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Analysis)，即主成分分析方法，是一种使用最广泛的数据降维算法。PCA的主要思想是将n维特征映射到k维上，这k维是全新的正交特征也被称为主成分，是在原有n维特征的基础上重新构造出来的k维特征。PCA的工作就是从原始的空间中顺序地找一组相互正交的坐标轴，新的坐标轴的选择与数据本身是密切相关的。其中，第一个新坐标轴选择是原始数据中方差最大的方向，第二个新坐标轴选取是与第一个坐标轴正交的平面中使得方差最大的，第三个轴是与第1,2个轴正交的平面中方差最大的。依次类推，可以得到n个这样的坐标轴。通过这种方式获得的新的坐标轴，我们发现，大部分方差都包含在前面k个坐标轴中，后面的坐标轴所含的方差几乎为0。于是，我们可以忽略余下的坐标轴，只保留前面k个含有绝大部分方差的坐标轴。事实上，这相当于只保留包含绝大部分方差的维度特征，而忽略包含方差几乎为0的特征维度，实现对数据特征的降维处理。",[1096,1100,1101,1102,1106,1107,1110],{},"PCA 的目标是把高维数据投影到少数几个方向上,既保留尽量多的信息,又大幅降低维度。它有两个等价的视角:",[1103,1104,1105],"strong",{},"最大方差投影","和",[1103,1108,1109],{},"最小重构误差","——两条路推出来的最优方向是同一组特征向量。",[1112,1113,1115],"h3",{"id":1114},"视角一从最大方差考虑","*视角一:从最大方差考虑",[1096,1117,1118,1119,1305,1306,1441,1442,1445,1446,1599,1600,1631,1632,1744,1745,1748],{},"给定训练样本 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x_n",[1120,1172,1175],{"className":1173,"ariaHidden":1152},[1174],"katex-html",[1120,1176,1179,1184,1242,1246,1251,1255,1258,1261,1264],{"className":1177},[1178],"base",[1120,1180],{"className":1181,"style":1183},[1182],"strut","height:0.625em;vertical-align:-0.1944em;",[1120,1185,1188,1192],{"className":1186},[1187],"mord",[1120,1189,1144],{"className":1190},[1187,1191],"mathnormal",[1120,1193,1196],{"className":1194},[1195],"msupsub",[1120,1197,1201,1233],{"className":1198},[1199,1200],"vlist-t","vlist-t2",[1120,1202,1205,1228],{"className":1203},[1204],"vlist-r",[1120,1206,1210],{"className":1207,"style":1209},[1208],"vlist","height:0.3011em;",[1120,1211,1213,1218],{"style":1212},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1120,1214],{"className":1215,"style":1217},[1216],"pstrut","height:2.7em;",[1120,1219,1225],{"className":1220},[1221,1222,1223,1224],"sizing","reset-size6","size3","mtight",[1120,1226,1148],{"className":1227},[1187,1224],[1120,1229,1232],{"className":1230},[1231],"vlist-s","​",[1120,1234,1236],{"className":1235},[1204],[1120,1237,1240],{"className":1238,"style":1239},[1208],"height:0.15em;",[1120,1241],{},[1120,1243,1153],{"className":1244},[1245],"mpunct",[1120,1247],{"className":1248,"style":1250},[1249],"mspace","margin-right:0.1667em;",[1120,1252,1156],{"className":1253},[1254],"minner",[1120,1256],{"className":1257,"style":1250},[1249],[1120,1259,1153],{"className":1260},[1245],[1120,1262],{"className":1263,"style":1250},[1249],[1120,1265,1267,1270],{"className":1266},[1187],[1120,1268,1144],{"className":1269},[1187,1191],[1120,1271,1273],{"className":1272},[1195],[1120,1274,1276,1297],{"className":1275},[1199,1200],[1120,1277,1279,1294],{"className":1278},[1204],[1120,1280,1283],{"className":1281,"style":1282},[1208],"height:0.1514em;",[1120,1284,1285,1288],{"style":1212},[1120,1286],{"className":1287,"style":1217},[1216],[1120,1289,1291],{"className":1290},[1221,1222,1223,1224],[1120,1292,1165],{"className":1293},[1187,1191,1224],[1120,1295,1232],{"className":1296},[1231],[1120,1298,1300],{"className":1299},[1204],[1120,1301,1303],{"className":1302,"style":1239},[1208],[1120,1304],{},",其中 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\\mathbb{R}^p",[1120,1341,1343,1402],{"className":1342,"ariaHidden":1152},[1174],[1120,1344,1346,1350,1391,1395,1399],{"className":1345},[1178],[1120,1347],{"className":1348,"style":1349},[1182],"height:0.6891em;vertical-align:-0.15em;",[1120,1351,1353,1356],{"className":1352},[1187],[1120,1354,1144],{"className":1355},[1187,1191],[1120,1357,1359],{"className":1358},[1195],[1120,1360,1362,1383],{"className":1361},[1199,1200],[1120,1363,1365,1380],{"className":1364},[1204],[1120,1366,1369],{"className":1367,"style":1368},[1208],"height:0.3117em;",[1120,1370,1371,1374],{"style":1212},[1120,1372],{"className":1373,"style":1217},[1216],[1120,1375,1377],{"className":1376},[1221,1222,1223,1224],[1120,1378,1324],{"className":1379},[1187,1191,1224],[1120,1381,1232],{"className":1382},[1231],[1120,1384,1386],{"className":1385},[1204],[1120,1387,1389],{"className":1388,"style":1239},[1208],[1120,1390],{},[1120,1392],{"className":1393,"style":1394},[1249],"margin-right:0.2778em;",[1120,1396,1327],{"className":1397},[1398],"mrel",[1120,1400],{"className":1401,"style":1394},[1249],[1120,1403,1405,1409],{"className":1404},[1178],[1120,1406],{"className":1407,"style":1408},[1182],"height:0.6889em;",[1120,1410,1412,1416],{"className":1411},[1187],[1120,1413,1334],{"className":1414},[1187,1415],"mathbb",[1120,1417,1419],{"className":1418},[1195],[1120,1420,1422],{"className":1421},[1199],[1120,1423,1425],{"className":1424},[1204],[1120,1426,1429],{"className":1427,"style":1428},[1208],"height:0.6644em;",[1120,1430,1432,1435],{"style":1431},"top:-3.063em;margin-right:0.05em;",[1120,1433],{"className":1434,"style":1217},[1216],[1120,1436,1438],{"className":1437},[1221,1222,1223,1224],[1120,1439,1096],{"className":1440},[1187,1191,1224],"。假设数据已被",[1103,1443,1444],{},"标准化(中心化)",",使得 ",[1120,1447,1449,1480],{"className":1448},[1123],[1120,1450,1452],{"className":1451},[1127],[1129,1453,1454],{"xmlns":1131},[1133,1455,1456,1477],{},[1136,1457,1458,1465,1471,1474],{},[1139,1459,1460,1463],{},[1150,1461,1462],{},"∑",[1142,1464,1324],{},[1139,1466,1467,1469],{},[1142,1468,1144],{},[1142,1470,1324],{},[1150,1472,1473],{},"=",[1146,1475,1476],{},"0",[1167,1478,1479],{"encoding":1169},"\\sum_i x_i = 0",[1120,1481,1483,1589],{"className":1482,"ariaHidden":1152},[1174],[1120,1484,1486,1490,1537,1540,1580,1583,1586],{"className":1485},[1178],[1120,1487],{"className":1488,"style":1489},[1182],"height:1.0497em;vertical-align:-0.2997em;",[1120,1491,1494,1500],{"className":1492},[1493],"mop",[1120,1495,1462],{"className":1496,"style":1499},[1493,1497,1498],"op-symbol","small-op","position:relative;top:0em;",[1120,1501,1503],{"className":1502},[1195],[1120,1504,1506,1528],{"className":1505},[1199,1200],[1120,1507,1509,1525],{"className":1508},[1204],[1120,1510,1513],{"className":1511,"style":1512},[1208],"height:0.162em;",[1120,1514,1516,1519],{"style":1515},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[1120,1517],{"className":1518,"style":1217},[1216],[1120,1520,1522],{"className":1521},[1221,1222,1223,1224],[1120,1523,1324],{"className":1524},[1187,1191,1224],[1120,1526,1232],{"className":1527},[1231],[1120,1529,1531],{"className":1530},[1204],[1120,1532,1535],{"className":1533,"style":1534},[1208],"height:0.2997em;",[1120,1536],{},[1120,1538],{"className":1539,"style":1250},[1249],[1120,1541,1543,1546],{"className":1542},[1187],[1120,1544,1144],{"className":1545},[1187,1191],[1120,1547,1549],{"className":1548},[1195],[1120,1550,1552,1572],{"className":1551},[1199,1200],[1120,1553,1555,1569],{"className":1554},[1204],[1120,1556,1558],{"className":1557,"style":1368},[1208],[1120,1559,1560,1563],{"style":1212},[1120,1561],{"className":1562,"style":1217},[1216],[1120,1564,1566],{"className":1565},[1221,1222,1223,1224],[1120,1567,1324],{"className":1568},[1187,1191,1224],[1120,1570,1232],{"className":1571},[1231],[1120,1573,1575],{"className":1574},[1204],[1120,1576,1578],{"className":1577,"style":1239},[1208],[1120,1579],{},[1120,1581],{"className":1582,"style":1394},[1249],[1120,1584,1473],{"className":1585},[1398],[1120,1587],{"className":1588,"style":1394},[1249],[1120,1590,1592,1596],{"className":1591},[1178],[1120,1593],{"className":1594,"style":1595},[1182],"height:0.6444em;",[1120,1597,1476],{"className":1598},[1187],"。PCA 想找一个方向 ",[1120,1601,1603,1617],{"className":1602},[1123],[1120,1604,1606],{"className":1605},[1127],[1129,1607,1608],{"xmlns":1131},[1133,1609,1610,1615],{},[1136,1611,1612],{},[1142,1613,1614],{},"w",[1167,1616,1614],{"encoding":1169},[1120,1618,1620],{"className":1619,"ariaHidden":1152},[1174],[1120,1621,1623,1627],{"className":1622},[1178],[1120,1624],{"className":1625,"style":1626},[1182],"height:0.4306em;",[1120,1628,1614],{"className":1629,"style":1630},[1187,1191],"margin-right:0.0269em;","(单位向量),使得原始数据在该方向上的投影 ",[1120,1633,1635,1661],{"className":1634},[1123],[1120,1636,1638],{"className":1637},[1127],[1129,1639,1640],{"xmlns":1131},[1133,1641,1642,1658],{},[1136,1643,1644,1652],{},[1329,1645,1646,1648],{},[1142,1647,1614],{},[1142,1649,1651],{"mathvariant":1650},"normal","⊤",[1139,1653,1654,1656],{},[1142,1655,1144],{},[1142,1657,1324],{},[1167,1659,1660],{"encoding":1169},"w^{\\top}x_i",[1120,1662,1664],{"className":1663,"ariaHidden":1152},[1174],[1120,1665,1667,1671,1704],{"className":1666},[1178],[1120,1668],{"className":1669,"style":1670},[1182],"height:0.9991em;vertical-align:-0.15em;",[1120,1672,1674,1677],{"className":1673},[1187],[1120,1675,1614],{"className":1676,"style":1630},[1187,1191],[1120,1678,1680],{"className":1679},[1195],[1120,1681,1683],{"className":1682},[1199],[1120,1684,1686],{"className":1685},[1204],[1120,1687,1690],{"className":1688,"style":1689},[1208],"height:0.8491em;",[1120,1691,1692,1695],{"style":1431},[1120,1693],{"className":1694,"style":1217},[1216],[1120,1696,1698],{"className":1697},[1221,1222,1223,1224],[1120,1699,1701],{"className":1700},[1187,1224],[1120,1702,1651],{"className":1703},[1187,1224],[1120,1705,1707,1710],{"className":1706},[1187],[1120,1708,1144],{"className":1709},[1187,1191],[1120,1711,1713],{"className":1712},[1195],[1120,1714,1716,1736],{"className":1715},[1199,1200],[1120,1717,1719,1733],{"className":1718},[1204],[1120,1720,1722],{"className":1721,"style":1368},[1208],[1120,1723,1724,1727],{"style":1212},[1120,1725],{"className":1726,"style":1217},[1216],[1120,1728,1730],{"className":1729},[1221,1222,1223,1224],[1120,1731,1324],{"className":1732},[1187,1191,1224],[1120,1734,1232],{"className":1735},[1231],[1120,1737,1739],{"className":1738},[1204],[1120,1740,1742],{"className":1741,"style":1239},[1208],[1120,1743],{}," 的",[1103,1746,1747],{},"方差最大","。",[1096,1750,1751],{},"投影后的方差为",[1096,1753,1754],{},[1120,1755,1757,1869],{"className":1756},[1123],[1120,1758,1760],{"className":1759},[1127],[1129,1761,1762],{"xmlns":1131},[1133,1763,1764,1866],{},[1136,1765,1766,1773,1788,1792,1798,1804,1812,1814,1820,1823,1829,1835,1841,1849,1851,1853,1855,1861,1864],{},[1767,1768,1769,1771],"mfrac",{},[1146,1770,1148],{},[1142,1772,1165],{},[1774,1775,1776,1778,1786],"msubsup",{},[1150,1777,1462],{},[1136,1779,1780,1782,1784],{},[1142,1781,1324],{},[1150,1783,1473],{},[1146,1785,1148],{},[1142,1787,1165],{},[1150,1789,1791],{"stretchy":1790},"false","(",[1329,1793,1794,1796],{},[1142,1795,1614],{},[1142,1797,1651],{"mathvariant":1650},[1139,1799,1800,1802],{},[1142,1801,1144],{},[1142,1803,1324],{},[1329,1805,1806,1809],{},[1150,1807,1808],{"stretchy":1790},")",[1146,1810,1811],{},"2",[1150,1813,1473],{},[1329,1815,1816,1818],{},[1142,1817,1614],{},[1142,1819,1651],{"mathvariant":1650},[1150,1821,1791],{"fence":1790,"stretchy":1152,"minsize":1822,"maxsize":1822},"1.8em",[1767,1824,1825,1827],{},[1146,1826,1148],{},[1142,1828,1165],{},[1139,1830,1831,1833],{},[1150,1832,1462],{},[1142,1834,1324],{},[1139,1836,1837,1839],{},[1142,1838,1144],{},[1142,1840,1324],{},[1774,1842,1843,1845,1847],{},[1142,1844,1144],{},[1142,1846,1324],{},[1142,1848,1651],{"mathvariant":1650},[1150,1850,1808],{"fence":1790,"stretchy":1152,"minsize":1822,"maxsize":1822},[1142,1852,1614],{},[1150,1854,1473],{},[1329,1856,1857,1859],{},[1142,1858,1614],{},[1142,1860,1651],{"mathvariant":1650},[1142,1862,1863],{"mathvariant":1650},"Σ",[1142,1865,1614],{},[1167,1867,1868],{"encoding":1169},"\\frac{1}{n}\\sum_{i=1}^{n}(w^{\\top}x_i)^2 = w^{\\top}\\Big(\\frac{1}{n}\\sum_i x_i x_i^{\\top}\\Big)w = w^{\\top}\\Sigma w",[1120,1870,1872,2140,2415],{"className":1871,"ariaHidden":1152},[1174],[1120,1873,1875,1879,1958,1961,2026,2029,2061,2101,2131,2134,2137],{"className":1874},[1178],[1120,1876],{"className":1877,"style":1878},[1182],"height:1.1941em;vertical-align:-0.345em;",[1120,1880,1882,1887,1954],{"className":1881},[1187],[1120,1883],{"className":1884},[1885,1886],"mopen","nulldelimiter",[1120,1888,1890],{"className":1889},[1767],[1120,1891,1893,1945],{"className":1892},[1199,1200],[1120,1894,1896,1942],{"className":1895},[1204],[1120,1897,1900,1916,1927],{"className":1898,"style":1899},[1208],"height:0.8451em;",[1120,1901,1903,1907],{"style":1902},"top:-2.655em;",[1120,1904],{"className":1905,"style":1906},[1216],"height:3em;",[1120,1908,1910],{"className":1909},[1221,1222,1223,1224],[1120,1911,1913],{"className":1912},[1187,1224],[1120,1914,1165],{"className":1915},[1187,1191,1224],[1120,1917,1919,1922],{"style":1918},"top:-3.23em;",[1120,1920],{"className":1921,"style":1906},[1216],[1120,1923],{"className":1924,"style":1926},[1925],"frac-line","border-bottom-width:0.04em;",[1120,1928,1930,1933],{"style":1929},"top:-3.394em;",[1120,1931],{"className":1932,"style":1906},[1216],[1120,1934,1936],{"className":1935},[1221,1222,1223,1224],[1120,1937,1939],{"className":1938},[1187,1224],[1120,1940,1148],{"className":1941},[1187,1224],[1120,1943,1232],{"className":1944},[1231],[1120,1946,1948],{"className":1947},[1204],[1120,1949,1952],{"className":1950,"style":1951},[1208],"height:0.345em;",[1120,1953],{},[1120,1955],{"className":1956},[1957,1886],"mclose",[1120,1959],{"className":1960,"style":1250},[1249],[1120,1962,1964,1967],{"className":1963},[1493],[1120,1965,1462],{"className":1966,"style":1499},[1493,1497,1498],[1120,1968,1970],{"className":1969},[1195],[1120,1971,1973,2018],{"className":1972},[1199,1200],[1120,1974,1976,2015],{"className":1975},[1204],[1120,1977,1980,2000],{"className":1978,"style":1979},[1208],"height:0.8043em;",[1120,1981,1982,1985],{"style":1515},[1120,1983],{"className":1984,"style":1217},[1216],[1120,1986,1988],{"className":1987},[1221,1222,1223,1224],[1120,1989,1991,1994,1997],{"className":1990},[1187,1224],[1120,1992,1324],{"className":1993},[1187,1191,1224],[1120,1995,1473],{"className":1996},[1398,1224],[1120,1998,1148],{"className":1999},[1187,1224],[1120,2001,2003,2006],{"style":2002},"top:-3.2029em;margin-right:0.05em;",[1120,2004],{"className":2005,"style":1217},[1216],[1120,2007,2009],{"className":2008},[1221,1222,1223,1224],[1120,2010,2012],{"className":2011},[1187,1224],[1120,2013,1165],{"className":2014},[1187,1191,1224],[1120,2016,1232],{"className":2017},[1231],[1120,2019,2021],{"className":2020},[1204],[1120,2022,2024],{"className":2023,"style":1534},[1208],[1120,2025],{},[1120,2027,1791],{"className":2028},[1885],[1120,2030,2032,2035],{"className":2031},[1187],[1120,2033,1614],{"className":2034,"style":1630},[1187,1191],[1120,2036,2038],{"className":2037},[1195],[1120,2039,2041],{"className":2040},[1199],[1120,2042,2044],{"className":2043},[1204],[1120,2045,2047],{"className":2046,"style":1689},[1208],[1120,2048,2049,2052],{"style":1431},[1120,2050],{"className":2051,"style":1217},[1216],[1120,2053,2055],{"className":2054},[1221,1222,1223,1224],[1120,2056,2058],{"className":2057},[1187,1224],[1120,2059,1651],{"className":2060},[1187,1224],[1120,2062,2064,2067],{"className":2063},[1187],[1120,2065,1144],{"className":2066},[1187,1191],[1120,2068,2070],{"className":2069},[1195],[1120,2071,2073,2093],{"className":2072},[1199,1200],[1120,2074,2076,2090],{"className":2075},[1204],[1120,2077,2079],{"className":2078,"style":1368},[1208],[1120,2080,2081,2084],{"style":1212},[1120,2082],{"className":2083,"style":1217},[1216],[1120,2085,2087],{"className":2086},[1221,1222,1223,1224],[1120,2088,1324],{"className":2089},[1187,1191,1224],[1120,2091,1232],{"className":2092},[1231],[1120,2094,2096],{"className":2095},[1204],[1120,2097,2099],{"className":2098,"style":1239},[1208],[1120,2100],{},[1120,2102,2104,2107],{"className":2103},[1957],[1120,2105,1808],{"className":2106},[1957],[1120,2108,2110],{"className":2109},[1195],[1120,2111,2113],{"className":2112},[1199],[1120,2114,2116],{"className":2115},[1204],[1120,2117,2120],{"className":2118,"style":2119},[1208],"height:0.8141em;",[1120,2121,2122,2125],{"style":1431},[1120,2123],{"className":2124,"style":1217},[1216],[1120,2126,2128],{"className":2127},[1221,1222,1223,1224],[1120,2129,1811],{"className":2130},[1187,1224],[1120,2132],{"className":2133,"style":1394},[1249],[1120,2135,1473],{"className":2136},[1398],[1120,2138],{"className":2139,"style":1394},[1249],[1120,2141,2143,2147,2179,2187,2255,2258,2298,2301,2341,2397,2403,2406,2409,2412],{"className":2142},[1178],[1120,2144],{"className":2145,"style":2146},[1182],"height:1.8em;vertical-align:-0.65em;",[1120,2148,2150,2153],{"className":2149},[1187],[1120,2151,1614],{"className":2152,"style":1630},[1187,1191],[1120,2154,2156],{"className":2155},[1195],[1120,2157,2159],{"className":2158},[1199],[1120,2160,2162],{"className":2161},[1204],[1120,2163,2165],{"className":2164,"style":1689},[1208],[1120,2166,2167,2170],{"style":1431},[1120,2168],{"className":2169,"style":1217},[1216],[1120,2171,2173],{"className":2172},[1221,1222,1223,1224],[1120,2174,2176],{"className":2175},[1187,1224],[1120,2177,1651],{"className":2178},[1187,1224],[1120,2180,2182],{"className":2181},[1187],[1120,2183,1791],{"className":2184},[2185,2186],"delimsizing","size2",[1120,2188,2190,2193,2252],{"className":2189},[1187],[1120,2191],{"className":2192},[1885,1886],[1120,2194,2196],{"className":2195},[1767],[1120,2197,2199,2244],{"className":2198},[1199,1200],[1120,2200,2202,2241],{"className":2201},[1204],[1120,2203,2205,2219,2227],{"className":2204,"style":1899},[1208],[1120,2206,2207,2210],{"style":1902},[1120,2208],{"className":2209,"style":1906},[1216],[1120,2211,2213],{"className":2212},[1221,1222,1223,1224],[1120,2214,2216],{"className":2215},[1187,1224],[1120,2217,1165],{"className":2218},[1187,1191,1224],[1120,2220,2221,2224],{"style":1918},[1120,2222],{"className":2223,"style":1906},[1216],[1120,2225],{"className":2226,"style":1926},[1925],[1120,2228,2229,2232],{"style":1929},[1120,2230],{"className":2231,"style":1906},[1216],[1120,2233,2235],{"className":2234},[1221,1222,1223,1224],[1120,2236,2238],{"className":2237},[1187,1224],[1120,2239,1148],{"className":2240},[1187,1224],[1120,2242,1232],{"className":2243},[1231],[1120,2245,2247],{"className":2246},[1204],[1120,2248,2250],{"className":2249,"style":1951},[1208],[1120,2251],{},[1120,2253],{"className":2254},[1957,1886],[1120,2256],{"className":2257,"style":1250},[1249],[1120,2259,2261,2264],{"className":2260},[1493],[1120,2262,1462],{"className":2263,"style":1499},[1493,1497,1498],[1120,2265,2267],{"className":2266},[1195],[1120,2268,2270,2290],{"className":2269},[1199,1200],[1120,2271,2273,2287],{"className":2272},[1204],[1120,2274,2276],{"className":2275,"style":1512},[1208],[1120,2277,2278,2281],{"style":1515},[1120,2279],{"className":2280,"style":1217},[1216],[1120,2282,2284],{"className":2283},[1221,1222,1223,1224],[1120,2285,1324],{"className":2286},[1187,1191,1224],[1120,2288,1232],{"className":2289},[1231],[1120,2291,2293],{"className":2292},[1204],[1120,2294,2296],{"className":2295,"style":1534},[1208],[1120,2297],{},[1120,2299],{"className":2300,"style":1250},[1249],[1120,2302,2304,2307],{"className":2303},[1187],[1120,2305,1144],{"className":2306},[1187,1191],[1120,2308,2310],{"className":2309},[1195],[1120,2311,2313,2333],{"className":2312},[1199,1200],[1120,2314,2316,2330],{"className":2315},[1204],[1120,2317,2319],{"className":2318,"style":1368},[1208],[1120,2320,2321,2324],{"style":1212},[1120,2322],{"className":2323,"style":1217},[1216],[1120,2325,2327],{"className":2326},[1221,1222,1223,1224],[1120,2328,1324],{"className":2329},[1187,1191,1224],[1120,2331,1232],{"className":2332},[1231],[1120,2334,2336],{"className":2335},[1204],[1120,2337,2339],{"className":2338,"style":1239},[1208],[1120,2340],{},[1120,2342,2344,2347],{"className":2343},[1187],[1120,2345,1144],{"className":2346},[1187,1191],[1120,2348,2350],{"className":2349},[1195],[1120,2351,2353,2388],{"className":2352},[1199,1200],[1120,2354,2356,2385],{"className":2355},[1204],[1120,2357,2359,2371],{"className":2358,"style":1689},[1208],[1120,2360,2362,2365],{"style":2361},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[1120,2363],{"className":2364,"style":1217},[1216],[1120,2366,2368],{"className":2367},[1221,1222,1223,1224],[1120,2369,1324],{"className":2370},[1187,1191,1224],[1120,2372,2373,2376],{"style":1431},[1120,2374],{"className":2375,"style":1217},[1216],[1120,2377,2379],{"className":2378},[1221,1222,1223,1224],[1120,2380,2382],{"className":2381},[1187,1224],[1120,2383,1651],{"className":2384},[1187,1224],[1120,2386,1232],{"className":2387},[1231],[1120,2389,2391],{"className":2390},[1204],[1120,2392,2395],{"className":2393,"style":2394},[1208],"height:0.2587em;",[1120,2396],{},[1120,2398,2400],{"className":2399},[1187],[1120,2401,1808],{"className":2402},[2185,2186],[1120,2404,1614],{"className":2405,"style":1630},[1187,1191],[1120,2407],{"className":2408,"style":1394},[1249],[1120,2410,1473],{"className":2411},[1398],[1120,2413],{"className":2414,"style":1394},[1249],[1120,2416,2418,2421,2453,2456],{"className":2417},[1178],[1120,2419],{"className":2420,"style":1689},[1182],[1120,2422,2424,2427],{"className":2423},[1187],[1120,2425,1614],{"className":2426,"style":1630},[1187,1191],[1120,2428,2430],{"className":2429},[1195],[1120,2431,2433],{"className":2432},[1199],[1120,2434,2436],{"className":2435},[1204],[1120,2437,2439],{"className":2438,"style":1689},[1208],[1120,2440,2441,2444],{"style":1431},[1120,2442],{"className":2443,"style":1217},[1216],[1120,2445,2447],{"className":2446},[1221,1222,1223,1224],[1120,2448,2450],{"className":2449},[1187,1224],[1120,2451,1651],{"className":2452},[1187,1224],[1120,2454,1863],{"className":2455},[1187],[1120,2457,1614],{"className":2458,"style":1630},[1187,1191],[1096,2460,2461,2462,2743,2744,2772],{},"其中 ",[1120,2463,2465,2507],{"className":2464},[1123],[1120,2466,2468],{"className":2467},[1127],[1129,2469,2470],{"xmlns":1131},[1133,2471,2472,2504],{},[1136,2473,2474,2476,2478,2484,2490,2496],{},[1142,2475,1863],{"mathvariant":1650},[1150,2477,1473],{},[1767,2479,2480,2482],{},[1146,2481,1148],{},[1142,2483,1165],{},[1139,2485,2486,2488],{},[1150,2487,1462],{},[1142,2489,1324],{},[1139,2491,2492,2494],{},[1142,2493,1144],{},[1142,2495,1324],{},[1774,2497,2498,2500,2502],{},[1142,2499,1144],{},[1142,2501,1324],{},[1142,2503,1651],{"mathvariant":1650},[1167,2505,2506],{"encoding":1169},"\\Sigma = \\frac{1}{n}\\sum_i x_i x_i^{\\top}",[1120,2508,2510,2529],{"className":2509,"ariaHidden":1152},[1174],[1120,2511,2513,2517,2520,2523,2526],{"className":2512},[1178],[1120,2514],{"className":2515,"style":2516},[1182],"height:0.6833em;",[1120,2518,1863],{"className":2519},[1187],[1120,2521],{"className":2522,"style":1394},[1249],[1120,2524,1473],{"className":2525},[1398],[1120,2527],{"className":2528,"style":1394},[1249],[1120,2530,2532,2535,2603,2606,2646,2649,2689],{"className":2531},[1178],[1120,2533],{"className":2534,"style":1878},[1182],[1120,2536,2538,2541,2600],{"className":2537},[1187],[1120,2539],{"className":2540},[1885,1886],[1120,2542,2544],{"className":2543},[1767],[1120,2545,2547,2592],{"className":2546},[1199,1200],[1120,2548,2550,2589],{"className":2549},[1204],[1120,2551,2553,2567,2575],{"className":2552,"style":1899},[1208],[1120,2554,2555,2558],{"style":1902},[1120,2556],{"className":2557,"style":1906},[1216],[1120,2559,2561],{"className":2560},[1221,1222,1223,1224],[1120,2562,2564],{"className":2563},[1187,1224],[1120,2565,1165],{"className":2566},[1187,1191,1224],[1120,2568,2569,2572],{"style":1918},[1120,2570],{"className":2571,"style":1906},[1216],[1120,2573],{"className":2574,"style":1926},[1925],[1120,2576,2577,2580],{"style":1929},[1120,2578],{"className":2579,"style":1906},[1216],[1120,2581,2583],{"className":2582},[1221,1222,1223,1224],[1120,2584,2586],{"className":2585},[1187,1224],[1120,2587,1148],{"className":2588},[1187,1224],[1120,2590,1232],{"className":2591},[1231],[1120,2593,2595],{"className":2594},[1204],[1120,2596,2598],{"className":2597,"style":1951},[1208],[1120,2599],{},[1120,2601],{"className":2602},[1957,1886],[1120,2604],{"className":2605,"style":1250},[1249],[1120,2607,2609,2612],{"className":2608},[1493],[1120,2610,1462],{"className":2611,"style":1499},[1493,1497,1498],[1120,2613,2615],{"className":2614},[1195],[1120,2616,2618,2638],{"className":2617},[1199,1200],[1120,2619,2621,2635],{"className":2620},[1204],[1120,2622,2624],{"className":2623,"style":1512},[1208],[1120,2625,2626,2629],{"style":1515},[1120,2627],{"className":2628,"style":1217},[1216],[1120,2630,2632],{"className":2631},[1221,1222,1223,1224],[1120,2633,1324],{"className":2634},[1187,1191,1224],[1120,2636,1232],{"className":2637},[1231],[1120,2639,2641],{"className":2640},[1204],[1120,2642,2644],{"className":2643,"style":1534},[1208],[1120,2645],{},[1120,2647],{"className":2648,"style":1250},[1249],[1120,2650,2652,2655],{"className":2651},[1187],[1120,2653,1144],{"className":2654},[1187,1191],[1120,2656,2658],{"className":2657},[1195],[1120,2659,2661,2681],{"className":2660},[1199,1200],[1120,2662,2664,2678],{"className":2663},[1204],[1120,2665,2667],{"className":2666,"style":1368},[1208],[1120,2668,2669,2672],{"style":1212},[1120,2670],{"className":2671,"style":1217},[1216],[1120,2673,2675],{"className":2674},[1221,1222,1223,1224],[1120,2676,1324],{"className":2677},[1187,1191,1224],[1120,2679,1232],{"className":2680},[1231],[1120,2682,2684],{"className":2683},[1204],[1120,2685,2687],{"className":2686,"style":1239},[1208],[1120,2688],{},[1120,2690,2692,2695],{"className":2691},[1187],[1120,2693,1144],{"className":2694},[1187,1191],[1120,2696,2698],{"className":2697},[1195],[1120,2699,2701,2735],{"className":2700},[1199,1200],[1120,2702,2704,2732],{"className":2703},[1204],[1120,2705,2707,2718],{"className":2706,"style":1689},[1208],[1120,2708,2709,2712],{"style":2361},[1120,2710],{"className":2711,"style":1217},[1216],[1120,2713,2715],{"className":2714},[1221,1222,1223,1224],[1120,2716,1324],{"className":2717},[1187,1191,1224],[1120,2719,2720,2723],{"style":1431},[1120,2721],{"className":2722,"style":1217},[1216],[1120,2724,2726],{"className":2725},[1221,1222,1223,1224],[1120,2727,2729],{"className":2728},[1187,1224],[1120,2730,1651],{"className":2731},[1187,1224],[1120,2733,1232],{"className":2734},[1231],[1120,2736,2738],{"className":2737},[1204],[1120,2739,2741],{"className":2740,"style":2394},[1208],[1120,2742],{}," 是样本协方差矩阵。于是优化问题写成(约束 ",[1120,2745,2747,2760],{"className":2746},[1123],[1120,2748,2750],{"className":2749},[1127],[1129,2751,2752],{"xmlns":1131},[1133,2753,2754,2758],{},[1136,2755,2756],{},[1142,2757,1614],{},[1167,2759,1614],{"encoding":1169},[1120,2761,2763],{"className":2762,"ariaHidden":1152},[1174],[1120,2764,2766,2769],{"className":2765},[1178],[1120,2767],{"className":2768,"style":1626},[1182],[1120,2770,1614],{"className":2771,"style":1630},[1187,1191]," 为单位向量):",[1096,2774,2775],{},[1120,2776,2778,2836],{"className":2777},[1123],[1120,2779,2781],{"className":2780},[1127],[1129,2782,2783],{"xmlns":1131},[1133,2784,2785,2833],{},[1136,2786,2787,2799,2803,2809,2811,2813,2816,2819,2821,2827,2829,2831],{},[1139,2788,2789,2797],{},[1136,2790,2791,2794],{},[1142,2792,2793],{},"max",[1150,2795,2796],{},"⁡",[1142,2798,1614],{},[2800,2801,2802],"mtext",{}," ",[1329,2804,2805,2807],{},[1142,2806,1614],{},[1142,2808,1651],{"mathvariant":1650},[1142,2810,1863],{"mathvariant":1650},[1142,2812,1614],{},[1249,2814],{"width":2815},"1em",[2800,2817,2818],{},"s.t.",[1249,2820],{"width":2815},[1329,2822,2823,2825],{},[1142,2824,1614],{},[1142,2826,1651],{"mathvariant":1650},[1142,2828,1614],{},[1150,2830,1473],{},[1146,2832,1148],{},[1167,2834,2835],{"encoding":1169},"\\max_{w}\\ w^{\\top}\\Sigma w \\quad \\text{s.t.}\\quad w^{\\top}w = 1",[1120,2837,2839,2991],{"className":2838,"ariaHidden":1152},[1174],[1120,2840,2842,2845,2889,2892,2895,2927,2930,2933,2937,2944,2947,2979,2982,2985,2988],{"className":2841},[1178],[1120,2843],{"className":2844,"style":1670},[1182],[1120,2846,2848,2851],{"className":2847},[1493],[1120,2849,2793],{"className":2850},[1493],[1120,2852,2854],{"className":2853},[1195],[1120,2855,2857,2881],{"className":2856},[1199,1200],[1120,2858,2860,2878],{"className":2859},[1204],[1120,2861,2863],{"className":2862,"style":1282},[1208],[1120,2864,2866,2869],{"style":2865},"top:-2.55em;margin-right:0.05em;",[1120,2867],{"className":2868,"style":1217},[1216],[1120,2870,2872],{"className":2871},[1221,1222,1223,1224],[1120,2873,2875],{"className":2874},[1187,1224],[1120,2876,1614],{"className":2877,"style":1630},[1187,1191,1224],[1120,2879,1232],{"className":2880},[1231],[1120,2882,2884],{"className":2883},[1204],[1120,2885,2887],{"className":2886,"style":1239},[1208],[1120,2888],{},[1120,2890,2802],{"className":2891},[1249],[1120,2893],{"className":2894,"style":1250},[1249],[1120,2896,2898,2901],{"className":2897},[1187],[1120,2899,1614],{"className":2900,"style":1630},[1187,1191],[1120,2902,2904],{"className":2903},[1195],[1120,2905,2907],{"className":2906},[1199],[1120,2908,2910],{"className":2909},[1204],[1120,2911,2913],{"className":2912,"style":1689},[1208],[1120,2914,2915,2918],{"style":1431},[1120,2916],{"className":2917,"style":1217},[1216],[1120,2919,2921],{"className":2920},[1221,1222,1223,1224],[1120,2922,2924],{"className":2923},[1187,1224],[1120,2925,1651],{"className":2926},[1187,1224],[1120,2928,1863],{"className":2929},[1187],[1120,2931,1614],{"className":2932,"style":1630},[1187,1191],[1120,2934],{"className":2935,"style":2936},[1249],"margin-right:1em;",[1120,2938,2941],{"className":2939},[1187,2940],"text",[1120,2942,2818],{"className":2943},[1187],[1120,2945],{"className":2946,"style":2936},[1249],[1120,2948,2950,2953],{"className":2949},[1187],[1120,2951,1614],{"className":2952,"style":1630},[1187,1191],[1120,2954,2956],{"className":2955},[1195],[1120,2957,2959],{"className":2958},[1199],[1120,2960,2962],{"className":2961},[1204],[1120,2963,2965],{"className":2964,"style":1689},[1208],[1120,2966,2967,2970],{"style":1431},[1120,2968],{"className":2969,"style":1217},[1216],[1120,2971,2973],{"className":2972},[1221,1222,1223,1224],[1120,2974,2976],{"className":2975},[1187,1224],[1120,2977,1651],{"className":2978},[1187,1224],[1120,2980,1614],{"className":2981,"style":1630},[1187,1191],[1120,2983],{"className":2984,"style":1394},[1249],[1120,2986,1473],{"className":2987},[1398],[1120,2989],{"className":2990,"style":1394},[1249],[1120,2992,2994,2997],{"className":2993},[1178],[1120,2995],{"className":2996,"style":1595},[1182],[1120,2998,1148],{"className":2999},[1187],[1096,3001,3002,3003,3006,3007,3206,3207,3235],{},"用",[1103,3004,3005],{},"拉格朗日乘子法",",构造 ",[1120,3008,3010,3059],{"className":3009},[1123],[1120,3011,3013],{"className":3012},[1127],[1129,3014,3015],{"xmlns":1131},[1133,3016,3017,3056],{},[1136,3018,3019,3022,3024,3030,3032,3034,3037,3040,3042,3048,3050,3052,3054],{},[1142,3020,3021],{},"L",[1150,3023,1473],{},[1329,3025,3026,3028],{},[1142,3027,1614],{},[1142,3029,1651],{"mathvariant":1650},[1142,3031,1863],{"mathvariant":1650},[1142,3033,1614],{},[1150,3035,3036],{},"−",[1142,3038,3039],{},"λ",[1150,3041,1791],{"stretchy":1790},[1329,3043,3044,3046],{},[1142,3045,1614],{},[1142,3047,1651],{"mathvariant":1650},[1142,3049,1614],{},[1150,3051,3036],{},[1146,3053,1148],{},[1150,3055,1808],{"stretchy":1790},[1167,3057,3058],{"encoding":1169},"L = w^{\\top}\\Sigma w - \\lambda(w^{\\top}w - 1)",[1120,3060,3062,3080,3136,3193],{"className":3061,"ariaHidden":1152},[1174],[1120,3063,3065,3068,3071,3074,3077],{"className":3064},[1178],[1120,3066],{"className":3067,"style":2516},[1182],[1120,3069,3021],{"className":3070},[1187,1191],[1120,3072],{"className":3073,"style":1394},[1249],[1120,3075,1473],{"className":3076},[1398],[1120,3078],{"className":3079,"style":1394},[1249],[1120,3081,3083,3087,3119,3122,3125,3129,3133],{"className":3082},[1178],[1120,3084],{"className":3085,"style":3086},[1182],"height:0.9324em;vertical-align:-0.0833em;",[1120,3088,3090,3093],{"className":3089},[1187],[1120,3091,1614],{"className":3092,"style":1630},[1187,1191],[1120,3094,3096],{"className":3095},[1195],[1120,3097,3099],{"className":3098},[1199],[1120,3100,3102],{"className":3101},[1204],[1120,3103,3105],{"className":3104,"style":1689},[1208],[1120,3106,3107,3110],{"style":1431},[1120,3108],{"className":3109,"style":1217},[1216],[1120,3111,3113],{"className":3112},[1221,1222,1223,1224],[1120,3114,3116],{"className":3115},[1187,1224],[1120,3117,1651],{"className":3118},[1187,1224],[1120,3120,1863],{"className":3121},[1187],[1120,3123,1614],{"className":3124,"style":1630},[1187,1191],[1120,3126],{"className":3127,"style":3128},[1249],"margin-right:0.2222em;",[1120,3130,3036],{"className":3131},[3132],"mbin",[1120,3134],{"className":3135,"style":3128},[1249],[1120,3137,3139,3143,3146,3149,3181,3184,3187,3190],{"className":3138},[1178],[1120,3140],{"className":3141,"style":3142},[1182],"height:1.0991em;vertical-align:-0.25em;",[1120,3144,3039],{"className":3145},[1187,1191],[1120,3147,1791],{"className":3148},[1885],[1120,3150,3152,3155],{"className":3151},[1187],[1120,3153,1614],{"className":3154,"style":1630},[1187,1191],[1120,3156,3158],{"className":3157},[1195],[1120,3159,3161],{"className":3160},[1199],[1120,3162,3164],{"className":3163},[1204],[1120,3165,3167],{"className":3166,"style":1689},[1208],[1120,3168,3169,3172],{"style":1431},[1120,3170],{"className":3171,"style":1217},[1216],[1120,3173,3175],{"className":3174},[1221,1222,1223,1224],[1120,3176,3178],{"className":3177},[1187,1224],[1120,3179,1651],{"className":3180},[1187,1224],[1120,3182,1614],{"className":3183,"style":1630},[1187,1191],[1120,3185],{"className":3186,"style":3128},[1249],[1120,3188,3036],{"className":3189},[3132],[1120,3191],{"className":3192,"style":3128},[1249],[1120,3194,3196,3200,3203],{"className":3195},[1178],[1120,3197],{"className":3198,"style":3199},[1182],"height:1em;vertical-align:-0.25em;",[1120,3201,1148],{"className":3202},[1187],[1120,3204,1808],{"className":3205},[1957],",对 ",[1120,3208,3210,3223],{"className":3209},[1123],[1120,3211,3213],{"className":3212},[1127],[1129,3214,3215],{"xmlns":1131},[1133,3216,3217,3221],{},[1136,3218,3219],{},[1142,3220,1614],{},[1167,3222,1614],{"encoding":1169},[1120,3224,3226],{"className":3225,"ariaHidden":1152},[1174],[1120,3227,3229,3232],{"className":3228},[1178],[1120,3230],{"className":3231,"style":1626},[1182],[1120,3233,1614],{"className":3234,"style":1630},[1187,1191]," 求偏导并令其为零:",[1096,3237,3238],{},[1120,3239,3241,3305],{"className":3240},[1123],[1120,3242,3244],{"className":3243},[1127],[1129,3245,3246],{"xmlns":1131},[1133,3247,3248,3302],{},[1136,3249,3250,3265,3267,3269,3271,3273,3275,3277,3279,3281,3283,3285,3287,3290,3292,3294,3296,3298,3300],{},[1767,3251,3252,3259],{},[1136,3253,3254,3257],{},[1142,3255,3256],{"mathvariant":1650},"∂",[1142,3258,3021],{},[1136,3260,3261,3263],{},[1142,3262,3256],{"mathvariant":1650},[1142,3264,1614],{},[1150,3266,1473],{},[1146,3268,1811],{},[1142,3270,1863],{"mathvariant":1650},[1142,3272,1614],{},[1150,3274,3036],{},[1146,3276,1811],{},[1142,3278,3039],{},[1142,3280,1614],{},[1150,3282,1473],{},[1146,3284,1476],{},[2800,3286,2802],{},[1150,3288,3289],{},"⟹",[2800,3291,2802],{},[1142,3293,1863],{"mathvariant":1650},[1142,3295,1614],{},[1150,3297,1473],{},[1142,3299,3039],{},[1142,3301,1614],{},[1167,3303,3304],{"encoding":1169},"\\frac{\\partial L}{\\partial w} = 2\\Sigma w - 2\\lambda w = 0 \\ \\Longrightarrow\\ \\Sigma w = \\lambda w",[1120,3306,3308,3400,3423,3448,3473,3494],{"className":3307,"ariaHidden":1152},[1174],[1120,3309,3311,3315,3391,3394,3397],{"className":3310},[1178],[1120,3312],{"className":3313,"style":3314},[1182],"height:1.2251em;vertical-align:-0.345em;",[1120,3316,3318,3321,3388],{"className":3317},[1187],[1120,3319],{"className":3320},[1885,1886],[1120,3322,3324],{"className":3323},[1767],[1120,3325,3327,3380],{"className":3326},[1199,1200],[1120,3328,3330,3377],{"className":3329},[1204],[1120,3331,3334,3352,3360],{"className":3332,"style":3333},[1208],"height:0.8801em;",[1120,3335,3336,3339],{"style":1902},[1120,3337],{"className":3338,"style":1906},[1216],[1120,3340,3342],{"className":3341},[1221,1222,1223,1224],[1120,3343,3345,3349],{"className":3344},[1187,1224],[1120,3346,3256],{"className":3347,"style":3348},[1187,1224],"margin-right:0.0556em;",[1120,3350,1614],{"className":3351,"style":1630},[1187,1191,1224],[1120,3353,3354,3357],{"style":1918},[1120,3355],{"className":3356,"style":1906},[1216],[1120,3358],{"className":3359,"style":1926},[1925],[1120,3361,3362,3365],{"style":1929},[1120,3363],{"className":3364,"style":1906},[1216],[1120,3366,3368],{"className":3367},[1221,1222,1223,1224],[1120,3369,3371,3374],{"className":3370},[1187,1224],[1120,3372,3256],{"className":3373,"style":3348},[1187,1224],[1120,3375,3021],{"className":3376},[1187,1191,1224],[1120,3378,1232],{"className":3379},[1231],[1120,3381,3383],{"className":3382},[1204],[1120,3384,3386],{"className":3385,"style":1951},[1208],[1120,3387],{},[1120,3389],{"className":3390},[1957,1886],[1120,3392],{"className":3393,"style":1394},[1249],[1120,3395,1473],{"className":3396},[1398],[1120,3398],{"className":3399,"style":1394},[1249],[1120,3401,3403,3407,3411,3414,3417,3420],{"className":3402},[1178],[1120,3404],{"className":3405,"style":3406},[1182],"height:0.7667em;vertical-align:-0.0833em;",[1120,3408,3410],{"className":3409},[1187],"2Σ",[1120,3412,1614],{"className":3413,"style":1630},[1187,1191],[1120,3415],{"className":3416,"style":3128},[1249],[1120,3418,3036],{"className":3419},[3132],[1120,3421],{"className":3422,"style":3128},[1249],[1120,3424,3426,3430,3433,3436,3439,3442,3445],{"className":3425},[1178],[1120,3427],{"className":3428,"style":3429},[1182],"height:0.6944em;",[1120,3431,1811],{"className":3432},[1187],[1120,3434,3039],{"className":3435},[1187,1191],[1120,3437,1614],{"className":3438,"style":1630},[1187,1191],[1120,3440],{"className":3441,"style":1394},[1249],[1120,3443,1473],{"className":3444},[1398],[1120,3446],{"className":3447,"style":1394},[1249],[1120,3449,3451,3455,3458,3461,3464,3467,3470],{"className":3450},[1178],[1120,3452],{"className":3453,"style":3454},[1182],"height:0.6684em;vertical-align:-0.024em;",[1120,3456,1476],{"className":3457},[1187],[1120,3459,2802],{"className":3460},[1249],[1120,3462],{"className":3463,"style":1394},[1249],[1120,3465,3289],{"className":3466},[1398],[1120,3468,2802],{"className":3469},[1249],[1120,3471],{"className":3472,"style":1394},[1249],[1120,3474,3476,3479,3482,3485,3488,3491],{"className":3475},[1178],[1120,3477],{"className":3478,"style":2516},[1182],[1120,3480,1863],{"className":3481},[1187],[1120,3483,1614],{"className":3484,"style":1630},[1187,1191],[1120,3486],{"className":3487,"style":1394},[1249],[1120,3489,1473],{"className":3490},[1398],[1120,3492],{"className":3493,"style":1394},[1249],[1120,3495,3497,3500,3503],{"className":3496},[1178],[1120,3498],{"className":3499,"style":3429},[1182],[1120,3501,3039],{"className":3502},[1187,1191],[1120,3504,1614],{"className":3505,"style":1630},[1187,1191],[1096,3507,3508,3509,3512,3513,3541,3542,1744,3571,3574,3575,3669],{},"这正是",[1103,3510,3511],{},"特征方程","。所以最优的 ",[1120,3514,3516,3529],{"className":3515},[1123],[1120,3517,3519],{"className":3518},[1127],[1129,3520,3521],{"xmlns":1131},[1133,3522,3523,3527],{},[1136,3524,3525],{},[1142,3526,1614],{},[1167,3528,1614],{"encoding":1169},[1120,3530,3532],{"className":3531,"ariaHidden":1152},[1174],[1120,3533,3535,3538],{"className":3534},[1178],[1120,3536],{"className":3537,"style":1626},[1182],[1120,3539,1614],{"className":3540,"style":1630},[1187,1191]," 就是协方差矩阵 ",[1120,3543,3545,3559],{"className":3544},[1123],[1120,3546,3548],{"className":3547},[1127],[1129,3549,3550],{"xmlns":1131},[1133,3551,3552,3556],{},[1136,3553,3554],{},[1142,3555,1863],{"mathvariant":1650},[1167,3557,3558],{"encoding":1169},"\\Sigma",[1120,3560,3562],{"className":3561,"ariaHidden":1152},[1174],[1120,3563,3565,3568],{"className":3564},[1178],[1120,3566],{"className":3567,"style":2516},[1182],[1120,3569,1863],{"className":3570},[1187],[1103,3572,3573],{},"最大特征值所对应的特征向量","(此时方差 ",[1120,3576,3578,3604],{"className":3577},[1123],[1120,3579,3581],{"className":3580},[1127],[1129,3582,3583],{"xmlns":1131},[1133,3584,3585,3601],{},[1136,3586,3587,3593,3595,3597,3599],{},[1329,3588,3589,3591],{},[1142,3590,1614],{},[1142,3592,1651],{"mathvariant":1650},[1142,3594,1863],{"mathvariant":1650},[1142,3596,1614],{},[1150,3598,1473],{},[1142,3600,3039],{},[1167,3602,3603],{"encoding":1169},"w^{\\top}\\Sigma w = \\lambda",[1120,3605,3607,3660],{"className":3606,"ariaHidden":1152},[1174],[1120,3608,3610,3613,3645,3648,3651,3654,3657],{"className":3609},[1178],[1120,3611],{"className":3612,"style":1689},[1182],[1120,3614,3616,3619],{"className":3615},[1187],[1120,3617,1614],{"className":3618,"style":1630},[1187,1191],[1120,3620,3622],{"className":3621},[1195],[1120,3623,3625],{"className":3624},[1199],[1120,3626,3628],{"className":3627},[1204],[1120,3629,3631],{"className":3630,"style":1689},[1208],[1120,3632,3633,3636],{"style":1431},[1120,3634],{"className":3635,"style":1217},[1216],[1120,3637,3639],{"className":3638},[1221,1222,1223,1224],[1120,3640,3642],{"className":3641},[1187,1224],[1120,3643,1651],{"className":3644},[1187,1224],[1120,3646,1863],{"className":3647},[1187],[1120,3649,1614],{"className":3650,"style":1630},[1187,1191],[1120,3652],{"className":3653,"style":1394},[1249],[1120,3655,1473],{"className":3656},[1398],[1120,3658],{"className":3659,"style":1394},[1249],[1120,3661,3663,3666],{"className":3662},[1178],[1120,3664],{"className":3665,"style":3429},[1182],[1120,3667,3039],{"className":3668},[1187,1191]," 取最大)。",[1096,3671,3672,3675,3676,3748,3749,3752],{},[1103,3673,3674],{},"求第二个主成分",":要求它在最大化方差的同时,与第一个主成分 ",[1120,3677,3679,3697],{"className":3678},[1123],[1120,3680,3682],{"className":3681},[1127],[1129,3683,3684],{"xmlns":1131},[1133,3685,3686,3694],{},[1136,3687,3688],{},[1139,3689,3690,3692],{},[1142,3691,1614],{},[1146,3693,1148],{},[1167,3695,3696],{"encoding":1169},"w_1",[1120,3698,3700],{"className":3699,"ariaHidden":1152},[1174],[1120,3701,3703,3707],{"className":3702},[1178],[1120,3704],{"className":3705,"style":3706},[1182],"height:0.5806em;vertical-align:-0.15em;",[1120,3708,3710,3713],{"className":3709},[1187],[1120,3711,1614],{"className":3712,"style":1630},[1187,1191],[1120,3714,3716],{"className":3715},[1195],[1120,3717,3719,3740],{"className":3718},[1199,1200],[1120,3720,3722,3737],{"className":3721},[1204],[1120,3723,3725],{"className":3724,"style":1209},[1208],[1120,3726,3728,3731],{"style":3727},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[1120,3729],{"className":3730,"style":1217},[1216],[1120,3732,3734],{"className":3733},[1221,1222,1223,1224],[1120,3735,1148],{"className":3736},[1187,1224],[1120,3738,1232],{"className":3739},[1231],[1120,3741,3743],{"className":3742},[1204],[1120,3744,3746],{"className":3745,"style":1239},[1208],[1120,3747],{}," ",[1103,3750,3751],{},"正交","。优化问题为",[1096,3754,3755],{},[1120,3756,3758,3848],{"className":3757},[1123],[1120,3759,3761],{"className":3760},[1127],[1129,3762,3763],{"xmlns":1131},[1133,3764,3765,3845],{},[1136,3766,3767,3781,3783,3791,3793,3799,3801,3803,3805,3813,3819,3821,3823,3825,3827,3835,3841,3843],{},[1139,3768,3769,3775],{},[1136,3770,3771,3773],{},[1142,3772,2793],{},[1150,3774,2796],{},[1139,3776,3777,3779],{},[1142,3778,1614],{},[1146,3780,1811],{},[2800,3782,2802],{},[1774,3784,3785,3787,3789],{},[1142,3786,1614],{},[1146,3788,1811],{},[1142,3790,1651],{"mathvariant":1650},[1142,3792,1863],{"mathvariant":1650},[1139,3794,3795,3797],{},[1142,3796,1614],{},[1146,3798,1811],{},[1249,3800],{"width":2815},[2800,3802,2818],{},[1249,3804],{"width":2815},[1774,3806,3807,3809,3811],{},[1142,3808,1614],{},[1146,3810,1811],{},[1142,3812,1651],{"mathvariant":1650},[1139,3814,3815,3817],{},[1142,3816,1614],{},[1146,3818,1811],{},[1150,3820,1473],{},[1146,3822,1148],{},[1150,3824,1153],{"separator":1152},[2800,3826,2802],{},[1774,3828,3829,3831,3833],{},[1142,3830,1614],{},[1146,3832,1811],{},[1142,3834,1651],{"mathvariant":1650},[1139,3836,3837,3839],{},[1142,3838,1614],{},[1146,3840,1148],{},[1150,3842,1473],{},[1146,3844,1476],{},[1167,3846,3847],{"encoding":1169},"\\max_{w_2}\\ w_2^{\\top}\\Sigma w_2 \\quad \\text{s.t.}\\quad w_2^{\\top}w_2 = 1,\\ w_2^{\\top}w_1 = 0",[1120,3849,3851,4165,4287],{"className":3850,"ariaHidden":1152},[1174],[1120,3852,3854,3858,3945,3948,3951,4007,4010,4050,4053,4059,4062,4116,4156,4159,4162],{"className":3853},[1178],[1120,3855],{"className":3856,"style":3857},[1182],"height:1.0992em;vertical-align:-0.2501em;",[1120,3859,3861,3864],{"className":3860},[1493],[1120,3862,2793],{"className":3863},[1493],[1120,3865,3867],{"className":3866},[1195],[1120,3868,3870,3936],{"className":3869},[1199,1200],[1120,3871,3873,3933],{"className":3872},[1204],[1120,3874,3876],{"className":3875,"style":1282},[1208],[1120,3877,3878,3881],{"style":2865},[1120,3879],{"className":3880,"style":1217},[1216],[1120,3882,3884],{"className":3883},[1221,1222,1223,1224],[1120,3885,3887],{"className":3886},[1187,1224],[1120,3888,3890,3893],{"className":3889},[1187,1224],[1120,3891,1614],{"className":3892,"style":1630},[1187,1191,1224],[1120,3894,3896],{"className":3895},[1195],[1120,3897,3899,3924],{"className":3898},[1199,1200],[1120,3900,3902,3921],{"className":3901},[1204],[1120,3903,3906],{"className":3904,"style":3905},[1208],"height:0.3173em;",[1120,3907,3909,3913],{"style":3908},"top:-2.357em;margin-left:-0.0269em;margin-right:0.0714em;",[1120,3910],{"className":3911,"style":3912},[1216],"height:2.5em;",[1120,3914,3918],{"className":3915},[1221,3916,3917,1224],"reset-size3","size1",[1120,3919,1811],{"className":3920},[1187,1224],[1120,3922,1232],{"className":3923},[1231],[1120,3925,3927],{"className":3926},[1204],[1120,3928,3931],{"className":3929,"style":3930},[1208],"height:0.143em;",[1120,3932],{},[1120,3934,1232],{"className":3935},[1231],[1120,3937,3939],{"className":3938},[1204],[1120,3940,3943],{"className":3941,"style":3942},[1208],"height:0.2501em;",[1120,3944],{},[1120,3946,2802],{"className":3947},[1249],[1120,3949],{"className":3950,"style":1250},[1249],[1120,3952,3954,3957],{"className":3953},[1187],[1120,3955,1614],{"className":3956,"style":1630},[1187,1191],[1120,3958,3960],{"className":3959},[1195],[1120,3961,3963,3998],{"className":3962},[1199,1200],[1120,3964,3966,3995],{"className":3965},[1204],[1120,3967,3969,3981],{"className":3968,"style":1689},[1208],[1120,3970,3972,3975],{"style":3971},"top:-2.4519em;margin-left:-0.0269em;margin-right:0.05em;",[1120,3973],{"className":3974,"style":1217},[1216],[1120,3976,3978],{"className":3977},[1221,1222,1223,1224],[1120,3979,1811],{"className":3980},[1187,1224],[1120,3982,3983,3986],{"style":1431},[1120,3984],{"className":3985,"style":1217},[1216],[1120,3987,3989],{"className":3988},[1221,1222,1223,1224],[1120,3990,3992],{"className":3991},[1187,1224],[1120,3993,1651],{"className":3994},[1187,1224],[1120,3996,1232],{"className":3997},[1231],[1120,3999,4001],{"className":4000},[1204],[1120,4002,4005],{"className":4003,"style":4004},[1208],"height:0.2481em;",[1120,4006],{},[1120,4008,1863],{"className":4009},[1187],[1120,4011,4013,4016],{"className":4012},[1187],[1120,4014,1614],{"className":4015,"style":1630},[1187,1191],[1120,4017,4019],{"className":4018},[1195],[1120,4020,4022,4042],{"className":4021},[1199,1200],[1120,4023,4025,4039],{"className":4024},[1204],[1120,4026,4028],{"className":4027,"style":1209},[1208],[1120,4029,4030,4033],{"style":3727},[1120,4031],{"className":4032,"style":1217},[1216],[1120,4034,4036],{"className":4035},[1221,1222,1223,1224],[1120,4037,1811],{"className":4038},[1187,1224],[1120,4040,1232],{"className":4041},[1231],[1120,4043,4045],{"className":4044},[1204],[1120,4046,4048],{"className":4047,"style":1239},[1208],[1120,4049],{},[1120,4051],{"className":4052,"style":2936},[1249],[1120,4054,4056],{"className":4055},[1187,2940],[1120,4057,2818],{"className":4058},[1187],[1120,4060],{"className":4061,"style":2936},[1249],[1120,4063,4065,4068],{"className":4064},[1187],[1120,4066,1614],{"className":4067,"style":1630},[1187,1191],[1120,4069,4071],{"className":4070},[1195],[1120,4072,4074,4108],{"className":4073},[1199,1200],[1120,4075,4077,4105],{"className":4076},[1204],[1120,4078,4080,4091],{"className":4079,"style":1689},[1208],[1120,4081,4082,4085],{"style":3971},[1120,4083],{"className":4084,"style":1217},[1216],[1120,4086,4088],{"className":4087},[1221,1222,1223,1224],[1120,4089,1811],{"className":4090},[1187,1224],[1120,4092,4093,4096],{"style":1431},[1120,4094],{"className":4095,"style":1217},[1216],[1120,4097,4099],{"className":4098},[1221,1222,1223,1224],[1120,4100,4102],{"className":4101},[1187,1224],[1120,4103,1651],{"className":4104},[1187,1224],[1120,4106,1232],{"className":4107},[1231],[1120,4109,4111],{"className":4110},[1204],[1120,4112,4114],{"className":4113,"style":4004},[1208],[1120,4115],{},[1120,4117,4119,4122],{"className":4118},[1187],[1120,4120,1614],{"className":4121,"style":1630},[1187,1191],[1120,4123,4125],{"className":4124},[1195],[1120,4126,4128,4148],{"className":4127},[1199,1200],[1120,4129,4131,4145],{"className":4130},[1204],[1120,4132,4134],{"className":4133,"style":1209},[1208],[1120,4135,4136,4139],{"style":3727},[1120,4137],{"className":4138,"style":1217},[1216],[1120,4140,4142],{"className":4141},[1221,1222,1223,1224],[1120,4143,1811],{"className":4144},[1187,1224],[1120,4146,1232],{"className":4147},[1231],[1120,4149,4151],{"className":4150},[1204],[1120,4152,4154],{"className":4153,"style":1239},[1208],[1120,4155],{},[1120,4157],{"className":4158,"style":1394},[1249],[1120,4160,1473],{"className":4161},[1398],[1120,4163],{"className":4164,"style":1394},[1249],[1120,4166,4168,4172,4175,4178,4181,4184,4238,4278,4281,4284],{"className":4167},[1178],[1120,4169],{"className":4170,"style":4171},[1182],"height:1.0972em;vertical-align:-0.2481em;",[1120,4173,1148],{"className":4174},[1187],[1120,4176,1153],{"className":4177},[1245],[1120,4179,2802],{"className":4180},[1249],[1120,4182],{"className":4183,"style":1250},[1249],[1120,4185,4187,4190],{"className":4186},[1187],[1120,4188,1614],{"className":4189,"style":1630},[1187,1191],[1120,4191,4193],{"className":4192},[1195],[1120,4194,4196,4230],{"className":4195},[1199,1200],[1120,4197,4199,4227],{"className":4198},[1204],[1120,4200,4202,4213],{"className":4201,"style":1689},[1208],[1120,4203,4204,4207],{"style":3971},[1120,4205],{"className":4206,"style":1217},[1216],[1120,4208,4210],{"className":4209},[1221,1222,1223,1224],[1120,4211,1811],{"className":4212},[1187,1224],[1120,4214,4215,4218],{"style":1431},[1120,4216],{"className":4217,"style":1217},[1216],[1120,4219,4221],{"className":4220},[1221,1222,1223,1224],[1120,4222,4224],{"className":4223},[1187,1224],[1120,4225,1651],{"className":4226},[1187,1224],[1120,4228,1232],{"className":4229},[1231],[1120,4231,4233],{"className":4232},[1204],[1120,4234,4236],{"className":4235,"style":4004},[1208],[1120,4237],{},[1120,4239,4241,4244],{"className":4240},[1187],[1120,4242,1614],{"className":4243,"style":1630},[1187,1191],[1120,4245,4247],{"className":4246},[1195],[1120,4248,4250,4270],{"className":4249},[1199,1200],[1120,4251,4253,4267],{"className":4252},[1204],[1120,4254,4256],{"className":4255,"style":1209},[1208],[1120,4257,4258,4261],{"style":3727},[1120,4259],{"className":4260,"style":1217},[1216],[1120,4262,4264],{"className":4263},[1221,1222,1223,1224],[1120,4265,1148],{"className":4266},[1187,1224],[1120,4268,1232],{"className":4269},[1231],[1120,4271,4273],{"className":4272},[1204],[1120,4274,4276],{"className":4275,"style":1239},[1208],[1120,4277],{},[1120,4279],{"className":4280,"style":1394},[1249],[1120,4282,1473],{"className":4283},[1398],[1120,4285],{"className":4286,"style":1394},[1249],[1120,4288,4290,4293],{"className":4289},[1178],[1120,4291],{"className":4292,"style":1595},[1182],[1120,4294,1476],{"className":4295},[1187],[1096,4297,4298,4299,4485,4486,4556,4557,4560,4561,4590,4591,4619,4620,4648],{},"用带两个乘子的拉格朗日函数求导,可以得到 ",[1120,4300,4302,4336],{"className":4301},[1123],[1120,4303,4305],{"className":4304},[1127],[1129,4306,4307],{"xmlns":1131},[1133,4308,4309,4333],{},[1136,4310,4311,4313,4319,4321,4327],{},[1142,4312,1863],{"mathvariant":1650},[1139,4314,4315,4317],{},[1142,4316,1614],{},[1146,4318,1811],{},[1150,4320,1473],{},[1139,4322,4323,4325],{},[1142,4324,3039],{},[1146,4326,1811],{},[1139,4328,4329,4331],{},[1142,4330,1614],{},[1146,4332,1811],{},[1167,4334,4335],{"encoding":1169},"\\Sigma w_2 = \\lambda_2 w_2",[1120,4337,4339,4398],{"className":4338,"ariaHidden":1152},[1174],[1120,4340,4342,4346,4349,4389,4392,4395],{"className":4341},[1178],[1120,4343],{"className":4344,"style":4345},[1182],"height:0.8333em;vertical-align:-0.15em;",[1120,4347,1863],{"className":4348},[1187],[1120,4350,4352,4355],{"className":4351},[1187],[1120,4353,1614],{"className":4354,"style":1630},[1187,1191],[1120,4356,4358],{"className":4357},[1195],[1120,4359,4361,4381],{"className":4360},[1199,1200],[1120,4362,4364,4378],{"className":4363},[1204],[1120,4365,4367],{"className":4366,"style":1209},[1208],[1120,4368,4369,4372],{"style":3727},[1120,4370],{"className":4371,"style":1217},[1216],[1120,4373,4375],{"className":4374},[1221,1222,1223,1224],[1120,4376,1811],{"className":4377},[1187,1224],[1120,4379,1232],{"className":4380},[1231],[1120,4382,4384],{"className":4383},[1204],[1120,4385,4387],{"className":4386,"style":1239},[1208],[1120,4388],{},[1120,4390],{"className":4391,"style":1394},[1249],[1120,4393,1473],{"className":4394},[1398],[1120,4396],{"className":4397,"style":1394},[1249],[1120,4399,4401,4405,4445],{"className":4400},[1178],[1120,4402],{"className":4403,"style":4404},[1182],"height:0.8444em;vertical-align:-0.15em;",[1120,4406,4408,4411],{"className":4407},[1187],[1120,4409,3039],{"className":4410},[1187,1191],[1120,4412,4414],{"className":4413},[1195],[1120,4415,4417,4437],{"className":4416},[1199,1200],[1120,4418,4420,4434],{"className":4419},[1204],[1120,4421,4423],{"className":4422,"style":1209},[1208],[1120,4424,4425,4428],{"style":1212},[1120,4426],{"className":4427,"style":1217},[1216],[1120,4429,4431],{"className":4430},[1221,1222,1223,1224],[1120,4432,1811],{"className":4433},[1187,1224],[1120,4435,1232],{"className":4436},[1231],[1120,4438,4440],{"className":4439},[1204],[1120,4441,4443],{"className":4442,"style":1239},[1208],[1120,4444],{},[1120,4446,4448,4451],{"className":4447},[1187],[1120,4449,1614],{"className":4450,"style":1630},[1187,1191],[1120,4452,4454],{"className":4453},[1195],[1120,4455,4457,4477],{"className":4456},[1199,1200],[1120,4458,4460,4474],{"className":4459},[1204],[1120,4461,4463],{"className":4462,"style":1209},[1208],[1120,4464,4465,4468],{"style":3727},[1120,4466],{"className":4467,"style":1217},[1216],[1120,4469,4471],{"className":4470},[1221,1222,1223,1224],[1120,4472,1811],{"className":4473},[1187,1224],[1120,4475,1232],{"className":4476},[1231],[1120,4478,4480],{"className":4479},[1204],[1120,4481,4483],{"className":4482,"style":1239},[1208],[1120,4484],{},",即 ",[1120,4487,4489,4507],{"className":4488},[1123],[1120,4490,4492],{"className":4491},[1127],[1129,4493,4494],{"xmlns":1131},[1133,4495,4496,4504],{},[1136,4497,4498],{},[1139,4499,4500,4502],{},[1142,4501,1614],{},[1146,4503,1811],{},[1167,4505,4506],{"encoding":1169},"w_2",[1120,4508,4510],{"className":4509,"ariaHidden":1152},[1174],[1120,4511,4513,4516],{"className":4512},[1178],[1120,4514],{"className":4515,"style":3706},[1182],[1120,4517,4519,4522],{"className":4518},[1187],[1120,4520,1614],{"className":4521,"style":1630},[1187,1191],[1120,4523,4525],{"className":4524},[1195],[1120,4526,4528,4548],{"className":4527},[1199,1200],[1120,4529,4531,4545],{"className":4530},[1204],[1120,4532,4534],{"className":4533,"style":1209},[1208],[1120,4535,4536,4539],{"style":3727},[1120,4537],{"className":4538,"style":1217},[1216],[1120,4540,4542],{"className":4541},[1221,1222,1223,1224],[1120,4543,1811],{"className":4544},[1187,1224],[1120,4546,1232],{"className":4547},[1231],[1120,4549,4551],{"className":4550},[1204],[1120,4552,4554],{"className":4553,"style":1239},[1208],[1120,4555],{}," 是",[1103,4558,4559],{},"第二大特征值对应的特征向量","。依此类推,前 ",[1120,4562,4564,4578],{"className":4563},[1123],[1120,4565,4567],{"className":4566},[1127],[1129,4568,4569],{"xmlns":1131},[1133,4570,4571,4576],{},[1136,4572,4573],{},[1142,4574,4575],{},"d",[1167,4577,4575],{"encoding":1169},[1120,4579,4581],{"className":4580,"ariaHidden":1152},[1174],[1120,4582,4584,4587],{"className":4583},[1178],[1120,4585],{"className":4586,"style":3429},[1182],[1120,4588,4575],{"className":4589},[1187,1191]," 个主成分就是 ",[1120,4592,4594,4607],{"className":4593},[1123],[1120,4595,4597],{"className":4596},[1127],[1129,4598,4599],{"xmlns":1131},[1133,4600,4601,4605],{},[1136,4602,4603],{},[1142,4604,1863],{"mathvariant":1650},[1167,4606,3558],{"encoding":1169},[1120,4608,4610],{"className":4609,"ariaHidden":1152},[1174],[1120,4611,4613,4616],{"className":4612},[1178],[1120,4614],{"className":4615,"style":2516},[1182],[1120,4617,1863],{"className":4618},[1187]," 前 ",[1120,4621,4623,4636],{"className":4622},[1123],[1120,4624,4626],{"className":4625},[1127],[1129,4627,4628],{"xmlns":1131},[1133,4629,4630,4634],{},[1136,4631,4632],{},[1142,4633,4575],{},[1167,4635,4575],{"encoding":1169},[1120,4637,4639],{"className":4638,"ariaHidden":1152},[1174],[1120,4640,4642,4645],{"className":4641},[1178],[1120,4643],{"className":4644,"style":3429},[1182],[1120,4646,4575],{"className":4647},[1187,1191]," 大特征值对应的特征向量。",[1112,4650,4652],{"id":4651},"视角二从最小重构误差考虑","视角二:从最小重构误差考虑",[1096,4654,4655,4656,3748,4659,1791,4814,5030,5031,5059],{},"换个角度:找一组",[1103,4657,4658],{},"正交基底",[1120,4660,4662,4699],{"className":4661},[1123],[1120,4663,4665],{"className":4664},[1127],[1129,4666,4667],{"xmlns":1131},[1133,4668,4669,4696],{},[1136,4670,4671,4674,4681,4683,4685,4687,4693],{},[1150,4672,4673],{"stretchy":1790},"{",[1139,4675,4676,4679],{},[1142,4677,4678],{},"u",[1146,4680,1148],{},[1150,4682,1153],{"separator":1152},[1150,4684,1156],{},[1150,4686,1153],{"separator":1152},[1139,4688,4689,4691],{},[1142,4690,4678],{},[1142,4692,1096],{},[1150,4694,4695],{"stretchy":1790},"}",[1167,4697,4698],{"encoding":1169},"\\{u_1, \\dots, u_p\\}",[1120,4700,4702],{"className":4701,"ariaHidden":1152},[1174],[1120,4703,4705,4709,4712,4752,4755,4758,4761,4764,4767,4770,4811],{"className":4704},[1178],[1120,4706],{"className":4707,"style":4708},[1182],"height:1.0361em;vertical-align:-0.2861em;",[1120,4710,4673],{"className":4711},[1885],[1120,4713,4715,4718],{"className":4714},[1187],[1120,4716,4678],{"className":4717},[1187,1191],[1120,4719,4721],{"className":4720},[1195],[1120,4722,4724,4744],{"className":4723},[1199,1200],[1120,4725,4727,4741],{"className":4726},[1204],[1120,4728,4730],{"className":4729,"style":1209},[1208],[1120,4731,4732,4735],{"style":1212},[1120,4733],{"className":4734,"style":1217},[1216],[1120,4736,4738],{"className":4737},[1221,1222,1223,1224],[1120,4739,1148],{"className":4740},[1187,1224],[1120,4742,1232],{"className":4743},[1231],[1120,4745,4747],{"className":4746},[1204],[1120,4748,4750],{"className":4749,"style":1239},[1208],[1120,4751],{},[1120,4753,1153],{"className":4754},[1245],[1120,4756],{"className":4757,"style":1250},[1249],[1120,4759,1156],{"className":4760},[1254],[1120,4762],{"className":4763,"style":1250},[1249],[1120,4765,1153],{"className":4766},[1245],[1120,4768],{"className":4769,"style":1250},[1249],[1120,4771,4773,4776],{"className":4772},[1187],[1120,4774,4678],{"className":4775},[1187,1191],[1120,4777,4779],{"className":4778},[1195],[1120,4780,4782,4802],{"className":4781},[1199,1200],[1120,4783,4785,4799],{"className":4784},[1204],[1120,4786,4788],{"className":4787,"style":1282},[1208],[1120,4789,4790,4793],{"style":1212},[1120,4791],{"className":4792,"style":1217},[1216],[1120,4794,4796],{"className":4795},[1221,1222,1223,1224],[1120,4797,1096],{"className":4798},[1187,1191,1224],[1120,4800,1232],{"className":4801},[1231],[1120,4803,4805],{"className":4804},[1204],[1120,4806,4809],{"className":4807,"style":4808},[1208],"height:0.2861em;",[1120,4810],{},[1120,4812,4695],{"className":4813},[1957],[1120,4815,4817,4858],{"className":4816},[1123],[1120,4818,4820],{"className":4819},[1127],[1129,4821,4822],{"xmlns":1131},[1133,4823,4824,4855],{},[1136,4825,4826,4835,4842,4844],{},[1774,4827,4828,4830,4833],{},[1142,4829,4678],{},[1142,4831,4832],{},"j",[1142,4834,1651],{"mathvariant":1650},[1139,4836,4837,4839],{},[1142,4838,4678],{},[1142,4840,4841],{},"k",[1150,4843,1473],{},[1139,4845,4846,4849],{},[1142,4847,4848],{},"δ",[1136,4850,4851,4853],{},[1142,4852,4832],{},[1142,4854,4841],{},[1167,4856,4857],{"encoding":1169},"u_j^{\\top}u_k = \\delta_{jk}",[1120,4859,4861,4975],{"className":4860,"ariaHidden":1152},[1174],[1120,4862,4864,4868,4924,4966,4969,4972],{"className":4863},[1178],[1120,4865],{"className":4866,"style":4867},[1182],"height:1.2439em;vertical-align:-0.3948em;",[1120,4869,4871,4874],{"className":4870},[1187],[1120,4872,4678],{"className":4873},[1187,1191],[1120,4875,4877],{"className":4876},[1195],[1120,4878,4880,4915],{"className":4879},[1199,1200],[1120,4881,4883,4912],{"className":4882},[1204],[1120,4884,4886,4898],{"className":4885,"style":1689},[1208],[1120,4887,4888,4891],{"style":2361},[1120,4889],{"className":4890,"style":1217},[1216],[1120,4892,4894],{"className":4893},[1221,1222,1223,1224],[1120,4895,4832],{"className":4896,"style":4897},[1187,1191,1224],"margin-right:0.0572em;",[1120,4899,4900,4903],{"style":1431},[1120,4901],{"className":4902,"style":1217},[1216],[1120,4904,4906],{"className":4905},[1221,1222,1223,1224],[1120,4907,4909],{"className":4908},[1187,1224],[1120,4910,1651],{"className":4911},[1187,1224],[1120,4913,1232],{"className":4914},[1231],[1120,4916,4918],{"className":4917},[1204],[1120,4919,4922],{"className":4920,"style":4921},[1208],"height:0.3948em;",[1120,4923],{},[1120,4925,4927,4930],{"className":4926},[1187],[1120,4928,4678],{"className":4929},[1187,1191],[1120,4931,4933],{"className":4932},[1195],[1120,4934,4936,4958],{"className":4935},[1199,1200],[1120,4937,4939,4955],{"className":4938},[1204],[1120,4940,4943],{"className":4941,"style":4942},[1208],"height:0.3361em;",[1120,4944,4945,4948],{"style":1212},[1120,4946],{"className":4947,"style":1217},[1216],[1120,4949,4951],{"className":4950},[1221,1222,1223,1224],[1120,4952,4841],{"className":4953,"style":4954},[1187,1191,1224],"margin-right:0.0315em;",[1120,4956,1232],{"className":4957},[1231],[1120,4959,4961],{"className":4960},[1204],[1120,4962,4964],{"className":4963,"style":1239},[1208],[1120,4965],{},[1120,4967],{"className":4968,"style":1394},[1249],[1120,4970,1473],{"className":4971},[1398],[1120,4973],{"className":4974,"style":1394},[1249],[1120,4976,4978,4982],{"className":4977},[1178],[1120,4979],{"className":4980,"style":4981},[1182],"height:0.9805em;vertical-align:-0.2861em;",[1120,4983,4985,4989],{"className":4984},[1187],[1120,4986,4848],{"className":4987,"style":4988},[1187,1191],"margin-right:0.0379em;",[1120,4990,4992],{"className":4991},[1195],[1120,4993,4995,5022],{"className":4994},[1199,1200],[1120,4996,4998,5019],{"className":4997},[1204],[1120,4999,5001],{"className":5000,"style":4942},[1208],[1120,5002,5004,5007],{"style":5003},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[1120,5005],{"className":5006,"style":1217},[1216],[1120,5008,5010],{"className":5009},[1221,1222,1223,1224],[1120,5011,5013,5016],{"className":5012},[1187,1224],[1120,5014,4832],{"className":5015,"style":4897},[1187,1191,1224],[1120,5017,4841],{"className":5018,"style":4954},[1187,1191,1224],[1120,5020,1232],{"className":5021},[1231],[1120,5023,5025],{"className":5024},[1204],[1120,5026,5028],{"className":5027,"style":4808},[1208],[1120,5029],{},"),把每个样本用这组基表示。只保留前 ",[1120,5032,5034,5047],{"className":5033},[1123],[1120,5035,5037],{"className":5036},[1127],[1129,5038,5039],{"xmlns":1131},[1133,5040,5041,5045],{},[1136,5042,5043],{},[1142,5044,4575],{},[1167,5046,4575],{"encoding":1169},[1120,5048,5050],{"className":5049,"ariaHidden":1152},[1174],[1120,5051,5053,5056],{"className":5052},[1178],[1120,5054],{"className":5055,"style":3429},[1182],[1120,5057,4575],{"className":5058},[1187,1191]," 个基方向做投影,得到近似(重构):",[1096,5061,5062],{},[1120,5063,5065,5132],{"className":5064},[1123],[1120,5066,5068],{"className":5067},[1127],[1129,5069,5070],{"xmlns":1131},[1133,5071,5072,5129],{},[1136,5073,5074,5086,5088,5102,5104,5112,5118,5120,5123],{},[1139,5075,5076,5084],{},[5077,5078,5079,5081],"mover",{"accent":1152},[1142,5080,1144],{},[1150,5082,5083],{},"^",[1142,5085,1324],{},[1150,5087,1473],{},[1774,5089,5090,5092,5100],{},[1150,5091,1462],{},[1136,5093,5094,5096,5098],{},[1142,5095,4832],{},[1150,5097,1473],{},[1146,5099,1148],{},[1142,5101,4575],{},[1150,5103,1791],{"stretchy":1790},[1774,5105,5106,5108,5110],{},[1142,5107,4678],{},[1142,5109,4832],{},[1142,5111,1651],{"mathvariant":1650},[1139,5113,5114,5116],{},[1142,5115,1144],{},[1142,5117,1324],{},[1150,5119,1808],{"stretchy":1790},[2800,5121,5122],{}," ",[1139,5124,5125,5127],{},[1142,5126,4678],{},[1142,5128,4832],{},[1167,5130,5131],{"encoding":1169},"\\hat{x}_i = 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className":5425,"style":4897},[1187,1191,1224],[1120,5427,1232],{"className":5428},[1231],[1120,5430,5432],{"className":5431},[1204],[1120,5433,5435],{"className":5434,"style":4808},[1208],[1120,5436],{},[1096,5438,5439,5440,5443],{},"PCA 尝试",[1103,5441,5442],{},"最小化重构误差","——原始点与其低维重构之间的平方距离之和:",[1096,5445,5446],{},[1120,5447,5449,5526],{"className":5448},[1123],[1120,5450,5452],{"className":5451},[1127],[1129,5453,5454],{"xmlns":1131},[1133,5455,5456,5523],{},[1136,5457,5458,5479,5481,5495,5499,5505,5507,5517],{},[1139,5459,5460,5467],{},[1136,5461,5462,5465],{},[1142,5463,5464],{},"min",[1150,5466,2796],{},[1136,5468,5469,5471,5477],{},[1150,5470,4673],{"stretchy":1790},[1139,5472,5473,5475],{},[1142,5474,4678],{},[1142,5476,4832],{},[1150,5478,4695],{"stretchy":1790},[2800,5480,2802],{},[1774,5482,5483,5485,5493],{},[1150,5484,1462],{},[1136,5486,5487,5489,5491],{},[1142,5488,1324],{},[1150,5490,1473],{},[1146,5492,1148],{},[1142,5494,1165],{},[1150,5496,5498],{"fence":1790,"stretchy":1152,"minsize":5497,"maxsize":5497},"1.2em","∥",[1139,5500,5501,5503],{},[1142,5502,1144],{},[1142,5504,1324],{},[1150,5506,3036],{},[1139,5508,5509,5515],{},[5077,5510,5511,5513],{"accent":1152},[1142,5512,1144],{},[1150,5514,5083],{},[1142,5516,1324],{},[1329,5518,5519,5521],{},[1150,5520,5498],{"fence":1790,"stretchy":1152,"minsize":5497,"maxsize":5497},[1146,5522,1811],{},[1167,5524,5525],{"encoding":1169},"\\min_{\\{u_j\\}}\\ \\sum_{i=1}^{n}\\big\\|x_i - \\hat{x}_i\\big\\|^2",[1120,5527,5529,5798],{"className":5528,"ariaHidden":1152},[1174],[1120,5530,5532,5536,5628,5631,5634,5697,5700,5749,5789,5792,5795],{"className":5531},[1178],[1120,5533],{"className":5534,"style":5535},[1182],"height:1.2275em;vertical-align:-0.3775em;",[1120,5537,5539,5542],{"className":5538},[1493],[1120,5540,5464],{"className":5541},[1493],[1120,5543,5545],{"className":5544},[1195],[1120,5546,5548,5619],{"className":5547},[1199,1200],[1120,5549,5551,5616],{"className":5550},[1204],[1120,5552,5555],{"className":5553,"style":5554},[1208],"height:0.3448em;",[1120,5556,5558,5561],{"style":5557},"top:-2.5198em;margin-right:0.05em;",[1120,5559],{"className":5560,"style":1217},[1216],[1120,5562,5564],{"className":5563},[1221,1222,1223,1224],[1120,5565,5567,5570,5613],{"className":5566},[1187,1224],[1120,5568,4673],{"className":5569},[1885,1224],[1120,5571,5573,5576],{"className":5572},[1187,1224],[1120,5574,4678],{"className":5575},[1187,1191,1224],[1120,5577,5579],{"className":5578},[1195],[1120,5580,5582,5604],{"className":5581},[1199,1200],[1120,5583,5585,5601],{"className":5584},[1204],[1120,5586,5589],{"className":5587,"style":5588},[1208],"height:0.3281em;",[1120,5590,5592,5595],{"style":5591},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[1120,5593],{"className":5594,"style":3912},[1216],[1120,5596,5598],{"className":5597},[1221,3916,3917,1224],[1120,5599,4832],{"className":5600,"style":4897},[1187,1191,1224],[1120,5602,1232],{"className":5603},[1231],[1120,5605,5607],{"className":5606},[1204],[1120,5608,5611],{"className":5609,"style":5610},[1208],"height:0.2819em;",[1120,5612],{},[1120,5614,4695],{"className":5615},[1957,1224],[1120,5617,1232],{"className":5618},[1231],[1120,5620,5622],{"className":5621},[1204],[1120,5623,5626],{"className":5624,"style":5625},[1208],"height:0.3775em;",[1120,5627],{},[1120,5629,2802],{"className":5630},[1249],[1120,5632],{"className":5633,"style":1250},[1249],[1120,5635,5637,5640],{"className":5636},[1493],[1120,5638,1462],{"className":5639,"style":1499},[1493,1497,1498],[1120,5641,5643],{"className":5642},[1195],[1120,5644,5646,5689],{"className":5645},[1199,1200],[1120,5647,5649,5686],{"className":5648},[1204],[1120,5650,5652,5672],{"className":5651,"style":1979},[1208],[1120,5653,5654,5657],{"style":1515},[1120,5655],{"className":5656,"style":1217},[1216],[1120,5658,5660],{"className":5659},[1221,1222,1223,1224],[1120,5661,5663,5666,5669],{"className":5662},[1187,1224],[1120,5664,1324],{"className":5665},[1187,1191,1224],[1120,5667,1473],{"className":5668},[1398,1224],[1120,5670,1148],{"className":5671},[1187,1224],[1120,5673,5674,5677],{"style":2002},[1120,5675],{"className":5676,"style":1217},[1216],[1120,5678,5680],{"className":5679},[1221,1222,1223,1224],[1120,5681,5683],{"className":5682},[1187,1224],[1120,5684,1165],{"className":5685},[1187,1191,1224],[1120,5687,1232],{"className":5688},[1231],[1120,5690,5692],{"className":5691},[1204],[1120,5693,5695],{"className":5694,"style":1534},[1208],[1120,5696],{},[1120,5698],{"className":5699,"style":1250},[1249],[1120,5701,5703],{"className":5702},[1187],[1120,5704,5707],{"className":5705},[2185,5706],"mult",[1120,5708,5710,5740],{"className":5709},[1199,1200],[1120,5711,5713,5737],{"className":5712},[1204],[1120,5714,5717],{"className":5715,"style":5716},[1208],"height:0.85em;",[1120,5718,5720,5724],{"style":5719},"top:-2.85em;",[1120,5721],{"className":5722,"style":5723},[1216],"height:3.2em;",[1120,5725,5727],{"style":5726},"width:0.556em;height:1.2em;",[5728,5729,5733],"svg",{"xmlns":5730,"width":5731,"height":5497,"viewBox":5732},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.556em","0 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h43z",[1120,5738,1232],{"className":5739},[1231],[1120,5741,5743],{"className":5742},[1204],[1120,5744,5747],{"className":5745,"style":5746},[1208],"height:0.35em;",[1120,5748],{},[1120,5750,5752,5755],{"className":5751},[1187],[1120,5753,1144],{"className":5754},[1187,1191],[1120,5756,5758],{"className":5757},[1195],[1120,5759,5761,5781],{"className":5760},[1199,1200],[1120,5762,5764,5778],{"className":5763},[1204],[1120,5765,5767],{"className":5766,"style":1368},[1208],[1120,5768,5769,5772],{"style":1212},[1120,5770],{"className":5771,"style":1217},[1216],[1120,5773,5775],{"className":5774},[1221,1222,1223,1224],[1120,5776,1324],{"className":5777},[1187,1191,1224],[1120,5779,1232],{"className":5780},[1231],[1120,5782,5784],{"className":5783},[1204],[1120,5785,5787],{"className":5786,"style":1239},[1208],[1120,5788],{},[1120,5790],{"className":5791,"style":3128},[1249],[1120,5793,3036],{"className":5794},[3132],[1120,5796],{"className":5797,"style":3128},[1249],[1120,5799,5801,5805,5873],{"className":5800},[1178],[1120,5802],{"className":5803,"style":5804},[1182],"height:1.404em;vertical-align:-0.35em;",[1120,5806,5808,5839],{"className":5807},[1187],[1120,5809,5811],{"className":5810},[1187,5147],[1120,5812,5814],{"className":5813},[1199],[1120,5815,5817],{"className":5816},[1204],[1120,5818,5820,5828],{"className":5819,"style":3429},[1208],[1120,5821,5822,5825],{"style":5159},[1120,5823],{"className":5824,"style":1906},[1216],[1120,5826,1144],{"className":5827},[1187,1191],[1120,5829,5830,5833],{"style":5159},[1120,5831],{"className":5832,"style":1906},[1216],[1120,5834,5836],{"className":5835,"style":5175},[5174],[1120,5837,5083],{"className":5838},[1187],[1120,5840,5842],{"className":5841},[1195],[1120,5843,5845,5865],{"className":5844},[1199,1200],[1120,5846,5848,5862],{"className":5847},[1204],[1120,5849,5851],{"className":5850,"style":1368},[1208],[1120,5852,5853,5856],{"style":1212},[1120,5854],{"className":5855,"style":1217},[1216],[1120,5857,5859],{"className":5858},[1221,1222,1223,1224],[1120,5860,1324],{"className":5861},[1187,1191,1224],[1120,5863,1232],{"className":5864},[1231],[1120,5866,5868],{"className":5867},[1204],[1120,5869,5871],{"className":5870,"style":1239},[1208],[1120,5872],{},[1120,5874,5876,5913],{"className":5875},[1187],[1120,5877,5879],{"className":5878},[1187],[1120,5880,5882],{"className":5881},[2185,5706],[1120,5883,5885,5905],{"className":5884},[1199,1200],[1120,5886,5888,5902],{"className":5887},[1204],[1120,5889,5891],{"className":5890,"style":5716},[1208],[1120,5892,5893,5896],{"style":5719},[1120,5894],{"className":5895,"style":5723},[1216],[1120,5897,5898],{"style":5726},[5728,5899,5900],{"xmlns":5730,"width":5731,"height":5497,"viewBox":5732},[5734,5901],{"d":5736},[1120,5903,1232],{"className":5904},[1231],[1120,5906,5908],{"className":5907},[1204],[1120,5909,5911],{"className":5910,"style":5746},[1208],[1120,5912],{},[1120,5914,5916],{"className":5915},[1195],[1120,5917,5919],{"className":5918},[1199],[1120,5920,5922],{"className":5921},[1204],[1120,5923,5926],{"className":5924,"style":5925},[1208],"height:1.054em;",[1120,5927,5929,5932],{"style":5928},"top:-3.3029em;margin-right:0.05em;",[1120,5930],{"className":5931,"style":1217},[1216],[1120,5933,5935],{"className":5934},[1221,1222,1223,1224],[1120,5936,1811],{"className":5937},[1187,1224],[1096,5939,5940,5941,5973],{},"推导后可以证明,这等价于",[1103,5942,5943,5944,5972],{},"取协方差矩阵前 ",[1120,5945,5947,5960],{"className":5946},[1123],[1120,5948,5950],{"className":5949},[1127],[1129,5951,5952],{"xmlns":1131},[1133,5953,5954,5958],{},[1136,5955,5956],{},[1142,5957,4575],{},[1167,5959,4575],{"encoding":1169},[1120,5961,5963],{"className":5962,"ariaHidden":1152},[1174],[1120,5964,5966,5969],{"className":5965},[1178],[1120,5967],{"className":5968,"style":3429},[1182],[1120,5970,4575],{"className":5971},[1187,1191]," 个最大特征值对应的特征向量","作为基底。也就是说,\"投影后方差最大\"与\"重构误差最小\"选出的是同一组方向——这就是 PCA 两个视角的统一。",[5975,5976],"hr",{},[1091,5978,5980],{"id":5979},"二正则化框架下的降维","二、正则化框架下的降维",[1096,5982,5983,5984,5987,5988,5991],{},"除了 PCA 这种\"换基投影\"的降维,还可以在",[1103,5985,5986],{},"正则化框架","下做降维:通过在损失里加惩罚项,让部分参数被压成 0,从而自动做",[1103,5989,5990],{},"变量选择","(相当于在原始特征维度上降维)。",[1096,5993,5994],{},"基于正则化框架的方法尝试解决如下问题:",[1096,5996,5997],{},[1120,5998,6000,6096],{"className":5999},[1123],[1120,6001,6003],{"className":6002},[1127],[1129,6004,6005],{"xmlns":1131},[1133,6006,6007,6093],{},[1136,6008,6009,6020,6022,6028,6042,6044,6051,6053,6059,6061,6069,6071,6077,6080,6082,6084,6087,6089,6091],{},[1139,6010,6011,6017],{},[1136,6012,6013,6015],{},[1142,6014,5464],{},[1150,6016,2796],{},[1142,6018,6019],{},"β",[2800,6021,2802],{},[1767,6023,6024,6026],{},[1146,6025,1148],{},[1146,6027,1811],{},[1774,6029,6030,6032,6040],{},[1150,6031,1462],{},[1136,6033,6034,6036,6038],{},[1142,6035,1324],{},[1150,6037,1473],{},[1146,6039,1148],{},[1142,6041,1165],{},[1150,6043,1791],{"fence":1790,"stretchy":1152,"minsize":5497,"maxsize":5497},[1139,6045,6046,6049],{},[1142,6047,6048],{},"y",[1142,6050,1324],{},[1150,6052,3036],{},[1139,6054,6055,6057],{},[1142,6056,6019],{},[1146,6058,1476],{},[1150,6060,3036],{},[1774,6062,6063,6065,6067],{},[1142,6064,1144],{},[1142,6066,1324],{},[1142,6068,1651],{"mathvariant":1650},[1142,6070,6019],{},[1329,6072,6073,6075],{},[1150,6074,1808],{"fence":1790,"stretchy":1152,"minsize":5497,"maxsize":5497},[1146,6076,1811],{},[1150,6078,6079],{},"+",[1142,6081,3039],{},[2800,6083,5122],{},[1142,6085,6086],{},"P",[1150,6088,1791],{"stretchy":1790},[1142,6090,6019],{},[1150,6092,1808],{"stretchy":1790},[1167,6094,6095],{"encoding":1169},"\\min_{\\beta}\\ \\frac{1}{2}\\sum_{i=1}^{n}\\big(y_i - \\beta_0 - x_i^{\\top}\\beta\\big)^2 + \\lambda\\, P(\\beta)",[1120,6097,6099,6350,6407,6511],{"className":6098,"ariaHidden":1152},[1174],[1120,6100,6102,6106,6150,6153,6156,6224,6227,6290,6293,6299,6341,6344,6347],{"className":6101},[1178],[1120,6103],{"className":6104,"style":6105},[1182],"height:1.2em;vertical-align:-0.35em;",[1120,6107,6109,6112],{"className":6108},[1493],[1120,6110,5464],{"className":6111},[1493],[1120,6113,6115],{"className":6114},[1195],[1120,6116,6118,6142],{"className":6117},[1199,1200],[1120,6119,6121,6139],{"className":6120},[1204],[1120,6122,6124],{"className":6123,"style":4942},[1208],[1120,6125,6126,6129],{"style":2865},[1120,6127],{"className":6128,"style":1217},[1216],[1120,6130,6132],{"className":6131},[1221,1222,1223,1224],[1120,6133,6135],{"className":6134},[1187,1224],[1120,6136,6019],{"className":6137,"style":6138},[1187,1191,1224],"margin-right:0.0528em;",[1120,6140,1232],{"className":6141},[1231],[1120,6143,6145],{"className":6144},[1204],[1120,6146,6148],{"className":6147,"style":4808},[1208],[1120,6149],{},[1120,6151,2802],{"className":6152},[1249],[1120,6154],{"className":6155,"style":1250},[1249],[1120,6157,6159,6162,6221],{"className":6158},[1187],[1120,6160],{"className":6161},[1885,1886],[1120,6163,6165],{"className":6164},[1767],[1120,6166,6168,6213],{"className":6167},[1199,1200],[1120,6169,6171,6210],{"className":6170},[1204],[1120,6172,6174,6188,6196],{"className":6173,"style":1899},[1208],[1120,6175,6176,6179],{"style":1902},[1120,6177],{"className":6178,"style":1906},[1216],[1120,6180,6182],{"className":6181},[1221,1222,1223,1224],[1120,6183,6185],{"className":6184},[1187,1224],[1120,6186,1811],{"className":6187},[1187,1224],[1120,6189,6190,6193],{"style":1918},[1120,6191],{"className":6192,"style":1906},[1216],[1120,6194],{"className":6195,"style":1926},[1925],[1120,6197,6198,6201],{"style":1929},[1120,6199],{"className":6200,"style":1906},[1216],[1120,6202,6204],{"className":6203},[1221,1222,1223,1224],[1120,6205,6207],{"className":6206},[1187,1224],[1120,6208,1148],{"className":6209},[1187,1224],[1120,6211,1232],{"className":6212},[1231],[1120,6214,6216],{"className":6215},[1204],[1120,6217,6219],{"className":6218,"style":1951},[1208],[1120,6220],{},[1120,6222],{"className":6223},[1957,1886],[1120,6225],{"className":6226,"style":1250},[1249],[1120,6228,6230,6233],{"className":6229},[1493],[1120,6231,1462],{"className":6232,"style":1499},[1493,1497,1498],[1120,6234,6236],{"className":6235},[1195],[1120,6237,6239,6282],{"className":6238},[1199,1200],[1120,6240,6242,6279],{"className":6241},[1204],[1120,6243,6245,6265],{"className":6244,"style":1979},[1208],[1120,6246,6247,6250],{"style":1515},[1120,6248],{"className":6249,"style":1217},[1216],[1120,6251,6253],{"className":6252},[1221,1222,1223,1224],[1120,6254,6256,6259,6262],{"className":6255},[1187,1224],[1120,6257,1324],{"className":6258},[1187,1191,1224],[1120,6260,1473],{"className":6261},[1398,1224],[1120,6263,1148],{"className":6264},[1187,1224],[1120,6266,6267,6270],{"style":2002},[1120,6268],{"className":6269,"style":1217},[1216],[1120,6271,6273],{"className":6272},[1221,1222,1223,1224],[1120,6274,6276],{"className":6275},[1187,1224],[1120,6277,1165],{"className":6278},[1187,1191,1224],[1120,6280,1232],{"className":6281},[1231],[1120,6283,6285],{"className":6284},[1204],[1120,6286,6288],{"className":6287,"style":1534},[1208],[1120,6289],{},[1120,6291],{"className":6292,"style":1250},[1249],[1120,6294,6296],{"className":6295},[1187],[1120,6297,1791],{"className":6298},[2185,3917],[1120,6300,6302,6306],{"className":6301},[1187],[1120,6303,6048],{"className":6304,"style":6305},[1187,1191],"margin-right:0.0359em;",[1120,6307,6309],{"className":6308},[1195],[1120,6310,6312,6333],{"className":6311},[1199,1200],[1120,6313,6315,6330],{"className":6314},[1204],[1120,6316,6318],{"className":6317,"style":1368},[1208],[1120,6319,6321,6324],{"style":6320},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[1120,6322],{"className":6323,"style":1217},[1216],[1120,6325,6327],{"className":6326},[1221,1222,1223,1224],[1120,6328,1324],{"className":6329},[1187,1191,1224],[1120,6331,1232],{"className":6332},[1231],[1120,6334,6336],{"className":6335},[1204],[1120,6337,6339],{"className":6338,"style":1239},[1208],[1120,6340],{},[1120,6342],{"className":6343,"style":3128},[1249],[1120,6345,3036],{"className":6346},[3132],[1120,6348],{"className":6349,"style":3128},[1249],[1120,6351,6353,6357,6398,6401,6404],{"className":6352},[1178],[1120,6354],{"className":6355,"style":6356},[1182],"height:0.8889em;vertical-align:-0.1944em;",[1120,6358,6360,6363],{"className":6359},[1187],[1120,6361,6019],{"className":6362,"style":6138},[1187,1191],[1120,6364,6366],{"className":6365},[1195],[1120,6367,6369,6390],{"className":6368},[1199,1200],[1120,6370,6372,6387],{"className":6371},[1204],[1120,6373,6375],{"className":6374,"style":1209},[1208],[1120,6376,6378,6381],{"style":6377},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[1120,6379],{"className":6380,"style":1217},[1216],[1120,6382,6384],{"className":6383},[1221,1222,1223,1224],[1120,6385,1476],{"className":6386},[1187,1224],[1120,6388,1232],{"className":6389},[1231],[1120,6391,6393],{"className":6392},[1204],[1120,6394,6396],{"className":6395,"style":1239},[1208],[1120,6397],{},[1120,6399],{"className":6400,"style":3128},[1249],[1120,6402,3036],{"className":6403},[3132],[1120,6405],{"className":6406,"style":3128},[1249],[1120,6408,6410,6413,6467,6470,6502,6505,6508],{"className":6409},[1178],[1120,6411],{"className":6412,"style":5804},[1182],[1120,6414,6416,6419],{"className":6415},[1187],[1120,6417,1144],{"className":6418},[1187,1191],[1120,6420,6422],{"className":6421},[1195],[1120,6423,6425,6459],{"className":6424},[1199,1200],[1120,6426,6428,6456],{"className":6427},[1204],[1120,6429,6431,6442],{"className":6430,"style":1689},[1208],[1120,6432,6433,6436],{"style":2361},[1120,6434],{"className":6435,"style":1217},[1216],[1120,6437,6439],{"className":6438},[1221,1222,1223,1224],[1120,6440,1324],{"className":6441},[1187,1191,1224],[1120,6443,6444,6447],{"style":1431},[1120,6445],{"className":6446,"style":1217},[1216],[1120,6448,6450],{"className":6449},[1221,1222,1223,1224],[1120,6451,6453],{"className":6452},[1187,1224],[1120,6454,1651],{"className":6455},[1187,1224],[1120,6457,1232],{"className":6458},[1231],[1120,6460,6462],{"className":6461},[1204],[1120,6463,6465],{"className":6464,"style":2394},[1208],[1120,6466],{},[1120,6468,6019],{"className":6469,"style":6138},[1187,1191],[1120,6471,6473,6479],{"className":6472},[1187],[1120,6474,6476],{"className":6475},[1187],[1120,6477,1808],{"className":6478},[2185,3917],[1120,6480,6482],{"className":6481},[1195],[1120,6483,6485],{"className":6484},[1199],[1120,6486,6488],{"className":6487},[1204],[1120,6489,6491],{"className":6490,"style":5925},[1208],[1120,6492,6493,6496],{"style":5928},[1120,6494],{"className":6495,"style":1217},[1216],[1120,6497,6499],{"className":6498},[1221,1222,1223,1224],[1120,6500,1811],{"className":6501},[1187,1224],[1120,6503],{"className":6504,"style":3128},[1249],[1120,6506,6079],{"className":6507},[3132],[1120,6509],{"className":6510,"style":3128},[1249],[1120,6512,6514,6517,6520,6523,6527,6530,6533],{"className":6513},[1178],[1120,6515],{"className":6516,"style":3199},[1182],[1120,6518,3039],{"className":6519},[1187,1191],[1120,6521],{"className":6522,"style":1250},[1249],[1120,6524,6086],{"className":6525,"style":6526},[1187,1191],"margin-right:0.1389em;",[1120,6528,1791],{"className":6529},[1885],[1120,6531,6019],{"className":6532,"style":6138},[1187,1191],[1120,6534,1808],{"className":6535},[1957],[1096,6537,6538,6539,6542,6543,6613,6614,6643],{},"可以通过",[1103,6540,6541],{},"中心化","把常数项 ",[1120,6544,6546,6564],{"className":6545},[1123],[1120,6547,6549],{"className":6548},[1127],[1129,6550,6551],{"xmlns":1131},[1133,6552,6553,6561],{},[1136,6554,6555],{},[1139,6556,6557,6559],{},[1142,6558,6019],{},[1146,6560,1476],{},[1167,6562,6563],{"encoding":1169},"\\beta_0",[1120,6565,6567],{"className":6566,"ariaHidden":1152},[1174],[1120,6568,6570,6573],{"className":6569},[1178],[1120,6571],{"className":6572,"style":6356},[1182],[1120,6574,6576,6579],{"className":6575},[1187],[1120,6577,6019],{"className":6578,"style":6138},[1187,1191],[1120,6580,6582],{"className":6581},[1195],[1120,6583,6585,6605],{"className":6584},[1199,1200],[1120,6586,6588,6602],{"className":6587},[1204],[1120,6589,6591],{"className":6590,"style":1209},[1208],[1120,6592,6593,6596],{"style":6377},[1120,6594],{"className":6595,"style":1217},[1216],[1120,6597,6599],{"className":6598},[1221,1222,1223,1224],[1120,6600,1476],{"className":6601},[1187,1224],[1120,6603,1232],{"className":6604},[1231],[1120,6606,6608],{"className":6607},[1204],[1120,6609,6611],{"className":6610,"style":1239},[1208],[1120,6612],{}," 移除(中心化后截距为 0),进而得到只含 ",[1120,6615,6617,6631],{"className":6616},[1123],[1120,6618,6620],{"className":6619},[1127],[1129,6621,6622],{"xmlns":1131},[1133,6623,6624,6628],{},[1136,6625,6626],{},[1142,6627,6019],{},[1167,6629,6630],{"encoding":1169},"\\beta",[1120,6632,6634],{"className":6633,"ariaHidden":1152},[1174],[1120,6635,6637,6640],{"className":6636},[1178],[1120,6638],{"className":6639,"style":6356},[1182],[1120,6641,6019],{"className":6642,"style":6138},[1187,1191]," 的形式。",[1096,6645,6646,6649,6650,6720],{},[1103,6647,6648],{},"Lasso 估计","取惩罚项为 ",[1120,6651,6653,6671],{"className":6652},[1123],[1120,6654,6656],{"className":6655},[1127],[1129,6657,6658],{"xmlns":1131},[1133,6659,6660,6668],{},[1136,6661,6662],{},[1139,6663,6664,6666],{},[1142,6665,3021],{},[1146,6667,1148],{},[1167,6669,6670],{"encoding":1169},"L_1",[1120,6672,6674],{"className":6673,"ariaHidden":1152},[1174],[1120,6675,6677,6680],{"className":6676},[1178],[1120,6678],{"className":6679,"style":4345},[1182],[1120,6681,6683,6686],{"className":6682},[1187],[1120,6684,3021],{"className":6685},[1187,1191],[1120,6687,6689],{"className":6688},[1195],[1120,6690,6692,6712],{"className":6691},[1199,1200],[1120,6693,6695,6709],{"className":6694},[1204],[1120,6696,6698],{"className":6697,"style":1209},[1208],[1120,6699,6700,6703],{"style":1212},[1120,6701],{"className":6702,"style":1217},[1216],[1120,6704,6706],{"className":6705},[1221,1222,1223,1224],[1120,6707,1148],{"className":6708},[1187,1224],[1120,6710,1232],{"className":6711},[1231],[1120,6713,6715],{"className":6714},[1204],[1120,6716,6718],{"className":6717,"style":1239},[1208],[1120,6719],{}," 范数,即",[1096,6722,6723],{},[1120,6724,6726,6805],{"className":6725},[1123],[1120,6727,6729],{"className":6728},[1127],[1129,6730,6731],{"xmlns":1131},[1133,6732,6733,6802],{},[1136,6734,6735,6746,6748,6751,6753,6763,6765,6771,6773,6775,6777,6780,6782,6788,6790,6792,6794,6796],{},[1329,6736,6737,6743],{},[5077,6738,6739,6741],{"accent":1152},[1142,6740,6019],{},[1150,6742,5083],{},[2800,6744,6745],{},"Lasso",[1150,6747,1473],{},[1142,6749,6750],{},"arg",[1150,6752,2796],{},[1139,6754,6755,6761],{},[1136,6756,6757,6759],{},[1142,6758,5464],{},[1150,6760,2796],{},[1142,6762,6019],{},[2800,6764,2802],{},[1767,6766,6767,6769],{},[1146,6768,1148],{},[1146,6770,1811],{},[1142,6772,5498],{"mathvariant":1650},[1142,6774,6048],{},[1150,6776,3036],{},[1142,6778,6779],{},"X",[1142,6781,6019],{},[1329,6783,6784,6786],{},[1142,6785,5498],{"mathvariant":1650},[1146,6787,1811],{},[1150,6789,6079],{},[1142,6791,3039],{},[1142,6793,5498],{"mathvariant":1650},[1142,6795,6019],{},[1139,6797,6798,6800],{},[1142,6799,5498],{"mathvariant":1650},[1146,6801,1148],{},[1167,6803,6804],{"encoding":1169},"\\hat\\beta^{\\text{Lasso}} = \\arg\\min_{\\beta}\\ \\frac{1}{2}\\|y - X\\beta\\|^2 + \\lambda\\|\\beta\\|_1",[1120,6806,6808,6903,7053,7105],{"className":6807,"ariaHidden":1152},[1174],[1120,6809,6811,6815,6894,6897,6900],{"className":6810},[1178],[1120,6812],{"className":6813,"style":6814},[1182],"height:1.1523em;vertical-align:-0.1944em;",[1120,6816,6818,6864],{"className":6817},[1187],[1120,6819,6821],{"className":6820},[1187,5147],[1120,6822,6824,6855],{"className":6823},[1199,1200],[1120,6825,6827,6852],{"className":6826},[1204],[1120,6828,6831,6839],{"className":6829,"style":6830},[1208],"height:0.9579em;",[1120,6832,6833,6836],{"style":5159},[1120,6834],{"className":6835,"style":1906},[1216],[1120,6837,6019],{"className":6838,"style":6138},[1187,1191],[1120,6840,6842,6845],{"style":6841},"top:-3.2634em;",[1120,6843],{"className":6844,"style":1906},[1216],[1120,6846,6849],{"className":6847,"style":6848},[5174],"left:-0.1667em;",[1120,6850,5083],{"className":6851},[1187],[1120,6853,1232],{"className":6854},[1231],[1120,6856,6858],{"className":6857},[1204],[1120,6859,6862],{"className":6860,"style":6861},[1208],"height:0.1944em;",[1120,6863],{},[1120,6865,6867],{"className":6866},[1195],[1120,6868,6870],{"className":6869},[1199],[1120,6871,6873],{"className":6872},[1204],[1120,6874,6877],{"className":6875,"style":6876},[1208],"height:0.8413em;",[1120,6878,6879,6882],{"style":1431},[1120,6880],{"className":6881,"style":1217},[1216],[1120,6883,6885],{"className":6884},[1221,1222,1223,1224],[1120,6886,6888],{"className":6887},[1187,1224],[1120,6889,6891],{"className":6890},[1187,2940,1224],[1120,6892,6745],{"className":6893},[1187,1224],[1120,6895],{"className":6896,"style":1394},[1249],[1120,6898,1473],{"className":6899},[1398],[1120,6901],{"className":6902,"style":1394},[1249],[1120,6904,6906,6910,6918,6921,6964,6967,6970,7038,7041,7044,7047,7050],{"className":6905},[1178],[1120,6907],{"className":6908,"style":6909},[1182],"height:1.1901em;vertical-align:-0.345em;",[1120,6911,6913,6914],{"className":6912},[1493],"ar",[1120,6915,6917],{"style":6916},"margin-right:0.0139em;","g",[1120,6919],{"className":6920,"style":1250},[1249],[1120,6922,6924,6927],{"className":6923},[1493],[1120,6925,5464],{"className":6926},[1493],[1120,6928,6930],{"className":6929},[1195],[1120,6931,6933,6956],{"className":6932},[1199,1200],[1120,6934,6936,6953],{"className":6935},[1204],[1120,6937,6939],{"className":6938,"style":4942},[1208],[1120,6940,6941,6944],{"style":2865},[1120,6942],{"className":6943,"style":1217},[1216],[1120,6945,6947],{"className":6946},[1221,1222,1223,1224],[1120,6948,6950],{"className":6949},[1187,1224],[1120,6951,6019],{"className":6952,"style":6138},[1187,1191,1224],[1120,6954,1232],{"className":6955},[1231],[1120,6957,6959],{"className":6958},[1204],[1120,6960,6962],{"className":6961,"style":4808},[1208],[1120,6963],{},[1120,6965,2802],{"className":6966},[1249],[1120,6968],{"className":6969,"style":1250},[1249],[1120,6971,6973,6976,7035],{"className":6972},[1187],[1120,6974],{"className":6975},[1885,1886],[1120,6977,6979],{"className":6978},[1767],[1120,6980,6982,7027],{"className":6981},[1199,1200],[1120,6983,6985,7024],{"className":6984},[1204],[1120,6986,6988,7002,7010],{"className":6987,"style":1899},[1208],[1120,6989,6990,6993],{"style":1902},[1120,6991],{"className":6992,"style":1906},[1216],[1120,6994,6996],{"className":6995},[1221,1222,1223,1224],[1120,6997,6999],{"className":6998},[1187,1224],[1120,7000,1811],{"className":7001},[1187,1224],[1120,7003,7004,7007],{"style":1918},[1120,7005],{"className":7006,"style":1906},[1216],[1120,7008],{"className":7009,"style":1926},[1925],[1120,7011,7012,7015],{"style":1929},[1120,7013],{"className":7014,"style":1906},[1216],[1120,7016,7018],{"className":7017},[1221,1222,1223,1224],[1120,7019,7021],{"className":7020},[1187,1224],[1120,7022,1148],{"className":7023},[1187,1224],[1120,7025,1232],{"className":7026},[1231],[1120,7028,7030],{"className":7029},[1204],[1120,7031,7033],{"className":7032,"style":1951},[1208],[1120,7034],{},[1120,7036],{"className":7037},[1957,1886],[1120,7039,5498],{"className":7040},[1187],[1120,7042,6048],{"className":7043,"style":6305},[1187,1191],[1120,7045],{"className":7046,"style":3128},[1249],[1120,7048,3036],{"className":7049},[3132],[1120,7051],{"className":7052,"style":3128},[1249],[1120,7054,7056,7060,7064,7067,7096,7099,7102],{"className":7055},[1178],[1120,7057],{"className":7058,"style":7059},[1182],"height:1.0641em;vertical-align:-0.25em;",[1120,7061,6779],{"className":7062,"style":7063},[1187,1191],"margin-right:0.0785em;",[1120,7065,6019],{"className":7066,"style":6138},[1187,1191],[1120,7068,7070,7073],{"className":7069},[1187],[1120,7071,5498],{"className":7072},[1187],[1120,7074,7076],{"className":7075},[1195],[1120,7077,7079],{"className":7078},[1199],[1120,7080,7082],{"className":7081},[1204],[1120,7083,7085],{"className":7084,"style":2119},[1208],[1120,7086,7087,7090],{"style":1431},[1120,7088],{"className":7089,"style":1217},[1216],[1120,7091,7093],{"className":7092},[1221,1222,1223,1224],[1120,7094,1811],{"className":7095},[1187,1224],[1120,7097],{"className":7098,"style":3128},[1249],[1120,7100,6079],{"className":7101},[3132],[1120,7103],{"className":7104,"style":3128},[1249],[1120,7106,7108,7111,7114,7117,7120],{"className":7107},[1178],[1120,7109],{"className":7110,"style":3199},[1182],[1120,7112,3039],{"className":7113},[1187,1191],[1120,7115,5498],{"className":7116},[1187],[1120,7118,6019],{"className":7119,"style":6138},[1187,1191],[1120,7121,7123,7126],{"className":7122},[1187],[1120,7124,5498],{"className":7125},[1187],[1120,7127,7129],{"className":7128},[1195],[1120,7130,7132,7152],{"className":7131},[1199,1200],[1120,7133,7135,7149],{"className":7134},[1204],[1120,7136,7138],{"className":7137,"style":1209},[1208],[1120,7139,7140,7143],{"style":1212},[1120,7141],{"className":7142,"style":1217},[1216],[1120,7144,7146],{"className":7145},[1221,1222,1223,1224],[1120,7147,1148],{"className":7148},[1187,1224],[1120,7150,1232],{"className":7151},[1231],[1120,7153,7155],{"className":7154},[1204],[1120,7156,7158],{"className":7157,"style":1239},[1208],[1120,7159],{},[1096,7161,7162,7163,7343,7344,1748],{},"它等价于带约束形式 ",[1120,7164,7166,7204],{"className":7165},[1123],[1120,7167,7169],{"className":7168},[1127],[1129,7170,7171],{"xmlns":1131},[1133,7172,7173,7201],{},[1136,7174,7175,7177,7179,7185,7187,7189,7191,7193,7195],{},[1142,7176,5464],{},[1150,7178,2796],{},[1767,7180,7181,7183],{},[1146,7182,1148],{},[1146,7184,1811],{},[1142,7186,5498],{"mathvariant":1650},[1142,7188,6048],{},[1150,7190,3036],{},[1142,7192,6779],{},[1142,7194,6019],{},[1329,7196,7197,7199],{},[1142,7198,5498],{"mathvariant":1650},[1146,7200,1811],{},[1167,7202,7203],{"encoding":1169},"\\min \\frac{1}{2}\\|y-X\\beta\\|^2",[1120,7205,7207,7302],{"className":7206,"ariaHidden":1152},[1174],[1120,7208,7210,7213,7216,7219,7287,7290,7293,7296,7299],{"className":7209},[1178],[1120,7211],{"className":7212,"style":6909},[1182],[1120,7214,5464],{"className":7215},[1493],[1120,7217],{"className":7218,"style":1250},[1249],[1120,7220,7222,7225,7284],{"className":7221},[1187],[1120,7223],{"className":7224},[1885,1886],[1120,7226,7228],{"className":7227},[1767],[1120,7229,7231,7276],{"className":7230},[1199,1200],[1120,7232,7234,7273],{"className":7233},[1204],[1120,7235,7237,7251,7259],{"className":7236,"style":1899},[1208],[1120,7238,7239,7242],{"style":1902},[1120,7240],{"className":7241,"style":1906},[1216],[1120,7243,7245],{"className":7244},[1221,1222,1223,1224],[1120,7246,7248],{"className":7247},[1187,1224],[1120,7249,1811],{"className":7250},[1187,1224],[1120,7252,7253,7256],{"style":1918},[1120,7254],{"className":7255,"style":1906},[1216],[1120,7257],{"className":7258,"style":1926},[1925],[1120,7260,7261,7264],{"style":1929},[1120,7262],{"className":7263,"style":1906},[1216],[1120,7265,7267],{"className":7266},[1221,1222,1223,1224],[1120,7268,7270],{"className":7269},[1187,1224],[1120,7271,1148],{"className":7272},[1187,1224],[1120,7274,1232],{"className":7275},[1231],[1120,7277,7279],{"className":7278},[1204],[1120,7280,7282],{"className":7281,"style":1951},[1208],[1120,7283],{},[1120,7285],{"className":7286},[1957,1886],[1120,7288,5498],{"className":7289},[1187],[1120,7291,6048],{"className":7292,"style":6305},[1187,1191],[1120,7294],{"className":7295,"style":3128},[1249],[1120,7297,3036],{"className":7298},[3132],[1120,7300],{"className":7301,"style":3128},[1249],[1120,7303,7305,7308,7311,7314],{"className":7304},[1178],[1120,7306],{"className":7307,"style":7059},[1182],[1120,7309,6779],{"className":7310,"style":7063},[1187,1191],[1120,7312,6019],{"className":7313,"style":6138},[1187,1191],[1120,7315,7317,7320],{"className":7316},[1187],[1120,7318,5498],{"className":7319},[1187],[1120,7321,7323],{"className":7322},[1195],[1120,7324,7326],{"className":7325},[1199],[1120,7327,7329],{"className":7328},[1204],[1120,7330,7332],{"className":7331,"style":2119},[1208],[1120,7333,7334,7337],{"style":1431},[1120,7335],{"className":7336,"style":1217},[1216],[1120,7338,7340],{"className":7339},[1221,1222,1223,1224],[1120,7341,1811],{"className":7342},[1187,1224]," s.t. ",[1120,7345,7347,7375],{"className":7346},[1123],[1120,7348,7350],{"className":7349},[1127],[1129,7351,7352],{"xmlns":1131},[1133,7353,7354,7372],{},[1136,7355,7356,7358,7360,7366,7369],{},[1142,7357,5498],{"mathvariant":1650},[1142,7359,6019],{},[1139,7361,7362,7364],{},[1142,7363,5498],{"mathvariant":1650},[1146,7365,1148],{},[1150,7367,7368],{},"≤",[1142,7370,7371],{},"t",[1167,7373,7374],{"encoding":1169},"\\|\\beta\\|_1 \\le t",[1120,7376,7378,7439],{"className":7377,"ariaHidden":1152},[1174],[1120,7379,7381,7384,7387,7390,7430,7433,7436],{"className":7380},[1178],[1120,7382],{"className":7383,"style":3199},[1182],[1120,7385,5498],{"className":7386},[1187],[1120,7388,6019],{"className":7389,"style":6138},[1187,1191],[1120,7391,7393,7396],{"className":7392},[1187],[1120,7394,5498],{"className":7395},[1187],[1120,7397,7399],{"className":7398},[1195],[1120,7400,7402,7422],{"className":7401},[1199,1200],[1120,7403,7405,7419],{"className":7404},[1204],[1120,7406,7408],{"className":7407,"style":1209},[1208],[1120,7409,7410,7413],{"style":1212},[1120,7411],{"className":7412,"style":1217},[1216],[1120,7414,7416],{"className":7415},[1221,1222,1223,1224],[1120,7417,1148],{"className":7418},[1187,1224],[1120,7420,1232],{"className":7421},[1231],[1120,7423,7425],{"className":7424},[1204],[1120,7426,7428],{"className":7427,"style":1239},[1208],[1120,7429],{},[1120,7431],{"className":7432,"style":1394},[1249],[1120,7434,7368],{"className":7435},[1398],[1120,7437],{"className":7438,"style":1394},[1249],[1120,7440,7442,7446],{"className":7441},[1178],[1120,7443],{"className":7444,"style":7445},[1182],"height:0.6151em;",[1120,7447,7371],{"className":7448},[1187,1191],[1112,7450,7452],{"id":7451},"l1-正则化产生稀疏性的原理","L1 正则化产生稀疏性的原理",[1096,7454,7455],{},[1103,7456,7457],{},"为什么 L1 正则能让模型参数变稀疏(很多恰好为 0)?",[1096,7459,7460,7461,7545,7546,7597],{},"考虑最简单的一维情形。原始目标函数加上 L1 正则项后,目标函数变成 ",[1120,7462,7464,7496],{"className":7463},[1123],[1120,7465,7467],{"className":7466},[1127],[1129,7468,7469],{"xmlns":1131},[1133,7470,7471,7493],{},[1136,7472,7473,7476,7478,7480,7482,7484,7486,7489,7491],{},[1142,7474,7475],{},"f",[1150,7477,1791],{"stretchy":1790},[1142,7479,6019],{},[1150,7481,1808],{"stretchy":1790},[1150,7483,6079],{},[1142,7485,3039],{},[1142,7487,7488],{"mathvariant":1650},"∣",[1142,7490,6019],{},[1142,7492,7488],{"mathvariant":1650},[1167,7494,7495],{"encoding":1169},"f(\\beta) + \\lambda|\\beta|",[1120,7497,7499,7527],{"className":7498,"ariaHidden":1152},[1174],[1120,7500,7502,7505,7509,7512,7515,7518,7521,7524],{"className":7501},[1178],[1120,7503],{"className":7504,"style":3199},[1182],[1120,7506,7475],{"className":7507,"style":7508},[1187,1191],"margin-right:0.1076em;",[1120,7510,1791],{"className":7511},[1885],[1120,7513,6019],{"className":7514,"style":6138},[1187,1191],[1120,7516,1808],{"className":7517},[1957],[1120,7519],{"className":7520,"style":3128},[1249],[1120,7522,6079],{"className":7523},[3132],[1120,7525],{"className":7526,"style":3128},[1249],[1120,7528,7530,7533,7536,7539,7542],{"className":7529},[1178],[1120,7531],{"className":7532,"style":3199},[1182],[1120,7534,3039],{"className":7535},[1187,1191],[1120,7537,7488],{"className":7538},[1187],[1120,7540,6019],{"className":7541,"style":6138},[1187,1191],[1120,7543,7488],{"className":7544},[1187],"。此时最小值点常常恰好落在 ",[1120,7547,7549,7567],{"className":7548},[1123],[1120,7550,7552],{"className":7551},[1127],[1129,7553,7554],{"xmlns":1131},[1133,7555,7556,7564],{},[1136,7557,7558,7560,7562],{},[1142,7559,6019],{},[1150,7561,1473],{},[1146,7563,1476],{},[1167,7565,7566],{"encoding":1169},"\\beta = 0",[1120,7568,7570,7588],{"className":7569,"ariaHidden":1152},[1174],[1120,7571,7573,7576,7579,7582,7585],{"className":7572},[1178],[1120,7574],{"className":7575,"style":6356},[1182],[1120,7577,6019],{"className":7578,"style":6138},[1187,1191],[1120,7580],{"className":7581,"style":1394},[1249],[1120,7583,1473],{"className":7584},[1398],[1120,7586],{"className":7587,"style":1394},[1249],[1120,7589,7591,7594],{"className":7590},[1178],[1120,7592],{"className":7593,"style":1595},[1182],[1120,7595,1476],{"className":7596},[1187]," 处,于是产生了稀疏性。",[1096,7599,7600,7601,7674,7675,7714,7715,7763,7764,7796,7797,7851],{},"原因很直观:加入 L1 正则项后,对带正则项的目标函数求导,",[1103,7602,7603,7604,7639,7640],{},"正则项部分产生的导数在原点左边是 ",[1120,7605,7607,7623],{"className":7606},[1123],[1120,7608,7610],{"className":7609},[1127],[1129,7611,7612],{"xmlns":1131},[1133,7613,7614,7620],{},[1136,7615,7616,7618],{},[1150,7617,3036],{},[1142,7619,3039],{},[1167,7621,7622],{"encoding":1169},"-\\lambda",[1120,7624,7626],{"className":7625,"ariaHidden":1152},[1174],[1120,7627,7629,7633,7636],{"className":7628},[1178],[1120,7630],{"className":7631,"style":7632},[1182],"height:0.7778em;vertical-align:-0.0833em;",[1120,7634,3036],{"className":7635},[1187],[1120,7637,3039],{"className":7638},[1187,1191],"、在原点右边是 ",[1120,7641,7643,7659],{"className":7642},[1123],[1120,7644,7646],{"className":7645},[1127],[1129,7647,7648],{"xmlns":1131},[1133,7649,7650,7656],{},[1136,7651,7652,7654],{},[1150,7653,6079],{},[1142,7655,3039],{},[1167,7657,7658],{"encoding":1169},"+\\lambda",[1120,7660,7662],{"className":7661,"ariaHidden":1152},[1174],[1120,7663,7665,7668,7671],{"className":7664},[1178],[1120,7666],{"className":7667,"style":7632},[1182],[1120,7669,6079],{"className":7670},[1187],[1120,7672,3039],{"className":7673},[1187,1191],"(因为 ",[1120,7676,7678,7696],{"className":7677},[1123],[1120,7679,7681],{"className":7680},[1127],[1129,7682,7683],{"xmlns":1131},[1133,7684,7685,7693],{},[1136,7686,7687,7689,7691],{},[1142,7688,7488],{"mathvariant":1650},[1142,7690,6019],{},[1142,7692,7488],{"mathvariant":1650},[1167,7694,7695],{"encoding":1169},"|\\beta|",[1120,7697,7699],{"className":7698,"ariaHidden":1152},[1174],[1120,7700,7702,7705,7708,7711],{"className":7701},[1178],[1120,7703],{"className":7704,"style":3199},[1182],[1120,7706,7488],{"className":7707},[1187],[1120,7709,6019],{"className":7710,"style":6138},[1187,1191],[1120,7712,7488],{"className":7713},[1187]," 的导数是 ",[1120,7716,7718,7739],{"className":7717},[1123],[1120,7719,7721],{"className":7720},[1127],[1129,7722,7723],{"xmlns":1131},[1133,7724,7725,7736],{},[1136,7726,7727,7730,7732,7734],{},[2800,7728,7729],{},"sign",[1150,7731,1791],{"stretchy":1790},[1142,7733,6019],{},[1150,7735,1808],{"stretchy":1790},[1167,7737,7738],{"encoding":1169},"\\text{sign}(\\beta)",[1120,7740,7742],{"className":7741,"ariaHidden":1152},[1174],[1120,7743,7745,7748,7754,7757,7760],{"className":7744},[1178],[1120,7746],{"className":7747,"style":3199},[1182],[1120,7749,7751],{"className":7750},[1187,2940],[1120,7752,7729],{"className":7753},[1187],[1120,7755,1791],{"className":7756},[1885],[1120,7758,6019],{"className":7759,"style":6138},[1187,1191],[1120,7761,1808],{"className":7762},[1957],")。因此,只要原目标函数在原点处导数的",[1103,7765,7766,7767],{},"绝对值小于 ",[1120,7768,7770,7784],{"className":7769},[1123],[1120,7771,7773],{"className":7772},[1127],[1129,7774,7775],{"xmlns":1131},[1133,7776,7777,7781],{},[1136,7778,7779],{},[1142,7780,3039],{},[1167,7782,7783],{"encoding":1169},"\\lambda",[1120,7785,7787],{"className":7786,"ariaHidden":1152},[1174],[1120,7788,7790,7793],{"className":7789},[1178],[1120,7791],{"className":7792,"style":3429},[1182],[1120,7794,3039],{"className":7795},[1187,1191],",那么带正则项的目标函数在原点左边始终递减、在原点右边始终递增——最小值点自然就卡在",[1103,7798,7799,7800],{},"原点 ",[1120,7801,7803,7821],{"className":7802},[1123],[1120,7804,7806],{"className":7805},[1127],[1129,7807,7808],{"xmlns":1131},[1133,7809,7810,7818],{},[1136,7811,7812,7814,7816],{},[1142,7813,6019],{},[1150,7815,1473],{},[1146,7817,1476],{},[1167,7819,7820],{"encoding":1169},"\\beta=0",[1120,7822,7824,7842],{"className":7823,"ariaHidden":1152},[1174],[1120,7825,7827,7830,7833,7836,7839],{"className":7826},[1178],[1120,7828],{"className":7829,"style":6356},[1182],[1120,7831,6019],{"className":7832,"style":6138},[1187,1191],[1120,7834],{"className":7835,"style":1394},[1249],[1120,7837,1473],{"className":7838},[1398],[1120,7840],{"className":7841,"style":1394},[1249],[1120,7843,7845,7848],{"className":7844},[1178],[1120,7846],{"className":7847,"style":1595},[1182],[1120,7849,1476],{"className":7850},[1187]," 处,该参数被直接压到 0。",[1096,7853,7854,7855,7920,7921,7924],{},"对比之下,L2 正则项 ",[1120,7856,7858,7878],{"className":7857},[1123],[1120,7859,7861],{"className":7860},[1127],[1129,7862,7863],{"xmlns":1131},[1133,7864,7865,7875],{},[1136,7866,7867,7869],{},[1142,7868,3039],{},[1329,7870,7871,7873],{},[1142,7872,6019],{},[1146,7874,1811],{},[1167,7876,7877],{"encoding":1169},"\\lambda\\beta^2",[1120,7879,7881],{"className":7880,"ariaHidden":1152},[1174],[1120,7882,7884,7888,7891],{"className":7883},[1178],[1120,7885],{"className":7886,"style":7887},[1182],"height:1.0085em;vertical-align:-0.1944em;",[1120,7889,3039],{"className":7890},[1187,1191],[1120,7892,7894,7897],{"className":7893},[1187],[1120,7895,6019],{"className":7896,"style":6138},[1187,1191],[1120,7898,7900],{"className":7899},[1195],[1120,7901,7903],{"className":7902},[1199],[1120,7904,7906],{"className":7905},[1204],[1120,7907,7909],{"className":7908,"style":2119},[1208],[1120,7910,7911,7914],{"style":1431},[1120,7912],{"className":7913,"style":1217},[1216],[1120,7915,7917],{"className":7916},[1221,1222,1223,1224],[1120,7918,1811],{"className":7919},[1187,1224]," 在原点的导数是 0(平滑),不会制造这种\"尖角\",所以 L2 只会把参数压小、不会压到恰好为 0。这就是 ",[1103,7922,7923],{},"L1 稀疏、L2 不稀疏","的根本原因。",[5975,7926],{},[1091,7928,7930],{"id":7929},"三总结","三、总结",[7932,7933,7934,7941,8157,8163,8172,8178],"ul",{},[7935,7936,7937,7940],"li",{},[1103,7938,7939],{},"PCA 两视角等价",":最大方差投影 ⟺ 最小重构误差,选出的都是协方差矩阵 Σ 的前 d 大特征值对应的特征向量。",[7935,7942,7943,7946,7947,7343,8029,8090,8091,8152,8153,8156],{},[1103,7944,7945],{},"最大方差推导",":",[1120,7948,7950,7976],{"className":7949},[1123],[1120,7951,7953],{"className":7952},[1127],[1129,7954,7955],{"xmlns":1131},[1133,7956,7957,7973],{},[1136,7958,7959,7961,7963,7969,7971],{},[1142,7960,2793],{},[1150,7962,2796],{},[1329,7964,7965,7967],{},[1142,7966,1614],{},[1142,7968,1651],{"mathvariant":1650},[1142,7970,1863],{"mathvariant":1650},[1142,7972,1614],{},[1167,7974,7975],{"encoding":1169},"\\max w^{\\top}\\Sigma w",[1120,7977,7979],{"className":7978,"ariaHidden":1152},[1174],[1120,7980,7982,7985,7988,7991,8023,8026],{"className":7981},[1178],[1120,7983],{"className":7984,"style":1689},[1182],[1120,7986,2793],{"className":7987},[1493],[1120,7989],{"className":7990,"style":1250},[1249],[1120,7992,7994,7997],{"className":7993},[1187],[1120,7995,1614],{"className":7996,"style":1630},[1187,1191],[1120,7998,8000],{"className":7999},[1195],[1120,8001,8003],{"className":8002},[1199],[1120,8004,8006],{"className":8005},[1204],[1120,8007,8009],{"className":8008,"style":1689},[1208],[1120,8010,8011,8014],{"style":1431},[1120,8012],{"className":8013,"style":1217},[1216],[1120,8015,8017],{"className":8016},[1221,1222,1223,1224],[1120,8018,8020],{"className":8019},[1187,1224],[1120,8021,1651],{"className":8022},[1187,1224],[1120,8024,1863],{"className":8025},[1187],[1120,8027,1614],{"className":8028,"style":1630},[1187,1191],[1120,8030,8032,8054],{"className":8031},[1123],[1120,8033,8035],{"className":8034},[1127],[1129,8036,8037],{"xmlns":1131},[1133,8038,8039,8051],{},[1136,8040,8041,8043,8045,8047,8049],{},[1142,8042,5498],{"mathvariant":1650},[1142,8044,1614],{},[1142,8046,5498],{"mathvariant":1650},[1150,8048,1473],{},[1146,8050,1148],{},[1167,8052,8053],{"encoding":1169},"\\|w\\|=1",[1120,8055,8057,8081],{"className":8056,"ariaHidden":1152},[1174],[1120,8058,8060,8063,8066,8069,8072,8075,8078],{"className":8059},[1178],[1120,8061],{"className":8062,"style":3199},[1182],[1120,8064,5498],{"className":8065},[1187],[1120,8067,1614],{"className":8068,"style":1630},[1187,1191],[1120,8070,5498],{"className":8071},[1187],[1120,8073],{"className":8074,"style":1394},[1249],[1120,8076,1473],{"className":8077},[1398],[1120,8079],{"className":8080,"style":1394},[1249],[1120,8082,8084,8087],{"className":8083},[1178],[1120,8085],{"className":8086,"style":1595},[1182],[1120,8088,1148],{"className":8089},[1187]," → 拉格朗日求导得 ",[1120,8092,8094,8116],{"className":8093},[1123],[1120,8095,8097],{"className":8096},[1127],[1129,8098,8099],{"xmlns":1131},[1133,8100,8101,8113],{},[1136,8102,8103,8105,8107,8109,8111],{},[1142,8104,1863],{"mathvariant":1650},[1142,8106,1614],{},[1150,8108,1473],{},[1142,8110,3039],{},[1142,8112,1614],{},[1167,8114,8115],{"encoding":1169},"\\Sigma w=\\lambda w",[1120,8117,8119,8140],{"className":8118,"ariaHidden":1152},[1174],[1120,8120,8122,8125,8128,8131,8134,8137],{"className":8121},[1178],[1120,8123],{"className":8124,"style":2516},[1182],[1120,8126,1863],{"className":8127},[1187],[1120,8129,1614],{"className":8130,"style":1630},[1187,1191],[1120,8132],{"className":8133,"style":1394},[1249],[1120,8135,1473],{"className":8136},[1398],[1120,8138],{"className":8139,"style":1394},[1249],[1120,8141,8143,8146,8149],{"className":8142},[1178],[1120,8144],{"className":8145,"style":3429},[1182],[1120,8147,3039],{"className":8148},[1187,1191],[1120,8150,1614],{"className":8151,"style":1630},[1187,1191]," → 最优方向是",[1103,8154,8155],{},"最大特征值的特征向量",",方差就等于该特征值。",[7935,8158,8159,8162],{},[1103,8160,8161],{},"主成分排序",":第 k 个主成分 = 第 k 大特征值对应的特征向量,且与前面的主成分正交。",[7935,8164,8165,8168,8169,1748],{},[1103,8166,8167],{},"前提",":数据要先",[1103,8170,8171],{},"中心化(标准化)",[7935,8173,8174,8177],{},[1103,8175,8176],{},"正则化降维",":在损失上加惩罚做变量选择;Lasso = 平方损失 + λ‖β‖₁。",[7935,8179,8180,8183],{},[1103,8181,8182],{},"L1 稀疏原理",":|β| 的导数在原点左 −λ、右 +λ;只要原目标在原点导数绝对值 \u003C λ,最小值就卡在 β=0,产生稀疏。L2 平滑无尖角,只压小不压零。",{"title":8185,"searchDepth":8186,"depth":8186,"links":8187},"",4,[8188,8194,8197],{"id":1093,"depth":8189,"text":1094,"children":8190},2,[8191,8193],{"id":1114,"depth":8192,"text":1115},3,{"id":4651,"depth":8192,"text":4652},{"id":5979,"depth":8189,"text":5980,"children":8195},[8196],{"id":7451,"depth":8192,"text":7452},{"id":7929,"depth":8189,"text":7930},"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-139df0ed493532834526fe22fa33c9cc_1440w.jpg","PCA 两个等价视角推导、Lasso 正则化与 L1 稀疏性原理","md",true,{"uuid":8203,"slots":8204},"f3561dd0-e86a-11f0-8fd1-914b1152d8ed",{},5,{"title":1085,"description":8199},"posts\u002F分布式处理与计算\u002F分布式处理与计算：数据降维",[323,339,340,341],"dVajdpRAxFqIEuD5bWVXNRV2jq7mATeuS9A1bAt1m5Y",1790443282961]