109},[108],[38,36457,36435],{"className":36458},[113],[38,36460],{"className":36461,"style":109},[108],[38,36463,36465,36468],{"className":36464},[93],[38,36466],{"className":36467,"style":2429},[97],[38,36469,683],{"className":36470},[102],[38,36472,36474],{"className":36473,"translate":42},[621],[38,36475,36477,36546],{"className":36476,"translate":42},[41],[38,36478,36480],{"className":36479},[46],[48,36481,36482],{"xmlns":50,"display":630},[52,36483,36484,36543],{},[55,36485,36486,36488,36528,36530,36532,36534,36537,36539,36541],{},[58,36487,25818],{"mathvariant":70},[55,36489,36490,36492,36522,36524,36526],{},[62,36491,293],{"fence":89},[55,36493,36494,36496,36498,36500,36502,36504,36510,36512,36514,36520],{},[62,36495,7695],{"fence":89},[58,36497,35479],{"mathvariant":333},[62,36499,293],{"stretchy":292},[58,36501,286],{},[62,36503,305],{"separator":89},[295,36505,36506,36508],{},[58,36507,286],{},[300,36509,683],{},[62,36511,330],{"stretchy":292},[62,36513,710],{},[636,36515,36516,36518],{},[58,36517,25958],{},[300,36519,312],{},[62,36521,7695],{"fence":89},[62,36523,2879],{},[58,36525,36432],{},[62,36527,330],{"fence":89},[62,36529,77],{},[300,36531,302],{},[108,36533],{"width":4479},[3265,36535,36536],{},"当 ",[58,36538,25826],{},[62,36540,77],{},[58,36542,21550],{"mathvariant":333},[81,36544,36545],{"encoding":83},"\\mathbb{P}\\left( \\left| \\angle(x, x_0) - \\frac{\\pi}{2} \\right| \\leq \\epsilon \\right)\n\\to 1 \\quad \\text{当 } d \\to \\infty",[38,36547,36549,36813,36847],{"className":36548,"ariaHidden":89},[88],[38,36550,36552,36556,36559,36562,36804,36807,36810],{"className":36551},[93],[38,36553],{"className":36554,"style":36555},[97],"height:1.836em;vertical-align:-0.686em;",[38,36557,25818],{"className":36558},[102,130],[38,36560],{"className":36561,"style":425},[108],[38,36563,36565,36572,36786,36789,36792,36795,36798],{"className":36564},[475],[38,36566,36568],{"className":36567,"style":1481},[371,1480],[38,36569,293],{"className":36570},[1485,36571],"size2",[38,36573,36575,36620,36623,36626,36629,36632,36635,36675,36678,36681,36684,36687,36749],{"className":36574},[475],[38,36576,36578],{"className":36577},[371],[38,36579,36581],{"className":36580},[1485,4820],[38,36582,36584,36611],{"className":36583},[138,384],[38,36585,36587,36608],{"className":36586},[142],[38,36588,36591],{"className":36589,"style":36590},[146],"height:1.15em;",[38,36592,36594,36597],{"style":36593},"top:-3.15em;",[38,36595],{"className":36596,"style":20401},[154],[38,36598,36600],{"style":36599},"width:0.333em;height:1.8em;",[4842,36601,36605],{"xmlns":4844,"width":36602,"height":36603,"viewBox":36604},"0.333em","1.8em","0 0 333 1800",[4849,36606],{"d":36607},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[38,36609,408],{"className":36610},[407],[38,36612,36614],{"className":36613},[142],[38,36615,36618],{"className":36616,"style":36617},[146],"height:0.65em;",[38,36619],{},[38,36621,35479],{"className":36622},[102],[38,36624,293],{"className":36625},[371],[38,36627,286],{"className":36628},[102,103],[38,36630,305],{"className":36631},[421],[38,36633],{"className":36634,"style":425},[108],[38,36636,36638,36641],{"className":36637},[102],[38,36639,286],{"className":36640},[102,103],[38,36642,36644],{"className":36643},[134],[38,36645,36647,36667],{"className":36646},[138,384],[38,36648,36650,36664],{"className":36649},[142],[38,36651,36653],{"className":36652,"style":391},[146],[38,36654,36655,36658],{"style":394},[38,36656],{"className":36657,"style":155},[154],[38,36659,36661],{"className":36660},[159,160,161,162],[38,36662,683],{"className":36663},[102,162],[38,36665,408],{"className":36666},[407],[38,36668,36670],{"className":36669},[142],[38,36671,36673],{"className":36672,"style":415},[146],[38,36674],{},[38,36676,330],{"className":36677},[529],[38,36679],{"className":36680,"style":995},[108],[38,36682,710],{"className":36683},[999],[38,36685],{"className":36686,"style":995},[108],[38,36688,36690,36693,36746],{"className":36689},[102],[38,36691],{"className":36692},[371,740],[38,36694,36696],{"className":36695},[636],[38,36697,36699,36738],{"className":36698},[138,384],[38,36700,36702,36735],{"className":36701},[142],[38,36703,36705,36716,36724],{"className":36704,"style":36367},[146],[38,36706,36707,36710],{"style":756},[38,36708],{"className":36709,"style":760},[154],[38,36711,36713],{"className":36712},[102],[38,36714,312],{"className":36715},[102],[38,36717,36718,36721],{"style":811},[38,36719],{"className":36720,"style":760},[154],[38,36722],{"className":36723,"style":819},[818],[38,36725,36726,36729],{"style":822},[38,36727],{"className":36728,"style":760},[154],[38,36730,36732],{"className":36731},[102],[38,36733,25958],{"className":36734,"style":3420},[102,103],[38,36736,408],{"className":36737},[407],[38,36739,36741],{"className":36740},[142],[38,36742,36744],{"className":36743,"style":1079},[146],[38,36745],{},[38,36747],{"className":36748},[529,740],[38,36750,36752],{"className":36751},[529],[38,36753,36755],{"className":36754},[1485,4820],[38,36756,36758,36778],{"className":36757},[138,384],[38,36759,36761,36775],{"className":36760},[142],[38,36762,36764],{"className":36763,"style":36590},[146],[38,36765,36766,36769],{"style":36593},[38,36767],{"className":36768,"style":20401},[154],[38,36770,36771],{"style":36599},[4842,36772,36773],{"xmlns":4844,"width":36602,"height":36603,"viewBox":36604},[4849,36774],{"d":36607},[38,36776,408],{"className":36777},[407],[38,36779,36781],{"className":36780},[142],[38,36782,36784],{"className":36783,"style":36617},[146],[38,36785],{},[38,36787],{"className":36788,"style":109},[108],[38,36790,2879],{"className":36791},[113],[38,36793],{"className":36794,"style":109},[108],[38,36796,36432],{"className":36797},[102,103],[38,36799,36801],{"className":36800,"style":1481},[529,1480],[38,36802,330],{"className":36803},[1485,36571],[38,36805],{"className":36806,"style":109},[108],[38,36808,77],{"className":36809},[113],[38,36811],{"className":36812,"style":109},[108],[38,36814,36816,36819,36822,36825,36835,36838,36841,36844],{"className":36815},[93],[38,36817],{"className":36818,"style":889},[97],[38,36820,302],{"className":36821},[102],[38,36823],{"className":36824,"style":10740},[108],[38,36826,36828,36832],{"className":36827},[102,22531],[38,36829,36831],{"className":36830},[102,22535],"当",[38,36833,15653],{"className":36834},[102],[38,36836,25826],{"className":36837},[102,103],[38,36839],{"className":36840,"style":109},[108],[38,36842,77],{"className":36843},[113],[38,36845],{"className":36846,"style":109},[108],[38,36848,36850,36853],{"className":36849},[93],[38,36851],{"className":36852,"style":4312},[97],[38,36854,21550],{"className":36855},[102],[11,36857,36858],{},"这是正曲率空间上测度集中现象的一种体现。",[11,36860,36861],{},"这种几何解释可以描述为「赤道坍缩」。在欧氏空间中，样本集中在薄外壳中；而在球面上，样本则集中在赤道附近的细环带中。",[11,36863,36864],{},"这可以看作是曲率引起的测度再分布。",[31,36866,36867],{"id":36867},"正曲率下的角度距离",[11,36869,36870],{},"设：",[38,36872,36874],{"className":36873,"translate":42},[621],[38,36875,36877,36916],{"className":36876,"translate":42},[41],[38,36878,36880],{"className":36879},[46],[48,36881,36882],{"xmlns":50,"display":630},[52,36883,36884,36913],{},[55,36885,36886,36888,36890,36892,36894,36897,36899,36911],{},[58,36887,286],{},[62,36889,305],{"separator":89},[58,36891,26054],{},[62,36893,30522],{},[3265,36895,36896],{},"Uniform",[62,36898,293],{"stretchy":292},[66,36900,36901,36903],{},[58,36902,25953],{"mathvariant":70},[55,36904,36905,36907,36909],{},[58,36906,25826],{},[62,36908,710],{},[300,36910,302],{},[62,36912,330],{"stretchy":292},[81,36914,36915],{"encoding":83},"x, y \\sim \\text{Uniform}(\\mathbb{S}^{d-1})",[38,36917,36919,36946],{"className":36918,"ariaHidden":89},[88],[38,36920,36922,36925,36928,36931,36934,36937,36940,36943],{"className":36921},[93],[38,36923],{"className":36924,"style":3505},[97],[38,36926,286],{"className":36927},[102,103],[38,36929,305],{"className":36930},[421],[38,36932],{"className":36933,"style":425},[108],[38,36935,26054],{"className":36936,"style":3420},[102,103],[38,36938],{"className":36939,"style":109},[108],[38,36941,30522],{"className":36942},[113],[38,36944],{"className":36945,"style":109},[108],[38,36947,36949,36952,36958,36961,36999],{"className":36948},[93],[38,36950],{"className":36951,"style":2277},[97],[38,36953,36955],{"className":36954},[102,22531],[38,36956,36896],{"className":36957},[102],[38,36959,293],{"className":36960},[371],[38,36962,36964,36967],{"className":36963},[102],[38,36965,25953],{"className":36966},[102,130],[38,36968,36970],{"className":36969},[134],[38,36971,36973],{"className":36972},[138],[38,36974,36976],{"className":36975},[142],[38,36977,36979],{"className":36978,"style":2308},[146],[38,36980,36981,36984],{"style":1892},[38,36982],{"className":36983,"style":155},[154],[38,36985,36987],{"className":36986},[159,160,161,162],[38,36988,36990,36993,36996],{"className":36989},[102,162],[38,36991,25826],{"className":36992},[102,103,162],[38,36994,710],{"className":36995},[999,162],[38,36997,302],{"className":36998},[102,162],[38,37000,330],{"className":37001},[529],[11,37003,37004],{},"则它们的内积满足：",[38,37006,37008],{"className":37007,"translate":42},[621],[38,37009,37011,37053],{"className":37010,"translate":42},[41],[38,37012,37014],{"className":37013},[46],[48,37015,37016],{"xmlns":50,"display":630},[52,37017,37018,37050],{},[55,37019,37020,37022,37024,37026,37028,37030,37032,37034],{},[62,37021,31837],{"stretchy":292},[58,37023,286],{},[62,37025,305],{"separator":89},[58,37027,26054],{},[62,37029,31846],{"stretchy":292},[62,37031,30522],{},[58,37033,14701],{"mathvariant":30528},[55,37035,37036,37038,37040,37042,37048],{},[62,37037,293],{"fence":89},[300,37039,683],{},[62,37041,305],{"separator":89},[636,37043,37044,37046],{},[300,37045,302],{},[58,37047,25826],{},[62,37049,330],{"fence":89},[81,37051,37052],{"encoding":83},"\\langle x, y \\rangle \\sim \\mathcal{N}\\left(0, \\frac{1}{d}\\right)",[38,37054,37056,37089],{"className":37055,"ariaHidden":89},[88],[38,37057,37059,37062,37065,37068,37071,37074,37077,37080,37083,37086],{"className":37058},[93],[38,37060],{"className":37061,"style":1439},[97],[38,37063,31837],{"className":37064},[371],[38,37066,286],{"className":37067},[102,103],[38,37069,305],{"className":37070},[421],[38,37072],{"className":37073,"style":425},[108],[38,37075,26054],{"className":37076,"style":3420},[102,103],[38,37078,31846],{"className":37079},[529],[38,37081],{"className":37082,"style":109},[108],[38,37084,30522],{"className":37085},[113],[38,37087],{"className":37088,"style":109},[108],[38,37090,37092,37096,37099,37102],{"className":37091},[93],[38,37093],{"className":37094,"style":37095},[97],"height:2.4em;vertical-align:-0.95em;",[38,37097,14701],{"className":37098,"style":30597},[102,30596],[38,37100],{"className":37101,"style":425},[108],[38,37103,37105,37111,37114,37117,37120,37182],{"className":37104},[475],[38,37106,37108],{"className":37107,"style":1481},[371,1480],[38,37109,293],{"className":37110},[1485,161],[38,37112,683],{"className":37113},[102],[38,37115,305],{"className":37116},[421],[38,37118],{"className":37119,"style":425},[108],[38,37121,37123,37126,37179],{"className":37122},[102],[38,37124],{"className":37125},[371,740],[38,37127,37129],{"className":37128},[636],[38,37130,37132,37171],{"className":37131},[138,384],[38,37133,37135,37168],{"className":37134},[142],[38,37136,37138,37149,37157],{"className":37137,"style":11324},[146],[38,37139,37140,37143],{"style":756},[38,37141],{"className":37142,"style":760},[154],[38,37144,37146],{"className":37145},[102],[38,37147,25826],{"className":37148},[102,103],[38,37150,37151,37154],{"style":811},[38,37152],{"className":37153,"style":760},[154],[38,37155],{"className":37156,"style":819},[818],[38,37158,37159,37162],{"style":822},[38,37160],{"className":37161,"style":760},[154],[38,37163,37165],{"className":37164},[102],[38,37166,302],{"className":37167},[102],[38,37169,408],{"className":37170},[407],[38,37172,37174],{"className":37173},[142],[38,37175,37177],{"className":37176,"style":1079},[146],[38,37178],{},[38,37180],{"className":37181},[529,740],[38,37183,37185],{"className":37184,"style":1481},[529,1480],[38,37186,330],{"className":37187},[1485,161],[11,37189,32987],{},[38,37191,37193],{"className":37192,"translate":42},[621],[38,37194,37196,37247],{"className":37195,"translate":42},[41],[38,37197,37199],{"className":37198},[46],[48,37200,37201],{"xmlns":50,"display":630},[52,37202,37203,37244],{},[55,37204,37205,37207,37209,37211,37213,37215,37217,37219,37221,37223,37225,37227,37229,37232,37234,37236,37238],{},[58,37206,34575],{},[62,37208,673],{},[58,37210,3109],{},[62,37212,289],{},[62,37214,31837],{"stretchy":292},[58,37216,286],{},[62,37218,305],{"separator":89},[58,37220,26054],{},[62,37222,31846],{"stretchy":292},[62,37224,31671],{},[300,37226,683],{},[108,37228],{"width":4479},[62,37230,37231],{},"⇒",[108,37233],{"width":4479},[58,37235,3109],{},[62,37237,31671],{},[636,37239,37240,37242],{},[58,37241,25958],{},[300,37243,312],{},[81,37245,37246],{"encoding":83},"\\cos\\theta = \\langle x, y \\rangle \\approx 0\n\\quad \\Rightarrow \\quad\n\\theta \\approx \\frac{\\pi}{2}",[38,37248,37250,37274,37307,37331,37349],{"className":37249,"ariaHidden":89},[88],[38,37251,37253,37256,37259,37262,37265,37268,37271],{"className":37252},[93],[38,37254],{"className":37255,"style":889},[97],[38,37257,34575],{"className":37258},[878],[38,37260],{"className":37261,"style":425},[108],[38,37263,3109],{"className":37264,"style":2500},[102,103],[38,37266],{"className":37267,"style":109},[108],[38,37269,289],{"className":37270},[113],[38,37272],{"className":37273,"style":109},[108],[38,37275,37277,37280,37283,37286,37289,37292,37295,37298,37301,37304],{"className":37276},[93],[38,37278],{"className":37279,"style":1439},[97],[38,37281,31837],{"className":37282},[371],[38,37284,286],{"className":37285},[102,103],[38,37287,305],{"className":37288},[421],[38,37290],{"className":37291,"style":425},[108],[38,37293,26054],{"className":37294,"style":3420},[102,103],[38,37296,31846],{"className":37297},[529],[38,37299],{"className":37300,"style":109},[108],[38,37302,31671],{"className":37303},[113],[38,37305],{"className":37306,"style":109},[108],[38,37308,37310,37313,37316,37319,37322,37325,37328],{"className":37309},[93],[38,37311],{"className":37312,"style":2429},[97],[38,37314,683],{"className":37315},[102],[38,37317],{"className":37318,"style":10740},[108],[38,37320],{"className":37321,"style":109},[108],[38,37323,37231],{"className":37324},[113],[38,37326],{"className":37327,"style":10740},[108],[38,37329],{"className":37330,"style":109},[108],[38,37332,37334,37337,37340,37343,37346],{"className":37333},[93],[38,37335],{"className":37336,"style":889},[97],[38,37338,3109],{"className":37339,"style":2500},[102,103],[38,37341],{"className":37342,"style":109},[108],[38,37344,31671],{"className":37345},[113],[38,37347],{"className":37348,"style":109},[108],[38,37350,37352,37355],{"className":37351},[93],[38,37353],{"className":37354,"style":36348},[97],[38,37356,37358,37361,37414],{"className":37357},[102],[38,37359],{"className":37360},[371,740],[38,37362,37364],{"className":37363},[636],[38,37365,37367,37406],{"className":37366},[138,384],[38,37368,37370,37403],{"className":37369},[142],[38,37371,37373,37384,37392],{"className":37372,"style":36367},[146],[38,37374,37375,37378],{"style":756},[38,37376],{"className":37377,"style":760},[154],[38,37379,37381],{"className":37380},[102],[38,37382,312],{"className":37383},[102],[38,37385,37386,37389],{"style":811},[38,37387],{"className":37388,"style":760},[154],[38,37390],{"className":37391,"style":819},[818],[38,37393,37394,37397],{"style":822},[38,37395],{"className":37396,"style":760},[154],[38,37398,37400],{"className":37399},[102],[38,37401,25958],{"className":37402,"style":3420},[102,103],[38,37404,408],{"className":37405},[407],[38,37407,37409],{"className":37408},[142],[38,37410,37412],{"className":37411,"style":1079},[146],[38,37413],{},[38,37415],{"className":37416},[529,740],[11,37418,37419],{},"于是我们得出结论：",[1252,37421,37422],{},[11,37423,37424],{},"高维球面上的两个随机点几乎总是正交的。",[11,37426,37427],{},"这意味着：",[38,37429,37431],{"className":37430,"translate":42},[621],[38,37432,37434,37492],{"className":37433,"translate":42},[41],[38,37435,37437],{"className":37436},[46],[48,37438,37439],{"xmlns":50,"display":630},[52,37440,37441,37489],{},[55,37442,37443,37450,37452,37454,37456,37458,37460,37462,37465,37467,37469,37471,37473,37475,37477,37479,37481,37483],{},[295,37444,37445,37447],{},[58,37446,25826],{},[3265,37448,37449],{},"测地",[62,37451,293],{"stretchy":292},[58,37453,286],{},[62,37455,305],{"separator":89},[58,37457,26054],{},[62,37459,330],{"stretchy":292},[62,37461,289],{},[58,37463,37464],{},"arccos",[62,37466,673],{},[62,37468,293],{"stretchy":292},[62,37470,31837],{"stretchy":292},[58,37472,286],{},[62,37474,305],{"separator":89},[58,37476,26054],{},[62,37478,31846],{"stretchy":292},[62,37480,330],{"stretchy":292},[62,37482,31671],{},[636,37484,37485,37487],{},[58,37486,25958],{},[300,37488,312],{},[81,37490,37491],{"encoding":83},"d_{\\text{测地}}(x, y) = \\arccos(\\langle x, y \\rangle) \\approx \\frac{\\pi}{2}",[38,37493,37495,37574,37612],{"className":37494,"ariaHidden":89},[88],[38,37496,37498,37501,37547,37550,37553,37556,37559,37562,37565,37568,37571],{"className":37497},[93],[38,37499],{"className":37500,"style":1439},[97],[38,37502,37504,37507],{"className":37503},[102],[38,37505,25826],{"className":37506},[102,103],[38,37508,37510],{"className":37509},[134],[38,37511,37513,37539],{"className":37512},[138,384],[38,37514,37516,37536],{"className":37515},[142],[38,37517,37519],{"className":37518,"style":7469},[146],[38,37520,37521,37524],{"style":394},[38,37522],{"className":37523,"style":155},[154],[38,37525,37527],{"className":37526},[159,160,161,162],[38,37528,37530],{"className":37529},[102,162],[38,37531,37533],{"className":37532},[102,22531,162],[38,37534,37449],{"className":37535},[102,22535,162],[38,37537,408],{"className":37538},[407],[38,37540,37542],{"className":37541},[142],[38,37543,37545],{"className":37544,"style":415},[146],[38,37546],{},[38,37548,293],{"className":37549},[371],[38,37551,286],{"className":37552},[102,103],[38,37554,305],{"className":37555},[421],[38,37557],{"className":37558,"style":425},[108],[38,37560,26054],{"className":37561,"style":3420},[102,103],[38,37563,330],{"className":37564},[529],[38,37566],{"className":37567,"style":109},[108],[38,37569,289],{"className":37570},[113],[38,37572],{"className":37573,"style":109},[108],[38,37575,37577,37580,37583,37587,37590,37593,37596,37599,37603,37606,37609],{"className":37576},[93],[38,37578],{"className":37579,"style":1439},[97],[38,37581,37464],{"className":37582},[878],[38,37584,37586],{"className":37585},[371],"(⟨",[38,37588,286],{"className":37589},[102,103],[38,37591,305],{"className":37592},[421],[38,37594],{"className":37595,"style":425},[108],[38,37597,26054],{"className":37598,"style":3420},[102,103],[38,37600,37602],{"className":37601},[529],"⟩)",[38,37604],{"className":37605,"style":109},[108],[38,37607,31671],{"className":37608},[113],[38,37610],{"className":37611,"style":109},[108],[38,37613,37615,37618],{"className":37614},[93],[38,37616],{"className":37617,"style":36348},[97],[38,37619,37621,37624,37677],{"className":37620},[102],[38,37622],{"className":37623},[371,740],[38,37625,37627],{"className":37626},[636],[38,37628,37630,37669],{"className":37629},[138,384],[38,37631,37633,37666],{"className":37632},[142],[38,37634,37636,37647,37655],{"className":37635,"style":36367},[146],[38,37637,37638,37641],{"style":756},[38,37639],{"className":37640,"style":760},[154],[38,37642,37644],{"className":37643},[102],[38,37645,312],{"className":37646},[102],[38,37648,37649,37652],{"style":811},[38,37650],{"className":37651,"style":760},[154],[38,37653],{"className":37654,"style":819},[818],[38,37656,37657,37660],{"style":822},[38,37658],{"className":37659,"style":760},[154],[38,37661,37663],{"className":37662},[102],[38,37664,25958],{"className":37665,"style":3420},[102,103],[38,37667,408],{"className":37668},[407],[38,37670,37672],{"className":37671},[142],[38,37673,37675],{"className":37674,"style":1079},[146],[38,37676],{},[38,37678],{"className":37679},[529,740],[31,37681,37682],{"id":37682},"测地度量下的距离集中",[11,37684,37685,37686,37760],{},"与欧氏距离不同，",[38,37687,37689,37713],{"className":37688,"translate":42},[41],[38,37690,37692],{"className":37691},[46],[48,37693,37694],{"xmlns":50},[52,37695,37696,37710],{},[55,37697,37698],{},[66,37699,37700,37702],{},[58,37701,25953],{"mathvariant":70},[55,37703,37704,37706,37708],{},[58,37705,25826],{},[62,37707,710],{},[300,37709,302],{},[81,37711,37712],{"encoding":83},"\\mathbb{S}^{d-1}",[38,37714,37716],{"className":37715,"ariaHidden":89},[88],[38,37717,37719,37722],{"className":37718},[93],[38,37720],{"className":37721,"style":545},[97],[38,37723,37725,37728],{"className":37724},[102],[38,37726,25953],{"className":37727},[102,130],[38,37729,37731],{"className":37730},[134],[38,37732,37734],{"className":37733},[138],[38,37735,37737],{"className":37736},[142],[38,37738,37740],{"className":37739,"style":545},[146],[38,37741,37742,37745],{"style":150},[38,37743],{"className":37744,"style":155},[154],[38,37746,37748],{"className":37747},[159,160,161,162],[38,37749,37751,37754,37757],{"className":37750},[102,162],[38,37752,25826],{"className":37753},[102,103,162],[38,37755,710],{"className":37756},[999,162],[38,37758,302],{"className":37759},[102,162]," 上的自然度量是测地距离：",[38,37762,37764],{"className":37763,"translate":42},[621],[38,37765,37767,37815],{"className":37766,"translate":42},[41],[38,37768,37770],{"className":37769},[46],[48,37771,37772],{"xmlns":50,"display":630},[52,37773,37774,37812],{},[55,37775,37776,37782,37784,37786,37788,37790,37792,37794,37796,37798,37800,37802,37804,37806,37808,37810],{},[295,37777,37778,37780],{},[58,37779,25826],{},[58,37781,25953],{"mathvariant":70},[62,37783,293],{"stretchy":292},[58,37785,286],{},[62,37787,305],{"separator":89},[58,37789,26054],{},[62,37791,330],{"stretchy":292},[62,37793,289],{},[58,37795,37464],{},[62,37797,673],{},[62,37799,293],{"stretchy":292},[62,37801,31837],{"stretchy":292},[58,37803,286],{},[62,37805,305],{"separator":89},[58,37807,26054],{},[62,37809,31846],{"stretchy":292},[62,37811,330],{"stretchy":292},[81,37813,37814],{"encoding":83},"d_{\\mathbb{S}}(x, y) = \\arccos(\\langle x, y \\rangle)",[38,37816,37818,37895],{"className":37817,"ariaHidden":89},[88],[38,37819,37821,37824,37868,37871,37874,37877,37880,37883,37886,37889,37892],{"className":37820},[93],[38,37822],{"className":37823,"style":1439},[97],[38,37825,37827,37830],{"className":37826},[102],[38,37828,25826],{"className":37829},[102,103],[38,37831,37833],{"className":37832},[134],[38,37834,37836,37860],{"className":37835},[138,384],[38,37837,37839,37857],{"className":37838},[142],[38,37840,37843],{"className":37841,"style":37842},[146],"height:0.3322em;",[38,37844,37845,37848],{"style":394},[38,37846],{"className":37847,"style":155},[154],[38,37849,37851],{"className":37850},[159,160,161,162],[38,37852,37854],{"className":37853},[102,162],[38,37855,25953],{"className":37856},[102,130,162],[38,37858,408],{"className":37859},[407],[38,37861,37863],{"className":37862},[142],[38,37864,37866],{"className":37865,"style":415},[146],[38,37867],{},[38,37869,293],{"className":37870},[371],[38,37872,286],{"className":37873},[102,103],[38,37875,305],{"className":37876},[421],[38,37878],{"className":37879,"style":425},[108],[38,37881,26054],{"className":37882,"style":3420},[102,103],[38,37884,330],{"className":37885},[529],[38,37887],{"className":37888,"style":109},[108],[38,37890,289],{"className":37891},[113],[38,37893],{"className":37894,"style":109},[108],[38,37896,37898,37901,37904,37907,37910,37913,37916,37919],{"className":37897},[93],[38,37899],{"className":37900,"style":1439},[97],[38,37902,37464],{"className":37903},[878],[38,37905,37586],{"className":37906},[371],[38,37908,286],{"className":37909},[102,103],[38,37911,305],{"className":37912},[421],[38,37914],{"className":37915,"style":425},[108],[38,37917,26054],{"className":37918,"style":3420},[102,103],[38,37920,37602],{"className":37921},[529],[11,37923,32540],{},[7370,37925,37926,38002],{},[7373,37927,37928],{},[38,37929,37931,37957],{"className":37930,"translate":42},[41],[38,37932,37934],{"className":37933},[46],[48,37935,37936],{"xmlns":50},[52,37937,37938,37954],{},[55,37939,37940,37942,37944,37946,37948,37950,37952],{},[62,37941,31837],{"stretchy":292},[58,37943,286],{},[62,37945,305],{"separator":89},[58,37947,26054],{},[62,37949,31846],{"stretchy":292},[62,37951,77],{},[300,37953,683],{},[81,37955,37956],{"encoding":83},"\\langle x, y \\rangle \\to 0",[38,37958,37960,37993],{"className":37959,"ariaHidden":89},[88],[38,37961,37963,37966,37969,37972,37975,37978,37981,37984,37987,37990],{"className":37962},[93],[38,37964],{"className":37965,"style":1439},[97],[38,37967,31837],{"className":37968},[371],[38,37970,286],{"className":37971},[102,103],[38,37973,305],{"className":37974},[421],[38,37976],{"className":37977,"style":425},[108],[38,37979,26054],{"className":37980,"style":3420},[102,103],[38,37982,31846],{"className":37983},[529],[38,37985],{"className":37986,"style":109},[108],[38,37988,77],{"className":37989},[113],[38,37991],{"className":37992,"style":109},[108],[38,37994,37996,37999],{"className":37995},[93],[38,37997],{"className":37998,"style":2429},[97],[38,38000,683],{"className":38001},[102],[7373,38003,38004,38005],{},"波动为 ",[38,38006,38008,38034],{"className":38007,"translate":42},[41],[38,38009,38011],{"className":38010},[46],[48,38012,38013],{"xmlns":50},[52,38014,38015,38031],{},[55,38016,38017,38019,38021,38023,38025,38029],{},[58,38018,13105],{},[62,38020,293],{"stretchy":292},[300,38022,302],{},[58,38024,18490],{"mathvariant":333},[20042,38026,38027],{},[58,38028,25826],{},[62,38030,330],{"stretchy":292},[81,38032,38033],{"encoding":83},"O(1\u002F\\sqrt{d})",[38,38035,38037],{"className":38036,"ariaHidden":89},[88],[38,38038,38040,38043,38046,38049,38052,38099],{"className":38039},[93],[38,38041],{"className":38042,"style":33130},[97],[38,38044,13105],{"className":38045,"style":2500},[102,103],[38,38047,293],{"className":38048},[371],[38,38050,18766],{"className":38051},[102],[38,38053,38055],{"className":38054},[102,20187],[38,38056,38058,38091],{"className":38057},[138,384],[38,38059,38061,38088],{"className":38060},[142],[38,38062,38064,38076],{"className":38063,"style":31725},[146],[38,38065,38067,38070],{"className":38066,"style":913},[20201],[38,38068],{"className":38069,"style":760},[154],[38,38071,38073],{"className":38072,"style":21076},[102],[38,38074,25826],{"className":38075},[102,103],[38,38077,38078,38081],{"style":31740},[38,38079],{"className":38080,"style":760},[154],[38,38082,38084],{"className":38083,"style":21101},[20343],[4842,38085,38086],{"xmlns":4844,"width":20347,"height":21104,"viewBox":21105,"preserveAspectRatio":20350},[4849,38087],{"d":21108},[38,38089,408],{"className":38090},[407],[38,38092,38094],{"className":38093},[142],[38,38095,38097],{"className":38096,"style":31760},[146],[38,38098],{},[38,38100,330],{"className":38101},[529],[11,38103,29656],{},[38,38105,38107],{"className":38106,"translate":42},[621],[38,38108,38110,38148],{"className":38109,"translate":42},[41],[38,38111,38113],{"className":38112},[46],[48,38114,38115],{"xmlns":50,"display":630},[52,38116,38117,38145],{},[55,38118,38119,38121,38123,38129,38131,38133,38135,38137,38139,38141,38143],{},[3265,38120,32842],{},[62,38122,293],{"stretchy":292},[295,38124,38125,38127],{},[58,38126,25826],{},[58,38128,25953],{"mathvariant":70},[62,38130,293],{"stretchy":292},[58,38132,286],{},[62,38134,305],{"separator":89},[58,38136,26054],{},[62,38138,330],{"stretchy":292},[62,38140,330],{"stretchy":292},[62,38142,77],{},[300,38144,683],{},[81,38146,38147],{"encoding":83},"\\text{Var}(d_{\\mathbb{S}}(x, y)) \\to 0",[38,38149,38151,38236],{"className":38150,"ariaHidden":89},[88],[38,38152,38154,38157,38163,38166,38209,38212,38215,38218,38221,38224,38227,38230,38233],{"className":38153},[93],[38,38155],{"className":38156,"style":1439},[97],[38,38158,38160],{"className":38159},[102,22531],[38,38161,32842],{"className":38162},[102],[38,38164,293],{"className":38165},[371],[38,38167,38169,38172],{"className":38168},[102],[38,38170,25826],{"className":38171},[102,103],[38,38173,38175],{"className":38174},[134],[38,38176,38178,38201],{"className":38177},[138,384],[38,38179,38181,38198],{"className":38180},[142],[38,38182,38184],{"className":38183,"style":37842},[146],[38,38185,38186,38189],{"style":394},[38,38187],{"className":38188,"style":155},[154],[38,38190,38192],{"className":38191},[159,160,161,162],[38,38193,38195],{"className":38194},[102,162],[38,38196,25953],{"className":38197},[102,130,162],[38,38199,408],{"className":38200},[407],[38,38202,38204],{"className":38203},[142],[38,38205,38207],{"className":38206,"style":415},[146],[38,38208],{},[38,38210,293],{"className":38211},[371],[38,38213,286],{"className":38214},[102,103],[38,38216,305],{"className":38217},[421],[38,38219],{"className":38220,"style":425},[108],[38,38222,26054],{"className":38223,"style":3420},[102,103],[38,38225,35883],{"className":38226},[529],[38,38228],{"className":38229,"style":109},[108],[38,38231,77],{"className":38232},[113],[38,38234],{"className":38235,"style":109},[108],[38,38237,38239,38242],{"className":38238},[93],[38,38240],{"className":38241,"style":2429},[97],[38,38243,683],{"className":38244},[102],[11,38246,38247,38248,1964],{},"因此，测地距离同样会集中，即几乎所有点对之间的距离都等于 ",[38,38249,38251,38269],{"className":38250,"translate":42},[41],[38,38252,38254],{"className":38253},[46],[48,38255,38256],{"xmlns":50},[52,38257,38258,38266],{},[55,38259,38260],{},[636,38261,38262,38264],{},[58,38263,25958],{},[300,38265,312],{},[81,38267,38268],{"encoding":83},"\\frac{\\pi}{2}",[38,38270,38272],{"className":38271,"ariaHidden":89},[88],[38,38273,38275,38279],{"className":38274},[93],[38,38276],{"className":38277,"style":38278},[97],"height:1.0404em;vertical-align:-0.345em;",[38,38280,38282,38285,38345],{"className":38281},[102],[38,38283],{"className":38284},[371,740],[38,38286,38288],{"className":38287},[636],[38,38289,38291,38337],{"className":38290},[138,384],[38,38292,38294,38334],{"className":38293},[142],[38,38295,38298,38312,38320],{"className":38296,"style":38297},[146],"height:0.6954em;",[38,38299,38300,38303],{"style":17472},[38,38301],{"className":38302,"style":760},[154],[38,38304,38306],{"className":38305},[159,160,161,162],[38,38307,38309],{"className":38308},[102,162],[38,38310,312],{"className":38311},[102,162],[38,38313,38314,38317],{"style":811},[38,38315],{"className":38316,"style":760},[154],[38,38318],{"className":38319,"style":819},[818],[38,38321,38322,38325],{"style":17586},[38,38323],{"className":38324,"style":760},[154],[38,38326,38328],{"className":38327},[159,160,161,162],[38,38329,38331],{"className":38330},[102,162],[38,38332,25958],{"className":38333,"style":3420},[102,103,162],[38,38335,408],{"className":38336},[407],[38,38338,38340],{"className":38339},[142],[38,38341,38343],{"className":38342,"style":27393},[146],[38,38344],{},[38,38346],{"className":38347},[529,740],[31,38349,38350],{"id":38350},"曲率放大集中",[11,38352,38353],{},"一个重要的洞见是：正曲率并不会消除维数灾难，而是重塑它，并且常常强化它。",[11,38355,38356],{},"为什么？",[7370,38358,38359,38362,38365],{},[7373,38360,38361],{},"球面是紧致的，不存在径向扩散",[7373,38363,38364],{},"曲率迫使测地线「折返」",[7373,38366,38367],{},"体积增长比欧氏空间更慢",[11,38369,38370],{},"结果是：样本集中得更加紧密，角度判别变得更加困难，并且大多数方向变得难以区分。",[11,38372,38373,38374,38471],{},"在正曲率空间中：测度集中在赤道区域，角度集中在 ",[38,38375,38377,38394],{"className":38376,"translate":42},[41],[38,38378,38380],{"className":38379},[46],[48,38381,38382],{"xmlns":50},[52,38383,38384,38392],{},[55,38385,38386],{},[636,38387,38388,38390],{},[58,38389,25958],{},[300,38391,312],{},[81,38393,38268],{"encoding":83},[38,38395,38397],{"className":38396,"ariaHidden":89},[88],[38,38398,38400,38403],{"className":38399},[93],[38,38401],{"className":38402,"style":38278},[97],[38,38404,38406,38409,38468],{"className":38405},[102],[38,38407],{"className":38408},[371,740],[38,38410,38412],{"className":38411},[636],[38,38413,38415,38460],{"className":38414},[138,384],[38,38416,38418,38457],{"className":38417},[142],[38,38419,38421,38435,38443],{"className":38420,"style":38297},[146],[38,38422,38423,38426],{"style":17472},[38,38424],{"className":38425,"style":760},[154],[38,38427,38429],{"className":38428},[159,160,161,162],[38,38430,38432],{"className":38431},[102,162],[38,38433,312],{"className":38434},[102,162],[38,38436,38437,38440],{"style":811},[38,38438],{"className":38439,"style":760},[154],[38,38441],{"className":38442,"style":819},[818],[38,38444,38445,38448],{"style":17586},[38,38446],{"className":38447,"style":760},[154],[38,38449,38451],{"className":38450},[159,160,161,162],[38,38452,38454],{"className":38453},[102,162],[38,38455,25958],{"className":38456,"style":3420},[102,103,162],[38,38458,408],{"className":38459},[407],[38,38461,38463],{"className":38462},[142],[38,38464,38466],{"className":38465,"style":27393},[146],[38,38467],{},[38,38469],{"className":38470},[529,740]," 附近，测地距离几乎变为常数。维数灾难依然存在，但它以角度坍缩的形式表现，而非径向壳层集中。",{"title":14638,"searchDepth":14639,"depth":14639,"links":38473},[38474,38478],{"id":25498,"depth":14639,"text":25498,"children":38475},[38476,38477],{"id":30435,"depth":14644,"text":30435},{"id":34349,"depth":14644,"text":34349},{"id":35048,"depth":14639,"text":35048,"children":38479},[38480,38481,38482,38483],{"id":35285,"depth":14644,"text":35285},{"id":36867,"depth":14644,"text":36867},{"id":37682,"depth":14644,"text":37682},{"id":38350,"depth":14644,"text":38350},{},"\u002Fblog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space",{"title":25487,"description":25492},"blog\u002F2026\u002F2026-04-10-dimension-curse-on-zero-and-positive-curvature-space","维数灾难并非某种特定的几何结构，而是由高维空间内在性质所引发的一类现象。",[14675,14676],"2026-04-10T00:00:00+08:00","-Jkaht4YhjPjMVNnKKE3IswYDiK4xwXCbXLPTdjg4_M",{"left":38493,"top":38493,"width":38494,"height":38494,"rotate":38493,"vFlip":14664,"hFlip":14664,"body":38495},0,24,"\u003Cg 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