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21:40:26",{"text":11,"minutes":144,"time":145,"words":146},2.385,143100,477,{"stem":148,"path":149,"title":150,"date":78,"collapsed":9,"readingTime":151},"posts\u002FUGCourses\u002F2.sophomore-autumn\u002Findex","\u002Fugcourses\u002Fsophomore-autumn","大二秋冬",{"text":80,"minutes":81,"time":81,"words":81},{"stem":153,"path":154,"title":155,"date":156,"collapsed":9,"readingTime":157},"posts\u002FUGCourses\u002F2.sophomore-autumn\u002Fsolid-state-physics","\u002F2024\u002F09\u002F11\u002Fsolid-state-physics","固体物理基础","2024-09-11 19:15:26",{"text":158,"minutes":159,"time":160,"words":161},"4 min read",3.21,192600,642,{"stem":163,"path":164,"title":165,"date":166,"collapsed":9,"readingTime":167},"posts\u002FUGCourses\u002F3.sophomore-spring\u002FAIFrontier","\u002Fugcourses\u002Fsophomore-spring\u002Faifrontier","人工智能前沿","2025-06-12 14:58:16",{"text":168,"minutes":169,"time":170,"words":171},"25 min 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19:39:00",{"text":60,"minutes":200,"time":201,"words":202},0.535,32100,107,{"stem":204,"path":205,"title":206,"date":207,"collapsed":9,"readingTime":208},"posts\u002FUGCourses\u002F4.junior-autumn\u002F1.microelectronic-devices\u002FLecture10","\u002Fugcourses\u002Fjunior-autumn\u002Fmicroelectronic-devices\u002Flecture10","Lecture 10：从 MOS 电容到 MOSFET","2026-09-10 00:10:00",{"text":209,"minutes":210,"time":211,"words":212},"18 min read",17.615,1056900,3523,{"stem":214,"path":215,"title":216,"date":217,"collapsed":9,"readingTime":218},"posts\u002FUGCourses\u002F4.junior-autumn\u002F1.microelectronic-devices\u002FLecture11","\u002Fugcourses\u002Fjunior-autumn\u002Fmicroelectronic-devices\u002Flecture11","Lecture 11：亚阈值效应与 MOSFET 尺寸缩放","2026-09-10 00:20:00",{"text":219,"minutes":220,"time":221,"words":222},"23 min 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00:50:00",{"text":21,"minutes":51,"time":52,"words":53},{"stem":340,"path":341,"title":342,"date":198,"collapsed":343,"readingTime":344},"posts\u002FUGCourses\u002F4.junior-autumn\u002F1.microelectronic-devices\u002Findex","\u002Fugcourses\u002Fjunior-autumn\u002Fmicroelectronic-devices","微电子器件",true,{"text":158,"minutes":345,"time":346,"words":347},3.055,183300,611,{"stem":349,"path":350,"title":351,"date":352,"collapsed":9,"readingTime":353},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F1.HistoryMetaCharacters","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fhistorymetacharacters","History & Meta Characters","2026-09-10 20:25:44",{"text":269,"minutes":354,"time":355,"words":356},9.36,561600,1872,{"stem":358,"path":359,"title":360,"date":361,"collapsed":9,"readingTime":362},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F10.PastExams\u002F1.ExamAutumn2223","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fpastexams\u002Fexamautumn2223","22-23 秋冬历年卷","2026-09-10 20:26:49",{"text":319,"minutes":363,"time":364,"words":365},20.295,1217700,4059,{"stem":367,"path":368,"title":369,"date":370,"collapsed":9,"readingTime":371},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F10.PastExams\u002F2.Exam2021","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fpastexams\u002Fexam2021","20-21 历年卷","2026-09-10 20:26:50",{"text":289,"minutes":290,"time":291,"words":292},{"stem":373,"path":374,"title":375,"date":376,"collapsed":9,"readingTime":377},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F10.PastExams\u002F3.ExamRecall1","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fpastexams\u002Fexamrecall1","不知名回忆卷 1","2026-09-10 20:26:51",{"text":116,"minutes":378,"time":379,"words":380},4.215,252900,843,{"stem":382,"path":383,"title":384,"date":385,"collapsed":9,"readingTime":386},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F10.PastExams\u002F4.ExamRecall23","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fpastexams\u002Fexamrecall23","不知名回忆卷 2\u002F3","2026-09-10 20:26:52",{"text":387,"minutes":388,"time":389,"words":390},"12 min read",11.235,674100,2247,{"stem":392,"path":393,"title":394,"date":395,"collapsed":9,"readingTime":396},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F11.Detailed\u002F1.Chapter1-Introduction","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fdetailed\u002Fchapter1-introduction","Chapter 1：Introduction","2026-09-10 20:27:26",{"text":397,"minutes":398,"time":399,"words":400},"19 min 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20:27:54",{"text":209,"minutes":454,"time":455,"words":456},17.85,1071000,3570,{"stem":458,"path":459,"title":460,"date":461,"collapsed":9,"readingTime":462},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F11.Detailed\u002F8.Chapter8-CProgrammingInUnix","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fdetailed\u002Fchapter8-cprogramminginunix","Chapter 8：C Programming in Unix","2026-09-10 20:27:57",{"text":463,"minutes":464,"time":465,"words":466},"17 min read",16.855,1011300,3371,{"stem":468,"path":469,"title":470,"date":471,"collapsed":9,"readingTime":472},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F2.LinuxUnix","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Flinuxunix","Linux & Unix","2026-09-10 20:25:46",{"text":41,"minutes":473,"time":474,"words":475},29.33,1759800,5866,{"stem":477,"path":478,"title":479,"date":480,"collapsed":9,"readingTime":481},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F3.Vi","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fvi","Vi","2026-09-10 20:25:47",{"text":329,"minutes":482,"time":483,"words":484},7.95,477000,1590,{"stem":486,"path":487,"title":488,"date":489,"collapsed":9,"readingTime":490},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F4.ViCopy","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fvicopy","Vi（详细版）","2026-09-10 20:25:49",{"text":279,"minutes":491,"time":492,"words":493},13.785,827100,2757,{"stem":495,"path":496,"title":497,"date":498,"collapsed":9,"readingTime":499},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F5.Shells","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fshells","Shells","2026-09-10 20:25:51",{"text":500,"minutes":501,"time":502,"words":503},"22 min read",21.21,1272600,4242,{"stem":505,"path":506,"title":507,"date":508,"collapsed":9,"readingTime":509},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F6.Network","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fnetwork","Network","2026-09-10 20:25:54",{"text":319,"minutes":510,"time":511,"words":512},20.595,1235700,4119,{"stem":514,"path":515,"title":516,"date":517,"collapsed":9,"readingTime":518},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F7.SystemAdministration","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fsystemadministration","System Administration","2026-09-10 20:26:44",{"text":329,"minutes":519,"time":520,"words":521},7.12,427200,1424,{"stem":523,"path":524,"title":525,"date":526,"collapsed":9,"readingTime":527},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F8.XWindow","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fxwindow","XWindow","2026-09-10 20:26:45",{"text":97,"minutes":528,"time":529,"words":530},5.105,306300,1021,{"stem":532,"path":533,"title":534,"date":535,"collapsed":9,"readingTime":536},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002F9.SystemDevelopmentTools","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad\u002Fsystemdevelopmenttools","System Development Tools","2026-09-10 20:26:47",{"text":387,"minutes":537,"time":538,"words":539},11.35,681000,2270,{"stem":541,"path":542,"title":543,"date":544,"collapsed":343,"readingTime":545},"posts\u002FUGCourses\u002F4.junior-autumn\u002F2.ICCAD\u002Findex","\u002Fugcourses\u002Fjunior-autumn\u002Ficcad","面向 ICCAD 的软件基础技术","2026-09-10 12:37:00",{"text":60,"minutes":546,"time":547,"words":548},0.185,11100,37,{"stem":550,"path":551,"title":552,"date":553,"collapsed":9,"readingTime":554},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002FFeiQilai1","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl\u002Ffeiqilai1","飞起来1.0","2026-09-10 12:51:49",{"text":555,"minutes":556,"time":557,"words":558},"105 min read",104.655,6279300,20931,{"stem":560,"path":561,"title":562,"date":563,"collapsed":9,"readingTime":564},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002FFeiQilai2","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl\u002Ffeiqilai2","飞起来2.0","2026-09-10 12:52:06",{"text":565,"minutes":566,"time":567,"words":568},"29 min read",28.755,1725300,5751,{"stem":570,"path":571,"title":572,"date":573,"collapsed":9,"readingTime":574},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002FFeiQilai3","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl\u002Ffeiqilai3","飞起来3.0","2026-09-10 12:52:12",{"text":116,"minutes":575,"time":576,"words":577},4.49,269400,898,{"stem":579,"path":580,"title":581,"date":582,"collapsed":9,"readingTime":583},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002FLecture13","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl\u002Flecture13","Lecture 13：浮点数标准","2026-09-10 12:51:27",{"text":11,"minutes":584,"time":585,"words":586},2.575,154500,515,{"stem":588,"path":589,"title":590,"date":591,"collapsed":9,"readingTime":592},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002FLecture6","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl\u002Flecture6","Lecture 6：Behavioral Modeling，非阻塞赋值","2026-09-10 12:51:12",{"text":97,"minutes":593,"time":594,"words":595},5.34,320400,1068,{"stem":597,"path":598,"title":599,"date":600,"collapsed":9,"readingTime":601},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002FPPTCorrespondence","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl\u002Fpptcorrespondence","与PPT的对应","2026-09-10 12:52:18",{"text":269,"minutes":602,"time":603,"words":604},9.275,556500,1855,{"stem":606,"path":607,"title":608,"date":544,"collapsed":343,"readingTime":609},"posts\u002FUGCourses\u002F4.junior-autumn\u002F3.HDL\u002Findex","\u002Fugcourses\u002Fjunior-autumn\u002Fhdl","硬件描述语言原理与应用",{"text":60,"minutes":61,"time":62,"words":63},{"stem":611,"path":612,"title":613,"date":614,"collapsed":9,"readingTime":615},"posts\u002FUGCourses\u002F4.junior-autumn\u002F4.analogICDesign\u002F1.BeforeMidterm","\u002Fugcourses\u002Fjunior-autumn\u002Fanalogicdesign\u002Fbeforemidterm","期中前","2026-09-10 20:28:52",{"text":616,"minutes":617,"time":618,"words":619},"11 min read",10.835,650100,2167,{"stem":621,"path":622,"title":623,"date":624,"collapsed":9,"readingTime":625},"posts\u002FUGCourses\u002F4.junior-autumn\u002F4.analogICDesign\u002F2.MidtermExam2425","\u002Fugcourses\u002Fjunior-autumn\u002Fanalogicdesign\u002Fmidtermexam2425","24-25 期中","2026-09-10 20:29:00",{"text":21,"minutes":626,"time":627,"words":628},1.22,73200,244,{"stem":630,"path":631,"title":632,"date":633,"collapsed":9,"readingTime":634},"posts\u002FUGCourses\u002F4.junior-autumn\u002F4.analogICDesign\u002F3.FinalExamReview","\u002Fugcourses\u002Fjunior-autumn\u002Fanalogicdesign\u002Ffinalexamreview","期末考试整理","2026-09-10 20:29:09",{"text":387,"minutes":635,"time":636,"words":637},11.36,681600,2272,{"stem":639,"path":640,"title":641,"date":642,"collapsed":343,"readingTime":643},"posts\u002FUGCourses\u002F4.junior-autumn\u002F4.analogICDesign\u002Findex","\u002Fugcourses\u002Fjunior-autumn\u002Fanalogicdesign","模拟集成电路设计","2026-09-10 20:00:00",{"text":80,"minutes":81,"time":81,"words":81},{"stem":645,"path":646,"title":647,"date":78,"collapsed":9,"readingTime":648},"posts\u002FUGCourses\u002F4.junior-autumn\u002Findex","\u002Fugcourses\u002Fjunior-autumn","大三秋冬",{"text":80,"minutes":81,"time":81,"words":81},{"stem":650,"path":651,"title":652,"date":653,"collapsed":9,"readingTime":654},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F1.Chapter3-Device","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter3-device","Chapter 3：器件 Device","2026-09-10 21:14:41",{"text":249,"minutes":655,"time":656,"words":657},6.055,363300,1211,{"stem":659,"path":660,"title":661,"date":653,"collapsed":9,"readingTime":662},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F10.Chapter12-Arithmetic","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter12-arithmetic","Chapter 12：算术运算电路 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期中试题",{"text":80,"minutes":81,"time":81,"words":81},{"stem":693,"path":694,"title":695,"date":653,"collapsed":9,"readingTime":696},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F13.PastExams\u002F3.FinalPolished","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fpastexams\u002Ffinalpolished","期末历年卷（AI 润色版）",{"text":21,"minutes":697,"time":698,"words":699},1.575,94500,315,{"stem":701,"path":702,"title":703,"date":653,"collapsed":9,"readingTime":704},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F14.ExamRecall2026","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fexamrecall2026","数字集成电路回忆 2026",{"text":60,"minutes":705,"time":706,"words":707},0.49,29400,98,{"stem":709,"path":710,"title":711,"date":653,"collapsed":9,"readingTime":712},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F2.Chapter4-Wire","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter4-wire","Chapter 4：导线 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Gate",{"text":31,"minutes":729,"time":730,"words":731},8.635,518100,1727,{"stem":733,"path":734,"title":735,"date":653,"collapsed":9,"readingTime":736},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F5.Chapter7-SequentialLogic","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter7-sequentiallogic","Chapter 7：时序逻辑 Sequential Logic",{"text":463,"minutes":737,"time":738,"words":739},16.645,998700,3329,{"stem":741,"path":742,"title":743,"date":653,"collapsed":9,"readingTime":744},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F6.Chapter8-Speed","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter8-speed","Chapter 8：延迟与速度 Speed",{"text":745,"minutes":746,"time":747,"words":748},"33 min read",32.885,1973100,6577,{"stem":750,"path":751,"title":752,"date":653,"collapsed":9,"readingTime":753},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F7.Chapter9-Power","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter9-power","Chapter 9：功耗 Power",{"text":754,"minutes":755,"time":756,"words":757},"31 min read",31.005,1860300,6201,{"stem":759,"path":760,"title":761,"date":653,"collapsed":9,"readingTime":762},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F8.Chapter10-Interconnect","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter10-interconnect","Chapter 10：互连 Interconnect",{"text":463,"minutes":763,"time":764,"words":765},16.32,979200,3264,{"stem":767,"path":768,"title":769,"date":653,"collapsed":9,"readingTime":770},"posts\u002FUGCourses\u002F5.junior-spring\u002F1.digitalICDesign\u002F9.Chapter11-Timing","\u002Fugcourses\u002Fjunior-spring\u002Fdigitalicdesign\u002Fchapter11-timing","Chapter 11：时序 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read",43.985,2639100,8797,{"stem":824,"path":825,"title":826,"date":653,"collapsed":9,"readingTime":827},"posts\u002FUGCourses\u002F5.junior-spring\u002F2.computerOrganization\u002F14.L15","\u002Fugcourses\u002Fjunior-spring\u002Fcomputerorganization\u002Fl15","L15：期末复习问答",{"text":269,"minutes":828,"time":829,"words":830},9.105,546300,1821,{"stem":832,"path":833,"title":834,"date":653,"collapsed":9,"readingTime":835},"posts\u002FUGCourses\u002F5.junior-spring\u002F2.computerOrganization\u002F2.L2-Binary","\u002Fugcourses\u002Fjunior-spring\u002Fcomputerorganization\u002Fl2-binary","L2：二进制数制",{"text":80,"minutes":81,"time":81,"words":81},{"stem":837,"path":838,"title":839,"date":653,"collapsed":9,"readingTime":840},"posts\u002FUGCourses\u002F5.junior-spring\u002F2.computerOrganization\u002F3.L3-CMemory","\u002Fugcourses\u002Fjunior-spring\u002Fcomputerorganization\u002Fl3-cmemory","L3：C 语言与内存模型",{"text":841,"minutes":842,"time":843,"words":844},"98 min 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5,1436,1438],{},[1071,1437,1102],{},[1071,1439,1093],{},[1128,1441,1442],{"encoding":1130},"y_i",[1052,1444,1446],{"className":1445,"ariaHidden":1096},[1135],[1052,1447,1449,1453],{"className":1448},[1139],[1052,1450],{"className":1451,"style":1452},[1143],"height:0.625em;vertical-align:-0.1944em;",[1052,1454,1456,1459],{"className":1455},[1148],[1052,1457,1102],{"className":1458,"style":1245},[1148,1149],[1052,1460,1462],{"className":1461},[1184],[1052,1463,1465,1485],{"className":1464},[1188,1189],[1052,1466,1468,1482],{"className":1467},[1193],[1052,1469,1471],{"className":1470,"style":1198},[1197],[1052,1472,1473,1476],{"style":1260},[1052,1474],{"className":1475,"style":1206},[1205],[1052,1477,1479],{"className":1478},[1210,1211,1212,1213],[1052,1480,1093],{"className":1481},[1148,1149,1213],[1052,1483,1221],{"className":1484},[1220],[1052,1486,1488],{"className":1487},[1193],[1052,1489,1491],{"className":1490,"style":1228},[1197],[1052,1492],{},"的映射，其中D被称为训练集，n是训练样例的数量。",[996,1495,1496,1497,1500],{},"监督学习算法从假设空间（hypothesisspace）学习得到一个",[1020,1498,1499],{},"最优映射函数f（又称决策函数）","，映射函数f将输入数据映射到语义标注空间，实现数据的分类和识别。",[1045,1502,1504],{"id":1503},"有监督学习训练集验证集测试集",[1020,1505,1506],{},"有监督学习：训练集、验证集、测试集",[1015,1508,1509,1516,1522],{},[983,1510,1511,1512,1515],{},"在",[1020,1513,1514],{},"训练集","上完成模型参数优化",[983,1517,1518,1519],{},"将训练集中⼀部分数据作为",[1020,1520,1521],{},"验证集（validation set）",[983,1523,1524,1525,1528],{},"最后在",[1020,1526,1527],{},"测试集","上进⾏测试，将测试结果作为模型性能最终结果",[996,1530,1531],{},[1532,1533,1534],"mark",{},"要注意的是，训练集、验证集和测试集所包含数据之间没有任何交叉",[1042,1536,1037],{"id":1037},[996,1538,1539,1540,1691],{},"无监督学习则是直接从无标签数据",[1052,1541,1543,1588],{"className":1542},[1055],[1052,1544,1546],{"className":1545},[1059],[1061,1547,1548],{"xmlns":1063},[1065,1549,1550,1585],{},[1068,1551,1552,1554,1560,1563,1565,1567,1569,1571,1573,1577,1579,1581,1583],{},[1075,1553,1081],{"stretchy":1080},[1086,1555,1556,1558],{},[1071,1557,1090],{},[1071,1559,1093],{},[1075,1561,1562],{"separator":1096},";",[1075,1564,1097],{"separator":1096},[1071,1566,1093],{},[1075,1568,1077],{},[1121,1570,1123],{},[1075,1572,1097],{"separator":1096},[1071,1574,1576],{"mathvariant":1575},"normal",".",[1071,1578,1576],{"mathvariant":1575},[1075,1580,1097],{"separator":1096},[1071,1582,1126],{},[1075,1584,1113],{"stretchy":1080},[1128,1586,1587],{"encoding":1130},"\\{x_i;,i=1,..,n\\}",[1052,1589,1591,1660],{"className":1590,"ariaHidden":1096},[1135],[1052,1592,1594,1598,1601,1641,1645,1648,1651,1654,1657],{"className":1593},[1139],[1052,1595],{"className":1596,"style":1597},[1143],"height:1em;vertical-align:-0.25em;",[1052,1599,1081],{"className":1600},[1173],[1052,1602,1604,1607],{"className":1603},[1148],[1052,1605,1090],{"className":1606},[1148,1149],[1052,1608,1610],{"className":1609},[1184],[1052,1611,1613,1633],{"className":1612},[1188,1189],[1052,1614,1616,1630],{"className":1615},[1193],[1052,1617,1619],{"className":1618,"style":1198},[1197],[1052,1620,1621,1624],{"style":1201},[1052,1622],{"className":1623,"style":1206},[1205],[1052,1625,1627],{"className":1626},[1210,1211,1212,1213],[1052,1628,1093],{"className":1629},[1148,1149,1213],[1052,1631,1221],{"className":1632},[1220],[1052,1634,1636],{"className":1635},[1193],[1052,1637,1639],{"className":1638,"style":1228},[1197],[1052,1640],{},[1052,1642,1644],{"className":1643},[1234],";,",[1052,1646],{"className":1647,"style":1238},[1154],[1052,1649,1093],{"className":1650},[1148,1149],[1052,1652],{"className":1653,"style":1155},[1154],[1052,1655,1077],{"className":1656},[1159],[1052,1658],{"className":1659,"style":1155},[1154],[1052,1661,1663,1666,1669,1672,1675,1679,1682,1685,1688],{"className":1662},[1139],[1052,1664],{"className":1665,"style":1597},[1143],[1052,1667,1123],{"className":1668},[1148],[1052,1670,1097],{"className":1671},[1234],[1052,1673],{"className":1674,"style":1238},[1154],[1052,1676,1678],{"className":1677},[1148],"..",[1052,1680,1097],{"className":1681},[1234],[1052,1683],{"className":1684,"style":1238},[1154],[1052,1686,1126],{"className":1687},[1148,1149],[1052,1689,1113],{"className":1690},[1284],"出发学习映射函数，而半监督学习在学习映射函数过程中使用的一部分数据有标签、一部分数据没有标签。",[1010,1693,1695,1696],{"id":1694},"二模型评估与参数估计手段损失函数","二、",[1020,1697,1698],{},"模型评估与参数估计手段：损失函数",[1042,1700,1702],{"id":1701},"泛化能力generalization",[1020,1703,1704],{},"泛化能力（generalization）",[996,1706,1707],{},"在机器学习中，需要保证模型在训练集上所取得性能与在测试集上所取得性能保持一致，即模型具有泛化能力（generalization）。",[996,1709,1710],{},[1711,1712],"img",{"alt":1711,"src":1713},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-01.png",[996,1715,1716],{},[1711,1717],{"alt":1711,"src":1718},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-02.png",[1042,1720,1722],{"id":1721},"经验风险与期望风险",[1020,1723,1721],{},[1045,1725,1727],{"id":1726},"经验风险",[1020,1728,1726],{},[996,1730,1731,1732,1763,1764,1766,1767,1858],{},"映射函数",[1052,1733,1735,1749],{"className":1734},[1055],[1052,1736,1738],{"className":1737},[1059],[1061,1739,1740],{"xmlns":1063},[1065,1741,1742,1747],{},[1068,1743,1744],{},[1071,1745,1746],{},"f",[1128,1748,1746],{"encoding":1130},[1052,1750,1752],{"className":1751,"ariaHidden":1096},[1135],[1052,1753,1755,1759],{"className":1754},[1139],[1052,1756],{"className":1757,"style":1758},[1143],"height:0.8889em;vertical-align:-0.1944em;",[1052,1760,1746],{"className":1761,"style":1762},[1148,1149],"margin-right:0.1076em;","在训练集上所产⽣损失⼀般被称为",[1020,1765,1726],{}," ",[1052,1768,1770,1797],{"className":1769},[1055],[1052,1771,1773],{"className":1772},[1059],[1061,1774,1775],{"xmlns":1063},[1065,1776,1777,1794],{},[1068,1778,1779],{},[1086,1780,1781,1784],{},[1071,1782,1783],{"mathvariant":1575},"ℜ",[1068,1785,1786,1789,1792],{},[1071,1787,1788],{},"e",[1071,1790,1791],{},"m",[1071,1793,996],{},[1128,1795,1796],{"encoding":1130},"ℜ_{emp}",[1052,1798,1800],{"className":1799,"ariaHidden":1096},[1135],[1052,1801,1803,1807],{"className":1802},[1139],[1052,1804],{"className":1805,"style":1806},[1143],"height:0.9805em;vertical-align:-0.2861em;",[1052,1808,1810,1813],{"className":1809},[1148],[1052,1811,1783],{"className":1812},[1148],[1052,1814,1816],{"className":1815},[1184],[1052,1817,1819,1849],{"className":1818},[1188,1189],[1052,1820,1822,1846],{"className":1821},[1193],[1052,1823,1826],{"className":1824,"style":1825},[1197],"height:0.1514em;",[1052,1827,1828,1831],{"style":1201},[1052,1829],{"className":1830,"style":1206},[1205],[1052,1832,1834],{"className":1833},[1210,1211,1212,1213],[1052,1835,1837,1840,1843],{"className":1836},[1148,1213],[1052,1838,1788],{"className":1839},[1148,1149,1213],[1052,1841,1791],{"className":1842},[1148,1149,1213],[1052,1844,996],{"className":1845},[1148,1149,1213],[1052,1847,1221],{"className":1848},[1220],[1052,1850,1852],{"className":1851},[1193],[1052,1853,1856],{"className":1854,"style":1855},[1197],"height:0.2861em;",[1052,1857],{},"(empirical risk)。。",[996,1860,1861],{},"经验风险越小说明模型对训练集数据拟合程度越好。经验风险被定义为：",[1052,1863,1866],{"className":1864},[1865],"katex-display",[1052,1867,1869,1940],{"className":1868},[1055],[1052,1870,1872],{"className":1871},[1059],[1061,1873,1875],{"xmlns":1063,"display":1874},"block",[1065,1876,1877,1937],{},[1068,1878,1879,1886,1902,1905,1908,1911,1913,1915,1921,1923,1925,1927,1933,1935],{},[1880,1881,1882,1884],"mfrac",{},[1121,1883,1123],{},[1071,1885,1126],{},[1887,1888,1889,1892,1900],"munderover",{},[1075,1890,1891],{},"∑",[1068,1893,1894,1896,1898],{},[1071,1895,1093],{},[1075,1897,1077],{},[1121,1899,1123],{},[1071,1901,1126],{},[1071,1903,1904],{},"L",[1071,1906,1907],{},"o",[1071,1909,1910],{},"s",[1071,1912,1910],{},[1075,1914,1084],{"stretchy":1080},[1086,1916,1917,1919],{},[1071,1918,1102],{},[1071,1920,1093],{},[1075,1922,1097],{"separator":1096},[1071,1924,1746],{},[1075,1926,1084],{"stretchy":1080},[1086,1928,1929,1931],{},[1071,1930,1090],{},[1071,1932,1093],{},[1075,1934,1107],{"stretchy":1080},[1075,1936,1107],{"stretchy":1080},[1128,1938,1939],{"encoding":1130},"\\frac{1}{n}\\sum_{i = 1}^n Loss(y_i,f(x_i))",[1052,1941,1943],{"className":1942,"ariaHidden":1096},[1135],[1052,1944,1946,1950,2021,2024,2098,2101,2104,2108,2111,2151,2154,2157,2160,2163,2203],{"className":1945},[1139],[1052,1947],{"className":1948,"style":1949},[1143],"height:2.9291em;vertical-align:-1.2777em;",[1052,1951,1953,1957,2018],{"className":1952},[1148],[1052,1954],{"className":1955},[1173,1956],"nulldelimiter",[1052,1958,1960],{"className":1959},[1880],[1052,1961,1963,2009],{"className":1962},[1188,1189],[1052,1964,1966,2006],{"className":1965},[1193],[1052,1967,1970,1983,1994],{"className":1968,"style":1969},[1197],"height:1.3214em;",[1052,1971,1973,1977],{"style":1972},"top:-2.314em;",[1052,1974],{"className":1975,"style":1976},[1205],"height:3em;",[1052,1978,1980],{"className":1979},[1148],[1052,1981,1126],{"className":1982},[1148,1149],[1052,1984,1986,1989],{"style":1985},"top:-3.23em;",[1052,1987],{"className":1988,"style":1976},[1205],[1052,1990],{"className":1991,"style":1993},[1992],"frac-line","border-bottom-width:0.04em;",[1052,1995,1997,2000],{"style":1996},"top:-3.677em;",[1052,1998],{"className":1999,"style":1976},[1205],[1052,2001,2003],{"className":2002},[1148],[1052,2004,1123],{"className":2005},[1148],[1052,2007,1221],{"className":2008},[1220],[1052,2010,2012],{"className":2011},[1193],[1052,2013,2016],{"className":2014,"style":2015},[1197],"height:0.686em;",[1052,2017],{},[1052,2019],{"className":2020},[1284,1956],[1052,2022],{"className":2023,"style":1238},[1154],[1052,2025,2029],{"className":2026},[2027,2028],"mop","op-limits",[1052,2030,2032,2089],{"className":2031},[1188,1189],[1052,2033,2035,2086],{"className":2034},[1193],[1052,2036,2039,2061,2074],{"className":2037,"style":2038},[1197],"height:1.6514em;",[1052,2040,2042,2046],{"style":2041},"top:-1.8723em;margin-left:0em;",[1052,2043],{"className":2044,"style":2045},[1205],"height:3.05em;",[1052,2047,2049],{"className":2048},[1210,1211,1212,1213],[1052,2050,2052,2055,2058],{"className":2051},[1148,1213],[1052,2053,1093],{"className":2054},[1148,1149,1213],[1052,2056,1077],{"className":2057},[1159,1213],[1052,2059,1123],{"className":2060},[1148,1213],[1052,2062,2064,2067],{"style":2063},"top:-3.05em;",[1052,2065],{"className":2066,"style":2045},[1205],[1052,2068,2069],{},[1052,2070,1891],{"className":2071},[2027,2072,2073],"op-symbol","large-op",[1052,2075,2077,2080],{"style":2076},"top:-4.3em;margin-left:0em;",[1052,2078],{"className":2079,"style":2045},[1205],[1052,2081,2083],{"className":2082},[1210,1211,1212,1213],[1052,2084,1126],{"className":2085},[1148,1149,1213],[1052,2087,1221],{"className":2088},[1220],[1052,2090,2092],{"className":2091},[1193],[1052,2093,2096],{"className":2094,"style":2095},[1197],"height:1.2777em;",[1052,2097],{},[1052,2099],{"className":2100,"style":1238},[1154],[1052,2102,1904],{"className":2103},[1148,1149],[1052,2105,2107],{"className":2106},[1148,1149],"oss",[1052,2109,1084],{"className":2110},[1173],[1052,2112,2114,2117],{"className":2113},[1148],[1052,2115,1102],{"className":2116,"style":1245},[1148,1149],[1052,2118,2120],{"className":2119},[1184],[1052,2121,2123,2143],{"className":2122},[1188,1189],[1052,2124,2126,2140],{"className":2125},[1193],[1052,2127,2129],{"className":2128,"style":1198},[1197],[1052,2130,2131,2134],{"style":1260},[1052,2132],{"className":2133,"style":1206},[1205],[1052,2135,2137],{"className":2136},[1210,1211,1212,1213],[1052,2138,1093],{"className":2139},[1148,1149,1213],[1052,2141,1221],{"className":2142},[1220],[1052,2144,2146],{"className":2145},[1193],[1052,2147,2149],{"className":2148,"style":1228},[1197],[1052,2150],{},[1052,2152,1097],{"className":2153},[1234],[1052,2155],{"className":2156,"style":1238},[1154],[1052,2158,1746],{"className":2159,"style":1762},[1148,1149],[1052,2161,1084],{"className":2162},[1173],[1052,2164,2166,2169],{"className":2165},[1148],[1052,2167,1090],{"className":2168},[1148,1149],[1052,2170,2172],{"className":2171},[1184],[1052,2173,2175,2195],{"className":2174},[1188,1189],[1052,2176,2178,2192],{"className":2177},[1193],[1052,2179,2181],{"className":2180,"style":1198},[1197],[1052,2182,2183,2186],{"style":1201},[1052,2184],{"className":2185,"style":1206},[1205],[1052,2187,2189],{"className":2188},[1210,1211,1212,1213],[1052,2190,1093],{"className":2191},[1148,1149,1213],[1052,2193,1221],{"className":2194},[1220],[1052,2196,2198],{"className":2197},[1193],[1052,2199,2201],{"className":2200,"style":1228},[1197],[1052,2202],{},[1052,2204,2206],{"className":2205},[1284],"))",[1045,2208,2209],{"id":2209},"期望风险",[996,2211,2212,2213,2242,2243,2302],{},"如果知道某⼀任务包含的所有数据，则可以从所有数据中计算模型产⽣的损失，这⼀误差损失被称为期望风险",[1052,2214,2216,2229],{"className":2215},[1055],[1052,2217,2219],{"className":2218},[1059],[1061,2220,2221],{"xmlns":1063},[1065,2222,2223,2227],{},[1068,2224,2225],{},[1071,2226,1783],{"mathvariant":1575},[1128,2228,1783],{"encoding":1130},[1052,2230,2232],{"className":2231,"ariaHidden":1096},[1135],[1052,2233,2235,2239],{"className":2234},[1139],[1052,2236],{"className":2237,"style":2238},[1143],"height:0.6944em;",[1052,2240,1783],{"className":2241},[1148],"（expected risk）即真实风险或真实误差。记该任务中所有数据的联合分布为",[1052,2244,2246,2271],{"className":2245},[1055],[1052,2247,2249],{"className":2248},[1059],[1061,2250,2251],{"xmlns":1063},[1065,2252,2253,2268],{},[1068,2254,2255,2258,2260,2262,2264,2266],{},[1071,2256,2257],{},"P",[1075,2259,1084],{"stretchy":1080},[1071,2261,1090],{},[1075,2263,1097],{"separator":1096},[1071,2265,1102],{},[1075,2267,1107],{"stretchy":1080},[1128,2269,2270],{"encoding":1130},"P(x,y)",[1052,2272,2274],{"className":2273,"ariaHidden":1096},[1135],[1052,2275,2277,2280,2284,2287,2290,2293,2296,2299],{"className":2276},[1139],[1052,2278],{"className":2279,"style":1597},[1143],[1052,2281,2257],{"className":2282,"style":2283},[1148,1149],"margin-right:0.1389em;",[1052,2285,1084],{"className":2286},[1173],[1052,2288,1090],{"className":2289},[1148,1149],[1052,2291,1097],{"className":2292},[1234],[1052,2294],{"className":2295,"style":1238},[1154],[1052,2297,1102],{"className":2298,"style":1245},[1148,1149],[1052,2300,1107],{"className":2301},[1284],",期望风险被定义为",[1052,2304,2306],{"className":2305},[1865],[1052,2307,2309,2380],{"className":2308},[1055],[1052,2310,2312],{"className":2311},[1059],[1061,2313,2314],{"xmlns":1063,"display":1874},[1065,2315,2316,2377],{},[1068,2317,2318,2332,2334,2336,2338,2340,2342,2344,2346,2348,2350,2352,2354,2356,2358,2360,2362,2364,2366,2368,2371,2373,2375],{},[1086,2319,2320,2323],{},[1075,2321,2322],{},"∫",[1068,2324,2325,2327,2330],{},[1071,2326,1090],{},[1075,2328,2329],{},"×",[1071,2331,1102],{},[1071,2333,1904],{},[1071,2335,1907],{},[1071,2337,1910],{},[1071,2339,1910],{},[1075,2341,1084],{"stretchy":1080},[1071,2343,1102],{},[1075,2345,1097],{"separator":1096},[1071,2347,1746],{},[1075,2349,1084],{"stretchy":1080},[1071,2351,1090],{},[1075,2353,1107],{"stretchy":1080},[1075,2355,1107],{"stretchy":1080},[1071,2357,2257],{},[1075,2359,1084],{"stretchy":1080},[1071,2361,1090],{},[1075,2363,1097],{"separator":1096},[1071,2365,1102],{},[1075,2367,1107],{"stretchy":1080},[1071,2369,2370],{},"d",[1071,2372,1090],{},[1071,2374,2370],{},[1071,2376,1102],{},[1128,2378,2379],{"encoding":1130},"\\int_{x \\times y} Loss(y,f(x))P(x,y)dxdy",[1052,2381,2383],{"className":2382,"ariaHidden":1096},[1135],[1052,2384,2386,2390,2444,2447,2450,2453,2456,2459,2462,2465,2468,2471,2474,2477,2480,2483,2486,2489,2492,2495,2498,2501,2504,2507],{"className":2385},[1139],[1052,2387],{"className":2388,"style":2389},[1143],"height:2.4081em;vertical-align:-1.0481em;",[1052,2391,2393,2397],{"className":2392},[2027],[1052,2394,2322],{"className":2395,"style":2396},[2027,2072,2073],"margin-right:0.4445em;position:relative;top:-0.0011em;",[1052,2398,2400],{"className":2399},[1184],[1052,2401,2403,2435],{"className":2402},[1188,1189],[1052,2404,2406,2432],{"className":2405},[1193],[1052,2407,2410],{"className":2408,"style":2409},[1197],"height:-0.5036em;",[1052,2411,2413,2416],{"style":2412},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[1052,2414],{"className":2415,"style":1206},[1205],[1052,2417,2419],{"className":2418},[1210,1211,1212,1213],[1052,2420,2422,2425,2429],{"className":2421},[1148,1213],[1052,2423,1090],{"className":2424},[1148,1149,1213],[1052,2426,2329],{"className":2427},[2428,1213],"mbin",[1052,2430,1102],{"className":2431,"style":1245},[1148,1149,1213],[1052,2433,1221],{"className":2434},[1220],[1052,2436,2438],{"className":2437},[1193],[1052,2439,2442],{"className":2440,"style":2441},[1197],"height:1.0481em;",[1052,2443],{},[1052,2445],{"className":2446,"style":1238},[1154],[1052,2448,1904],{"className":2449},[1148,1149],[1052,2451,2107],{"className":2452},[1148,1149],[1052,2454,1084],{"className":2455},[1173],[1052,2457,1102],{"className":2458,"style":1245},[1148,1149],[1052,2460,1097],{"className":2461},[1234],[1052,2463],{"className":2464,"style":1238},[1154],[1052,2466,1746],{"className":2467,"style":1762},[1148,1149],[1052,2469,1084],{"className":2470},[1173],[1052,2472,1090],{"className":2473},[1148,1149],[1052,2475,2206],{"className":2476},[1284],[1052,2478,2257],{"className":2479,"style":2283},[1148,1149],[1052,2481,1084],{"className":2482},[1173],[1052,2484,1090],{"className":2485},[1148,1149],[1052,2487,1097],{"className":2488},[1234],[1052,2490],{"className":2491,"style":1238},[1154],[1052,2493,1102],{"className":2494,"style":1245},[1148,1149],[1052,2496,1107],{"className":2497},[1284],[1052,2499,2370],{"className":2500},[1148,1149],[1052,2502,1090],{"className":2503},[1148,1149],[1052,2505,2370],{"className":2506},[1148,1149],[1052,2508,1102],{"className":2509,"style":1245},[1148,1149],[993,2511,2512,2517],{},[996,2513,2514,2516],{},[1020,2515,1726],{}," ≈ 训练误差",[996,2518,2519,2521],{},[1020,2520,2209],{}," ≈ 真实的泛化误差",[1042,2523,2524],{"id":2524},"经验风险最小化",[996,2526,2527,2528,2531,2532,2536],{},"当然，由于无法事先就得到任何任务所对应的所有数据分布（如无法采取世界中所有人脸图像来写信完成人脸识别），使得",[1020,2529,2530],{},"计算期望风险这一目标可望不可及","。因此，机器学习中模型优化目标一般为",[1532,2533,2534],{},[1020,2535,2524],{},"（empirical risk minimization），虽然机器学习的目标是追求期望风险最小化，即不断提升模型泛化能力。",[1042,2538,2539],{"id":2539},"期望风险与经验风险的关系",[996,2541,2542,2543,2576,2577,2663],{},"期望风险 ",[1052,2544,2546,2562],{"className":2545},[1055],[1052,2547,2549],{"className":2548},[1059],[1061,2550,2551],{"xmlns":1063},[1065,2552,2553,2559],{},[1068,2554,2555],{},[1071,2556,2558],{"mathvariant":2557},"fraktur","R",[1128,2560,2561],{"encoding":1130},"\\mathfrak{R}",[1052,2563,2565],{"className":2564,"ariaHidden":1096},[1135],[1052,2566,2568,2572],{"className":2567},[1139],[1052,2569],{"className":2570,"style":2571},[1143],"height:0.6914em;",[1052,2573,2558],{"className":2574},[1148,2575],"mathfrak"," 与经验风险 ",[1052,2578,2580,2604],{"className":2579},[1055],[1052,2581,2583],{"className":2582},[1059],[1061,2584,2585],{"xmlns":1063},[1065,2586,2587,2601],{},[1068,2588,2589],{},[1086,2590,2591,2593],{},[1071,2592,2558],{"mathvariant":2557},[1068,2594,2595,2597,2599],{},[1071,2596,1788],{},[1071,2598,1791],{},[1071,2600,996],{},[1128,2602,2603],{"encoding":1130},"\\mathfrak{R}_{emp}",[1052,2605,2607],{"className":2606,"ariaHidden":1096},[1135],[1052,2608,2610,2614],{"className":2609},[1139],[1052,2611],{"className":2612,"style":2613},[1143],"height:0.9775em;vertical-align:-0.2861em;",[1052,2615,2617,2620],{"className":2616},[1148],[1052,2618,2558],{"className":2619},[1148,2575],[1052,2621,2623],{"className":2622},[1184],[1052,2624,2626,2655],{"className":2625},[1188,1189],[1052,2627,2629,2652],{"className":2628},[1193],[1052,2630,2632],{"className":2631,"style":1825},[1197],[1052,2633,2634,2637],{"style":1201},[1052,2635],{"className":2636,"style":1206},[1205],[1052,2638,2640],{"className":2639},[1210,1211,1212,1213],[1052,2641,2643,2646,2649],{"className":2642},[1148,1213],[1052,2644,1788],{"className":2645},[1148,1149,1213],[1052,2647,1791],{"className":2648},[1148,1149,1213],[1052,2650,996],{"className":2651},[1148,1149,1213],[1052,2653,1221],{"className":2654},[1220],[1052,2656,2658],{"className":2657},[1193],[1052,2659,2661],{"className":2660,"style":1855},[1197],[1052,2662],{}," 之间存在如下关系：",[996,2665,2666],{},[1711,2667],{"alt":1711,"src":2668},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-03.png",[996,2670,2671,2710],{},[1052,2672,2674,2693],{"className":2673},[1055],[1052,2675,2677],{"className":2676},[1059],[1061,2678,2679],{"xmlns":1063},[1065,2680,2681,2690],{},[1068,2682,2683,2685,2688],{},[1071,2684,1788],{},[1071,2686,2687],{},"r",[1071,2689,2687],{},[1128,2691,2692],{"encoding":1130},"err",[1052,2694,2696],{"className":2695,"ariaHidden":1096},[1135],[1052,2697,2699,2703,2707],{"className":2698},[1139],[1052,2700],{"className":2701,"style":2702},[1143],"height:0.4306em;",[1052,2704,2706],{"className":2705,"style":1150},[1148,1149],"er",[1052,2708,2687],{"className":2709,"style":1150},[1148,1149],[1020,2711,2712],{},"为误差项",[996,2714,2715,2716,2751,2752,2755,2756,2759,2760,2763,2764,2848,2849,2884,2885,2913,2914,2919],{},"其中 ",[1052,2717,2719,2736],{"className":2718},[1055],[1052,2720,2722],{"className":2721},[1059],[1061,2723,2724],{"xmlns":1063},[1065,2725,2726,2734],{},[1068,2727,2728,2730,2732],{},[1071,2729,1788],{},[1071,2731,2687],{},[1071,2733,2687],{},[1128,2735,2692],{"encoding":1130},[1052,2737,2739],{"className":2738,"ariaHidden":1096},[1135],[1052,2740,2742,2745,2748],{"className":2741},[1139],[1052,2743],{"className":2744,"style":2702},[1143],[1052,2746,2706],{"className":2747,"style":1150},[1148,1149],[1052,2749,2687],{"className":2750,"style":1150},[1148,1149]," 取值与机器学习",[1020,2753,2754],{},"模型的复杂程度","和",[1020,2757,2758],{},"训练集样本数目","有关。在模型训练过程中，",[1532,2761,2762],{},"如果使用同一批训练数据反复训练，模型会变得越复杂","，虽然经验风险 ",[1052,2765,2767,2790],{"className":2766},[1055],[1052,2768,2770],{"className":2769},[1059],[1061,2771,2772],{"xmlns":1063},[1065,2773,2774,2788],{},[1068,2775,2776],{},[1086,2777,2778,2780],{},[1071,2779,2558],{"mathvariant":2557},[1068,2781,2782,2784,2786],{},[1071,2783,1788],{},[1071,2785,1791],{},[1071,2787,996],{},[1128,2789,2603],{"encoding":1130},[1052,2791,2793],{"className":2792,"ariaHidden":1096},[1135],[1052,2794,2796,2799],{"className":2795},[1139],[1052,2797],{"className":2798,"style":2613},[1143],[1052,2800,2802,2805],{"className":2801},[1148],[1052,2803,2558],{"className":2804},[1148,2575],[1052,2806,2808],{"className":2807},[1184],[1052,2809,2811,2840],{"className":2810},[1188,1189],[1052,2812,2814,2837],{"className":2813},[1193],[1052,2815,2817],{"className":2816,"style":1825},[1197],[1052,2818,2819,2822],{"style":1201},[1052,2820],{"className":2821,"style":1206},[1205],[1052,2823,2825],{"className":2824},[1210,1211,1212,1213],[1052,2826,2828,2831,2834],{"className":2827},[1148,1213],[1052,2829,1788],{"className":2830},[1148,1149,1213],[1052,2832,1791],{"className":2833},[1148,1149,1213],[1052,2835,996],{"className":2836},[1148,1149,1213],[1052,2838,1221],{"className":2839},[1220],[1052,2841,2843],{"className":2842},[1193],[1052,2844,2846],{"className":2845,"style":1855},[1197],[1052,2847],{}," 会降低，但是 ",[1052,2850,2852,2869],{"className":2851},[1055],[1052,2853,2855],{"className":2854},[1059],[1061,2856,2857],{"xmlns":1063},[1065,2858,2859,2867],{},[1068,2860,2861,2863,2865],{},[1071,2862,1788],{},[1071,2864,2687],{},[1071,2866,2687],{},[1128,2868,2692],{"encoding":1130},[1052,2870,2872],{"className":2871,"ariaHidden":1096},[1135],[1052,2873,2875,2878,2881],{"className":2874},[1139],[1052,2876],{"className":2877,"style":2702},[1143],[1052,2879,2706],{"className":2880,"style":1150},[1148,1149],[1052,2882,2687],{"className":2883,"style":1150},[1148,1149]," 取值会越大，导致期望风险 ",[1052,2886,2888,2901],{"className":2887},[1055],[1052,2889,2891],{"className":2890},[1059],[1061,2892,2893],{"xmlns":1063},[1065,2894,2895,2899],{},[1068,2896,2897],{},[1071,2898,2558],{"mathvariant":2557},[1128,2900,2561],{"encoding":1130},[1052,2902,2904],{"className":2903,"ariaHidden":1096},[1135],[1052,2905,2907,2910],{"className":2906},[1139],[1052,2908],{"className":2909,"style":2571},[1143],[1052,2911,2558],{"className":2912},[1148,2575]," 增加。这一现象被称为",[1532,2915,2916],{},[1020,2917,2918],{},"过学习（过拟合）（overfitting）","。",[996,2921,2922],{},"让我为您详细解释模型泛化能力与经验风险、期望风险之间的关系：",[1042,2924,2925],{"id":2925},"四种典型情况分析",[996,2927,2928],{},"根据表格中的四种情况，我们可以看到：",[2930,2931,2932,2952],"table",{},[2933,2934,2935],"thead",{},[2936,2937,2938,2942,2944,2946,2949],"tr",{},[2939,2940,2941],"th",{},"情况类型",[2939,2943,1726],{},[2939,2945,2209],{},[2939,2947,2948],{},"模型泛化能力",[2939,2950,2951],{},"特点\u002F说明",[2953,2954,2955,2973,2989,3005],"tbody",{},[2936,2956,2957,2961,2964,2967,2970],{},[2958,2959,2960],"td",{},"理想状态",[2958,2962,2963],{},"小（训练集上表现好）",[2958,2965,2966],{},"小（所有数据上表现好）",[2958,2968,2969],{},"泛化能力强",[2958,2971,2972],{},"这是我们最希望达到的状态",[2936,2974,2975,2978,2980,2983,2986],{},[2958,2976,2977],{},"过拟合",[2958,2979,2963],{},[2958,2981,2982],{},"大（所有数据上表现不好）",[2958,2984,2985],{},"过学习（模型过于复杂）",[2958,2987,2988],{},"典型的过拟合现象",[2936,2990,2991,2994,2997,2999,3002],{},[2958,2992,2993],{},"欠拟合",[2958,2995,2996],{},"大（训练集上表现不好）",[2958,2998,2982],{},[2958,3000,3001],{},"欠学习",[2958,3003,3004],{},"模型过于简单，无法捕捉数据规律",[2936,3006,3007,3010,3012,3014,3017],{},[2958,3008,3009],{},"异常情况",[2958,3011,2996],{},[2958,3013,2966],{},[2958,3015,3016],{},"\"神仙操作\"或\"贵族套餐\"",[2958,3018,3019],{},"这种情况在实际中很少见，可能表示训练数据有问题",[1042,3021,3022],{"id":3022},"结构风险最小化",[996,3024,3025,3026,3028],{},"为了解决过拟合问题，我们引入",[1020,3027,3022],{},"：",[1045,3030,3031],{"id":3031},"核心思想",[996,3033,3034,3035,3038],{},"在模型优化中引入恰当先验约束可提升模型性能。通过",[1020,3036,3037],{},"正则化","来降低模型复杂度，在最小化经验风险的同时，寻求降低模型复杂度的平衡。",[1045,3040,3041],{"id":3041},"数学表达",[1052,3043,3045],{"className":3044},[1865],[1052,3046,3048,3127],{"className":3047},[1055],[1052,3049,3051],{"className":3050},[1059],[1061,3052,3053],{"xmlns":1063,"display":1874},[1065,3054,3055,3124],{},[1068,3056,3057,3063,3077,3079,3081,3083,3085,3087,3093,3095,3097,3099,3105,3107,3109,3112,3115,3118,3120,3122],{},[1880,3058,3059,3061],{},[1121,3060,1123],{},[1071,3062,1126],{},[1887,3064,3065,3067,3075],{},[1075,3066,1891],{},[1068,3068,3069,3071,3073],{},[1071,3070,1093],{},[1075,3072,1077],{},[1121,3074,1123],{},[1071,3076,1126],{},[1071,3078,1904],{},[1071,3080,1907],{},[1071,3082,1910],{},[1071,3084,1910],{},[1075,3086,1084],{"stretchy":1080},[1086,3088,3089,3091],{},[1071,3090,1102],{},[1071,3092,1093],{},[1075,3094,1097],{"separator":1096},[1071,3096,1746],{},[1075,3098,1084],{"stretchy":1080},[1086,3100,3101,3103],{},[1071,3102,1090],{},[1071,3104,1093],{},[1075,3106,1107],{"stretchy":1080},[1075,3108,1107],{"stretchy":1080},[1075,3110,3111],{},"+",[1071,3113,3114],{},"λ",[1071,3116,3117],{},"J",[1075,3119,1084],{"stretchy":1080},[1071,3121,1746],{},[1075,3123,1107],{"stretchy":1080},[1128,3125,3126],{"encoding":1130},"\\frac{1}{n}\\sum_{i=1}^{n} Loss(y_i, f(x_i)) + \\lambda J(f)",[1052,3128,3130,3385],{"className":3129,"ariaHidden":1096},[1135],[1052,3131,3133,3136,3198,3201,3268,3271,3274,3277,3280,3320,3323,3326,3329,3332,3372,3375,3379,3382],{"className":3132},[1139],[1052,3134],{"className":3135,"style":1949},[1143],[1052,3137,3139,3142,3195],{"className":3138},[1148],[1052,3140],{"className":3141},[1173,1956],[1052,3143,3145],{"className":3144},[1880],[1052,3146,3148,3187],{"className":3147},[1188,1189],[1052,3149,3151,3184],{"className":3150},[1193],[1052,3152,3154,3165,3173],{"className":3153,"style":1969},[1197],[1052,3155,3156,3159],{"style":1972},[1052,3157],{"className":3158,"style":1976},[1205],[1052,3160,3162],{"className":3161},[1148],[1052,3163,1126],{"className":3164},[1148,1149],[1052,3166,3167,3170],{"style":1985},[1052,3168],{"className":3169,"style":1976},[1205],[1052,3171],{"className":3172,"style":1993},[1992],[1052,3174,3175,3178],{"style":1996},[1052,3176],{"className":3177,"style":1976},[1205],[1052,3179,3181],{"className":3180},[1148],[1052,3182,1123],{"className":3183},[1148],[1052,3185,1221],{"className":3186},[1220],[1052,3188,3190],{"className":3189},[1193],[1052,3191,3193],{"className":3192,"style":2015},[1197],[1052,3194],{},[1052,3196],{"className":3197},[1284,1956],[1052,3199],{"className":3200,"style":1238},[1154],[1052,3202,3204],{"className":3203},[2027,2028],[1052,3205,3207,3260],{"className":3206},[1188,1189],[1052,3208,3210,3257],{"className":3209},[1193],[1052,3211,3213,3233,3243],{"className":3212,"style":2038},[1197],[1052,3214,3215,3218],{"style":2041},[1052,3216],{"className":3217,"style":2045},[1205],[1052,3219,3221],{"className":3220},[1210,1211,1212,1213],[1052,3222,3224,3227,3230],{"className":3223},[1148,1213],[1052,3225,1093],{"className":3226},[1148,1149,1213],[1052,3228,1077],{"className":3229},[1159,1213],[1052,3231,1123],{"className":3232},[1148,1213],[1052,3234,3235,3238],{"style":2063},[1052,3236],{"className":3237,"style":2045},[1205],[1052,3239,3240],{},[1052,3241,1891],{"className":3242},[2027,2072,2073],[1052,3244,3245,3248],{"style":2076},[1052,3246],{"className":3247,"style":2045},[1205],[1052,3249,3251],{"className":3250},[1210,1211,1212,1213],[1052,3252,3254],{"className":3253},[1148,1213],[1052,3255,1126],{"className":3256},[1148,1149,1213],[1052,3258,1221],{"className":3259},[1220],[1052,3261,3263],{"className":3262},[1193],[1052,3264,3266],{"className":3265,"style":2095},[1197],[1052,3267],{},[1052,3269],{"className":3270,"style":1238},[1154],[1052,3272,1904],{"className":3273},[1148,1149],[1052,3275,2107],{"className":3276},[1148,1149],[1052,3278,1084],{"className":3279},[1173],[1052,3281,3283,3286],{"className":3282},[1148],[1052,3284,1102],{"className":3285,"style":1245},[1148,1149],[1052,3287,3289],{"className":3288},[1184],[1052,3290,3292,3312],{"className":3291},[1188,1189],[1052,3293,3295,3309],{"className":3294},[1193],[1052,3296,3298],{"className":3297,"style":1198},[1197],[1052,3299,3300,3303],{"style":1260},[1052,3301],{"className":3302,"style":1206},[1205],[1052,3304,3306],{"className":3305},[1210,1211,1212,1213],[1052,3307,1093],{"className":3308},[1148,1149,1213],[1052,3310,1221],{"className":3311},[1220],[1052,3313,3315],{"className":3314},[1193],[1052,3316,3318],{"className":3317,"style":1228},[1197],[1052,3319],{},[1052,3321,1097],{"className":3322},[1234],[1052,3324],{"className":3325,"style":1238},[1154],[1052,3327,1746],{"className":3328,"style":1762},[1148,1149],[1052,3330,1084],{"className":3331},[1173],[1052,3333,3335,3338],{"className":3334},[1148],[1052,3336,1090],{"className":3337},[1148,1149],[1052,3339,3341],{"className":3340},[1184],[1052,3342,3344,3364],{"className":3343},[1188,1189],[1052,3345,3347,3361],{"className":3346},[1193],[1052,3348,3350],{"className":3349,"style":1198},[1197],[1052,3351,3352,3355],{"style":1201},[1052,3353],{"className":3354,"style":1206},[1205],[1052,3356,3358],{"className":3357},[1210,1211,1212,1213],[1052,3359,1093],{"className":3360},[1148,1149,1213],[1052,3362,1221],{"className":3363},[1220],[1052,3365,3367],{"className":3366},[1193],[1052,3368,3370],{"className":3369,"style":1228},[1197],[1052,3371],{},[1052,3373,2206],{"className":3374},[1284],[1052,3376],{"className":3377,"style":3378},[1154],"margin-right:0.2222em;",[1052,3380,3111],{"className":3381},[2428],[1052,3383],{"className":3384,"style":3378},[1154],[1052,3386,3388,3391,3394,3398,3401,3404],{"className":3387},[1139],[1052,3389],{"className":3390,"style":1597},[1143],[1052,3392,3114],{"className":3393},[1148,1149],[1052,3395,3117],{"className":3396,"style":3397},[1148,1149],"margin-right:0.0962em;",[1052,3399,1084],{"className":3400},[1173],[1052,3402,1746],{"className":3403,"style":1762},[1148,1149],[1052,3405,1107],{"className":3406},[1284],[996,3408,3409],{},"其中：",[978,3411,3412,3415,3462],{},[983,3413,3414],{},"第一项：经验风险（拟合训练数据）",[983,3416,3417,3461],{},[1052,3418,3420,3440],{"className":3419},[1055],[1052,3421,3423],{"className":3422},[1059],[1061,3424,3425],{"xmlns":1063},[1065,3426,3427,3437],{},[1068,3428,3429,3431,3433,3435],{},[1071,3430,3117],{},[1075,3432,1084],{"stretchy":1080},[1071,3434,1746],{},[1075,3436,1107],{"stretchy":1080},[1128,3438,3439],{"encoding":1130},"J(f)",[1052,3441,3443],{"className":3442,"ariaHidden":1096},[1135],[1052,3444,3446,3449,3452,3455,3458],{"className":3445},[1139],[1052,3447],{"className":3448,"style":1597},[1143],[1052,3450,3117],{"className":3451,"style":3397},[1148,1149],[1052,3453,1084],{"className":3454},[1173],[1052,3456,1746],{"className":3457,"style":1762},[1148,1149],[1052,3459,1107],{"className":3460},[1284],"：正则化因子或惩罚项因子（控制模型复杂度）",[983,3463,3464,3493],{},[1052,3465,3467,3481],{"className":3466},[1055],[1052,3468,3470],{"className":3469},[1059],[1061,3471,3472],{"xmlns":1063},[1065,3473,3474,3478],{},[1068,3475,3476],{},[1071,3477,3114],{},[1128,3479,3480],{"encoding":1130},"\\lambda",[1052,3482,3484],{"className":3483,"ariaHidden":1096},[1135],[1052,3485,3487,3490],{"className":3486},[1139],[1052,3488],{"className":3489,"style":2238},[1143],[1052,3491,3114],{"className":3492},[1148,1149],"：调整惩罚强度的系数",[1045,3495,3496],{"id":3496},"简单有效原理",[996,3498,3499,3502],{},[1020,3500,3501],{},"\"如无必要，勿增实体\""," - 即\"简单有效原理\"",[996,3504,3505],{},"这个哲学原理在机器学习中的体现就是：",[978,3507,3508,3511,3514],{},[983,3509,3510],{},"在能够解释数据的前提下，选择最简单的模型",[983,3512,3513],{},"避免不必要的复杂性",[983,3515,3516],{},"通过约束（如约束模型参数稀疏等）使建模过程\"能够化繁为简、大巧不工\"",[1045,3518,3520],{"id":3519},"实际应用意义平衡拟合与泛化","实际应用意义——平衡拟合与泛化",[996,3522,3523],{},"结构风险最小化——防止过学习，结构风险最小化，求取二者的平衡",[1042,3525,3527],{"id":3526},"模型度量方法",[1020,3528,3526],{},[996,3530,3531],{},"以二分类问题（正类、负类）为例：",[996,3533,3534],{},"True、False表示真正结果，Positive、Negative表示预测，因此有TP、FP、FN、TN",[996,3536,3537],{},[1532,3538,3539],{},[1020,3540,3541],{},"但注意TP、FP、FN、TN，不代表TF就是真实类别，而PN确实是代表预测对错",[996,3543,3544],{},[1532,3545,3546],{},[1020,3547,3548],{},"T、F代表的是最后预测的对错",[996,3550,3551],{},[1532,3552,3553],{},[1020,3554,3555],{},"使用TP FN负负得正的出来的才是真实类别",[993,3557,3558],{},[996,3559,3560],{},[1020,3561,3562],{},"”纵坐标“是预测，”横坐标“是真实值",[996,3564,3565],{},[1711,3566],{"alt":1711,"src":3567},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-04.png",[1045,3569,3571,3572],{"id":3570},"准确率accuracy","准确率",[1020,3573,3574],{},"(accuracy)",[1052,3576,3578],{"className":3577},[1865],[1052,3579,3581,3627],{"className":3580},[1055],[1052,3582,3584],{"className":3583},[1059],[1061,3585,3586],{"xmlns":1063,"display":1874},[1065,3587,3588,3624],{},[1068,3589,3590,3593,3596,3598,3600],{},[1071,3591,3592],{},"A",[1071,3594,3595],{},"C",[1071,3597,3595],{},[1075,3599,1077],{},[1880,3601,3602,3616],{},[1068,3603,3604,3607,3609,3611,3613],{},[1071,3605,3606],{},"T",[1071,3608,2257],{},[1075,3610,3111],{},[1071,3612,3606],{},[1071,3614,3615],{},"N",[1068,3617,3618,3620,3622],{},[1071,3619,2257],{},[1075,3621,3111],{},[1071,3623,3615],{},[1128,3625,3626],{"encoding":1130},"ACC = \\frac{TP + TN}{P + N}",[1052,3628,3630,3655],{"className":3629,"ariaHidden":1096},[1135],[1052,3631,3633,3636,3639,3643,3646,3649,3652],{"className":3632},[1139],[1052,3634],{"className":3635,"style":1144},[1143],[1052,3637,3592],{"className":3638},[1148,1149],[1052,3640,3595],{"className":3641,"style":3642},[1148,1149],"margin-right:0.0715em;",[1052,3644,3595],{"className":3645,"style":3642},[1148,1149],[1052,3647],{"className":3648,"style":1155},[1154],[1052,3650,1077],{"className":3651},[1159],[1052,3653],{"className":3654,"style":1155},[1154],[1052,3656,3658,3662],{"className":3657},[1139],[1052,3659],{"className":3660,"style":3661},[1143],"height:2.1297em;vertical-align:-0.7693em;",[1052,3663,3665,3668,3754],{"className":3664},[1148],[1052,3666],{"className":3667},[1173,1956],[1052,3669,3671],{"className":3670},[1880],[1052,3672,3674,3745],{"className":3673},[1188,1189],[1052,3675,3677,3742],{"className":3676},[1193],[1052,3678,3681,3705,3713],{"className":3679,"style":3680},[1197],"height:1.3603em;",[1052,3682,3683,3686],{"style":1972},[1052,3684],{"className":3685,"style":1976},[1205],[1052,3687,3689,3692,3695,3698,3701],{"className":3688},[1148],[1052,3690,2257],{"className":3691,"style":2283},[1148,1149],[1052,3693],{"className":3694,"style":3378},[1154],[1052,3696,3111],{"className":3697},[2428],[1052,3699],{"className":3700,"style":3378},[1154],[1052,3702,3615],{"className":3703,"style":3704},[1148,1149],"margin-right:0.109em;",[1052,3706,3707,3710],{"style":1985},[1052,3708],{"className":3709,"style":1976},[1205],[1052,3711],{"className":3712,"style":1993},[1992],[1052,3714,3715,3718],{"style":1996},[1052,3716],{"className":3717,"style":1976},[1205],[1052,3719,3721,3724,3727,3730,3733,3736,3739],{"className":3720},[1148],[1052,3722,3606],{"className":3723,"style":2283},[1148,1149],[1052,3725,2257],{"className":3726,"style":2283},[1148,1149],[1052,3728],{"className":3729,"style":3378},[1154],[1052,3731,3111],{"className":3732},[2428],[1052,3734],{"className":3735,"style":3378},[1154],[1052,3737,3606],{"className":3738,"style":2283},[1148,1149],[1052,3740,3615],{"className":3741,"style":3704},[1148,1149],[1052,3743,1221],{"className":3744},[1220],[1052,3746,3748],{"className":3747},[1193],[1052,3749,3752],{"className":3750,"style":3751},[1197],"height:0.7693em;",[1052,3753],{},[1052,3755],{"className":3756},[1284,1956],[996,3758,3759],{},"这个事情很显然，判断正确的几率",[1045,3761,3763,3764],{"id":3762},"错误率error-rate","错误率",[1020,3765,3766],{},"（error rate）",[1052,3768,3770],{"className":3769},[1865],[1052,3771,3773,3830],{"className":3772},[1055],[1052,3774,3776],{"className":3775},[1059],[1061,3777,3778],{"xmlns":1063,"display":1874},[1065,3779,3780,3827],{},[1068,3781,3782,3784,3786,3788,3790,3792,3794,3797,3800,3802,3804],{},[1071,3783,1788],{},[1071,3785,2687],{},[1071,3787,2687],{},[1071,3789,1907],{},[1071,3791,2687],{},[1071,3793,2558],{},[1071,3795,3796],{},"a",[1071,3798,3799],{},"t",[1071,3801,1788],{},[1075,3803,1077],{},[1880,3805,3806,3819],{},[1068,3807,3808,3811,3813,3815,3817],{},[1071,3809,3810],{},"F",[1071,3812,2257],{},[1075,3814,3111],{},[1071,3816,3810],{},[1071,3818,3615],{},[1068,3820,3821,3823,3825],{},[1071,3822,2257],{},[1075,3824,3111],{},[1071,3826,3615],{},[1128,3828,3829],{"encoding":1130},"errorRate = \\frac{FP + FN}{P + N}",[1052,3831,3833,3871],{"className":3832,"ariaHidden":1096},[1135],[1052,3834,3836,3839,3842,3845,3849,3853,3856,3859,3862,3865,3868],{"className":3835},[1139],[1052,3837],{"className":3838,"style":1144},[1143],[1052,3840,2706],{"className":3841,"style":1150},[1148,1149],[1052,3843,2687],{"className":3844,"style":1150},[1148,1149],[1052,3846,3848],{"className":3847,"style":1150},[1148,1149],"or",[1052,3850,2558],{"className":3851,"style":3852},[1148,1149],"margin-right:0.0077em;",[1052,3854,3796],{"className":3855},[1148,1149],[1052,3857,3799],{"className":3858},[1148,1149],[1052,3860,1788],{"className":3861},[1148,1149],[1052,3863],{"className":3864,"style":1155},[1154],[1052,3866,1077],{"className":3867},[1159],[1052,3869],{"className":3870,"style":1155},[1154],[1052,3872,3874,3877],{"className":3873},[1139],[1052,3875],{"className":3876,"style":3661},[1143],[1052,3878,3880,3883,3966],{"className":3879},[1148],[1052,3881],{"className":3882},[1173,1956],[1052,3884,3886],{"className":3885},[1880],[1052,3887,3889,3958],{"className":3888},[1188,1189],[1052,3890,3892,3955],{"className":3891},[1193],[1052,3893,3895,3918,3926],{"className":3894,"style":3680},[1197],[1052,3896,3897,3900],{"style":1972},[1052,3898],{"className":3899,"style":1976},[1205],[1052,3901,3903,3906,3909,3912,3915],{"className":3902},[1148],[1052,3904,2257],{"className":3905,"style":2283},[1148,1149],[1052,3907],{"className":3908,"style":3378},[1154],[1052,3910,3111],{"className":3911},[2428],[1052,3913],{"className":3914,"style":3378},[1154],[1052,3916,3615],{"className":3917,"style":3704},[1148,1149],[1052,3919,3920,3923],{"style":1985},[1052,3921],{"className":3922,"style":1976},[1205],[1052,3924],{"className":3925,"style":1993},[1992],[1052,3927,3928,3931],{"style":1996},[1052,3929],{"className":3930,"style":1976},[1205],[1052,3932,3934,3937,3940,3943,3946,3949,3952],{"className":3933},[1148],[1052,3935,3810],{"className":3936,"style":2283},[1148,1149],[1052,3938,2257],{"className":3939,"style":2283},[1148,1149],[1052,3941],{"className":3942,"style":3378},[1154],[1052,3944,3111],{"className":3945},[2428],[1052,3947],{"className":3948,"style":3378},[1154],[1052,3950,3810],{"className":3951,"style":2283},[1148,1149],[1052,3953,3615],{"className":3954,"style":3704},[1148,1149],[1052,3956,1221],{"className":3957},[1220],[1052,3959,3961],{"className":3960},[1193],[1052,3962,3964],{"className":3963,"style":3751},[1197],[1052,3965],{},[1052,3967],{"className":3968},[1284,1956],[996,3970,3971,3972],{},"这个事情也很显然，",[1052,3973,3975,4018],{"className":3974},[1055],[1052,3976,3978],{"className":3977},[1059],[1061,3979,3980],{"xmlns":1063},[1065,3981,3982,4015],{},[1068,3983,3984,3986,3989,3991,3993,3995,3997,3999,4001,4003,4005,4007,4009,4011,4013],{},[1121,3985,1123],{},[1075,3987,3988],{},"−",[1071,3990,3592],{},[1071,3992,3595],{},[1071,3994,3595],{},[1075,3996,1077],{},[1071,3998,1788],{},[1071,4000,2687],{},[1071,4002,2687],{},[1071,4004,1907],{},[1071,4006,2687],{},[1071,4008,2558],{},[1071,4010,3796],{},[1071,4012,3799],{},[1071,4014,1788],{},[1128,4016,4017],{"encoding":1130},"1-ACC = errorRate",[1052,4019,4021,4040,4064],{"className":4020,"ariaHidden":1096},[1135],[1052,4022,4024,4028,4031,4034,4037],{"className":4023},[1139],[1052,4025],{"className":4026,"style":4027},[1143],"height:0.7278em;vertical-align:-0.0833em;",[1052,4029,1123],{"className":4030},[1148],[1052,4032],{"className":4033,"style":3378},[1154],[1052,4035,3988],{"className":4036},[2428],[1052,4038],{"className":4039,"style":3378},[1154],[1052,4041,4043,4046,4049,4052,4055,4058,4061],{"className":4042},[1139],[1052,4044],{"className":4045,"style":1144},[1143],[1052,4047,3592],{"className":4048},[1148,1149],[1052,4050,3595],{"className":4051,"style":3642},[1148,1149],[1052,4053,3595],{"className":4054,"style":3642},[1148,1149],[1052,4056],{"className":4057,"style":1155},[1154],[1052,4059,1077],{"className":4060},[1159],[1052,4062],{"className":4063,"style":1155},[1154],[1052,4065,4067,4070,4073,4076,4079,4082,4085,4088],{"className":4066},[1139],[1052,4068],{"className":4069,"style":1144},[1143],[1052,4071,2706],{"className":4072,"style":1150},[1148,1149],[1052,4074,2687],{"className":4075,"style":1150},[1148,1149],[1052,4077,3848],{"className":4078,"style":1150},[1148,1149],[1052,4080,2558],{"className":4081,"style":3852},[1148,1149],[1052,4083,3796],{"className":4084},[1148,1149],[1052,4086,3799],{"className":4087},[1148,1149],[1052,4089,1788],{"className":4090},[1148,1149],[1045,4092,4094],{"id":4093},"精确率真对了多少precision",[1020,4095,4096],{},"精确率——“真”对了多少（precision）",[993,4098,4099],{},[996,4100,4101,4102,4105,4106,4109,4110],{},"精确率",[1020,4103,4104],{},"也叫查准率，","即",[1532,4107,4108],{},"预测结果为正确里面有多少个实际也是正确的","，即",[1020,4111,4112],{},"真对了多少",[1052,4114,4116],{"className":4115},[1865],[1052,4117,4119,4172],{"className":4118},[1055],[1052,4120,4122],{"className":4121},[1059],[1061,4123,4124],{"xmlns":1063,"display":1874},[1065,4125,4126,4169],{},[1068,4127,4128,4130,4132,4134,4137,4139,4141,4143,4145,4147,4149],{},[1071,4129,996],{},[1071,4131,2687],{},[1071,4133,1788],{},[1071,4135,4136],{},"c",[1071,4138,1093],{},[1071,4140,1910],{},[1071,4142,1093],{},[1071,4144,1907],{},[1071,4146,1126],{},[1075,4148,1077],{},[1880,4150,4151,4157],{},[1068,4152,4153,4155],{},[1071,4154,3606],{},[1071,4156,2257],{},[1068,4158,4159,4161,4163,4165,4167],{},[1071,4160,3606],{},[1071,4162,2257],{},[1075,4164,3111],{},[1071,4166,3810],{},[1071,4168,2257],{},[1128,4170,4171],{"encoding":1130},"precision = \\frac{TP}{TP + FP}",[1052,4173,4175,4216],{"className":4174,"ariaHidden":1096},[1135],[1052,4176,4178,4182,4185,4188,4192,4195,4198,4201,4204,4207,4210,4213],{"className":4177},[1139],[1052,4179],{"className":4180,"style":4181},[1143],"height:0.854em;vertical-align:-0.1944em;",[1052,4183,996],{"className":4184},[1148,1149],[1052,4186,2687],{"className":4187,"style":1150},[1148,1149],[1052,4189,4191],{"className":4190},[1148,1149],"ec",[1052,4193,1093],{"className":4194},[1148,1149],[1052,4196,1910],{"className":4197},[1148,1149],[1052,4199,1093],{"className":4200},[1148,1149],[1052,4202,1907],{"className":4203},[1148,1149],[1052,4205,1126],{"className":4206},[1148,1149],[1052,4208],{"className":4209,"style":1155},[1154],[1052,4211,1077],{"className":4212},[1159],[1052,4214],{"className":4215,"style":1155},[1154],[1052,4217,4219,4222],{"className":4218},[1139],[1052,4220],{"className":4221,"style":3661},[1143],[1052,4223,4225,4228,4302],{"className":4224},[1148],[1052,4226],{"className":4227},[1173,1956],[1052,4229,4231],{"className":4230},[1880],[1052,4232,4234,4294],{"className":4233},[1188,1189],[1052,4235,4237,4291],{"className":4236},[1193],[1052,4238,4240,4269,4277],{"className":4239,"style":3680},[1197],[1052,4241,4242,4245],{"style":1972},[1052,4243],{"className":4244,"style":1976},[1205],[1052,4246,4248,4251,4254,4257,4260,4263,4266],{"className":4247},[1148],[1052,4249,3606],{"className":4250,"style":2283},[1148,1149],[1052,4252,2257],{"className":4253,"style":2283},[1148,1149],[1052,4255],{"className":4256,"style":3378},[1154],[1052,4258,3111],{"className":4259},[2428],[1052,4261],{"className":4262,"style":3378},[1154],[1052,4264,3810],{"className":4265,"style":2283},[1148,1149],[1052,4267,2257],{"className":4268,"style":2283},[1148,1149],[1052,4270,4271,4274],{"style":1985},[1052,4272],{"className":4273,"style":1976},[1205],[1052,4275],{"className":4276,"style":1993},[1992],[1052,4278,4279,4282],{"style":1996},[1052,4280],{"className":4281,"style":1976},[1205],[1052,4283,4285,4288],{"className":4284},[1148],[1052,4286,3606],{"className":4287,"style":2283},[1148,1149],[1052,4289,2257],{"className":4290,"style":2283},[1148,1149],[1052,4292,1221],{"className":4293},[1220],[1052,4295,4297],{"className":4296},[1193],[1052,4298,4300],{"className":4299,"style":3751},[1197],[1052,4301],{},[1052,4303],{"className":4304},[1284,1956],[1045,4306,4308],{"id":4307},"召回率找对了多少recall",[1020,4309,4310],{},"召回率——“找”对了多少（Recall）",[993,4312,4313],{},[996,4314,4315,4316,4319,4320,2919],{},"也叫",[1020,4317,4318],{},"查全率","，表示",[1020,4321,4322],{},"所有正例样本中被模型预测为正例的比例",[1052,4324,4326],{"className":4325},[1865],[1052,4327,4329,4376],{"className":4328},[1055],[1052,4330,4332],{"className":4331},[1059],[1061,4333,4334],{"xmlns":1063,"display":1874},[1065,4335,4336,4373],{},[1068,4337,4338,4340,4342,4344,4346,4349,4351,4353],{},[1071,4339,2558],{},[1071,4341,1788],{},[1071,4343,4136],{},[1071,4345,3796],{},[1071,4347,4348],{},"l",[1071,4350,4348],{},[1075,4352,1077],{},[1880,4354,4355,4361],{},[1068,4356,4357,4359],{},[1071,4358,3606],{},[1071,4360,2257],{},[1068,4362,4363,4365,4367,4369,4371],{},[1071,4364,3606],{},[1071,4366,2257],{},[1075,4368,3111],{},[1071,4370,3810],{},[1071,4372,3615],{},[1128,4374,4375],{"encoding":1130},"Recall = \\frac{TP}{TP+FN}",[1052,4377,4379,4410],{"className":4378,"ariaHidden":1096},[1135],[1052,4380,4382,4385,4388,4391,4394,4398,4401,4404,4407],{"className":4381},[1139],[1052,4383],{"className":4384,"style":2238},[1143],[1052,4386,2558],{"className":4387,"style":3852},[1148,1149],[1052,4389,4191],{"className":4390},[1148,1149],[1052,4392,3796],{"className":4393},[1148,1149],[1052,4395,4348],{"className":4396,"style":4397},[1148,1149],"margin-right:0.0197em;",[1052,4399,4348],{"className":4400,"style":4397},[1148,1149],[1052,4402],{"className":4403,"style":1155},[1154],[1052,4405,1077],{"className":4406},[1159],[1052,4408],{"className":4409,"style":1155},[1154],[1052,4411,4413,4416],{"className":4412},[1139],[1052,4414],{"className":4415,"style":3661},[1143],[1052,4417,4419,4422,4496],{"className":4418},[1148],[1052,4420],{"className":4421},[1173,1956],[1052,4423,4425],{"className":4424},[1880],[1052,4426,4428,4488],{"className":4427},[1188,1189],[1052,4429,4431,4485],{"className":4430},[1193],[1052,4432,4434,4463,4471],{"className":4433,"style":3680},[1197],[1052,4435,4436,4439],{"style":1972},[1052,4437],{"className":4438,"style":1976},[1205],[1052,4440,4442,4445,4448,4451,4454,4457,4460],{"className":4441},[1148],[1052,4443,3606],{"className":4444,"style":2283},[1148,1149],[1052,4446,2257],{"className":4447,"style":2283},[1148,1149],[1052,4449],{"className":4450,"style":3378},[1154],[1052,4452,3111],{"className":4453},[2428],[1052,4455],{"className":4456,"style":3378},[1154],[1052,4458,3810],{"className":4459,"style":2283},[1148,1149],[1052,4461,3615],{"className":4462,"style":3704},[1148,1149],[1052,4464,4465,4468],{"style":1985},[1052,4466],{"className":4467,"style":1976},[1205],[1052,4469],{"className":4470,"style":1993},[1992],[1052,4472,4473,4476],{"style":1996},[1052,4474],{"className":4475,"style":1976},[1205],[1052,4477,4479,4482],{"className":4478},[1148],[1052,4480,3606],{"className":4481,"style":2283},[1148,1149],[1052,4483,2257],{"className":4484,"style":2283},[1148,1149],[1052,4486,1221],{"className":4487},[1220],[1052,4489,4491],{"className":4490},[1193],[1052,4492,4494],{"className":4493,"style":3751},[1197],[1052,4495],{},[1052,4497],{"className":4498},[1284,1956],[996,4500,4501,4502,4506],{},"在实际应用中，精确率和召回率之间是相互矛盾的，比如可以将所有样本分类为正例使得召回率为100%而精确率极低。因此为了综合考虑精确率和召回率，可采用",[4503,4504,4505],"code",{"code":4505},"F1-Score"," 这一综合分类率：",[996,4508,4509],{},[1711,4510],{"alt":1711,"src":4511},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-05.png",[993,4513,4514],{},[996,4515,4516],{},[1020,4517,4518],{},"即调和平均数",[1010,4520,4522],{"id":4521},"二回归分析","二、回归分析",[1042,4524,4525],{"id":4525},"线性回归",[1045,4527,4528],{"id":4528},"一维线性回归",[996,4530,4531],{},[1711,4532],{"alt":1711,"src":4533},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-06.png",[996,4535,4536],{},[1711,4537],{"alt":1711,"src":4538},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-07.png",[1045,4540,4541],{"id":4541},"高维线性回归",[996,4543,4544],{},[1711,4545],{"alt":1711,"src":4546},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-08.png",[4548,4549,4550],"h6",{"id":4550},"矩阵形式",[996,4552,4553],{},"为了方便，我们使用矩阵来表示所有的训练数据和数据标签。",[1052,4555,4557],{"className":4556},[1865],[1052,4558,4560,4641],{"className":4559},[1055],[1052,4561,4563],{"className":4562},[1059],[1061,4564,4565],{"xmlns":1063,"display":1874},[1065,4566,4567,4638],{},[1068,4568,4569,4572,4574,4577,4583,4585,4587,4589,4591,4593,4599,4602,4604,4607,4610,4612,4614,4620,4622,4624,4626,4628,4630,4636],{},[1071,4570,4571],{},"X",[1075,4573,1077],{},[1075,4575,4576],{"stretchy":1080},"[",[1086,4578,4579,4581],{},[1071,4580,1090],{},[1121,4582,1123],{},[1075,4584,1097],{"separator":1096},[1071,4586,1576],{"mathvariant":1575},[1071,4588,1576],{"mathvariant":1575},[1071,4590,1576],{"mathvariant":1575},[1075,4592,1097],{"separator":1096},[1086,4594,4595,4597],{},[1071,4596,1090],{},[1071,4598,1791],{},[1075,4600,4601],{"stretchy":1080},"]",[1075,4603,1097],{"separator":1096},[1154,4605],{"width":4606},"1em",[1071,4608,1102],{"mathvariant":4609},"bold",[1075,4611,1077],{},[1075,4613,4576],{"stretchy":1080},[1086,4615,4616,4618],{},[1071,4617,1102],{},[1121,4619,1123],{},[1075,4621,1097],{"separator":1096},[1071,4623,1576],{"mathvariant":1575},[1071,4625,1576],{"mathvariant":1575},[1071,4627,1576],{"mathvariant":1575},[1075,4629,1097],{"separator":1096},[1086,4631,4632,4634],{},[1071,4633,1102],{},[1071,4635,1791],{},[1075,4637,4601],{"stretchy":1080},[1128,4639,4640],{"encoding":1130},"X = [x_1, ..., x_m], \\quad \\mathbf{y} = [y_1, ..., y_m]",[1052,4642,4644,4663,4796],{"className":4643,"ariaHidden":1096},[1135],[1052,4645,4647,4650,4654,4657,4660],{"className":4646},[1139],[1052,4648],{"className":4649,"style":1144},[1143],[1052,4651,4571],{"className":4652,"style":4653},[1148,1149],"margin-right:0.0785em;",[1052,4655],{"className":4656,"style":1155},[1154],[1052,4658,1077],{"className":4659},[1159],[1052,4661],{"className":4662,"style":1155},[1154],[1052,4664,4666,4669,4672,4713,4716,4719,4723,4726,4729,4769,4772,4775,4779,4782,4787,4790,4793],{"className":4665},[1139],[1052,4667],{"className":4668,"style":1597},[1143],[1052,4670,4576],{"className":4671},[1173],[1052,4673,4675,4678],{"className":4674},[1148],[1052,4676,1090],{"className":4677},[1148,1149],[1052,4679,4681],{"className":4680},[1184],[1052,4682,4684,4705],{"className":4683},[1188,1189],[1052,4685,4687,4702],{"className":4686},[1193],[1052,4688,4691],{"className":4689,"style":4690},[1197],"height:0.3011em;",[1052,4692,4693,4696],{"style":1201},[1052,4694],{"className":4695,"style":1206},[1205],[1052,4697,4699],{"className":4698},[1210,1211,1212,1213],[1052,4700,1123],{"className":4701},[1148,1213],[1052,4703,1221],{"className":4704},[1220],[1052,4706,4708],{"className":4707},[1193],[1052,4709,4711],{"className":4710,"style":1228},[1197],[1052,4712],{},[1052,4714,1097],{"className":4715},[1234],[1052,4717],{"className":4718,"style":1238},[1154],[1052,4720,4722],{"className":4721},[1148],"...",[1052,4724,1097],{"className":4725},[1234],[1052,4727],{"className":4728,"style":1238},[1154],[1052,4730,4732,4735],{"className":4731},[1148],[1052,4733,1090],{"className":4734},[1148,1149],[1052,4736,4738],{"className":4737},[1184],[1052,4739,4741,4761],{"className":4740},[1188,1189],[1052,4742,4744,4758],{"className":4743},[1193],[1052,4745,4747],{"className":4746,"style":1825},[1197],[1052,4748,4749,4752],{"style":1201},[1052,4750],{"className":4751,"style":1206},[1205],[1052,4753,4755],{"className":4754},[1210,1211,1212,1213],[1052,4756,1791],{"className":4757},[1148,1149,1213],[1052,4759,1221],{"className":4760},[1220],[1052,4762,4764],{"className":4763},[1193],[1052,4765,4767],{"className":4766,"style":1228},[1197],[1052,4768],{},[1052,4770,4601],{"className":4771},[1284],[1052,4773,1097],{"className":4774},[1234],[1052,4776],{"className":4777,"style":4778},[1154],"margin-right:1em;",[1052,4780],{"className":4781,"style":1238},[1154],[1052,4783,1102],{"className":4784,"style":4786},[1148,4785],"mathbf","margin-right:0.016em;",[1052,4788],{"className":4789,"style":1155},[1154],[1052,4791,1077],{"className":4792},[1159],[1052,4794],{"className":4795,"style":1155},[1154],[1052,4797,4799,4802,4805,4845,4848,4851,4854,4857,4860,4900],{"className":4798},[1139],[1052,4800],{"className":4801,"style":1597},[1143],[1052,4803,4576],{"className":4804},[1173],[1052,4806,4808,4811],{"className":4807},[1148],[1052,4809,1102],{"className":4810,"style":1245},[1148,1149],[1052,4812,4814],{"className":4813},[1184],[1052,4815,4817,4837],{"className":4816},[1188,1189],[1052,4818,4820,4834],{"className":4819},[1193],[1052,4821,4823],{"className":4822,"style":4690},[1197],[1052,4824,4825,4828],{"style":1260},[1052,4826],{"className":4827,"style":1206},[1205],[1052,4829,4831],{"className":4830},[1210,1211,1212,1213],[1052,4832,1123],{"className":4833},[1148,1213],[1052,4835,1221],{"className":4836},[1220],[1052,4838,4840],{"className":4839},[1193],[1052,4841,4843],{"className":4842,"style":1228},[1197],[1052,4844],{},[1052,4846,1097],{"className":4847},[1234],[1052,4849],{"className":4850,"style":1238},[1154],[1052,4852,4722],{"className":4853},[1148],[1052,4855,1097],{"className":4856},[1234],[1052,4858],{"className":4859,"style":1238},[1154],[1052,4861,4863,4866],{"className":4862},[1148],[1052,4864,1102],{"className":4865,"style":1245},[1148,1149],[1052,4867,4869],{"className":4868},[1184],[1052,4870,4872,4892],{"className":4871},[1188,1189],[1052,4873,4875,4889],{"className":4874},[1193],[1052,4876,4878],{"className":4877,"style":1825},[1197],[1052,4879,4880,4883],{"style":1260},[1052,4881],{"className":4882,"style":1206},[1205],[1052,4884,4886],{"className":4885},[1210,1211,1212,1213],[1052,4887,1791],{"className":4888},[1148,1149,1213],[1052,4890,1221],{"className":4891},[1220],[1052,4893,4895],{"className":4894},[1193],[1052,4896,4898],{"className":4897,"style":1228},[1197],[1052,4899],{},[1052,4901,4601],{"className":4902},[1284],[996,4904,4905,4906,4975,4976,5047],{},"其中每一个数据",[1052,4907,4909,4926],{"className":4908},[1055],[1052,4910,4912],{"className":4911},[1059],[1061,4913,4914],{"xmlns":1063},[1065,4915,4916,4924],{},[1068,4917,4918],{},[1086,4919,4920,4922],{},[1071,4921,1090],{},[1071,4923,1093],{},[1128,4925,1370],{"encoding":1130},[1052,4927,4929],{"className":4928,"ariaHidden":1096},[1135],[1052,4930,4932,4935],{"className":4931},[1139],[1052,4933],{"className":4934,"style":1380},[1143],[1052,4936,4938,4941],{"className":4937},[1148],[1052,4939,1090],{"className":4940},[1148,1149],[1052,4942,4944],{"className":4943},[1184],[1052,4945,4947,4967],{"className":4946},[1188,1189],[1052,4948,4950,4964],{"className":4949},[1193],[1052,4951,4953],{"className":4952,"style":1198},[1197],[1052,4954,4955,4958],{"style":1201},[1052,4956],{"className":4957,"style":1206},[1205],[1052,4959,4961],{"className":4960},[1210,1211,1212,1213],[1052,4962,1093],{"className":4963},[1148,1149,1213],[1052,4965,1221],{"className":4966},[1220],[1052,4968,4970],{"className":4969},[1193],[1052,4971,4973],{"className":4972,"style":1228},[1197],[1052,4974],{},"会扩展一个维度，其值为1，对应参数",[1052,4977,4979,4998],{"className":4978},[1055],[1052,4980,4982],{"className":4981},[1059],[1061,4983,4984],{"xmlns":1063},[1065,4985,4986,4995],{},[1068,4987,4988],{},[1086,4989,4990,4992],{},[1071,4991,3796],{},[1121,4993,4994],{},"0",[1128,4996,4997],{"encoding":1130},"a_0",[1052,4999,5001],{"className":5000,"ariaHidden":1096},[1135],[1052,5002,5004,5007],{"className":5003},[1139],[1052,5005],{"className":5006,"style":1380},[1143],[1052,5008,5010,5013],{"className":5009},[1148],[1052,5011,3796],{"className":5012},[1148,1149],[1052,5014,5016],{"className":5015},[1184],[1052,5017,5019,5039],{"className":5018},[1188,1189],[1052,5020,5022,5036],{"className":5021},[1193],[1052,5023,5025],{"className":5024,"style":4690},[1197],[1052,5026,5027,5030],{"style":1201},[1052,5028],{"className":5029,"style":1206},[1205],[1052,5031,5033],{"className":5032},[1210,1211,1212,1213],[1052,5034,4994],{"className":5035},[1148,1213],[1052,5037,1221],{"className":5038},[1220],[1052,5040,5042],{"className":5041},[1193],[1052,5043,5045],{"className":5044,"style":1228},[1197],[1052,5046],{},"。均方误差函数可以表示为：",[1052,5049,5051],{"className":5050},[1865],[1052,5052,5054,5117],{"className":5053},[1055],[1052,5055,5057],{"className":5056},[1059],[1061,5058,5059],{"xmlns":1063,"display":1874},[1065,5060,5061,5114],{},[1068,5062,5063,5069,5071,5073,5075,5077,5079,5081,5083,5090,5092,5098,5100,5102,5104,5110,5112],{},[1086,5064,5065,5067],{},[1071,5066,3117],{},[1071,5068,1791],{},[1075,5070,1084],{"stretchy":1080},[1071,5072,3796],{"mathvariant":4609},[1075,5074,1107],{"stretchy":1080},[1075,5076,1077],{},[1075,5078,1084],{"stretchy":1080},[1071,5080,1102],{"mathvariant":4609},[1075,5082,3988],{},[5084,5085,5086,5088],"msup",{},[1071,5087,4571],{},[1071,5089,3606],{},[1071,5091,3796],{"mathvariant":4609},[5084,5093,5094,5096],{},[1075,5095,1107],{"stretchy":1080},[1071,5097,3606],{},[1075,5099,1084],{"stretchy":1080},[1071,5101,1102],{"mathvariant":4609},[1075,5103,3988],{},[5084,5105,5106,5108],{},[1071,5107,4571],{},[1071,5109,3606],{},[1071,5111,3796],{"mathvariant":4609},[1075,5113,1107],{"stretchy":1080},[1128,5115,5116],{"encoding":1130},"J_m(\\mathbf{a}) = (\\mathbf{y} - X^T\\mathbf{a})^T(\\mathbf{y} - X^T\\mathbf{a})",[1052,5118,5120,5185,5206,5291],{"className":5119,"ariaHidden":1096},[1135],[1052,5121,5123,5126,5167,5170,5173,5176,5179,5182],{"className":5122},[1139],[1052,5124],{"className":5125,"style":1597},[1143],[1052,5127,5129,5132],{"className":5128},[1148],[1052,5130,3117],{"className":5131,"style":3397},[1148,1149],[1052,5133,5135],{"className":5134},[1184],[1052,5136,5138,5159],{"className":5137},[1188,1189],[1052,5139,5141,5156],{"className":5140},[1193],[1052,5142,5144],{"className":5143,"style":1825},[1197],[1052,5145,5147,5150],{"style":5146},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[1052,5148],{"className":5149,"style":1206},[1205],[1052,5151,5153],{"className":5152},[1210,1211,1212,1213],[1052,5154,1791],{"className":5155},[1148,1149,1213],[1052,5157,1221],{"className":5158},[1220],[1052,5160,5162],{"className":5161},[1193],[1052,5163,5165],{"className":5164,"style":1228},[1197],[1052,5166],{},[1052,5168,1084],{"className":5169},[1173],[1052,5171,3796],{"className":5172},[1148,4785],[1052,5174,1107],{"className":5175},[1284],[1052,5177],{"className":5178,"style":1155},[1154],[1052,5180,1077],{"className":5181},[1159],[1052,5183],{"className":5184,"style":1155},[1154],[1052,5186,5188,5191,5194,5197,5200,5203],{"className":5187},[1139],[1052,5189],{"className":5190,"style":1597},[1143],[1052,5192,1084],{"className":5193},[1173],[1052,5195,1102],{"className":5196,"style":4786},[1148,4785],[1052,5198],{"className":5199,"style":3378},[1154],[1052,5201,3988],{"className":5202},[2428],[1052,5204],{"className":5205,"style":3378},[1154],[1052,5207,5209,5213,5244,5247,5276,5279,5282,5285,5288],{"className":5208},[1139],[1052,5210],{"className":5211,"style":5212},[1143],"height:1.1413em;vertical-align:-0.25em;",[1052,5214,5216,5219],{"className":5215},[1148],[1052,5217,4571],{"className":5218,"style":4653},[1148,1149],[1052,5220,5222],{"className":5221},[1184],[1052,5223,5225],{"className":5224},[1188],[1052,5226,5228],{"className":5227},[1193],[1052,5229,5232],{"className":5230,"style":5231},[1197],"height:0.8913em;",[1052,5233,5235,5238],{"style":5234},"top:-3.113em;margin-right:0.05em;",[1052,5236],{"className":5237,"style":1206},[1205],[1052,5239,5241],{"className":5240},[1210,1211,1212,1213],[1052,5242,3606],{"className":5243,"style":2283},[1148,1149,1213],[1052,5245,3796],{"className":5246},[1148,4785],[1052,5248,5250,5253],{"className":5249},[1284],[1052,5251,1107],{"className":5252},[1284],[1052,5254,5256],{"className":5255},[1184],[1052,5257,5259],{"className":5258},[1188],[1052,5260,5262],{"className":5261},[1193],[1052,5263,5265],{"className":5264,"style":5231},[1197],[1052,5266,5267,5270],{"style":5234},[1052,5268],{"className":5269,"style":1206},[1205],[1052,5271,5273],{"className":5272},[1210,1211,1212,1213],[1052,5274,3606],{"className":5275,"style":2283},[1148,1149,1213],[1052,5277,1084],{"className":5278},[1173],[1052,5280,1102],{"className":5281,"style":4786},[1148,4785],[1052,5283],{"className":5284,"style":3378},[1154],[1052,5286,3988],{"className":5287},[2428],[1052,5289],{"className":5290,"style":3378},[1154],[1052,5292,5294,5297,5326,5329],{"className":5293},[1139],[1052,5295],{"className":5296,"style":5212},[1143],[1052,5298,5300,5303],{"className":5299},[1148],[1052,5301,4571],{"className":5302,"style":4653},[1148,1149],[1052,5304,5306],{"className":5305},[1184],[1052,5307,5309],{"className":5308},[1188],[1052,5310,5312],{"className":5311},[1193],[1052,5313,5315],{"className":5314,"style":5231},[1197],[1052,5316,5317,5320],{"style":5234},[1052,5318],{"className":5319,"style":1206},[1205],[1052,5321,5323],{"className":5322},[1210,1211,1212,1213],[1052,5324,3606],{"className":5325,"style":2283},[1148,1149,1213],[1052,5327,3796],{"className":5328},[1148,4785],[1052,5330,1107],{"className":5331},[1284],[996,5333,5334,5335,5420,5421,5451],{},"均方误差函数",[1052,5336,5338,5362],{"className":5337},[1055],[1052,5339,5341],{"className":5340},[1059],[1061,5342,5343],{"xmlns":1063},[1065,5344,5345,5359],{},[1068,5346,5347,5353,5355,5357],{},[1086,5348,5349,5351],{},[1071,5350,3117],{},[1071,5352,1126],{},[1075,5354,1084],{"stretchy":1080},[1071,5356,3796],{"mathvariant":4609},[1075,5358,1107],{"stretchy":1080},[1128,5360,5361],{"encoding":1130},"J_n(\\mathbf{a})",[1052,5363,5365],{"className":5364,"ariaHidden":1096},[1135],[1052,5366,5368,5371,5411,5414,5417],{"className":5367},[1139],[1052,5369],{"className":5370,"style":1597},[1143],[1052,5372,5374,5377],{"className":5373},[1148],[1052,5375,3117],{"className":5376,"style":3397},[1148,1149],[1052,5378,5380],{"className":5379},[1184],[1052,5381,5383,5403],{"className":5382},[1188,1189],[1052,5384,5386,5400],{"className":5385},[1193],[1052,5387,5389],{"className":5388,"style":1825},[1197],[1052,5390,5391,5394],{"style":5146},[1052,5392],{"className":5393,"style":1206},[1205],[1052,5395,5397],{"className":5396},[1210,1211,1212,1213],[1052,5398,1126],{"className":5399},[1148,1149,1213],[1052,5401,1221],{"className":5402},[1220],[1052,5404,5406],{"className":5405},[1193],[1052,5407,5409],{"className":5408,"style":1228},[1197],[1052,5410],{},[1052,5412,1084],{"className":5413},[1173],[1052,5415,3796],{"className":5416},[1148,4785],[1052,5418,1107],{"className":5419},[1284],"对所有参数",[1052,5422,5424,5438],{"className":5423},[1055],[1052,5425,5427],{"className":5426},[1059],[1061,5428,5429],{"xmlns":1063},[1065,5430,5431,5435],{},[1068,5432,5433],{},[1071,5434,3796],{"mathvariant":4609},[1128,5436,5437],{"encoding":1130},"\\mathbf{a}",[1052,5439,5441],{"className":5440,"ariaHidden":1096},[1135],[1052,5442,5444,5448],{"className":5443},[1139],[1052,5445],{"className":5446,"style":5447},[1143],"height:0.4444em;",[1052,5449,3796],{"className":5450},[1148,4785],"求导可得：",[1052,5453,5455],{"className":5454},[1865],[1052,5456,5458,5506],{"className":5457},[1055],[1052,5459,5461],{"className":5460},[1059],[1061,5462,5463],{"xmlns":1063,"display":1874},[1065,5464,5465,5503],{},[1068,5466,5467,5470,5472,5474,5476,5478,5480,5482,5485,5487,5489,5491,5493,5499,5501],{},[1071,5468,5469],{"mathvariant":1575},"∇",[1071,5471,3117],{},[1075,5473,1084],{"stretchy":1080},[1071,5475,3796],{"mathvariant":4609},[1075,5477,1107],{"stretchy":1080},[1075,5479,1077],{},[1075,5481,3988],{},[1121,5483,5484],{},"2",[1071,5486,4571],{},[1075,5488,1084],{"stretchy":1080},[1071,5490,1102],{"mathvariant":4609},[1075,5492,3988],{},[5084,5494,5495,5497],{},[1071,5496,4571],{},[1071,5498,3606],{},[1071,5500,3796],{"mathvariant":4609},[1075,5502,1107],{"stretchy":1080},[1128,5504,5505],{"encoding":1130},"\\nabla J(\\mathbf{a}) = -2X(\\mathbf{y} - X^T\\mathbf{a})",[1052,5507,5509,5539,5569],{"className":5508,"ariaHidden":1096},[1135],[1052,5510,5512,5515,5518,5521,5524,5527,5530,5533,5536],{"className":5511},[1139],[1052,5513],{"className":5514,"style":1597},[1143],[1052,5516,5469],{"className":5517},[1148],[1052,5519,3117],{"className":5520,"style":3397},[1148,1149],[1052,5522,1084],{"className":5523},[1173],[1052,5525,3796],{"className":5526},[1148,4785],[1052,5528,1107],{"className":5529},[1284],[1052,5531],{"className":5532,"style":1155},[1154],[1052,5534,1077],{"className":5535},[1159],[1052,5537],{"className":5538,"style":1155},[1154],[1052,5540,5542,5545,5548,5551,5554,5557,5560,5563,5566],{"className":5541},[1139],[1052,5543],{"className":5544,"style":1597},[1143],[1052,5546,3988],{"className":5547},[1148],[1052,5549,5484],{"className":5550},[1148],[1052,5552,4571],{"className":5553,"style":4653},[1148,1149],[1052,5555,1084],{"className":5556},[1173],[1052,5558,1102],{"className":5559,"style":4786},[1148,4785],[1052,5561],{"className":5562,"style":3378},[1154],[1052,5564,3988],{"className":5565},[2428],[1052,5567],{"className":5568,"style":3378},[1154],[1052,5570,5572,5575,5604,5607],{"className":5571},[1139],[1052,5573],{"className":5574,"style":5212},[1143],[1052,5576,5578,5581],{"className":5577},[1148],[1052,5579,4571],{"className":5580,"style":4653},[1148,1149],[1052,5582,5584],{"className":5583},[1184],[1052,5585,5587],{"className":5586},[1188],[1052,5588,5590],{"className":5589},[1193],[1052,5591,5593],{"className":5592,"style":5231},[1197],[1052,5594,5595,5598],{"style":5234},[1052,5596],{"className":5597,"style":1206},[1205],[1052,5599,5601],{"className":5600},[1210,1211,1212,1213],[1052,5602,3606],{"className":5603,"style":2283},[1148,1149,1213],[1052,5605,3796],{"className":5606},[1148,4785],[1052,5608,1107],{"className":5609},[1284],[996,5611,5612,5613,5697],{},"因为均方误差函数",[1052,5614,5616,5639],{"className":5615},[1055],[1052,5617,5619],{"className":5618},[1059],[1061,5620,5621],{"xmlns":1063},[1065,5622,5623,5637],{},[1068,5624,5625,5631,5633,5635],{},[1086,5626,5627,5629],{},[1071,5628,3117],{},[1071,5630,1126],{},[1075,5632,1084],{"stretchy":1080},[1071,5634,3796],{"mathvariant":4609},[1075,5636,1107],{"stretchy":1080},[1128,5638,5361],{"encoding":1130},[1052,5640,5642],{"className":5641,"ariaHidden":1096},[1135],[1052,5643,5645,5648,5688,5691,5694],{"className":5644},[1139],[1052,5646],{"className":5647,"style":1597},[1143],[1052,5649,5651,5654],{"className":5650},[1148],[1052,5652,3117],{"className":5653,"style":3397},[1148,1149],[1052,5655,5657],{"className":5656},[1184],[1052,5658,5660,5680],{"className":5659},[1188,1189],[1052,5661,5663,5677],{"className":5662},[1193],[1052,5664,5666],{"className":5665,"style":1825},[1197],[1052,5667,5668,5671],{"style":5146},[1052,5669],{"className":5670,"style":1206},[1205],[1052,5672,5674],{"className":5673},[1210,1211,1212,1213],[1052,5675,1126],{"className":5676},[1148,1149,1213],[1052,5678,1221],{"className":5679},[1220],[1052,5681,5683],{"className":5682},[1193],[1052,5684,5686],{"className":5685,"style":1228},[1197],[1052,5687],{},[1052,5689,1084],{"className":5690},[1173],[1052,5692,3796],{"className":5693},[1148,4785],[1052,5695,1107],{"className":5696},[1284],"是一个二次的凸函数，所以函数只存在一个极小值点，也同样是最小值点，",[996,5699,5700,5701,5773],{},"所以令",[1052,5702,5704,5730],{"className":5703},[1055],[1052,5705,5707],{"className":5706},[1059],[1061,5708,5709],{"xmlns":1063},[1065,5710,5711,5727],{},[1068,5712,5713,5715,5717,5719,5721,5723,5725],{},[1071,5714,5469],{"mathvariant":1575},[1071,5716,3117],{},[1075,5718,1084],{"stretchy":1080},[1071,5720,3796],{"mathvariant":4609},[1075,5722,1107],{"stretchy":1080},[1075,5724,1077],{},[1121,5726,4994],{},[1128,5728,5729],{"encoding":1130},"\\nabla J(\\mathbf{a}) = 0",[1052,5731,5733,5763],{"className":5732,"ariaHidden":1096},[1135],[1052,5734,5736,5739,5742,5745,5748,5751,5754,5757,5760],{"className":5735},[1139],[1052,5737],{"className":5738,"style":1597},[1143],[1052,5740,5469],{"className":5741},[1148],[1052,5743,3117],{"className":5744,"style":3397},[1148,1149],[1052,5746,1084],{"className":5747},[1173],[1052,5749,3796],{"className":5750},[1148,4785],[1052,5752,1107],{"className":5753},[1284],[1052,5755],{"className":5756,"style":1155},[1154],[1052,5758,1077],{"className":5759},[1159],[1052,5761],{"className":5762,"style":1155},[1154],[1052,5764,5766,5770],{"className":5765},[1139],[1052,5767],{"className":5768,"style":5769},[1143],"height:0.6444em;",[1052,5771,4994],{"className":5772},[1148],"可得：",[1052,5775,5777],{"className":5776},[1865],[1052,5778,5780,5808],{"className":5779},[1055],[1052,5781,5783],{"className":5782},[1059],[1061,5784,5785],{"xmlns":1063,"display":1874},[1065,5786,5787,5805],{},[1068,5788,5789,5791,5797,5799,5801,5803],{},[1071,5790,4571],{},[5084,5792,5793,5795],{},[1071,5794,4571],{},[1071,5796,3606],{},[1071,5798,3796],{"mathvariant":4609},[1075,5800,1077],{},[1071,5802,4571],{},[1071,5804,1102],{"mathvariant":4609},[1128,5806,5807],{"encoding":1130},"XX^T\\mathbf{a} = X\\mathbf{y}",[1052,5809,5811,5861],{"className":5810,"ariaHidden":1096},[1135],[1052,5812,5814,5817,5820,5849,5852,5855,5858],{"className":5813},[1139],[1052,5815],{"className":5816,"style":5231},[1143],[1052,5818,4571],{"className":5819,"style":4653},[1148,1149],[1052,5821,5823,5826],{"className":5822},[1148],[1052,5824,4571],{"className":5825,"style":4653},[1148,1149],[1052,5827,5829],{"className":5828},[1184],[1052,5830,5832],{"className":5831},[1188],[1052,5833,5835],{"className":5834},[1193],[1052,5836,5838],{"className":5837,"style":5231},[1197],[1052,5839,5840,5843],{"style":5234},[1052,5841],{"className":5842,"style":1206},[1205],[1052,5844,5846],{"className":5845},[1210,1211,1212,1213],[1052,5847,3606],{"className":5848,"style":2283},[1148,1149,1213],[1052,5850,3796],{"className":5851},[1148,4785],[1052,5853],{"className":5854,"style":1155},[1154],[1052,5856,1077],{"className":5857},[1159],[1052,5859],{"className":5860,"style":1155},[1154],[1052,5862,5864,5868,5871],{"className":5863},[1139],[1052,5865],{"className":5866,"style":5867},[1143],"height:0.8778em;vertical-align:-0.1944em;",[1052,5869,4571],{"className":5870,"style":4653},[1148,1149],[1052,5872,1102],{"className":5873,"style":4786},[1148,4785],[1052,5875,5877],{"className":5876},[1865],[1052,5878,5880,5920],{"className":5879},[1055],[1052,5881,5883],{"className":5882},[1059],[1061,5884,5885],{"xmlns":1063,"display":1874},[1065,5886,5887,5917],{},[1068,5888,5889,5891,5893,5895,5897,5903,5913,5915],{},[1071,5890,3796],{"mathvariant":4609},[1075,5892,1077],{},[1075,5894,1084],{"stretchy":1080},[1071,5896,4571],{},[5084,5898,5899,5901],{},[1071,5900,4571],{},[1071,5902,3606],{},[5084,5904,5905,5907],{},[1075,5906,1107],{"stretchy":1080},[1068,5908,5909,5911],{},[1075,5910,3988],{},[1121,5912,1123],{},[1071,5914,4571],{},[1071,5916,1102],{"mathvariant":4609},[1128,5918,5919],{"encoding":1130},"\\mathbf{a} = (XX^T)^{-1}X\\mathbf{y}",[1052,5921,5923,5941],{"className":5922,"ariaHidden":1096},[1135],[1052,5924,5926,5929,5932,5935,5938],{"className":5925},[1139],[1052,5927],{"className":5928,"style":5447},[1143],[1052,5930,3796],{"className":5931},[1148,4785],[1052,5933],{"className":5934,"style":1155},[1154],[1052,5936,1077],{"className":5937},[1159],[1052,5939],{"className":5940,"style":1155},[1154],[1052,5942,5944,5947,5950,5953,5982,6018,6021],{"className":5943},[1139],[1052,5945],{"className":5946,"style":5212},[1143],[1052,5948,1084],{"className":5949},[1173],[1052,5951,4571],{"className":5952,"style":4653},[1148,1149],[1052,5954,5956,5959],{"className":5955},[1148],[1052,5957,4571],{"className":5958,"style":4653},[1148,1149],[1052,5960,5962],{"className":5961},[1184],[1052,5963,5965],{"className":5964},[1188],[1052,5966,5968],{"className":5967},[1193],[1052,5969,5971],{"className":5970,"style":5231},[1197],[1052,5972,5973,5976],{"style":5234},[1052,5974],{"className":5975,"style":1206},[1205],[1052,5977,5979],{"className":5978},[1210,1211,1212,1213],[1052,5980,3606],{"className":5981,"style":2283},[1148,1149,1213],[1052,5983,5985,5988],{"className":5984},[1284],[1052,5986,1107],{"className":5987},[1284],[1052,5989,5991],{"className":5990},[1184],[1052,5992,5994],{"className":5993},[1188],[1052,5995,5997],{"className":5996},[1193],[1052,5998,6001],{"className":5999,"style":6000},[1197],"height:0.8641em;",[1052,6002,6003,6006],{"style":5234},[1052,6004],{"className":6005,"style":1206},[1205],[1052,6007,6009],{"className":6008},[1210,1211,1212,1213],[1052,6010,6012,6015],{"className":6011},[1148,1213],[1052,6013,3988],{"className":6014},[1148,1213],[1052,6016,1123],{"className":6017},[1148,1213],[1052,6019,4571],{"className":6020,"style":4653},[1148,1149],[1052,6022,1102],{"className":6023,"style":4786},[1148,4785],[1045,6025,6026],{"id":6026},"矩阵求梯度详解",[4548,6028,1047],{"id":6029},"基本概念-1",[996,6031,6032],{},"矩阵求梯度是指对标量函数相对于向量参数求偏导数，结果是一个与参数向量同维度的梯度向量。",[4548,6034,6035],{"id":6035},"常用矩阵求导公式",[996,6037,6038],{},"在推导前，我们需要掌握几个重要的矩阵求导公式：",[996,6040,6041],{},[1711,6042],{"alt":1711,"src":6043},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-09.png",[996,6045,6046],{},[1711,6047],{"alt":1711,"src":6048},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-10.png",[4548,6050,6051],{"id":6051},"具体推导过程",[996,6053,6054],{},"让我们详细推导均方误差函数的梯度：",[1052,6056,6058],{"className":6057},[1865],[1052,6059,6061,6122],{"className":6060},[1055],[1052,6062,6064],{"className":6063},[1059],[1061,6065,6066],{"xmlns":1063,"display":1874},[1065,6067,6068,6120],{},[1068,6069,6070,6076,6078,6080,6082,6084,6086,6088,6090,6096,6098,6104,6106,6108,6110,6116,6118],{},[1086,6071,6072,6074],{},[1071,6073,3117],{},[1071,6075,1791],{},[1075,6077,1084],{"stretchy":1080},[1071,6079,3796],{"mathvariant":4609},[1075,6081,1107],{"stretchy":1080},[1075,6083,1077],{},[1075,6085,1084],{"stretchy":1080},[1071,6087,1102],{"mathvariant":4609},[1075,6089,3988],{},[5084,6091,6092,6094],{},[1071,6093,4571],{},[1071,6095,3606],{},[1071,6097,3796],{"mathvariant":4609},[5084,6099,6100,6102],{},[1075,6101,1107],{"stretchy":1080},[1071,6103,3606],{},[1075,6105,1084],{"stretchy":1080},[1071,6107,1102],{"mathvariant":4609},[1075,6109,3988],{},[5084,6111,6112,6114],{},[1071,6113,4571],{},[1071,6115,3606],{},[1071,6117,3796],{"mathvariant":4609},[1075,6119,1107],{"stretchy":1080},[1128,6121,5116],{"encoding":1130},[1052,6123,6125,6189,6210,6292],{"className":6124,"ariaHidden":1096},[1135],[1052,6126,6128,6131,6171,6174,6177,6180,6183,6186],{"className":6127},[1139],[1052,6129],{"className":6130,"style":1597},[1143],[1052,6132,6134,6137],{"className":6133},[1148],[1052,6135,3117],{"className":6136,"style":3397},[1148,1149],[1052,6138,6140],{"className":6139},[1184],[1052,6141,6143,6163],{"className":6142},[1188,1189],[1052,6144,6146,6160],{"className":6145},[1193],[1052,6147,6149],{"className":6148,"style":1825},[1197],[1052,6150,6151,6154],{"style":5146},[1052,6152],{"className":6153,"style":1206},[1205],[1052,6155,6157],{"className":6156},[1210,1211,1212,1213],[1052,6158,1791],{"className":6159},[1148,1149,1213],[1052,6161,1221],{"className":6162},[1220],[1052,6164,6166],{"className":6165},[1193],[1052,6167,6169],{"className":6168,"style":1228},[1197],[1052,6170],{},[1052,6172,1084],{"className":6173},[1173],[1052,6175,3796],{"className":6176},[1148,4785],[1052,6178,1107],{"className":6179},[1284],[1052,6181],{"className":6182,"style":1155},[1154],[1052,6184,1077],{"className":6185},[1159],[1052,6187],{"className":6188,"style":1155},[1154],[1052,6190,6192,6195,6198,6201,6204,6207],{"className":6191},[1139],[1052,6193],{"className":6194,"style":1597},[1143],[1052,6196,1084],{"className":6197},[1173],[1052,6199,1102],{"className":6200,"style":4786},[1148,4785],[1052,6202],{"className":6203,"style":3378},[1154],[1052,6205,3988],{"className":6206},[2428],[1052,6208],{"className":6209,"style":3378},[1154],[1052,6211,6213,6216,6245,6248,6277,6280,6283,6286,6289],{"className":6212},[1139],[1052,6214],{"className":6215,"style":5212},[1143],[1052,6217,6219,6222],{"className":6218},[1148],[1052,6220,4571],{"className":6221,"style":4653},[1148,1149],[1052,6223,6225],{"className":6224},[1184],[1052,6226,6228],{"className":6227},[1188],[1052,6229,6231],{"className":6230},[1193],[1052,6232,6234],{"className":6233,"style":5231},[1197],[1052,6235,6236,6239],{"style":5234},[1052,6237],{"className":6238,"style":1206},[1205],[1052,6240,6242],{"className":6241},[1210,1211,1212,1213],[1052,6243,3606],{"className":6244,"style":2283},[1148,1149,1213],[1052,6246,3796],{"className":6247},[1148,4785],[1052,6249,6251,6254],{"className":6250},[1284],[1052,6252,1107],{"className":6253},[1284],[1052,6255,6257],{"className":6256},[1184],[1052,6258,6260],{"className":6259},[1188],[1052,6261,6263],{"className":6262},[1193],[1052,6264,6266],{"className":6265,"style":5231},[1197],[1052,6267,6268,6271],{"style":5234},[1052,6269],{"className":6270,"style":1206},[1205],[1052,6272,6274],{"className":6273},[1210,1211,1212,1213],[1052,6275,3606],{"className":6276,"style":2283},[1148,1149,1213],[1052,6278,1084],{"className":6279},[1173],[1052,6281,1102],{"className":6282,"style":4786},[1148,4785],[1052,6284],{"className":6285,"style":3378},[1154],[1052,6287,3988],{"className":6288},[2428],[1052,6290],{"className":6291,"style":3378},[1154],[1052,6293,6295,6298,6327,6330],{"className":6294},[1139],[1052,6296],{"className":6297,"style":5212},[1143],[1052,6299,6301,6304],{"className":6300},[1148],[1052,6302,4571],{"className":6303,"style":4653},[1148,1149],[1052,6305,6307],{"className":6306},[1184],[1052,6308,6310],{"className":6309},[1188],[1052,6311,6313],{"className":6312},[1193],[1052,6314,6316],{"className":6315,"style":5231},[1197],[1052,6317,6318,6321],{"style":5234},[1052,6319],{"className":6320,"style":1206},[1205],[1052,6322,6324],{"className":6323},[1210,1211,1212,1213],[1052,6325,3606],{"className":6326,"style":2283},[1148,1149,1213],[1052,6328,3796],{"className":6329},[1148,4785],[1052,6331,1107],{"className":6332},[1284],[4548,6334,6336],{"id":6335},"第一步展开二次型","第一步：展开二次型",[1052,6338,6340],{"className":6339},[1865],[1052,6341,6343,6404],{"className":6342},[1055],[1052,6344,6346],{"className":6345},[1059],[1061,6347,6348],{"xmlns":1063,"display":1874},[1065,6349,6350,6402],{},[1068,6351,6352,6358,6360,6362,6364,6366,6368,6370,6372,6378,6380,6386,6388,6390,6392,6398,6400],{},[1086,6353,6354,6356],{},[1071,6355,3117],{},[1071,6357,1791],{},[1075,6359,1084],{"stretchy":1080},[1071,6361,3796],{"mathvariant":4609},[1075,6363,1107],{"stretchy":1080},[1075,6365,1077],{},[1075,6367,1084],{"stretchy":1080},[1071,6369,1102],{"mathvariant":4609},[1075,6371,3988],{},[5084,6373,6374,6376],{},[1071,6375,4571],{},[1071,6377,3606],{},[1071,6379,3796],{"mathvariant":4609},[5084,6381,6382,6384],{},[1075,6383,1107],{"stretchy":1080},[1071,6385,3606],{},[1075,6387,1084],{"stretchy":1080},[1071,6389,1102],{"mathvariant":4609},[1075,6391,3988],{},[5084,6393,6394,6396],{},[1071,6395,4571],{},[1071,6397,3606],{},[1071,6399,3796],{"mathvariant":4609},[1075,6401,1107],{"stretchy":1080},[1128,6403,5116],{"encoding":1130},[1052,6405,6407,6471,6492,6574],{"className":6406,"ariaHidden":1096},[1135],[1052,6408,6410,6413,6453,6456,6459,6462,6465,6468],{"className":6409},[1139],[1052,6411],{"className":6412,"style":1597},[1143],[1052,6414,6416,6419],{"className":6415},[1148],[1052,6417,3117],{"className":6418,"style":3397},[1148,1149],[1052,6420,6422],{"className":6421},[1184],[1052,6423,6425,6445],{"className":6424},[1188,1189],[1052,6426,6428,6442],{"className":6427},[1193],[1052,6429,6431],{"className":6430,"style":1825},[1197],[1052,6432,6433,6436],{"style":5146},[1052,6434],{"className":6435,"style":1206},[1205],[1052,6437,6439],{"className":6438},[1210,1211,1212,1213],[1052,6440,1791],{"className":6441},[1148,1149,1213],[1052,6443,1221],{"className":6444},[1220],[1052,6446,6448],{"className":6447},[1193],[1052,6449,6451],{"className":6450,"style":1228},[1197],[1052,6452],{},[1052,6454,1084],{"className":6455},[1173],[1052,6457,3796],{"className":6458},[1148,4785],[1052,6460,1107],{"className":6461},[1284],[1052,6463],{"className":6464,"style":1155},[1154],[1052,6466,1077],{"className":6467},[1159],[1052,6469],{"className":6470,"style":1155},[1154],[1052,6472,6474,6477,6480,6483,6486,6489],{"className":6473},[1139],[1052,6475],{"className":6476,"style":1597},[1143],[1052,6478,1084],{"className":6479},[1173],[1052,6481,1102],{"className":6482,"style":4786},[1148,4785],[1052,6484],{"className":6485,"style":3378},[1154],[1052,6487,3988],{"className":6488},[2428],[1052,6490],{"className":6491,"style":3378},[1154],[1052,6493,6495,6498,6527,6530,6559,6562,6565,6568,6571],{"className":6494},[1139],[1052,6496],{"className":6497,"style":5212},[1143],[1052,6499,6501,6504],{"className":6500},[1148],[1052,6502,4571],{"className":6503,"style":4653},[1148,1149],[1052,6505,6507],{"className":6506},[1184],[1052,6508,6510],{"className":6509},[1188],[1052,6511,6513],{"className":6512},[1193],[1052,6514,6516],{"className":6515,"style":5231},[1197],[1052,6517,6518,6521],{"style":5234},[1052,6519],{"className":6520,"style":1206},[1205],[1052,6522,6524],{"className":6523},[1210,1211,1212,1213],[1052,6525,3606],{"className":6526,"style":2283},[1148,1149,1213],[1052,6528,3796],{"className":6529},[1148,4785],[1052,6531,6533,6536],{"className":6532},[1284],[1052,6534,1107],{"className":6535},[1284],[1052,6537,6539],{"className":6538},[1184],[1052,6540,6542],{"className":6541},[1188],[1052,6543,6545],{"className":6544},[1193],[1052,6546,6548],{"className":6547,"style":5231},[1197],[1052,6549,6550,6553],{"style":5234},[1052,6551],{"className":6552,"style":1206},[1205],[1052,6554,6556],{"className":6555},[1210,1211,1212,1213],[1052,6557,3606],{"className":6558,"style":2283},[1148,1149,1213],[1052,6560,1084],{"className":6561},[1173],[1052,6563,1102],{"className":6564,"style":4786},[1148,4785],[1052,6566],{"className":6567,"style":3378},[1154],[1052,6569,3988],{"className":6570},[2428],[1052,6572],{"className":6573,"style":3378},[1154],[1052,6575,6577,6580,6609,6612],{"className":6576},[1139],[1052,6578],{"className":6579,"style":5212},[1143],[1052,6581,6583,6586],{"className":6582},[1148],[1052,6584,4571],{"className":6585,"style":4653},[1148,1149],[1052,6587,6589],{"className":6588},[1184],[1052,6590,6592],{"className":6591},[1188],[1052,6593,6595],{"className":6594},[1193],[1052,6596,6598],{"className":6597,"style":5231},[1197],[1052,6599,6600,6603],{"style":5234},[1052,6601],{"className":6602,"style":1206},[1205],[1052,6604,6606],{"className":6605},[1210,1211,1212,1213],[1052,6607,3606],{"className":6608,"style":2283},[1148,1149,1213],[1052,6610,3796],{"className":6611},[1148,4785],[1052,6613,1107],{"className":6614},[1284],[1052,6616,6618],{"className":6617},[1865],[1052,6619,6621,6689],{"className":6620},[1055],[1052,6622,6624],{"className":6623},[1059],[1061,6625,6626],{"xmlns":1063,"display":1874},[1065,6627,6628,6686],{},[1068,6629,6630,6632,6638,6640,6642,6648,6654,6656,6658,6664,6666,6668,6670,6676,6678,6684],{},[1075,6631,1077],{},[5084,6633,6634,6636],{},[1071,6635,1102],{"mathvariant":4609},[1071,6637,3606],{},[1071,6639,1102],{"mathvariant":4609},[1075,6641,3988],{},[5084,6643,6644,6646],{},[1071,6645,1102],{"mathvariant":4609},[1071,6647,3606],{},[5084,6649,6650,6652],{},[1071,6651,4571],{},[1071,6653,3606],{},[1071,6655,3796],{"mathvariant":4609},[1075,6657,3988],{},[5084,6659,6660,6662],{},[1071,6661,3796],{"mathvariant":4609},[1071,6663,3606],{},[1071,6665,4571],{},[1071,6667,1102],{"mathvariant":4609},[1075,6669,3111],{},[5084,6671,6672,6674],{},[1071,6673,3796],{"mathvariant":4609},[1071,6675,3606],{},[1071,6677,4571],{},[5084,6679,6680,6682],{},[1071,6681,4571],{},[1071,6683,3606],{},[1071,6685,3796],{"mathvariant":4609},[1128,6687,6688],{"encoding":1130},"= \\mathbf{y}^T\\mathbf{y} - \\mathbf{y}^T X^T\\mathbf{a} - \\mathbf{a}^T X \\mathbf{y} + \\mathbf{a}^T X X^T \\mathbf{a}",[1052,6690,6692,6705,6753,6829,6879],{"className":6691,"ariaHidden":1096},[1135],[1052,6693,6695,6699,6702],{"className":6694},[1139],[1052,6696],{"className":6697,"style":6698},[1143],"height:0.3669em;",[1052,6700,1077],{"className":6701},[1159],[1052,6703],{"className":6704,"style":1155},[1154],[1052,6706,6708,6712,6741,6744,6747,6750],{"className":6707},[1139],[1052,6709],{"className":6710,"style":6711},[1143],"height:1.0858em;vertical-align:-0.1944em;",[1052,6713,6715,6718],{"className":6714},[1148],[1052,6716,1102],{"className":6717,"style":4786},[1148,4785],[1052,6719,6721],{"className":6720},[1184],[1052,6722,6724],{"className":6723},[1188],[1052,6725,6727],{"className":6726},[1193],[1052,6728,6730],{"className":6729,"style":5231},[1197],[1052,6731,6732,6735],{"style":5234},[1052,6733],{"className":6734,"style":1206},[1205],[1052,6736,6738],{"className":6737},[1210,1211,1212,1213],[1052,6739,3606],{"className":6740,"style":2283},[1148,1149,1213],[1052,6742,1102],{"className":6743,"style":4786},[1148,4785],[1052,6745],{"className":6746,"style":3378},[1154],[1052,6748,3988],{"className":6749},[2428],[1052,6751],{"className":6752,"style":3378},[1154],[1052,6754,6756,6759,6788,6817,6820,6823,6826],{"className":6755},[1139],[1052,6757],{"className":6758,"style":6711},[1143],[1052,6760,6762,6765],{"className":6761},[1148],[1052,6763,1102],{"className":6764,"style":4786},[1148,4785],[1052,6766,6768],{"className":6767},[1184],[1052,6769,6771],{"className":6770},[1188],[1052,6772,6774],{"className":6773},[1193],[1052,6775,6777],{"className":6776,"style":5231},[1197],[1052,6778,6779,6782],{"style":5234},[1052,6780],{"className":6781,"style":1206},[1205],[1052,6783,6785],{"className":6784},[1210,1211,1212,1213],[1052,6786,3606],{"className":6787,"style":2283},[1148,1149,1213],[1052,6789,6791,6794],{"className":6790},[1148],[1052,6792,4571],{"className":6793,"style":4653},[1148,1149],[1052,6795,6797],{"className":6796},[1184],[1052,6798,6800],{"className":6799},[1188],[1052,6801,6803],{"className":6802},[1193],[1052,6804,6806],{"className":6805,"style":5231},[1197],[1052,6807,6808,6811],{"style":5234},[1052,6809],{"className":6810,"style":1206},[1205],[1052,6812,6814],{"className":6813},[1210,1211,1212,1213],[1052,6815,3606],{"className":6816,"style":2283},[1148,1149,1213],[1052,6818,3796],{"className":6819},[1148,4785],[1052,6821],{"className":6822,"style":3378},[1154],[1052,6824,3988],{"className":6825},[2428],[1052,6827],{"className":6828,"style":3378},[1154],[1052,6830,6832,6835,6864,6867,6870,6873,6876],{"className":6831},[1139],[1052,6833],{"className":6834,"style":6711},[1143],[1052,6836,6838,6841],{"className":6837},[1148],[1052,6839,3796],{"className":6840},[1148,4785],[1052,6842,6844],{"className":6843},[1184],[1052,6845,6847],{"className":6846},[1188],[1052,6848,6850],{"className":6849},[1193],[1052,6851,6853],{"className":6852,"style":5231},[1197],[1052,6854,6855,6858],{"style":5234},[1052,6856],{"className":6857,"style":1206},[1205],[1052,6859,6861],{"className":6860},[1210,1211,1212,1213],[1052,6862,3606],{"className":6863,"style":2283},[1148,1149,1213],[1052,6865,4571],{"className":6866,"style":4653},[1148,1149],[1052,6868,1102],{"className":6869,"style":4786},[1148,4785],[1052,6871],{"className":6872,"style":3378},[1154],[1052,6874,3111],{"className":6875},[2428],[1052,6877],{"className":6878,"style":3378},[1154],[1052,6880,6882,6885,6914,6917,6946],{"className":6881},[1139],[1052,6883],{"className":6884,"style":5231},[1143],[1052,6886,6888,6891],{"className":6887},[1148],[1052,6889,3796],{"className":6890},[1148,4785],[1052,6892,6894],{"className":6893},[1184],[1052,6895,6897],{"className":6896},[1188],[1052,6898,6900],{"className":6899},[1193],[1052,6901,6903],{"className":6902,"style":5231},[1197],[1052,6904,6905,6908],{"style":5234},[1052,6906],{"className":6907,"style":1206},[1205],[1052,6909,6911],{"className":6910},[1210,1211,1212,1213],[1052,6912,3606],{"className":6913,"style":2283},[1148,1149,1213],[1052,6915,4571],{"className":6916,"style":4653},[1148,1149],[1052,6918,6920,6923],{"className":6919},[1148],[1052,6921,4571],{"className":6922,"style":4653},[1148,1149],[1052,6924,6926],{"className":6925},[1184],[1052,6927,6929],{"className":6928},[1188],[1052,6930,6932],{"className":6931},[1193],[1052,6933,6935],{"className":6934,"style":5231},[1197],[1052,6936,6937,6940],{"style":5234},[1052,6938],{"className":6939,"style":1206},[1205],[1052,6941,6943],{"className":6942},[1210,1211,1212,1213],[1052,6944,3606],{"className":6945,"style":2283},[1148,1149,1213],[1052,6947,3796],{"className":6948},[1148,4785],[4548,6950,6952],{"id":6951},"第二步利用对称性简化","第二步：利用对称性简化",[996,6954,6955,6956,7057,7058],{},"注意到 ",[1052,6957,6959,6985],{"className":6958},[1055],[1052,6960,6962],{"className":6961},[1059],[1061,6963,6964],{"xmlns":1063},[1065,6965,6966,6982],{},[1068,6967,6968,6974,6980],{},[5084,6969,6970,6972],{},[1071,6971,1102],{"mathvariant":4609},[1071,6973,3606],{},[5084,6975,6976,6978],{},[1071,6977,4571],{},[1071,6979,3606],{},[1071,6981,3796],{"mathvariant":4609},[1128,6983,6984],{"encoding":1130},"\\mathbf{y}^T X^T\\mathbf{a}",[1052,6986,6988],{"className":6987,"ariaHidden":1096},[1135],[1052,6989,6991,6995,7025,7054],{"className":6990},[1139],[1052,6992],{"className":6993,"style":6994},[1143],"height:1.0358em;vertical-align:-0.1944em;",[1052,6996,6998,7001],{"className":6997},[1148],[1052,6999,1102],{"className":7000,"style":4786},[1148,4785],[1052,7002,7004],{"className":7003},[1184],[1052,7005,7007],{"className":7006},[1188],[1052,7008,7010],{"className":7009},[1193],[1052,7011,7014],{"className":7012,"style":7013},[1197],"height:0.8413em;",[1052,7015,7016,7019],{"style":1327},[1052,7017],{"className":7018,"style":1206},[1205],[1052,7020,7022],{"className":7021},[1210,1211,1212,1213],[1052,7023,3606],{"className":7024,"style":2283},[1148,1149,1213],[1052,7026,7028,7031],{"className":7027},[1148],[1052,7029,4571],{"className":7030,"style":4653},[1148,1149],[1052,7032,7034],{"className":7033},[1184],[1052,7035,7037],{"className":7036},[1188],[1052,7038,7040],{"className":7039},[1193],[1052,7041,7043],{"className":7042,"style":7013},[1197],[1052,7044,7045,7048],{"style":1327},[1052,7046],{"className":7047,"style":1206},[1205],[1052,7049,7051],{"className":7050},[1210,1211,1212,1213],[1052,7052,3606],{"className":7053,"style":2283},[1148,1149,1213],[1052,7055,3796],{"className":7056},[1148,4785]," 是标量，所以： ",[1052,7059,7061,7123],{"className":7060},[1055],[1052,7062,7064],{"className":7063},[1059],[1061,7065,7066],{"xmlns":1063},[1065,7067,7068,7120],{},[1068,7069,7070,7076,7082,7084,7086,7088,7094,7100,7102,7108,7110,7116,7118],{},[5084,7071,7072,7074],{},[1071,7073,1102],{"mathvariant":4609},[1071,7075,3606],{},[5084,7077,7078,7080],{},[1071,7079,4571],{},[1071,7081,3606],{},[1071,7083,3796],{"mathvariant":4609},[1075,7085,1077],{},[1075,7087,1084],{"stretchy":1080},[5084,7089,7090,7092],{},[1071,7091,1102],{"mathvariant":4609},[1071,7093,3606],{},[5084,7095,7096,7098],{},[1071,7097,4571],{},[1071,7099,3606],{},[1071,7101,3796],{"mathvariant":4609},[5084,7103,7104,7106],{},[1075,7105,1107],{"stretchy":1080},[1071,7107,3606],{},[1075,7109,1077],{},[5084,7111,7112,7114],{},[1071,7113,3796],{"mathvariant":4609},[1071,7115,3606],{},[1071,7117,4571],{},[1071,7119,1102],{"mathvariant":4609},[1128,7121,7122],{"encoding":1130},"\\mathbf{y}^T X^T\\mathbf{a} = (\\mathbf{y}^T X^T\\mathbf{a})^T = \\mathbf{a}^T X \\mathbf{y}",[1052,7124,7126,7202,7311],{"className":7125,"ariaHidden":1096},[1135],[1052,7127,7129,7132,7161,7190,7193,7196,7199],{"className":7128},[1139],[1052,7130],{"className":7131,"style":6994},[1143],[1052,7133,7135,7138],{"className":7134},[1148],[1052,7136,1102],{"className":7137,"style":4786},[1148,4785],[1052,7139,7141],{"className":7140},[1184],[1052,7142,7144],{"className":7143},[1188],[1052,7145,7147],{"className":7146},[1193],[1052,7148,7150],{"className":7149,"style":7013},[1197],[1052,7151,7152,7155],{"style":1327},[1052,7153],{"className":7154,"style":1206},[1205],[1052,7156,7158],{"className":7157},[1210,1211,1212,1213],[1052,7159,3606],{"className":7160,"style":2283},[1148,1149,1213],[1052,7162,7164,7167],{"className":7163},[1148],[1052,7165,4571],{"className":7166,"style":4653},[1148,1149],[1052,7168,7170],{"className":7169},[1184],[1052,7171,7173],{"className":7172},[1188],[1052,7174,7176],{"className":7175},[1193],[1052,7177,7179],{"className":7178,"style":7013},[1197],[1052,7180,7181,7184],{"style":1327},[1052,7182],{"className":7183,"style":1206},[1205],[1052,7185,7187],{"className":7186},[1210,1211,1212,1213],[1052,7188,3606],{"className":7189,"style":2283},[1148,1149,1213],[1052,7191,3796],{"className":7192},[1148,4785],[1052,7194],{"className":7195,"style":1155},[1154],[1052,7197,1077],{"className":7198},[1159],[1052,7200],{"className":7201,"style":1155},[1154],[1052,7203,7205,7209,7212,7241,7270,7273,7302,7305,7308],{"className":7204},[1139],[1052,7206],{"className":7207,"style":7208},[1143],"height:1.0913em;vertical-align:-0.25em;",[1052,7210,1084],{"className":7211},[1173],[1052,7213,7215,7218],{"className":7214},[1148],[1052,7216,1102],{"className":7217,"style":4786},[1148,4785],[1052,7219,7221],{"className":7220},[1184],[1052,7222,7224],{"className":7223},[1188],[1052,7225,7227],{"className":7226},[1193],[1052,7228,7230],{"className":7229,"style":7013},[1197],[1052,7231,7232,7235],{"style":1327},[1052,7233],{"className":7234,"style":1206},[1205],[1052,7236,7238],{"className":7237},[1210,1211,1212,1213],[1052,7239,3606],{"className":7240,"style":2283},[1148,1149,1213],[1052,7242,7244,7247],{"className":7243},[1148],[1052,7245,4571],{"className":7246,"style":4653},[1148,1149],[1052,7248,7250],{"className":7249},[1184],[1052,7251,7253],{"className":7252},[1188],[1052,7254,7256],{"className":7255},[1193],[1052,7257,7259],{"className":7258,"style":7013},[1197],[1052,7260,7261,7264],{"style":1327},[1052,7262],{"className":7263,"style":1206},[1205],[1052,7265,7267],{"className":7266},[1210,1211,1212,1213],[1052,7268,3606],{"className":7269,"style":2283},[1148,1149,1213],[1052,7271,3796],{"className":7272},[1148,4785],[1052,7274,7276,7279],{"className":7275},[1284],[1052,7277,1107],{"className":7278},[1284],[1052,7280,7282],{"className":7281},[1184],[1052,7283,7285],{"className":7284},[1188],[1052,7286,7288],{"className":7287},[1193],[1052,7289,7291],{"className":7290,"style":7013},[1197],[1052,7292,7293,7296],{"style":1327},[1052,7294],{"className":7295,"style":1206},[1205],[1052,7297,7299],{"className":7298},[1210,1211,1212,1213],[1052,7300,3606],{"className":7301,"style":2283},[1148,1149,1213],[1052,7303],{"className":7304,"style":1155},[1154],[1052,7306,1077],{"className":7307},[1159],[1052,7309],{"className":7310,"style":1155},[1154],[1052,7312,7314,7317,7346,7349],{"className":7313},[1139],[1052,7315],{"className":7316,"style":6994},[1143],[1052,7318,7320,7323],{"className":7319},[1148],[1052,7321,3796],{"className":7322},[1148,4785],[1052,7324,7326],{"className":7325},[1184],[1052,7327,7329],{"className":7328},[1188],[1052,7330,7332],{"className":7331},[1193],[1052,7333,7335],{"className":7334,"style":7013},[1197],[1052,7336,7337,7340],{"style":1327},[1052,7338],{"className":7339,"style":1206},[1205],[1052,7341,7343],{"className":7342},[1210,1211,1212,1213],[1052,7344,3606],{"className":7345,"style":2283},[1148,1149,1213],[1052,7347,4571],{"className":7348,"style":4653},[1148,1149],[1052,7350,1102],{"className":7351,"style":4786},[1148,4785],[996,7353,7354,7355],{},"因此： ",[1052,7356,7358,7424],{"className":7357},[1055],[1052,7359,7361],{"className":7360},[1059],[1061,7362,7363],{"xmlns":1063},[1065,7364,7365,7421],{},[1068,7366,7367,7373,7375,7377,7379,7381,7387,7389,7391,7393,7399,7401,7403,7405,7411,7413,7419],{},[1086,7368,7369,7371],{},[1071,7370,3117],{},[1071,7372,1791],{},[1075,7374,1084],{"stretchy":1080},[1071,7376,3796],{"mathvariant":4609},[1075,7378,1107],{"stretchy":1080},[1075,7380,1077],{},[5084,7382,7383,7385],{},[1071,7384,1102],{"mathvariant":4609},[1071,7386,3606],{},[1071,7388,1102],{"mathvariant":4609},[1075,7390,3988],{},[1121,7392,5484],{},[5084,7394,7395,7397],{},[1071,7396,3796],{"mathvariant":4609},[1071,7398,3606],{},[1071,7400,4571],{},[1071,7402,1102],{"mathvariant":4609},[1075,7404,3111],{},[5084,7406,7407,7409],{},[1071,7408,3796],{"mathvariant":4609},[1071,7410,3606],{},[1071,7412,4571],{},[5084,7414,7415,7417],{},[1071,7416,4571],{},[1071,7418,3606],{},[1071,7420,3796],{"mathvariant":4609},[1128,7422,7423],{"encoding":1130},"J_m(\\mathbf{a}) = \\mathbf{y}^T\\mathbf{y} - 2\\mathbf{a}^T X \\mathbf{y} + \\mathbf{a}^T X X^T \\mathbf{a}",[1052,7425,7427,7491,7538,7591],{"className":7426,"ariaHidden":1096},[1135],[1052,7428,7430,7433,7473,7476,7479,7482,7485,7488],{"className":7429},[1139],[1052,7431],{"className":7432,"style":1597},[1143],[1052,7434,7436,7439],{"className":7435},[1148],[1052,7437,3117],{"className":7438,"style":3397},[1148,1149],[1052,7440,7442],{"className":7441},[1184],[1052,7443,7445,7465],{"className":7444},[1188,1189],[1052,7446,7448,7462],{"className":7447},[1193],[1052,7449,7451],{"className":7450,"style":1825},[1197],[1052,7452,7453,7456],{"style":5146},[1052,7454],{"className":7455,"style":1206},[1205],[1052,7457,7459],{"className":7458},[1210,1211,1212,1213],[1052,7460,1791],{"className":7461},[1148,1149,1213],[1052,7463,1221],{"className":7464},[1220],[1052,7466,7468],{"className":7467},[1193],[1052,7469,7471],{"className":7470,"style":1228},[1197],[1052,7472],{},[1052,7474,1084],{"className":7475},[1173],[1052,7477,3796],{"className":7478},[1148,4785],[1052,7480,1107],{"className":7481},[1284],[1052,7483],{"className":7484,"style":1155},[1154],[1052,7486,1077],{"className":7487},[1159],[1052,7489],{"className":7490,"style":1155},[1154],[1052,7492,7494,7497,7526,7529,7532,7535],{"className":7493},[1139],[1052,7495],{"className":7496,"style":6994},[1143],[1052,7498,7500,7503],{"className":7499},[1148],[1052,7501,1102],{"className":7502,"style":4786},[1148,4785],[1052,7504,7506],{"className":7505},[1184],[1052,7507,7509],{"className":7508},[1188],[1052,7510,7512],{"className":7511},[1193],[1052,7513,7515],{"className":7514,"style":7013},[1197],[1052,7516,7517,7520],{"style":1327},[1052,7518],{"className":7519,"style":1206},[1205],[1052,7521,7523],{"className":7522},[1210,1211,1212,1213],[1052,7524,3606],{"className":7525,"style":2283},[1148,1149,1213],[1052,7527,1102],{"className":7528,"style":4786},[1148,4785],[1052,7530],{"className":7531,"style":3378},[1154],[1052,7533,3988],{"className":7534},[2428],[1052,7536],{"className":7537,"style":3378},[1154],[1052,7539,7541,7544,7547,7576,7579,7582,7585,7588],{"className":7540},[1139],[1052,7542],{"className":7543,"style":6994},[1143],[1052,7545,5484],{"className":7546},[1148],[1052,7548,7550,7553],{"className":7549},[1148],[1052,7551,3796],{"className":7552},[1148,4785],[1052,7554,7556],{"className":7555},[1184],[1052,7557,7559],{"className":7558},[1188],[1052,7560,7562],{"className":7561},[1193],[1052,7563,7565],{"className":7564,"style":7013},[1197],[1052,7566,7567,7570],{"style":1327},[1052,7568],{"className":7569,"style":1206},[1205],[1052,7571,7573],{"className":7572},[1210,1211,1212,1213],[1052,7574,3606],{"className":7575,"style":2283},[1148,1149,1213],[1052,7577,4571],{"className":7578,"style":4653},[1148,1149],[1052,7580,1102],{"className":7581,"style":4786},[1148,4785],[1052,7583],{"className":7584,"style":3378},[1154],[1052,7586,3111],{"className":7587},[2428],[1052,7589],{"className":7590,"style":3378},[1154],[1052,7592,7594,7597,7626,7629,7658],{"className":7593},[1139],[1052,7595],{"className":7596,"style":7013},[1143],[1052,7598,7600,7603],{"className":7599},[1148],[1052,7601,3796],{"className":7602},[1148,4785],[1052,7604,7606],{"className":7605},[1184],[1052,7607,7609],{"className":7608},[1188],[1052,7610,7612],{"className":7611},[1193],[1052,7613,7615],{"className":7614,"style":7013},[1197],[1052,7616,7617,7620],{"style":1327},[1052,7618],{"className":7619,"style":1206},[1205],[1052,7621,7623],{"className":7622},[1210,1211,1212,1213],[1052,7624,3606],{"className":7625,"style":2283},[1148,1149,1213],[1052,7627,4571],{"className":7628,"style":4653},[1148,1149],[1052,7630,7632,7635],{"className":7631},[1148],[1052,7633,4571],{"className":7634,"style":4653},[1148,1149],[1052,7636,7638],{"className":7637},[1184],[1052,7639,7641],{"className":7640},[1188],[1052,7642,7644],{"className":7643},[1193],[1052,7645,7647],{"className":7646,"style":7013},[1197],[1052,7648,7649,7652],{"style":1327},[1052,7650],{"className":7651,"style":1206},[1205],[1052,7653,7655],{"className":7654},[1210,1211,1212,1213],[1052,7656,3606],{"className":7657,"style":2283},[1148,1149,1213],[1052,7659,3796],{"className":7660},[1148,4785],[4548,7662,7664],{"id":7663},"第三步对每一项求梯度","第三步：对每一项求梯度",[1015,7666,7667,7921,8213,8323],{},[983,7668,7669,3028,7672,7736,7737],{},[1020,7670,7671],{},"第一项",[1052,7673,7675,7695],{"className":7674},[1055],[1052,7676,7678],{"className":7677},[1059],[1061,7679,7680],{"xmlns":1063},[1065,7681,7682,7692],{},[1068,7683,7684,7690],{},[5084,7685,7686,7688],{},[1071,7687,1102],{"mathvariant":4609},[1071,7689,3606],{},[1071,7691,1102],{"mathvariant":4609},[1128,7693,7694],{"encoding":1130},"\\mathbf{y}^T\\mathbf{y}",[1052,7696,7698],{"className":7697,"ariaHidden":1096},[1135],[1052,7699,7701,7704,7733],{"className":7700},[1139],[1052,7702],{"className":7703,"style":6994},[1143],[1052,7705,7707,7710],{"className":7706},[1148],[1052,7708,1102],{"className":7709,"style":4786},[1148,4785],[1052,7711,7713],{"className":7712},[1184],[1052,7714,7716],{"className":7715},[1188],[1052,7717,7719],{"className":7718},[1193],[1052,7720,7722],{"className":7721,"style":7013},[1197],[1052,7723,7724,7727],{"style":1327},[1052,7725],{"className":7726,"style":1206},[1205],[1052,7728,7730],{"className":7729},[1210,1211,1212,1213],[1052,7731,3606],{"className":7732,"style":2283},[1148,1149,1213],[1052,7734,1102],{"className":7735,"style":4786},[1148,4785]," 是常数 ",[1052,7738,7740,7779],{"className":7739},[1055],[1052,7741,7743],{"className":7742},[1059],[1061,7744,7745],{"xmlns":1063},[1065,7746,7747,7776],{},[1068,7748,7749,7760,7762,7768,7770,7772,7774],{},[1880,7750,7751,7754],{},[1071,7752,7753],{"mathvariant":1575},"∂",[1068,7755,7756,7758],{},[1071,7757,7753],{"mathvariant":1575},[1071,7759,3796],{"mathvariant":4609},[1075,7761,1084],{"stretchy":1080},[5084,7763,7764,7766],{},[1071,7765,1102],{"mathvariant":4609},[1071,7767,3606],{},[1071,7769,1102],{"mathvariant":4609},[1075,7771,1107],{"stretchy":1080},[1075,7773,1077],{},[1121,7775,4994],{"mathvariant":4609},[1128,7777,7778],{"encoding":1130},"\\frac{\\partial}{\\partial \\mathbf{a}}(\\mathbf{y}^T\\mathbf{y}) = \\mathbf{0}",[1052,7780,7782,7912],{"className":7781,"ariaHidden":1096},[1135],[1052,7783,7785,7789,7865,7868,7897,7900,7903,7906,7909],{"className":7784},[1139],[1052,7786],{"className":7787,"style":7788},[1143],"height:1.2251em;vertical-align:-0.345em;",[1052,7790,7792,7795,7862],{"className":7791},[1148],[1052,7793],{"className":7794},[1173,1956],[1052,7796,7798],{"className":7797},[1880],[1052,7799,7801,7853],{"className":7800},[1188,1189],[1052,7802,7804,7850],{"className":7803},[1193],[1052,7805,7808,7827,7835],{"className":7806,"style":7807},[1197],"height:0.8801em;",[1052,7809,7811,7814],{"style":7810},"top:-2.655em;",[1052,7812],{"className":7813,"style":1976},[1205],[1052,7815,7817],{"className":7816},[1210,1211,1212,1213],[1052,7818,7820,7824],{"className":7819},[1148,1213],[1052,7821,7753],{"className":7822,"style":7823},[1148,1213],"margin-right:0.0556em;",[1052,7825,3796],{"className":7826},[1148,4785,1213],[1052,7828,7829,7832],{"style":1985},[1052,7830],{"className":7831,"style":1976},[1205],[1052,7833],{"className":7834,"style":1993},[1992],[1052,7836,7838,7841],{"style":7837},"top:-3.394em;",[1052,7839],{"className":7840,"style":1976},[1205],[1052,7842,7844],{"className":7843},[1210,1211,1212,1213],[1052,7845,7847],{"className":7846},[1148,1213],[1052,7848,7753],{"className":7849,"style":7823},[1148,1213],[1052,7851,1221],{"className":7852},[1220],[1052,7854,7856],{"className":7855},[1193],[1052,7857,7860],{"className":7858,"style":7859},[1197],"height:0.345em;",[1052,7861],{},[1052,7863],{"className":7864},[1284,1956],[1052,7866,1084],{"className":7867},[1173],[1052,7869,7871,7874],{"className":7870},[1148],[1052,7872,1102],{"className":7873,"style":4786},[1148,4785],[1052,7875,7877],{"className":7876},[1184],[1052,7878,7880],{"className":7879},[1188],[1052,7881,7883],{"className":7882},[1193],[1052,7884,7886],{"className":7885,"style":7013},[1197],[1052,7887,7888,7891],{"style":1327},[1052,7889],{"className":7890,"style":1206},[1205],[1052,7892,7894],{"className":7893},[1210,1211,1212,1213],[1052,7895,3606],{"className":7896,"style":2283},[1148,1149,1213],[1052,7898,1102],{"className":7899,"style":4786},[1148,4785],[1052,7901,1107],{"className":7902},[1284],[1052,7904],{"className":7905,"style":1155},[1154],[1052,7907,1077],{"className":7908},[1159],[1052,7910],{"className":7911,"style":1155},[1154],[1052,7913,7915,7918],{"className":7914},[1139],[1052,7916],{"className":7917,"style":5769},[1143],[1052,7919,4994],{"className":7920},[1148,4785],[983,7922,7923,3028,7926,8005,8006],{},[1020,7924,7925],{},"第二项",[1052,7927,7929,7955],{"className":7928},[1055],[1052,7930,7932],{"className":7931},[1059],[1061,7933,7934],{"xmlns":1063},[1065,7935,7936,7952],{},[1068,7937,7938,7940,7942,7948,7950],{},[1075,7939,3988],{},[1121,7941,5484],{},[5084,7943,7944,7946],{},[1071,7945,3796],{"mathvariant":4609},[1071,7947,3606],{},[1071,7949,4571],{},[1071,7951,1102],{"mathvariant":4609},[1128,7953,7954],{"encoding":1130},"-2\\mathbf{a}^T X \\mathbf{y}",[1052,7956,7958],{"className":7957,"ariaHidden":1096},[1135],[1052,7959,7961,7964,7967,7970,7999,8002],{"className":7960},[1139],[1052,7962],{"className":7963,"style":6994},[1143],[1052,7965,3988],{"className":7966},[1148],[1052,7968,5484],{"className":7969},[1148],[1052,7971,7973,7976],{"className":7972},[1148],[1052,7974,3796],{"className":7975},[1148,4785],[1052,7977,7979],{"className":7978},[1184],[1052,7980,7982],{"className":7981},[1188],[1052,7983,7985],{"className":7984},[1193],[1052,7986,7988],{"className":7987,"style":7013},[1197],[1052,7989,7990,7993],{"style":1327},[1052,7991],{"className":7992,"style":1206},[1205],[1052,7994,7996],{"className":7995},[1210,1211,1212,1213],[1052,7997,3606],{"className":7998,"style":2283},[1148,1149,1213],[1052,8000,4571],{"className":8001,"style":4653},[1148,1149],[1052,8003,1102],{"className":8004,"style":4786},[1148,4785]," 是线性项 ",[1052,8007,8009,8059],{"className":8008},[1055],[1052,8010,8012],{"className":8011},[1059],[1061,8013,8014],{"xmlns":1063},[1065,8015,8016,8056],{},[1068,8017,8018,8028,8030,8032,8034,8040,8042,8044,8046,8048,8050,8052,8054],{},[1880,8019,8020,8022],{},[1071,8021,7753],{"mathvariant":1575},[1068,8023,8024,8026],{},[1071,8025,7753],{"mathvariant":1575},[1071,8027,3796],{"mathvariant":4609},[1075,8029,1084],{"stretchy":1080},[1075,8031,3988],{},[1121,8033,5484],{},[5084,8035,8036,8038],{},[1071,8037,3796],{"mathvariant":4609},[1071,8039,3606],{},[1071,8041,4571],{},[1071,8043,1102],{"mathvariant":4609},[1075,8045,1107],{"stretchy":1080},[1075,8047,1077],{},[1075,8049,3988],{},[1121,8051,5484],{},[1071,8053,4571],{},[1071,8055,1102],{"mathvariant":4609},[1128,8057,8058],{"encoding":1130},"\\frac{\\partial}{\\partial \\mathbf{a}}(-2\\mathbf{a}^T X \\mathbf{y}) = -2X\\mathbf{y}",[1052,8060,8062,8195],{"className":8061,"ariaHidden":1096},[1135],[1052,8063,8065,8068,8139,8142,8145,8148,8177,8180,8183,8186,8189,8192],{"className":8064},[1139],[1052,8066],{"className":8067,"style":7788},[1143],[1052,8069,8071,8074,8136],{"className":8070},[1148],[1052,8072],{"className":8073},[1173,1956],[1052,8075,8077],{"className":8076},[1880],[1052,8078,8080,8128],{"className":8079},[1188,1189],[1052,8081,8083,8125],{"className":8082},[1193],[1052,8084,8086,8103,8111],{"className":8085,"style":7807},[1197],[1052,8087,8088,8091],{"style":7810},[1052,8089],{"className":8090,"style":1976},[1205],[1052,8092,8094],{"className":8093},[1210,1211,1212,1213],[1052,8095,8097,8100],{"className":8096},[1148,1213],[1052,8098,7753],{"className":8099,"style":7823},[1148,1213],[1052,8101,3796],{"className":8102},[1148,4785,1213],[1052,8104,8105,8108],{"style":1985},[1052,8106],{"className":8107,"style":1976},[1205],[1052,8109],{"className":8110,"style":1993},[1992],[1052,8112,8113,8116],{"style":7837},[1052,8114],{"className":8115,"style":1976},[1205],[1052,8117,8119],{"className":8118},[1210,1211,1212,1213],[1052,8120,8122],{"className":8121},[1148,1213],[1052,8123,7753],{"className":8124,"style":7823},[1148,1213],[1052,8126,1221],{"className":8127},[1220],[1052,8129,8131],{"className":8130},[1193],[1052,8132,8134],{"className":8133,"style":7859},[1197],[1052,8135],{},[1052,8137],{"className":8138},[1284,1956],[1052,8140,1084],{"className":8141},[1173],[1052,8143,3988],{"className":8144},[1148],[1052,8146,5484],{"className":8147},[1148],[1052,8149,8151,8154],{"className":8150},[1148],[1052,8152,3796],{"className":8153},[1148,4785],[1052,8155,8157],{"className":8156},[1184],[1052,8158,8160],{"className":8159},[1188],[1052,8161,8163],{"className":8162},[1193],[1052,8164,8166],{"className":8165,"style":7013},[1197],[1052,8167,8168,8171],{"style":1327},[1052,8169],{"className":8170,"style":1206},[1205],[1052,8172,8174],{"className":8173},[1210,1211,1212,1213],[1052,8175,3606],{"className":8176,"style":2283},[1148,1149,1213],[1052,8178,4571],{"className":8179,"style":4653},[1148,1149],[1052,8181,1102],{"className":8182,"style":4786},[1148,4785],[1052,8184,1107],{"className":8185},[1284],[1052,8187],{"className":8188,"style":1155},[1154],[1052,8190,1077],{"className":8191},[1159],[1052,8193],{"className":8194,"style":1155},[1154],[1052,8196,8198,8201,8204,8207,8210],{"className":8197},[1139],[1052,8199],{"className":8200,"style":5867},[1143],[1052,8202,3988],{"className":8203},[1148],[1052,8205,5484],{"className":8206},[1148],[1052,8208,4571],{"className":8209,"style":4653},[1148,1149],[1052,8211,1102],{"className":8212,"style":4786},[1148,4785],[983,8214,8215,3028,8218,8322],{},[1020,8216,8217],{},"第三项",[1052,8219,8221,8249],{"className":8220},[1055],[1052,8222,8224],{"className":8223},[1059],[1061,8225,8226],{"xmlns":1063},[1065,8227,8228,8246],{},[1068,8229,8230,8236,8238,8244],{},[5084,8231,8232,8234],{},[1071,8233,3796],{"mathvariant":4609},[1071,8235,3606],{},[1071,8237,4571],{},[5084,8239,8240,8242],{},[1071,8241,4571],{},[1071,8243,3606],{},[1071,8245,3796],{"mathvariant":4609},[1128,8247,8248],{"encoding":1130},"\\mathbf{a}^T X X^T \\mathbf{a}",[1052,8250,8252],{"className":8251,"ariaHidden":1096},[1135],[1052,8253,8255,8258,8287,8290,8319],{"className":8254},[1139],[1052,8256],{"className":8257,"style":7013},[1143],[1052,8259,8261,8264],{"className":8260},[1148],[1052,8262,3796],{"className":8263},[1148,4785],[1052,8265,8267],{"className":8266},[1184],[1052,8268,8270],{"className":8269},[1188],[1052,8271,8273],{"className":8272},[1193],[1052,8274,8276],{"className":8275,"style":7013},[1197],[1052,8277,8278,8281],{"style":1327},[1052,8279],{"className":8280,"style":1206},[1205],[1052,8282,8284],{"className":8283},[1210,1211,1212,1213],[1052,8285,3606],{"className":8286,"style":2283},[1148,1149,1213],[1052,8288,4571],{"className":8289,"style":4653},[1148,1149],[1052,8291,8293,8296],{"className":8292},[1148],[1052,8294,4571],{"className":8295,"style":4653},[1148,1149],[1052,8297,8299],{"className":8298},[1184],[1052,8300,8302],{"className":8301},[1188],[1052,8303,8305],{"className":8304},[1193],[1052,8306,8308],{"className":8307,"style":7013},[1197],[1052,8309,8310,8313],{"style":1327},[1052,8311],{"className":8312,"style":1206},[1205],[1052,8314,8316],{"className":8315},[1210,1211,1212,1213],[1052,8317,3606],{"className":8318,"style":2283},[1148,1149,1213],[1052,8320,3796],{"className":8321},[1148,4785]," 是二次项",[983,8324,8325,8326,8390,8391],{},"由于 ",[1052,8327,8329,8349],{"className":8328},[1055],[1052,8330,8332],{"className":8331},[1059],[1061,8333,8334],{"xmlns":1063},[1065,8335,8336,8346],{},[1068,8337,8338,8340],{},[1071,8339,4571],{},[5084,8341,8342,8344],{},[1071,8343,4571],{},[1071,8345,3606],{},[1128,8347,8348],{"encoding":1130},"XX^T",[1052,8350,8352],{"className":8351,"ariaHidden":1096},[1135],[1052,8353,8355,8358,8361],{"className":8354},[1139],[1052,8356],{"className":8357,"style":7013},[1143],[1052,8359,4571],{"className":8360,"style":4653},[1148,1149],[1052,8362,8364,8367],{"className":8363},[1148],[1052,8365,4571],{"className":8366,"style":4653},[1148,1149],[1052,8368,8370],{"className":8369},[1184],[1052,8371,8373],{"className":8372},[1188],[1052,8374,8376],{"className":8375},[1193],[1052,8377,8379],{"className":8378,"style":7013},[1197],[1052,8380,8381,8384],{"style":1327},[1052,8382],{"className":8383,"style":1206},[1205],[1052,8385,8387],{"className":8386},[1210,1211,1212,1213],[1052,8388,3606],{"className":8389,"style":2283},[1148,1149,1213]," 是对称矩阵，根据公式： ",[1052,8392,8394,8450],{"className":8393},[1055],[1052,8395,8397],{"className":8396},[1059],[1061,8398,8399],{"xmlns":1063},[1065,8400,8401,8447],{},[1068,8402,8403,8413,8415,8421,8423,8429,8431,8433,8435,8437,8439,8445],{},[1880,8404,8405,8407],{},[1071,8406,7753],{"mathvariant":1575},[1068,8408,8409,8411],{},[1071,8410,7753],{"mathvariant":1575},[1071,8412,3796],{"mathvariant":4609},[1075,8414,1084],{"stretchy":1080},[5084,8416,8417,8419],{},[1071,8418,3796],{"mathvariant":4609},[1071,8420,3606],{},[1071,8422,4571],{},[5084,8424,8425,8427],{},[1071,8426,4571],{},[1071,8428,3606],{},[1071,8430,3796],{"mathvariant":4609},[1075,8432,1107],{"stretchy":1080},[1075,8434,1077],{},[1121,8436,5484],{},[1071,8438,4571],{},[5084,8440,8441,8443],{},[1071,8442,4571],{},[1071,8444,3606],{},[1071,8446,3796],{"mathvariant":4609},[1128,8448,8449],{"encoding":1130},"\\frac{\\partial}{\\partial \\mathbf{a}}(\\mathbf{a}^T X X^T \\mathbf{a}) = 2XX^T\\mathbf{a}",[1052,8451,8453,8609],{"className":8452,"ariaHidden":1096},[1135],[1052,8454,8456,8459,8530,8533,8562,8565,8594,8597,8600,8603,8606],{"className":8455},[1139],[1052,8457],{"className":8458,"style":7788},[1143],[1052,8460,8462,8465,8527],{"className":8461},[1148],[1052,8463],{"className":8464},[1173,1956],[1052,8466,8468],{"className":8467},[1880],[1052,8469,8471,8519],{"className":8470},[1188,1189],[1052,8472,8474,8516],{"className":8473},[1193],[1052,8475,8477,8494,8502],{"className":8476,"style":7807},[1197],[1052,8478,8479,8482],{"style":7810},[1052,8480],{"className":8481,"style":1976},[1205],[1052,8483,8485],{"className":8484},[1210,1211,1212,1213],[1052,8486,8488,8491],{"className":8487},[1148,1213],[1052,8489,7753],{"className":8490,"style":7823},[1148,1213],[1052,8492,3796],{"className":8493},[1148,4785,1213],[1052,8495,8496,8499],{"style":1985},[1052,8497],{"className":8498,"style":1976},[1205],[1052,8500],{"className":8501,"style":1993},[1992],[1052,8503,8504,8507],{"style":7837},[1052,8505],{"className":8506,"style":1976},[1205],[1052,8508,8510],{"className":8509},[1210,1211,1212,1213],[1052,8511,8513],{"className":8512},[1148,1213],[1052,8514,7753],{"className":8515,"style":7823},[1148,1213],[1052,8517,1221],{"className":8518},[1220],[1052,8520,8522],{"className":8521},[1193],[1052,8523,8525],{"className":8524,"style":7859},[1197],[1052,8526],{},[1052,8528],{"className":8529},[1284,1956],[1052,8531,1084],{"className":8532},[1173],[1052,8534,8536,8539],{"className":8535},[1148],[1052,8537,3796],{"className":8538},[1148,4785],[1052,8540,8542],{"className":8541},[1184],[1052,8543,8545],{"className":8544},[1188],[1052,8546,8548],{"className":8547},[1193],[1052,8549,8551],{"className":8550,"style":7013},[1197],[1052,8552,8553,8556],{"style":1327},[1052,8554],{"className":8555,"style":1206},[1205],[1052,8557,8559],{"className":8558},[1210,1211,1212,1213],[1052,8560,3606],{"className":8561,"style":2283},[1148,1149,1213],[1052,8563,4571],{"className":8564,"style":4653},[1148,1149],[1052,8566,8568,8571],{"className":8567},[1148],[1052,8569,4571],{"className":8570,"style":4653},[1148,1149],[1052,8572,8574],{"className":8573},[1184],[1052,8575,8577],{"className":8576},[1188],[1052,8578,8580],{"className":8579},[1193],[1052,8581,8583],{"className":8582,"style":7013},[1197],[1052,8584,8585,8588],{"style":1327},[1052,8586],{"className":8587,"style":1206},[1205],[1052,8589,8591],{"className":8590},[1210,1211,1212,1213],[1052,8592,3606],{"className":8593,"style":2283},[1148,1149,1213],[1052,8595,3796],{"className":8596},[1148,4785],[1052,8598,1107],{"className":8599},[1284],[1052,8601],{"className":8602,"style":1155},[1154],[1052,8604,1077],{"className":8605},[1159],[1052,8607],{"className":8608,"style":1155},[1154],[1052,8610,8612,8615,8618,8621,8650],{"className":8611},[1139],[1052,8613],{"className":8614,"style":7013},[1143],[1052,8616,5484],{"className":8617},[1148],[1052,8619,4571],{"className":8620,"style":4653},[1148,1149],[1052,8622,8624,8627],{"className":8623},[1148],[1052,8625,4571],{"className":8626,"style":4653},[1148,1149],[1052,8628,8630],{"className":8629},[1184],[1052,8631,8633],{"className":8632},[1188],[1052,8634,8636],{"className":8635},[1193],[1052,8637,8639],{"className":8638,"style":7013},[1197],[1052,8640,8641,8644],{"style":1327},[1052,8642],{"className":8643,"style":1206},[1205],[1052,8645,8647],{"className":8646},[1210,1211,1212,1213],[1052,8648,3606],{"className":8649,"style":2283},[1148,1149,1213],[1052,8651,3796],{"className":8652},[1148,4785],[4548,8654,8656],{"id":8655},"第四步合并结果","第四步：合并结果",[1052,8658,8660],{"className":8659},[1865],[1052,8661,8663,8715],{"className":8662},[1055],[1052,8664,8666],{"className":8665},[1059],[1061,8667,8668],{"xmlns":1063,"display":1874},[1065,8669,8670,8712],{},[1068,8671,8672,8674,8680,8682,8684,8686,8688,8690,8692,8694,8696,8698,8700,8702,8704,8710],{},[1071,8673,5469],{"mathvariant":1575},[1086,8675,8676,8678],{},[1071,8677,3117],{},[1071,8679,1791],{},[1075,8681,1084],{"stretchy":1080},[1071,8683,3796],{"mathvariant":4609},[1075,8685,1107],{"stretchy":1080},[1075,8687,1077],{},[1121,8689,4994],{"mathvariant":4609},[1075,8691,3988],{},[1121,8693,5484],{},[1071,8695,4571],{},[1071,8697,1102],{"mathvariant":4609},[1075,8699,3111],{},[1121,8701,5484],{},[1071,8703,4571],{},[5084,8705,8706,8708],{},[1071,8707,4571],{},[1071,8709,3606],{},[1071,8711,3796],{"mathvariant":4609},[1128,8713,8714],{"encoding":1130},"\\nabla J_m(\\mathbf{a}) = \\mathbf{0} - 2X\\mathbf{y} + 2XX^T\\mathbf{a}",[1052,8716,8718,8785,8803,8827],{"className":8717,"ariaHidden":1096},[1135],[1052,8719,8721,8724,8727,8767,8770,8773,8776,8779,8782],{"className":8720},[1139],[1052,8722],{"className":8723,"style":1597},[1143],[1052,8725,5469],{"className":8726},[1148],[1052,8728,8730,8733],{"className":8729},[1148],[1052,8731,3117],{"className":8732,"style":3397},[1148,1149],[1052,8734,8736],{"className":8735},[1184],[1052,8737,8739,8759],{"className":8738},[1188,1189],[1052,8740,8742,8756],{"className":8741},[1193],[1052,8743,8745],{"className":8744,"style":1825},[1197],[1052,8746,8747,8750],{"style":5146},[1052,8748],{"className":8749,"style":1206},[1205],[1052,8751,8753],{"className":8752},[1210,1211,1212,1213],[1052,8754,1791],{"className":8755},[1148,1149,1213],[1052,8757,1221],{"className":8758},[1220],[1052,8760,8762],{"className":8761},[1193],[1052,8763,8765],{"className":8764,"style":1228},[1197],[1052,8766],{},[1052,8768,1084],{"className":8769},[1173],[1052,8771,3796],{"className":8772},[1148,4785],[1052,8774,1107],{"className":8775},[1284],[1052,8777],{"className":8778,"style":1155},[1154],[1052,8780,1077],{"className":8781},[1159],[1052,8783],{"className":8784,"style":1155},[1154],[1052,8786,8788,8791,8794,8797,8800],{"className":8787},[1139],[1052,8789],{"className":8790,"style":4027},[1143],[1052,8792,4994],{"className":8793},[1148,4785],[1052,8795],{"className":8796,"style":3378},[1154],[1052,8798,3988],{"className":8799},[2428],[1052,8801],{"className":8802,"style":3378},[1154],[1052,8804,8806,8809,8812,8815,8818,8821,8824],{"className":8805},[1139],[1052,8807],{"className":8808,"style":5867},[1143],[1052,8810,5484],{"className":8811},[1148],[1052,8813,4571],{"className":8814,"style":4653},[1148,1149],[1052,8816,1102],{"className":8817,"style":4786},[1148,4785],[1052,8819],{"className":8820,"style":3378},[1154],[1052,8822,3111],{"className":8823},[2428],[1052,8825],{"className":8826,"style":3378},[1154],[1052,8828,8830,8833,8836,8839,8868],{"className":8829},[1139],[1052,8831],{"className":8832,"style":5231},[1143],[1052,8834,5484],{"className":8835},[1148],[1052,8837,4571],{"className":8838,"style":4653},[1148,1149],[1052,8840,8842,8845],{"className":8841},[1148],[1052,8843,4571],{"className":8844,"style":4653},[1148,1149],[1052,8846,8848],{"className":8847},[1184],[1052,8849,8851],{"className":8850},[1188],[1052,8852,8854],{"className":8853},[1193],[1052,8855,8857],{"className":8856,"style":5231},[1197],[1052,8858,8859,8862],{"style":5234},[1052,8860],{"className":8861,"style":1206},[1205],[1052,8863,8865],{"className":8864},[1210,1211,1212,1213],[1052,8866,3606],{"className":8867,"style":2283},[1148,1149,1213],[1052,8869,3796],{"className":8870},[1148,4785],[1052,8872,8874],{"className":8873},[1865],[1052,8875,8877,8911],{"className":8876},[1055],[1052,8878,8880],{"className":8879},[1059],[1061,8881,8882],{"xmlns":1063,"display":1874},[1065,8883,8884,8908],{},[1068,8885,8886,8888,8890,8892,8898,8900,8902,8904,8906],{},[1075,8887,1077],{},[1121,8889,5484],{},[1071,8891,4571],{},[5084,8893,8894,8896],{},[1071,8895,4571],{},[1071,8897,3606],{},[1071,8899,3796],{"mathvariant":4609},[1075,8901,3988],{},[1121,8903,5484],{},[1071,8905,4571],{},[1071,8907,1102],{"mathvariant":4609},[1128,8909,8910],{"encoding":1130},"= 2XX^T\\mathbf{a} - 2X\\mathbf{y}",[1052,8912,8914,8926,8980],{"className":8913,"ariaHidden":1096},[1135],[1052,8915,8917,8920,8923],{"className":8916},[1139],[1052,8918],{"className":8919,"style":6698},[1143],[1052,8921,1077],{"className":8922},[1159],[1052,8924],{"className":8925,"style":1155},[1154],[1052,8927,8929,8933,8936,8939,8968,8971,8974,8977],{"className":8928},[1139],[1052,8930],{"className":8931,"style":8932},[1143],"height:0.9747em;vertical-align:-0.0833em;",[1052,8934,5484],{"className":8935},[1148],[1052,8937,4571],{"className":8938,"style":4653},[1148,1149],[1052,8940,8942,8945],{"className":8941},[1148],[1052,8943,4571],{"className":8944,"style":4653},[1148,1149],[1052,8946,8948],{"className":8947},[1184],[1052,8949,8951],{"className":8950},[1188],[1052,8952,8954],{"className":8953},[1193],[1052,8955,8957],{"className":8956,"style":5231},[1197],[1052,8958,8959,8962],{"style":5234},[1052,8960],{"className":8961,"style":1206},[1205],[1052,8963,8965],{"className":8964},[1210,1211,1212,1213],[1052,8966,3606],{"className":8967,"style":2283},[1148,1149,1213],[1052,8969,3796],{"className":8970},[1148,4785],[1052,8972],{"className":8973,"style":3378},[1154],[1052,8975,3988],{"className":8976},[2428],[1052,8978],{"className":8979,"style":3378},[1154],[1052,8981,8983,8986,8989,8992],{"className":8982},[1139],[1052,8984],{"className":8985,"style":5867},[1143],[1052,8987,5484],{"className":8988},[1148],[1052,8990,4571],{"className":8991,"style":4653},[1148,1149],[1052,8993,1102],{"className":8994,"style":4786},[1148,4785],[1052,8996,8998],{"className":8997},[1865],[1052,8999,9001,9035],{"className":9000},[1055],[1052,9002,9004],{"className":9003},[1059],[1061,9005,9006],{"xmlns":1063,"display":1874},[1065,9007,9008,9032],{},[1068,9009,9010,9012,9014,9016,9018,9024,9026,9028,9030],{},[1075,9011,1077],{},[1121,9013,5484],{},[1071,9015,4571],{},[1075,9017,1084],{"stretchy":1080},[5084,9019,9020,9022],{},[1071,9021,4571],{},[1071,9023,3606],{},[1071,9025,3796],{"mathvariant":4609},[1075,9027,3988],{},[1071,9029,1102],{"mathvariant":4609},[1075,9031,1107],{"stretchy":1080},[1128,9033,9034],{"encoding":1130},"= 2X(X^T\\mathbf{a} - \\mathbf{y})",[1052,9036,9038,9050,9106],{"className":9037,"ariaHidden":1096},[1135],[1052,9039,9041,9044,9047],{"className":9040},[1139],[1052,9042],{"className":9043,"style":6698},[1143],[1052,9045,1077],{"className":9046},[1159],[1052,9048],{"className":9049,"style":1155},[1154],[1052,9051,9053,9056,9059,9062,9065,9094,9097,9100,9103],{"className":9052},[1139],[1052,9054],{"className":9055,"style":5212},[1143],[1052,9057,5484],{"className":9058},[1148],[1052,9060,4571],{"className":9061,"style":4653},[1148,1149],[1052,9063,1084],{"className":9064},[1173],[1052,9066,9068,9071],{"className":9067},[1148],[1052,9069,4571],{"className":9070,"style":4653},[1148,1149],[1052,9072,9074],{"className":9073},[1184],[1052,9075,9077],{"className":9076},[1188],[1052,9078,9080],{"className":9079},[1193],[1052,9081,9083],{"className":9082,"style":5231},[1197],[1052,9084,9085,9088],{"style":5234},[1052,9086],{"className":9087,"style":1206},[1205],[1052,9089,9091],{"className":9090},[1210,1211,1212,1213],[1052,9092,3606],{"className":9093,"style":2283},[1148,1149,1213],[1052,9095,3796],{"className":9096},[1148,4785],[1052,9098],{"className":9099,"style":3378},[1154],[1052,9101,3988],{"className":9102},[2428],[1052,9104],{"className":9105,"style":3378},[1154],[1052,9107,9109,9112,9115],{"className":9108},[1139],[1052,9110],{"className":9111,"style":1597},[1143],[1052,9113,1102],{"className":9114,"style":4786},[1148,4785],[1052,9116,1107],{"className":9117},[1284],[1052,9119,9121],{"className":9120},[1865],[1052,9122,9124,9160],{"className":9123},[1055],[1052,9125,9127],{"className":9126},[1059],[1061,9128,9129],{"xmlns":1063,"display":1874},[1065,9130,9131,9157],{},[1068,9132,9133,9135,9137,9139,9141,9143,9145,9147,9153,9155],{},[1075,9134,1077],{},[1075,9136,3988],{},[1121,9138,5484],{},[1071,9140,4571],{},[1075,9142,1084],{"stretchy":1080},[1071,9144,1102],{"mathvariant":4609},[1075,9146,3988],{},[5084,9148,9149,9151],{},[1071,9150,4571],{},[1071,9152,3606],{},[1071,9154,3796],{"mathvariant":4609},[1075,9156,1107],{"stretchy":1080},[1128,9158,9159],{"encoding":1130},"= -2X(\\mathbf{y} - X^T\\mathbf{a})",[1052,9161,9163,9175,9205],{"className":9162,"ariaHidden":1096},[1135],[1052,9164,9166,9169,9172],{"className":9165},[1139],[1052,9167],{"className":9168,"style":6698},[1143],[1052,9170,1077],{"className":9171},[1159],[1052,9173],{"className":9174,"style":1155},[1154],[1052,9176,9178,9181,9184,9187,9190,9193,9196,9199,9202],{"className":9177},[1139],[1052,9179],{"className":9180,"style":1597},[1143],[1052,9182,3988],{"className":9183},[1148],[1052,9185,5484],{"className":9186},[1148],[1052,9188,4571],{"className":9189,"style":4653},[1148,1149],[1052,9191,1084],{"className":9192},[1173],[1052,9194,1102],{"className":9195,"style":4786},[1148,4785],[1052,9197],{"className":9198,"style":3378},[1154],[1052,9200,3988],{"className":9201},[2428],[1052,9203],{"className":9204,"style":3378},[1154],[1052,9206,9208,9211,9240,9243],{"className":9207},[1139],[1052,9209],{"className":9210,"style":5212},[1143],[1052,9212,9214,9217],{"className":9213},[1148],[1052,9215,4571],{"className":9216,"style":4653},[1148,1149],[1052,9218,9220],{"className":9219},[1184],[1052,9221,9223],{"className":9222},[1188],[1052,9224,9226],{"className":9225},[1193],[1052,9227,9229],{"className":9228,"style":5231},[1197],[1052,9230,9231,9234],{"style":5234},[1052,9232],{"className":9233,"style":1206},[1205],[1052,9235,9237],{"className":9236},[1210,1211,1212,1213],[1052,9238,3606],{"className":9239,"style":2283},[1148,1149,1213],[1052,9241,3796],{"className":9242},[1148,4785],[1052,9244,1107],{"className":9245},[1284],[4548,9247,9248],{"id":9248},"关键理解点",[1015,9250,9252],{"start":9251},5,[983,9253,9254],{},"维度匹配",[978,9256,9257,9341,9423,9506],{},[983,9258,9259,9287,9288,9340],{},[1052,9260,9262,9275],{"className":9261},[1055],[1052,9263,9265],{"className":9264},[1059],[1061,9266,9267],{"xmlns":1063},[1065,9268,9269,9273],{},[1068,9270,9271],{},[1071,9272,3796],{"mathvariant":4609},[1128,9274,5437],{"encoding":1130},[1052,9276,9278],{"className":9277,"ariaHidden":1096},[1135],[1052,9279,9281,9284],{"className":9280},[1139],[1052,9282],{"className":9283,"style":5447},[1143],[1052,9285,3796],{"className":9286},[1148,4785]," 是 ",[1052,9289,9291,9309],{"className":9290},[1055],[1052,9292,9294],{"className":9293},[1059],[1061,9295,9296],{"xmlns":1063},[1065,9297,9298,9306],{},[1068,9299,9300,9302,9304],{},[1071,9301,1126],{},[1075,9303,2329],{},[1121,9305,1123],{},[1128,9307,9308],{"encoding":1130},"n \\times 1",[1052,9310,9312,9331],{"className":9311,"ariaHidden":1096},[1135],[1052,9313,9315,9319,9322,9325,9328],{"className":9314},[1139],[1052,9316],{"className":9317,"style":9318},[1143],"height:0.6667em;vertical-align:-0.0833em;",[1052,9320,1126],{"className":9321},[1148,1149],[1052,9323],{"className":9324,"style":3378},[1154],[1052,9326,2329],{"className":9327},[2428],[1052,9329],{"className":9330,"style":3378},[1154],[1052,9332,9334,9337],{"className":9333},[1139],[1052,9335],{"className":9336,"style":5769},[1143],[1052,9338,1123],{"className":9339},[1148]," 向量",[983,9342,9343,9287,9371,9422],{},[1052,9344,9346,9359],{"className":9345},[1055],[1052,9347,9349],{"className":9348},[1059],[1061,9350,9351],{"xmlns":1063},[1065,9352,9353,9357],{},[1068,9354,9355],{},[1071,9356,4571],{},[1128,9358,4571],{"encoding":1130},[1052,9360,9362],{"className":9361,"ariaHidden":1096},[1135],[1052,9363,9365,9368],{"className":9364},[1139],[1052,9366],{"className":9367,"style":1144},[1143],[1052,9369,4571],{"className":9370,"style":4653},[1148,1149],[1052,9372,9374,9392],{"className":9373},[1055],[1052,9375,9377],{"className":9376},[1059],[1061,9378,9379],{"xmlns":1063},[1065,9380,9381,9389],{},[1068,9382,9383,9385,9387],{},[1071,9384,1126],{},[1075,9386,2329],{},[1071,9388,1791],{},[1128,9390,9391],{"encoding":1130},"n \\times m",[1052,9393,9395,9413],{"className":9394,"ariaHidden":1096},[1135],[1052,9396,9398,9401,9404,9407,9410],{"className":9397},[1139],[1052,9399],{"className":9400,"style":9318},[1143],[1052,9402,1126],{"className":9403},[1148,1149],[1052,9405],{"className":9406,"style":3378},[1154],[1052,9408,2329],{"className":9409},[2428],[1052,9411],{"className":9412,"style":3378},[1154],[1052,9414,9416,9419],{"className":9415},[1139],[1052,9417],{"className":9418,"style":2702},[1143],[1052,9420,1791],{"className":9421},[1148,1149]," 矩阵",[983,9424,9425,9287,9455,9340],{},[1052,9426,9428,9442],{"className":9427},[1055],[1052,9429,9431],{"className":9430},[1059],[1061,9432,9433],{"xmlns":1063},[1065,9434,9435,9439],{},[1068,9436,9437],{},[1071,9438,1102],{"mathvariant":4609},[1128,9440,9441],{"encoding":1130},"\\mathbf{y}",[1052,9443,9445],{"className":9444,"ariaHidden":1096},[1135],[1052,9446,9448,9452],{"className":9447},[1139],[1052,9449],{"className":9450,"style":9451},[1143],"height:0.6389em;vertical-align:-0.1944em;",[1052,9453,1102],{"className":9454,"style":4786},[1148,4785],[1052,9456,9458,9476],{"className":9457},[1055],[1052,9459,9461],{"className":9460},[1059],[1061,9462,9463],{"xmlns":1063},[1065,9464,9465,9473],{},[1068,9466,9467,9469,9471],{},[1071,9468,1791],{},[1075,9470,2329],{},[1121,9472,1123],{},[1128,9474,9475],{"encoding":1130},"m \\times 1",[1052,9477,9479,9497],{"className":9478,"ariaHidden":1096},[1135],[1052,9480,9482,9485,9488,9491,9494],{"className":9481},[1139],[1052,9483],{"className":9484,"style":9318},[1143],[1052,9486,1791],{"className":9487},[1148,1149],[1052,9489],{"className":9490,"style":3378},[1154],[1052,9492,2329],{"className":9493},[2428],[1052,9495],{"className":9496,"style":3378},[1154],[1052,9498,9500,9503],{"className":9499},[1139],[1052,9501],{"className":9502,"style":5769},[1143],[1052,9504,1123],{"className":9505},[1148],[983,9507,9508,9509,9599,9600,9340],{},"梯度 ",[1052,9510,9512,9538],{"className":9511},[1055],[1052,9513,9515],{"className":9514},[1059],[1061,9516,9517],{"xmlns":1063},[1065,9518,9519,9535],{},[1068,9520,9521,9523,9529,9531,9533],{},[1071,9522,5469],{"mathvariant":1575},[1086,9524,9525,9527],{},[1071,9526,3117],{},[1071,9528,1791],{},[1075,9530,1084],{"stretchy":1080},[1071,9532,3796],{"mathvariant":4609},[1075,9534,1107],{"stretchy":1080},[1128,9536,9537],{"encoding":1130},"\\nabla J_m(\\mathbf{a})",[1052,9539,9541],{"className":9540,"ariaHidden":1096},[1135],[1052,9542,9544,9547,9550,9590,9593,9596],{"className":9543},[1139],[1052,9545],{"className":9546,"style":1597},[1143],[1052,9548,5469],{"className":9549},[1148],[1052,9551,9553,9556],{"className":9552},[1148],[1052,9554,3117],{"className":9555,"style":3397},[1148,1149],[1052,9557,9559],{"className":9558},[1184],[1052,9560,9562,9582],{"className":9561},[1188,1189],[1052,9563,9565,9579],{"className":9564},[1193],[1052,9566,9568],{"className":9567,"style":1825},[1197],[1052,9569,9570,9573],{"style":5146},[1052,9571],{"className":9572,"style":1206},[1205],[1052,9574,9576],{"className":9575},[1210,1211,1212,1213],[1052,9577,1791],{"className":9578},[1148,1149,1213],[1052,9580,1221],{"className":9581},[1220],[1052,9583,9585],{"className":9584},[1193],[1052,9586,9588],{"className":9587,"style":1228},[1197],[1052,9589],{},[1052,9591,1084],{"className":9592},[1173],[1052,9594,3796],{"className":9595},[1148,4785],[1052,9597,1107],{"className":9598},[1284]," 也必须是 ",[1052,9601,9603,9620],{"className":9602},[1055],[1052,9604,9606],{"className":9605},[1059],[1061,9607,9608],{"xmlns":1063},[1065,9609,9610,9618],{},[1068,9611,9612,9614,9616],{},[1071,9613,1126],{},[1075,9615,2329],{},[1121,9617,1123],{},[1128,9619,9308],{"encoding":1130},[1052,9621,9623,9641],{"className":9622,"ariaHidden":1096},[1135],[1052,9624,9626,9629,9632,9635,9638],{"className":9625},[1139],[1052,9627],{"className":9628,"style":9318},[1143],[1052,9630,1126],{"className":9631},[1148,1149],[1052,9633],{"className":9634,"style":3378},[1154],[1052,9636,2329],{"className":9637},[2428],[1052,9639],{"className":9640,"style":3378},[1154],[1052,9642,9644,9647],{"className":9643},[1139],[1052,9645],{"className":9646,"style":5769},[1143],[1052,9648,1123],{"className":9649},[1148],[1015,9651,9653],{"start":9652},6,[983,9654,9655],{},"链式法则",[996,9657,9658,9659,9716,9717],{},"对于复合函数 ",[1052,9660,9662,9689],{"className":9661},[1055],[1052,9663,9665],{"className":9664},[1059],[1061,9666,9667],{"xmlns":1063},[1065,9668,9669,9686],{},[1068,9670,9671,9673,9675,9678,9680,9682,9684],{},[1071,9672,1746],{},[1075,9674,1084],{"stretchy":1080},[1071,9676,9677],{},"g",[1075,9679,1084],{"stretchy":1080},[1071,9681,1090],{"mathvariant":4609},[1075,9683,1107],{"stretchy":1080},[1075,9685,1107],{"stretchy":1080},[1128,9687,9688],{"encoding":1130},"f(g(\\mathbf{x}))",[1052,9690,9692],{"className":9691,"ariaHidden":1096},[1135],[1052,9693,9695,9698,9701,9704,9707,9710,9713],{"className":9694},[1139],[1052,9696],{"className":9697,"style":1597},[1143],[1052,9699,1746],{"className":9700,"style":1762},[1148,1149],[1052,9702,1084],{"className":9703},[1173],[1052,9705,9677],{"className":9706,"style":1245},[1148,1149],[1052,9708,1084],{"className":9709},[1173],[1052,9711,1090],{"className":9712},[1148,4785],[1052,9714,2206],{"className":9715},[1284],"，有： ",[1052,9718,9720,9779],{"className":9719},[1055],[1052,9721,9723],{"className":9722},[1059],[1061,9724,9725],{"xmlns":1063},[1065,9726,9727,9776],{},[1068,9728,9729,9743,9745,9759,9762],{},[1880,9730,9731,9737],{},[1068,9732,9733,9735],{},[1071,9734,7753],{"mathvariant":1575},[1071,9736,1746],{},[1068,9738,9739,9741],{},[1071,9740,7753],{"mathvariant":1575},[1071,9742,1090],{"mathvariant":4609},[1075,9744,1077],{},[1880,9746,9747,9753],{},[1068,9748,9749,9751],{},[1071,9750,7753],{"mathvariant":1575},[1071,9752,1746],{},[1068,9754,9755,9757],{},[1071,9756,7753],{"mathvariant":1575},[1071,9758,9677],{},[1075,9760,9761],{},"⋅",[1880,9763,9764,9770],{},[1068,9765,9766,9768],{},[1071,9767,7753],{"mathvariant":1575},[1071,9769,9677],{},[1068,9771,9772,9774],{},[1071,9773,7753],{"mathvariant":1575},[1071,9775,1090],{"mathvariant":4609},[1128,9777,9778],{"encoding":1130},"\\frac{\\partial f}{\\partial \\mathbf{x}} = \\frac{\\partial f}{\\partial g} \\cdot \\frac{\\partial g}{\\partial \\mathbf{x}}",[1052,9780,9782,9874,9965],{"className":9781,"ariaHidden":1096},[1135],[1052,9783,9785,9789,9865,9868,9871],{"className":9784},[1139],[1052,9786],{"className":9787,"style":9788},[1143],"height:1.2772em;vertical-align:-0.345em;",[1052,9790,9792,9795,9862],{"className":9791},[1148],[1052,9793],{"className":9794},[1173,1956],[1052,9796,9798],{"className":9797},[1880],[1052,9799,9801,9854],{"className":9800},[1188,1189],[1052,9802,9804,9851],{"className":9803},[1193],[1052,9805,9808,9825,9833],{"className":9806,"style":9807},[1197],"height:0.9322em;",[1052,9809,9810,9813],{"style":7810},[1052,9811],{"className":9812,"style":1976},[1205],[1052,9814,9816],{"className":9815},[1210,1211,1212,1213],[1052,9817,9819,9822],{"className":9818},[1148,1213],[1052,9820,7753],{"className":9821,"style":7823},[1148,1213],[1052,9823,1090],{"className":9824},[1148,4785,1213],[1052,9826,9827,9830],{"style":1985},[1052,9828],{"className":9829,"style":1976},[1205],[1052,9831],{"className":9832,"style":1993},[1992],[1052,9834,9836,9839],{"style":9835},"top:-3.4461em;",[1052,9837],{"className":9838,"style":1976},[1205],[1052,9840,9842],{"className":9841},[1210,1211,1212,1213],[1052,9843,9845,9848],{"className":9844},[1148,1213],[1052,9846,7753],{"className":9847,"style":7823},[1148,1213],[1052,9849,1746],{"className":9850,"style":1762},[1148,1149,1213],[1052,9852,1221],{"className":9853},[1220],[1052,9855,9857],{"className":9856},[1193],[1052,9858,9860],{"className":9859,"style":7859},[1197],[1052,9861],{},[1052,9863],{"className":9864},[1284,1956],[1052,9866],{"className":9867,"style":1155},[1154],[1052,9869,1077],{"className":9870},[1159],[1052,9872],{"className":9873,"style":1155},[1154],[1052,9875,9877,9881,9956,9959,9962],{"className":9876},[1139],[1052,9878],{"className":9879,"style":9880},[1143],"height:1.4133em;vertical-align:-0.4811em;",[1052,9882,9884,9887,9953],{"className":9883},[1148],[1052,9885],{"className":9886},[1173,1956],[1052,9888,9890],{"className":9889},[1880],[1052,9891,9893,9944],{"className":9892},[1188,1189],[1052,9894,9896,9941],{"className":9895},[1193],[1052,9897,9899,9916,9924],{"className":9898,"style":9807},[1197],[1052,9900,9901,9904],{"style":7810},[1052,9902],{"className":9903,"style":1976},[1205],[1052,9905,9907],{"className":9906},[1210,1211,1212,1213],[1052,9908,9910,9913],{"className":9909},[1148,1213],[1052,9911,7753],{"className":9912,"style":7823},[1148,1213],[1052,9914,9677],{"className":9915,"style":1245},[1148,1149,1213],[1052,9917,9918,9921],{"style":1985},[1052,9919],{"className":9920,"style":1976},[1205],[1052,9922],{"className":9923,"style":1993},[1992],[1052,9925,9926,9929],{"style":9835},[1052,9927],{"className":9928,"style":1976},[1205],[1052,9930,9932],{"className":9931},[1210,1211,1212,1213],[1052,9933,9935,9938],{"className":9934},[1148,1213],[1052,9936,7753],{"className":9937,"style":7823},[1148,1213],[1052,9939,1746],{"className":9940,"style":1762},[1148,1149,1213],[1052,9942,1221],{"className":9943},[1220],[1052,9945,9947],{"className":9946},[1193],[1052,9948,9951],{"className":9949,"style":9950},[1197],"height:0.4811em;",[1052,9952],{},[1052,9954],{"className":9955},[1284,1956],[1052,9957],{"className":9958,"style":3378},[1154],[1052,9960,9761],{"className":9961},[2428],[1052,9963],{"className":9964,"style":3378},[1154],[1052,9966,9968,9971],{"className":9967},[1139],[1052,9969],{"className":9970,"style":9788},[1143],[1052,9972,9974,9977,10042],{"className":9973},[1148],[1052,9975],{"className":9976},[1173,1956],[1052,9978,9980],{"className":9979},[1880],[1052,9981,9983,10034],{"className":9982},[1188,1189],[1052,9984,9986,10031],{"className":9985},[1193],[1052,9987,9989,10006,10014],{"className":9988,"style":9807},[1197],[1052,9990,9991,9994],{"style":7810},[1052,9992],{"className":9993,"style":1976},[1205],[1052,9995,9997],{"className":9996},[1210,1211,1212,1213],[1052,9998,10000,10003],{"className":9999},[1148,1213],[1052,10001,7753],{"className":10002,"style":7823},[1148,1213],[1052,10004,1090],{"className":10005},[1148,4785,1213],[1052,10007,10008,10011],{"style":1985},[1052,10009],{"className":10010,"style":1976},[1205],[1052,10012],{"className":10013,"style":1993},[1992],[1052,10015,10016,10019],{"style":9835},[1052,10017],{"className":10018,"style":1976},[1205],[1052,10020,10022],{"className":10021},[1210,1211,1212,1213],[1052,10023,10025,10028],{"className":10024},[1148,1213],[1052,10026,7753],{"className":10027,"style":7823},[1148,1213],[1052,10029,9677],{"className":10030,"style":1245},[1148,1149,1213],[1052,10032,1221],{"className":10033},[1220],[1052,10035,10037],{"className":10036},[1193],[1052,10038,10040],{"className":10039,"style":7859},[1197],[1052,10041],{},[1052,10043],{"className":10044},[1284,1956],[1015,10046,10048],{"start":10047},7,[983,10049,10050],{},"矩阵求导的本质",[996,10052,10053,10054],{},"矩阵求导实际上是对每个分量分别求偏导数： 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f}{\\partial \\mathbf{x}} = \\begin{bmatrix} \\frac{\\partial f}{\\partial x_1} \\ \\frac{\\partial f}{\\partial x_2} \\ \\dots \\ \\frac{\\partial f}{\\partial x_n} 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= \\mathbf{b}^T\\mathbf{a}",[1052,10742,10744,10791],{"className":10743,"ariaHidden":1096},[1135],[1052,10745,10747,10750,10779,10782,10785,10788],{"className":10746},[1139],[1052,10748],{"className":10749,"style":7013},[1143],[1052,10751,10753,10756],{"className":10752},[1148],[1052,10754,3796],{"className":10755},[1148,4785],[1052,10757,10759],{"className":10758},[1184],[1052,10760,10762],{"className":10761},[1188],[1052,10763,10765],{"className":10764},[1193],[1052,10766,10768],{"className":10767,"style":7013},[1197],[1052,10769,10770,10773],{"style":1327},[1052,10771],{"className":10772,"style":1206},[1205],[1052,10774,10776],{"className":10775},[1210,1211,1212,1213],[1052,10777,3606],{"className":10778,"style":2283},[1148,1149,1213],[1052,10780,10727],{"className":10781},[1148,4785],[1052,10783],{"className":10784,"style":1155},[1154],[1052,10786,1077],{"className":10787},[1159],[1052,10789],{"className":10790,"style":1155},[1154],[1052,10792,10794,10797,10826],{"className":10793},[1139],[1052,10795],{"className":10796,"style":7013},[1143],[1052,10798,10800,10803],{"className":10799},[1148],[1052,10801,10727],{"className":10802},[1148,4785],[1052,10804,10806],{"className":10805},[1184],[1052,10807,10809],{"className":10808},[1188],[1052,10810,10812],{"className":10811},[1193],[1052,10813,10815],{"className":10814,"style":7013},[1197],[1052,10816,10817,10820],{"style":1327},[1052,10818],{"className":10819,"style":1206},[1205],[1052,10821,10823],{"className":10822},[1210,1211,1212,1213],[1052,10824,3606],{"className":10825,"style":2283},[1148,1149,1213],[1052,10827,3796],{"className":10828},[1148,4785],"（当结果是标量时）",[983,10831,10832,10835],{},[1020,10833,10834],{},"分项处理","：把复杂函数分解为简单项的和",[983,10837,10838,10841],{},[1020,10839,10840],{},"检查维度","：每一步都要确保维度匹配",[996,10843,10844],{},"这样，通过系统的推导，我们就得到了均方误差函数的梯度表达式！",[1010,10846,10848],{"id":10847},"三前馈神经网络与参数优化","三、前馈神经网络与参数优化",[1045,10850,10852],{"id":10851},"_1人工神经网络概述","1、人工神经网络概述",[996,10854,10855],{},[1711,10856],{"alt":1711,"src":10857},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-11.png",[4548,10859,10860],{"id":10860},"感知机",[996,10862,10863],{},[1711,10864],{"alt":1711,"src":10865},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-12.png",[4548,10867,10868],{"id":10868},"前馈神经网络",[996,10870,10871],{},[1711,10872],{"alt":1711,"src":10873},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-13.png",[1045,10875,10877],{"id":10876},"_2激活函数","2、激活函数",[993,10879,10880],{},[996,10881,10882],{},"激活函数必须是【单调递增的】",[996,10884,10885],{},[1711,10886],{"alt":1711,"src":10887},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-14.png",[1010,10889,10891],{"id":10890},"四卷积神经网络","四、卷积神经网络",[993,10893,10894],{},[996,10895,10896],{},"一个很形象的可视化视频，从2min29s开始看就好了",[1042,10898,10899],{"id":10899},"一些杂项",[993,10901,10902],{},[996,10903,10904],{},"八股的概念",[996,10906,10907,10908,10911,10912,10915,10916,10919,10920],{},"图像卷积操作符合“",[1020,10909,10910],{},"视觉系统信息分层处理","”这一机制，即视觉感知是由低层细胞到高层细胞对原始输入信息不断抽象完成，",[1020,10913,10914],{},"更高层细胞拥有更高级的感受野","，并且对一些",[1020,10917,10918],{},"偏移、旋转","等具有一定的",[1020,10921,10922],{},"不变性",[1042,10924,1047],{"id":10925},"基本概念-2",[1045,10927,10928],{"id":10928},"感受野",[996,10930,10931,10933,10934,10937,10938,1511,10941,10944,10945,10948],{},[1020,10932,10928],{},"是卷积神经网络",[1020,10935,10936],{},"每一层输出的特征图","（feature map）上的",[1020,10939,10940],{},"像素点",[1020,10942,10943],{},"输入图像","上",[1020,10946,10947],{},"映射的区域大小，",[1532,10949,10950],{},[1020,10951,10952],{},"是特征图上一个点对应输入图像上的区域",[996,10954,10955,11010,11011,11013,11014,11017],{},[1052,10956,10958,10982],{"className":10957},[1055],[1052,10959,10961],{"className":10960},[1059],[1061,10962,10963],{"xmlns":1063},[1065,10964,10965,10979],{},[1068,10966,10967,10969,10972,10974,10977],{},[1075,10968,4576],{"stretchy":1080},[1071,10970,10971],{},"h",[1075,10973,1097],{"separator":1096},[1071,10975,10976],{},"w",[1075,10978,4601],{"stretchy":1080},[1128,10980,10981],{"encoding":1130},"[h,w]",[1052,10983,10985],{"className":10984,"ariaHidden":1096},[1135],[1052,10986,10988,10991,10994,10997,11000,11003,11007],{"className":10987},[1139],[1052,10989],{"className":10990,"style":1597},[1143],[1052,10992,4576],{"className":10993},[1173],[1052,10995,10971],{"className":10996},[1148,1149],[1052,10998,1097],{"className":10999},[1234],[1052,11001],{"className":11002,"style":1238},[1154],[1052,11004,10976],{"className":11005,"style":11006},[1148,1149],"margin-right:0.0269em;",[1052,11008,4601],{"className":11009},[1284],":",[1532,11012,10928],{},"大小\u002F",[1532,11015,11016],{},"卷积核","大小->对性能的影响",[996,11019,11020,11074],{},[1052,11021,11023,11046],{"className":11022},[1055],[1052,11024,11026],{"className":11025},[1059],[1061,11027,11028],{"xmlns":1063},[1065,11029,11030,11043],{},[1068,11031,11032,11034,11036,11038,11041],{},[1075,11033,1084],{"stretchy":1080},[1071,11035,1093],{},[1075,11037,1097],{"separator":1096},[1071,11039,11040],{},"j",[1075,11042,1107],{"stretchy":1080},[1128,11044,11045],{"encoding":1130},"(i,j)",[1052,11047,11049],{"className":11048,"ariaHidden":1096},[1135],[1052,11050,11052,11055,11058,11061,11064,11067,11071],{"className":11051},[1139],[1052,11053],{"className":11054,"style":1597},[1143],[1052,11056,1084],{"className":11057},[1173],[1052,11059,1093],{"className":11060},[1148,1149],[1052,11062,1097],{"className":11063},[1234],[1052,11065],{"className":11066,"style":1238},[1154],[1052,11068,11040],{"className":11069,"style":11070},[1148,1149],"margin-right:0.0572em;",[1052,11072,1107],{"className":11073},[1284],"：方向和距离->位移",[1045,11076,11077],{"id":11077},"下采样",[996,11079,11080,11081,11084],{},"图像卷积计算即对图像进行了",[1020,11082,11083],{},"下采样 (down sampling)","操作",[996,11086,11087],{},[1711,11088],{"alt":1711,"src":11089},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-15.png",[996,11091,11092],{},[1711,11093],{"alt":1711,"src":11094},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-16.png",[4548,11096,11097],{"id":11097},"下采样的减抽象",[993,11099,11100],{},[996,11101,11102],{},"感觉现场推就可以了",[996,11104,11105,11106,11159,11160,11211,11212,11284,11285,11315,11316,2919],{},"假设被卷积图像大小为",[1052,11107,11109,11128],{"className":11108},[1055],[1052,11110,11112],{"className":11111},[1059],[1061,11113,11114],{"xmlns":1063},[1065,11115,11116,11125],{},[1068,11117,11118,11121,11123],{},[1071,11119,11120],{},"W",[1075,11122,2329],{},[1071,11124,11120],{},[1128,11126,11127],{"encoding":1130},"W×W",[1052,11129,11131,11150],{"className":11130,"ariaHidden":1096},[1135],[1052,11132,11134,11138,11141,11144,11147],{"className":11133},[1139],[1052,11135],{"className":11136,"style":11137},[1143],"height:0.7667em;vertical-align:-0.0833em;",[1052,11139,11120],{"className":11140,"style":2283},[1148,1149],[1052,11142],{"className":11143,"style":3378},[1154],[1052,11145,2329],{"className":11146},[2428],[1052,11148],{"className":11149,"style":3378},[1154],[1052,11151,11153,11156],{"className":11152},[1139],[1052,11154],{"className":11155,"style":1144},[1143],[1052,11157,11120],{"className":11158,"style":2283},[1148,1149],"，卷积核大小为",[1052,11161,11163,11181],{"className":11162},[1055],[1052,11164,11166],{"className":11165},[1059],[1061,11167,11168],{"xmlns":1063},[1065,11169,11170,11178],{},[1068,11171,11172,11174,11176],{},[1071,11173,3810],{},[1075,11175,2329],{},[1071,11177,3810],{},[1128,11179,11180],{"encoding":1130},"F×F",[1052,11182,11184,11202],{"className":11183,"ariaHidden":1096},[1135],[1052,11185,11187,11190,11193,11196,11199],{"className":11186},[1139],[1052,11188],{"className":11189,"style":11137},[1143],[1052,11191,3810],{"className":11192,"style":2283},[1148,1149],[1052,11194],{"className":11195,"style":3378},[1154],[1052,11197,2329],{"className":11198},[2428],[1052,11200],{"className":11201,"style":3378},[1154],[1052,11203,11205,11208],{"className":11204},[1139],[1052,11206],{"className":11207,"style":1144},[1143],[1052,11209,3810],{"className":11210,"style":2283},[1148,1149],"，上下左右四个边缘填充像素行列数为",[1052,11213,11215,11244],{"className":11214},[1055],[1052,11216,11218],{"className":11217},[1059],[1061,11219,11220],{"xmlns":1063},[1065,11221,11222,11241],{},[1068,11223,11224,11226,11228,11231,11233,11236,11238],{},[1071,11225,2257],{},[1075,11227,1077],{},[1075,11229,11230],{"stretchy":1080},"⌊",[1071,11232,3810],{},[1071,11234,11235],{"mathvariant":1575},"\u002F",[1121,11237,5484],{},[1075,11239,11240],{"stretchy":1080},"⌋",[1128,11242,11243],{"encoding":1130},"P = ⌊F\u002F2⌋",[1052,11245,11247,11265],{"className":11246,"ariaHidden":1096},[1135],[1052,11248,11250,11253,11256,11259,11262],{"className":11249},[1139],[1052,11251],{"className":11252,"style":1144},[1143],[1052,11254,2257],{"className":11255,"style":2283},[1148,1149],[1052,11257],{"className":11258,"style":1155},[1154],[1052,11260,1077],{"className":11261},[1159],[1052,11263],{"className":11264,"style":1155},[1154],[1052,11266,11268,11271,11274,11277,11281],{"className":11267},[1139],[1052,11269],{"className":11270,"style":1597},[1143],[1052,11272,11230],{"className":11273},[1173],[1052,11275,3810],{"className":11276,"style":2283},[1148,1149],[1052,11278,11280],{"className":11279},[1148],"\u002F2",[1052,11282,11240],{"className":11283},[1284],"，步长为",[1052,11286,11288,11302],{"className":11287},[1055],[1052,11289,11291],{"className":11290},[1059],[1061,11292,11293],{"xmlns":1063},[1065,11294,11295,11300],{},[1068,11296,11297],{},[1071,11298,11299],{},"S",[1128,11301,11299],{"encoding":1130},[1052,11303,11305],{"className":11304,"ariaHidden":1096},[1135],[1052,11306,11308,11311],{"className":11307},[1139],[1052,11309],{"className":11310,"style":1144},[1143],[1052,11312,11299],{"className":11313,"style":11314},[1148,1149],"margin-right:0.0576em;","，则被卷积结果的分辨率是 ",[1052,11317,11319,11353],{"className":11318},[1055],[1052,11320,11322],{"className":11321},[1059],[1061,11323,11324],{"xmlns":1063},[1065,11325,11326,11350],{},[1068,11327,11328,11346,11348],{},[1880,11329,11330,11344],{},[1068,11331,11332,11334,11336,11338,11340,11342],{},[1071,11333,11120],{},[1075,11335,3988],{},[1071,11337,3810],{},[1075,11339,3111],{},[1121,11341,5484],{},[1071,11343,2257],{},[1071,11345,11299],{},[1075,11347,3111],{},[1121,11349,1123],{},[1128,11351,11352],{"encoding":1130},"\\frac{W-F+2P}{S} + 1",[1052,11354,11356,11456],{"className":11355,"ariaHidden":1096},[1135],[1052,11357,11359,11363,11447,11450,11453],{"className":11358},[1139],[1052,11360],{"className":11361,"style":11362},[1143],"height:1.2173em;vertical-align:-0.345em;",[1052,11364,11366,11369,11444],{"className":11365},[1148],[1052,11367],{"className":11368},[1173,1956],[1052,11370,11372],{"className":11371},[1880],[1052,11373,11375,11436],{"className":11374},[1188,1189],[1052,11376,11378,11433],{"className":11377},[1193],[1052,11379,11382,11396,11404],{"className":11380,"style":11381},[1197],"height:0.8723em;",[1052,11383,11384,11387],{"style":7810},[1052,11385],{"className":11386,"style":1976},[1205],[1052,11388,11390],{"className":11389},[1210,1211,1212,1213],[1052,11391,11393],{"className":11392},[1148,1213],[1052,11394,11299],{"className":11395,"style":11314},[1148,1149,1213],[1052,11397,11398,11401],{"style":1985},[1052,11399],{"className":11400,"style":1976},[1205],[1052,11402],{"className":11403,"style":1993},[1992],[1052,11405,11406,11409],{"style":7837},[1052,11407],{"className":11408,"style":1976},[1205],[1052,11410,11412],{"className":11411},[1210,1211,1212,1213],[1052,11413,11415,11418,11421,11424,11427,11430],{"className":11414},[1148,1213],[1052,11416,11120],{"className":11417,"style":2283},[1148,1149,1213],[1052,11419,3988],{"className":11420},[2428,1213],[1052,11422,3810],{"className":11423,"style":2283},[1148,1149,1213],[1052,11425,3111],{"className":11426},[2428,1213],[1052,11428,5484],{"className":11429},[1148,1213],[1052,11431,2257],{"className":11432,"style":2283},[1148,1149,1213],[1052,11434,1221],{"className":11435},[1220],[1052,11437,11439],{"className":11438},[1193],[1052,11440,11442],{"className":11441,"style":7859},[1197],[1052,11443],{},[1052,11445],{"className":11446},[1284,1956],[1052,11448],{"className":11449,"style":3378},[1154],[1052,11451,3111],{"className":11452},[2428],[1052,11454],{"className":11455,"style":3378},[1154],[1052,11457,11459,11462],{"className":11458},[1139],[1052,11460],{"className":11461,"style":5769},[1143],[1052,11463,1123],{"className":11464},[1148],[1042,11466,11468],{"id":11467},"padding填充","Padding（填充）",[993,11470,11471],{},[996,11472,11473],{},"填充可以补0或者复制",[996,11475,11476],{},[1711,11477],{"alt":1711,"src":11478},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-17.png",[1042,11480,11482],{"id":11481},"stride步长","Stride（步长）",[996,11484,11485],{},[1711,11486],{"alt":1711,"src":11487},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-18.png",[1042,11489,11491],{"id":11490},"dilation膨胀","Dilation（膨胀）",[996,11493,11494],{},[1711,11495],{"alt":1711,"src":11496},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-19.png",[1042,11498,11500],{"id":11499},"pooling池化","Pooling（池化）",[996,11502,11503,11504,11507],{},"池化操作是卷积神经网络中常用的",[1020,11505,11506],{},"下采样方法","，主要有最大池化（Max Pooling）、平均池化（Average Pooling）、 k-max 池化（K-Max Pooling）等",[996,11509,11510],{},[1711,11511],{"alt":1711,"src":11512},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-20.png",[1042,11514,11516],{"id":11515},"词嵌入word-embedding","词嵌入（Word embedding）",[996,11518,11519,11522],{},[1020,11520,11521],{},"词嵌入要达到的目的——",[1532,11523,11524],{},[1020,11525,11526],{},"经过word embedding，词意相近的两个词在特征空间会相隔比较近",[996,11528,11529],{},[1711,11530],{"alt":1711,"src":11531},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-21.png",[996,11533,11534],{},"通过将One-Hot编码表示的词与嵌入矩阵相乘就可以将高维稀疏的矩阵嵌入到一个低维稠密的矩阵中",[1010,11536,11538],{"id":11537},"五循环神经网络recurrent-neural-network-rnn","五、循环神经网络（Recurrent Neural Network, RNN）",[996,11540,11541],{},[1711,11542],{"alt":1711,"src":11543},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-22.png",[1042,11545,11547],{"id":11546},"_1基本原理","1、基本原理",[996,11549,11550,11551,11554,2755,11625,11628,11629,11716,11717,11720,11806,11809,11838,11841,11912,11913,11982,11983,12035,12036,12121,12122,12207],{},"循环神经网络的结构图如上所示。由\"循环\"两字可知，循环神经网络在处理数据过程中构成了一个循环体。对于一个序列数据，在每一时刻t，循环神经网络单元会",[1020,11552,11553],{},"读取当前输入数据",[1052,11555,11557,11575],{"className":11556},[1055],[1052,11558,11560],{"className":11559},[1059],[1061,11561,11562],{"xmlns":1063},[1065,11563,11564,11572],{},[1068,11565,11566],{},[1086,11567,11568,11570],{},[1071,11569,1090],{},[1071,11571,3799],{},[1128,11573,11574],{"encoding":1130},"x_t",[1052,11576,11578],{"className":11577,"ariaHidden":1096},[1135],[1052,11579,11581,11584],{"className":11580},[1139],[1052,11582],{"className":11583,"style":1380},[1143],[1052,11585,11587,11590],{"className":11586},[1148],[1052,11588,1090],{"className":11589},[1148,1149],[1052,11591,11593],{"className":11592},[1184],[1052,11594,11596,11617],{"className":11595},[1188,1189],[1052,11597,11599,11614],{"className":11598},[1193],[1052,11600,11603],{"className":11601,"style":11602},[1197],"height:0.2806em;",[1052,11604,11605,11608],{"style":1201},[1052,11606],{"className":11607,"style":1206},[1205],[1052,11609,11611],{"className":11610},[1210,1211,1212,1213],[1052,11612,3799],{"className":11613},[1148,1149,1213],[1052,11615,1221],{"className":11616},[1220],[1052,11618,11620],{"className":11619},[1193],[1052,11621,11623],{"className":11622,"style":1228},[1197],[1052,11624],{},[1020,11626,11627],{},"前一时刻","输入数据",[1052,11630,11632,11656],{"className":11631},[1055],[1052,11633,11635],{"className":11634},[1059],[1061,11636,11637],{"xmlns":1063},[1065,11638,11639,11653],{},[1068,11640,11641],{},[1086,11642,11643,11645],{},[1071,11644,1090],{},[1068,11646,11647,11649,11651],{},[1071,11648,3799],{},[1075,11650,3988],{},[1121,11652,1123],{},[1128,11654,11655],{"encoding":1130},"x_{t-1}",[1052,11657,11659],{"className":11658,"ariaHidden":1096},[1135],[1052,11660,11662,11666],{"className":11661},[1139],[1052,11663],{"className":11664,"style":11665},[1143],"height:0.6389em;vertical-align:-0.2083em;",[1052,11667,11669,11672],{"className":11668},[1148],[1052,11670,1090],{"className":11671},[1148,1149],[1052,11673,11675],{"className":11674},[1184],[1052,11676,11678,11707],{"className":11677},[1188,1189],[1052,11679,11681,11704],{"className":11680},[1193],[1052,11682,11684],{"className":11683,"style":4690},[1197],[1052,11685,11686,11689],{"style":1201},[1052,11687],{"className":11688,"style":1206},[1205],[1052,11690,11692],{"className":11691},[1210,1211,1212,1213],[1052,11693,11695,11698,11701],{"className":11694},[1148,1213],[1052,11696,3799],{"className":11697},[1148,1149,1213],[1052,11699,3988],{"className":11700},[2428,1213],[1052,11702,1123],{"className":11703},[1148,1213],[1052,11705,1221],{"className":11706},[1220],[1052,11708,11710],{"className":11709},[1193],[1052,11711,11714],{"className":11712,"style":11713},[1197],"height:0.2083em;",[1052,11715],{},"所",[1532,11718,11719],{},"对应的隐式编码结果",[1052,11721,11723,11747],{"className":11722},[1055],[1052,11724,11726],{"className":11725},[1059],[1061,11727,11728],{"xmlns":1063},[1065,11729,11730,11744],{},[1068,11731,11732],{},[1086,11733,11734,11736],{},[1071,11735,10971],{},[1068,11737,11738,11740,11742],{},[1071,11739,3799],{},[1075,11741,3988],{},[1121,11743,1123],{},[1128,11745,11746],{"encoding":1130},"h_{t-1}",[1052,11748,11750],{"className":11749,"ariaHidden":1096},[1135],[1052,11751,11753,11757],{"className":11752},[1139],[1052,11754],{"className":11755,"style":11756},[1143],"height:0.9028em;vertical-align:-0.2083em;",[1052,11758,11760,11763],{"className":11759},[1148],[1052,11761,10971],{"className":11762},[1148,1149],[1052,11764,11766],{"className":11765},[1184],[1052,11767,11769,11798],{"className":11768},[1188,1189],[1052,11770,11772,11795],{"className":11771},[1193],[1052,11773,11775],{"className":11774,"style":4690},[1197],[1052,11776,11777,11780],{"style":1201},[1052,11778],{"className":11779,"style":1206},[1205],[1052,11781,11783],{"className":11782},[1210,1211,1212,1213],[1052,11784,11786,11789,11792],{"className":11785},[1148,1213],[1052,11787,3799],{"className":11788},[1148,1149,1213],[1052,11790,3988],{"className":11791},[2428,1213],[1052,11793,1123],{"className":11794},[1148,1213],[1052,11796,1221],{"className":11797},[1220],[1052,11799,11801],{"className":11800},[1193],[1052,11802,11804],{"className":11803,"style":11713},[1197],[1052,11805],{},[1532,11807,11808],{},"一起生成",[1052,11810,11812,11825],{"className":11811},[1055],[1052,11813,11815],{"className":11814},[1059],[1061,11816,11817],{"xmlns":1063},[1065,11818,11819,11823],{},[1068,11820,11821],{},[1071,11822,3799],{},[1128,11824,3799],{"encoding":1130},[1052,11826,11828],{"className":11827,"ariaHidden":1096},[1135],[1052,11829,11831,11835],{"className":11830},[1139],[1052,11832],{"className":11833,"style":11834},[1143],"height:0.6151em;",[1052,11836,3799],{"className":11837},[1148,1149],[1532,11839,11840],{},"时刻的隐式编码结果",[1052,11842,11844,11862],{"className":11843},[1055],[1052,11845,11847],{"className":11846},[1059],[1061,11848,11849],{"xmlns":1063},[1065,11850,11851,11859],{},[1068,11852,11853],{},[1086,11854,11855,11857],{},[1071,11856,10971],{},[1071,11858,3799],{},[1128,11860,11861],{"encoding":1130},"h_t",[1052,11863,11865],{"className":11864,"ariaHidden":1096},[1135],[1052,11866,11868,11872],{"className":11867},[1139],[1052,11869],{"className":11870,"style":11871},[1143],"height:0.8444em;vertical-align:-0.15em;",[1052,11873,11875,11878],{"className":11874},[1148],[1052,11876,10971],{"className":11877},[1148,1149],[1052,11879,11881],{"className":11880},[1184],[1052,11882,11884,11904],{"className":11883},[1188,1189],[1052,11885,11887,11901],{"className":11886},[1193],[1052,11888,11890],{"className":11889,"style":11602},[1197],[1052,11891,11892,11895],{"style":1201},[1052,11893],{"className":11894,"style":1206},[1205],[1052,11896,11898],{"className":11897},[1210,1211,1212,1213],[1052,11899,3799],{"className":11900},[1148,1149,1213],[1052,11902,1221],{"className":11903},[1220],[1052,11905,11907],{"className":11906},[1193],[1052,11908,11910],{"className":11909,"style":1228},[1197],[1052,11911],{},"。接着将",[1052,11914,11916,11933],{"className":11915},[1055],[1052,11917,11919],{"className":11918},[1059],[1061,11920,11921],{"xmlns":1063},[1065,11922,11923,11931],{},[1068,11924,11925],{},[1086,11926,11927,11929],{},[1071,11928,10971],{},[1071,11930,3799],{},[1128,11932,11861],{"encoding":1130},[1052,11934,11936],{"className":11935,"ariaHidden":1096},[1135],[1052,11937,11939,11942],{"className":11938},[1139],[1052,11940],{"className":11941,"style":11871},[1143],[1052,11943,11945,11948],{"className":11944},[1148],[1052,11946,10971],{"className":11947},[1148,1149],[1052,11949,11951],{"className":11950},[1184],[1052,11952,11954,11974],{"className":11953},[1188,1189],[1052,11955,11957,11971],{"className":11956},[1193],[1052,11958,11960],{"className":11959,"style":11602},[1197],[1052,11961,11962,11965],{"style":1201},[1052,11963],{"className":11964,"style":1206},[1205],[1052,11966,11968],{"className":11967},[1210,1211,1212,1213],[1052,11969,3799],{"className":11970},[1148,1149,1213],[1052,11972,1221],{"className":11973},[1220],[1052,11975,11977],{"className":11976},[1193],[1052,11978,11980],{"className":11979,"style":1228},[1197],[1052,11981],{},"后传，去参与生成",[1052,11984,11986,12004],{"className":11985},[1055],[1052,11987,11989],{"className":11988},[1059],[1061,11990,11991],{"xmlns":1063},[1065,11992,11993,12001],{},[1068,11994,11995,11997,11999],{},[1071,11996,3799],{},[1075,11998,3111],{},[1121,12000,1123],{},[1128,12002,12003],{"encoding":1130},"t+1",[1052,12005,12007,12026],{"className":12006,"ariaHidden":1096},[1135],[1052,12008,12010,12014,12017,12020,12023],{"className":12009},[1139],[1052,12011],{"className":12012,"style":12013},[1143],"height:0.6984em;vertical-align:-0.0833em;",[1052,12015,3799],{"className":12016},[1148,1149],[1052,12018],{"className":12019,"style":3378},[1154],[1052,12021,3111],{"className":12022},[2428],[1052,12024],{"className":12025,"style":3378},[1154],[1052,12027,12029,12032],{"className":12028},[1139],[1052,12030],{"className":12031,"style":5769},[1143],[1052,12033,1123],{"className":12034},[1148],"时刻输入数据",[1052,12037,12039,12063],{"className":12038},[1055],[1052,12040,12042],{"className":12041},[1059],[1061,12043,12044],{"xmlns":1063},[1065,12045,12046,12060],{},[1068,12047,12048],{},[1086,12049,12050,12052],{},[1071,12051,1090],{},[1068,12053,12054,12056,12058],{},[1071,12055,3799],{},[1075,12057,3111],{},[1121,12059,1123],{},[1128,12061,12062],{"encoding":1130},"x_{t+1}",[1052,12064,12066],{"className":12065,"ariaHidden":1096},[1135],[1052,12067,12069,12072],{"className":12068},[1139],[1052,12070],{"className":12071,"style":11665},[1143],[1052,12073,12075,12078],{"className":12074},[1148],[1052,12076,1090],{"className":12077},[1148,1149],[1052,12079,12081],{"className":12080},[1184],[1052,12082,12084,12113],{"className":12083},[1188,1189],[1052,12085,12087,12110],{"className":12086},[1193],[1052,12088,12090],{"className":12089,"style":4690},[1197],[1052,12091,12092,12095],{"style":1201},[1052,12093],{"className":12094,"style":1206},[1205],[1052,12096,12098],{"className":12097},[1210,1211,1212,1213],[1052,12099,12101,12104,12107],{"className":12100},[1148,1213],[1052,12102,3799],{"className":12103},[1148,1149,1213],[1052,12105,3111],{"className":12106},[2428,1213],[1052,12108,1123],{"className":12109},[1148,1213],[1052,12111,1221],{"className":12112},[1220],[1052,12114,12116],{"className":12115},[1193],[1052,12117,12119],{"className":12118,"style":11713},[1197],[1052,12120],{},"的隐式编码",[1052,12123,12125,12149],{"className":12124},[1055],[1052,12126,12128],{"className":12127},[1059],[1061,12129,12130],{"xmlns":1063},[1065,12131,12132,12146],{},[1068,12133,12134],{},[1086,12135,12136,12138],{},[1071,12137,10971],{},[1068,12139,12140,12142,12144],{},[1071,12141,3799],{},[1075,12143,3111],{},[1121,12145,1123],{},[1128,12147,12148],{"encoding":1130},"h_{t+1}",[1052,12150,12152],{"className":12151,"ariaHidden":1096},[1135],[1052,12153,12155,12158],{"className":12154},[1139],[1052,12156],{"className":12157,"style":11756},[1143],[1052,12159,12161,12164],{"className":12160},[1148],[1052,12162,10971],{"className":12163},[1148,1149],[1052,12165,12167],{"className":12166},[1184],[1052,12168,12170,12199],{"className":12169},[1188,1189],[1052,12171,12173,12196],{"className":12172},[1193],[1052,12174,12176],{"className":12175,"style":4690},[1197],[1052,12177,12178,12181],{"style":1201},[1052,12179],{"className":12180,"style":1206},[1205],[1052,12182,12184],{"className":12183},[1210,1211,1212,1213],[1052,12185,12187,12190,12193],{"className":12186},[1148,1213],[1052,12188,3799],{"className":12189},[1148,1149,1213],[1052,12191,3111],{"className":12192},[2428,1213],[1052,12194,1123],{"className":12195},[1148,1213],[1052,12197,1221],{"className":12198},[1220],[1052,12200,12202],{"className":12201},[1193],[1052,12203,12205],{"className":12204,"style":11713},[1197],[1052,12206],{},"。如此循环处理，直至该序列数据被处理完毕。",[996,12209,12210],{},[1532,12211,12212],{},"当输入序列过长时，循环神经网络也容易出现梯度消失(gradient vanishing)或者梯度爆炸（gradient exploding）的问题",[1042,12214,12216],{"id":12215},"_2序列误差","2、序列误差",[996,12218,12219],{},[1711,12220],{"alt":1711,"src":12221},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-23.png",[1042,12223,12225],{"id":12224},"_3参数更新的问题","3、参数更新的问题",[996,12227,12228],{},[1711,12229],{"alt":1711,"src":12230},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-24.png",[1042,12232,12234],{"id":12233},"_4长短期记忆网络lstm","4、长短期记忆网络（LSTM）",[993,12236,12237],{},[996,12238,12239],{},[1532,12240,12241],{},[1020,12242,12243],{},"速记",[996,12245,12246,12247,12251,12252,12255],{},"长短期记忆网络引入了",[12248,12249,12250],"em",{},"记忆元","（memory cell），或简称为",[12248,12253,12254],{},"单元","（cell）。当前时间步的输入和前一个时间步的隐状态 作为数据送入长短期记忆网络的门中， 如图所示。 它们由三个具有sigmoid激活函数的全连接层处理， 以计算输入门、遗忘门和输出门的值。 因此，这三个门的值都在(0,1)的范围内。",[1045,12257,12259],{"id":12258},"输入门遗忘门和输出门","输入门、遗忘门和输出门",[996,12261,12262],{},[1711,12263],{"alt":1711,"src":12264},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-25.png",[996,12266,12267,12268,12410,12413,12569,12570,12708,12709,12847,12848,12987],{},"假设有h个隐藏单元，批量大小为n，输入数为d。因此，输入为",[1052,12269,12271,12304],{"className":12270},[1055],[1052,12272,12274],{"className":12273},[1059],[1061,12275,12276],{"xmlns":1063},[1065,12277,12278,12301],{},[1068,12279,12280,12286,12289],{},[1086,12281,12282,12284],{},[1071,12283,4571],{},[1071,12285,3799],{},[1075,12287,12288],{},"∈",[5084,12290,12291,12293],{},[1071,12292,2558],{"mathvariant":1575},[1068,12294,12295,12297,12299],{},[1071,12296,1126],{},[1075,12298,2329],{},[1071,12300,2370],{},[1128,12302,12303],{"encoding":1130},"X_t ∈ ℝ^{n×d}",[1052,12305,12307,12364],{"className":12306,"ariaHidden":1096},[1135],[1052,12308,12310,12314,12355,12358,12361],{"className":12309},[1139],[1052,12311],{"className":12312,"style":12313},[1143],"height:0.8333em;vertical-align:-0.15em;",[1052,12315,12317,12320],{"className":12316},[1148],[1052,12318,4571],{"className":12319,"style":4653},[1148,1149],[1052,12321,12323],{"className":12322},[1184],[1052,12324,12326,12347],{"className":12325},[1188,1189],[1052,12327,12329,12344],{"className":12328},[1193],[1052,12330,12332],{"className":12331,"style":11602},[1197],[1052,12333,12335,12338],{"style":12334},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1052,12336],{"className":12337,"style":1206},[1205],[1052,12339,12341],{"className":12340},[1210,1211,1212,1213],[1052,12342,3799],{"className":12343},[1148,1149,1213],[1052,12345,1221],{"className":12346},[1220],[1052,12348,12350],{"className":12349},[1193],[1052,12351,12353],{"className":12352,"style":1228},[1197],[1052,12354],{},[1052,12356],{"className":12357,"style":1155},[1154],[1052,12359,12288],{"className":12360},[1159],[1052,12362],{"className":12363,"style":1155},[1154],[1052,12365,12367,12371],{"className":12366},[1139],[1052,12368],{"className":12369,"style":12370},[1143],"height:0.8491em;",[1052,12372,12374,12378],{"className":12373},[1148],[1052,12375,2558],{"className":12376},[1148,12377],"amsrm",[1052,12379,12381],{"className":12380},[1184],[1052,12382,12384],{"className":12383},[1188],[1052,12385,12387],{"className":12386},[1193],[1052,12388,12390],{"className":12389,"style":12370},[1197],[1052,12391,12392,12395],{"style":1327},[1052,12393],{"className":12394,"style":1206},[1205],[1052,12396,12398],{"className":12397},[1210,1211,1212,1213],[1052,12399,12401,12404,12407],{"className":12400},[1148,1213],[1052,12402,1126],{"className":12403},[1148,1149,1213],[1052,12405,2329],{"className":12406},[2428,1213],[1052,12408,2370],{"className":12409},[1148,1149,1213],[12248,12411,12412],{},"，前一时间步的隐状态为",[1052,12414,12416,12455],{"className":12415},[1055],[1052,12417,12419],{"className":12418},[1059],[1061,12420,12421],{"xmlns":1063},[1065,12422,12423,12452],{},[1068,12424,12425,12438,12440],{},[1086,12426,12427,12430],{},[1071,12428,12429],{},"H",[1068,12431,12432,12434,12436],{},[1071,12433,3799],{},[1075,12435,3988],{},[1121,12437,1123],{},[1075,12439,12288],{},[5084,12441,12442,12444],{},[1071,12443,2558],{"mathvariant":1575},[1068,12445,12446,12448,12450],{},[1071,12447,1126],{},[1075,12449,2329],{},[1071,12451,10971],{},[1128,12453,12454],{"encoding":1130},"H_{t-1} ∈ ℝ^{n×h}",[1052,12456,12458,12525],{"className":12457,"ariaHidden":1096},[1135],[1052,12459,12461,12465,12516,12519,12522],{"className":12460},[1139],[1052,12462],{"className":12463,"style":12464},[1143],"height:0.8917em;vertical-align:-0.2083em;",[1052,12466,12468,12472],{"className":12467},[1148],[1052,12469,12429],{"className":12470,"style":12471},[1148,1149],"margin-right:0.0813em;",[1052,12473,12475],{"className":12474},[1184],[1052,12476,12478,12508],{"className":12477},[1188,1189],[1052,12479,12481,12505],{"className":12480},[1193],[1052,12482,12484],{"className":12483,"style":4690},[1197],[1052,12485,12487,12490],{"style":12486},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[1052,12488],{"className":12489,"style":1206},[1205],[1052,12491,12493],{"className":12492},[1210,1211,1212,1213],[1052,12494,12496,12499,12502],{"className":12495},[1148,1213],[1052,12497,3799],{"className":12498},[1148,1149,1213],[1052,12500,3988],{"className":12501},[2428,1213],[1052,12503,1123],{"className":12504},[1148,1213],[1052,12506,1221],{"className":12507},[1220],[1052,12509,12511],{"className":12510},[1193],[1052,12512,12514],{"className":12513,"style":11713},[1197],[1052,12515],{},[1052,12517],{"className":12518,"style":1155},[1154],[1052,12520,12288],{"className":12521},[1159],[1052,12523],{"className":12524,"style":1155},[1154],[1052,12526,12528,12531],{"className":12527},[1139],[1052,12529],{"className":12530,"style":12370},[1143],[1052,12532,12534,12537],{"className":12533},[1148],[1052,12535,2558],{"className":12536},[1148,12377],[1052,12538,12540],{"className":12539},[1184],[1052,12541,12543],{"className":12542},[1188],[1052,12544,12546],{"className":12545},[1193],[1052,12547,12549],{"className":12548,"style":12370},[1197],[1052,12550,12551,12554],{"style":1327},[1052,12552],{"className":12553,"style":1206},[1205],[1052,12555,12557],{"className":12556},[1210,1211,1212,1213],[1052,12558,12560,12563,12566],{"className":12559},[1148,1213],[1052,12561,1126],{"className":12562},[1148,1149,1213],[1052,12564,2329],{"className":12565},[2428,1213],[1052,12567,10971],{"className":12568},[1148,1149,1213],"。相应地，时间步t的门被定义如下：输入门是",[1052,12571,12573,12606],{"className":12572},[1055],[1052,12574,12576],{"className":12575},[1059],[1061,12577,12578],{"xmlns":1063},[1065,12579,12580,12603],{},[1068,12581,12582,12589,12591],{},[1086,12583,12584,12587],{},[1071,12585,12586],{},"I",[1071,12588,3799],{},[1075,12590,12288],{},[5084,12592,12593,12595],{},[1071,12594,2558],{"mathvariant":1575},[1068,12596,12597,12599,12601],{},[1071,12598,1126],{},[1075,12600,2329],{},[1071,12602,10971],{},[1128,12604,12605],{"encoding":1130},"I_t ∈ ℝ^{n×h}",[1052,12607,12609,12664],{"className":12608,"ariaHidden":1096},[1135],[1052,12610,12612,12615,12655,12658,12661],{"className":12611},[1139],[1052,12613],{"className":12614,"style":12313},[1143],[1052,12616,12618,12621],{"className":12617},[1148],[1052,12619,12586],{"className":12620,"style":4653},[1148,1149],[1052,12622,12624],{"className":12623},[1184],[1052,12625,12627,12647],{"className":12626},[1188,1189],[1052,12628,12630,12644],{"className":12629},[1193],[1052,12631,12633],{"className":12632,"style":11602},[1197],[1052,12634,12635,12638],{"style":12334},[1052,12636],{"className":12637,"style":1206},[1205],[1052,12639,12641],{"className":12640},[1210,1211,1212,1213],[1052,12642,3799],{"className":12643},[1148,1149,1213],[1052,12645,1221],{"className":12646},[1220],[1052,12648,12650],{"className":12649},[1193],[1052,12651,12653],{"className":12652,"style":1228},[1197],[1052,12654],{},[1052,12656],{"className":12657,"style":1155},[1154],[1052,12659,12288],{"className":12660},[1159],[1052,12662],{"className":12663,"style":1155},[1154],[1052,12665,12667,12670],{"className":12666},[1139],[1052,12668],{"className":12669,"style":12370},[1143],[1052,12671,12673,12676],{"className":12672},[1148],[1052,12674,2558],{"className":12675},[1148,12377],[1052,12677,12679],{"className":12678},[1184],[1052,12680,12682],{"className":12681},[1188],[1052,12683,12685],{"className":12684},[1193],[1052,12686,12688],{"className":12687,"style":12370},[1197],[1052,12689,12690,12693],{"style":1327},[1052,12691],{"className":12692,"style":1206},[1205],[1052,12694,12696],{"className":12695},[1210,1211,1212,1213],[1052,12697,12699,12702,12705],{"className":12698},[1148,1213],[1052,12700,1126],{"className":12701},[1148,1149,1213],[1052,12703,2329],{"className":12704},[2428,1213],[1052,12706,10971],{"className":12707},[1148,1149,1213],"，遗忘门是",[1052,12710,12712,12744],{"className":12711},[1055],[1052,12713,12715],{"className":12714},[1059],[1061,12716,12717],{"xmlns":1063},[1065,12718,12719,12741],{},[1068,12720,12721,12727,12729],{},[1086,12722,12723,12725],{},[1071,12724,3810],{},[1071,12726,3799],{},[1075,12728,12288],{},[5084,12730,12731,12733],{},[1071,12732,2558],{"mathvariant":1575},[1068,12734,12735,12737,12739],{},[1071,12736,1126],{},[1075,12738,2329],{},[1071,12740,10971],{},[1128,12742,12743],{"encoding":1130},"F_t ∈ ℝ^{n×h}",[1052,12745,12747,12803],{"className":12746,"ariaHidden":1096},[1135],[1052,12748,12750,12753,12794,12797,12800],{"className":12749},[1139],[1052,12751],{"className":12752,"style":12313},[1143],[1052,12754,12756,12759],{"className":12755},[1148],[1052,12757,3810],{"className":12758,"style":2283},[1148,1149],[1052,12760,12762],{"className":12761},[1184],[1052,12763,12765,12786],{"className":12764},[1188,1189],[1052,12766,12768,12783],{"className":12767},[1193],[1052,12769,12771],{"className":12770,"style":11602},[1197],[1052,12772,12774,12777],{"style":12773},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1052,12775],{"className":12776,"style":1206},[1205],[1052,12778,12780],{"className":12779},[1210,1211,1212,1213],[1052,12781,3799],{"className":12782},[1148,1149,1213],[1052,12784,1221],{"className":12785},[1220],[1052,12787,12789],{"className":12788},[1193],[1052,12790,12792],{"className":12791,"style":1228},[1197],[1052,12793],{},[1052,12795],{"className":12796,"style":1155},[1154],[1052,12798,12288],{"className":12799},[1159],[1052,12801],{"className":12802,"style":1155},[1154],[1052,12804,12806,12809],{"className":12805},[1139],[1052,12807],{"className":12808,"style":12370},[1143],[1052,12810,12812,12815],{"className":12811},[1148],[1052,12813,2558],{"className":12814},[1148,12377],[1052,12816,12818],{"className":12817},[1184],[1052,12819,12821],{"className":12820},[1188],[1052,12822,12824],{"className":12823},[1193],[1052,12825,12827],{"className":12826,"style":12370},[1197],[1052,12828,12829,12832],{"style":1327},[1052,12830],{"className":12831,"style":1206},[1205],[1052,12833,12835],{"className":12834},[1210,1211,1212,1213],[1052,12836,12838,12841,12844],{"className":12837},[1148,1213],[1052,12839,1126],{"className":12840},[1148,1149,1213],[1052,12842,2329],{"className":12843},[2428,1213],[1052,12845,10971],{"className":12846},[1148,1149,1213],"，输出门是",[1052,12849,12851,12884],{"className":12850},[1055],[1052,12852,12854],{"className":12853},[1059],[1061,12855,12856],{"xmlns":1063},[1065,12857,12858,12881],{},[1068,12859,12860,12867,12869],{},[1086,12861,12862,12865],{},[1071,12863,12864],{},"O",[1071,12866,3799],{},[1075,12868,12288],{},[5084,12870,12871,12873],{},[1071,12872,2558],{"mathvariant":1575},[1068,12874,12875,12877,12879],{},[1071,12876,1126],{},[1075,12878,2329],{},[1071,12880,10971],{},[1128,12882,12883],{"encoding":1130},"O_t ∈ ℝ^{n×h}",[1052,12885,12887,12943],{"className":12886,"ariaHidden":1096},[1135],[1052,12888,12890,12893,12934,12937,12940],{"className":12889},[1139],[1052,12891],{"className":12892,"style":12313},[1143],[1052,12894,12896,12899],{"className":12895},[1148],[1052,12897,12864],{"className":12898,"style":1150},[1148,1149],[1052,12900,12902],{"className":12901},[1184],[1052,12903,12905,12926],{"className":12904},[1188,1189],[1052,12906,12908,12923],{"className":12907},[1193],[1052,12909,12911],{"className":12910,"style":11602},[1197],[1052,12912,12914,12917],{"style":12913},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[1052,12915],{"className":12916,"style":1206},[1205],[1052,12918,12920],{"className":12919},[1210,1211,1212,1213],[1052,12921,3799],{"className":12922},[1148,1149,1213],[1052,12924,1221],{"className":12925},[1220],[1052,12927,12929],{"className":12928},[1193],[1052,12930,12932],{"className":12931,"style":1228},[1197],[1052,12933],{},[1052,12935],{"className":12936,"style":1155},[1154],[1052,12938,12288],{"className":12939},[1159],[1052,12941],{"className":12942,"style":1155},[1154],[1052,12944,12946,12949],{"className":12945},[1139],[1052,12947],{"className":12948,"style":12370},[1143],[1052,12950,12952,12955],{"className":12951},[1148],[1052,12953,2558],{"className":12954},[1148,12377],[1052,12956,12958],{"className":12957},[1184],[1052,12959,12961],{"className":12960},[1188],[1052,12962,12964],{"className":12963},[1193],[1052,12965,12967],{"className":12966,"style":12370},[1197],[1052,12968,12969,12972],{"style":1327},[1052,12970],{"className":12971,"style":1206},[1205],[1052,12973,12975],{"className":12974},[1210,1211,1212,1213],[1052,12976,12978,12981,12984],{"className":12977},[1148,1213],[1052,12979,1126],{"className":12980},[1148,1149,1213],[1052,12982,2329],{"className":12983},[2428,1213],[1052,12985,10971],{"className":12986},[1148,1149,1213],"。它们的计算方法如下：",[1052,12989,12991],{"className":12990},[1865],[1052,12992,12994,13071],{"className":12993},[1055],[1052,12995,12997],{"className":12996},[1059],[1061,12998,12999],{"xmlns":1063,"display":1874},[1065,13000,13001,13068],{},[1068,13002,13003,13009,13011,13014,13016,13022,13032,13034,13046,13056,13058,13064,13066],{},[1086,13004,13005,13007],{},[1071,13006,12586],{"mathvariant":4609},[1071,13008,3799],{},[1075,13010,1077],{},[1071,13012,13013],{},"σ",[1075,13015,1084],{"stretchy":1080},[1086,13017,13018,13020],{},[1071,13019,4571],{"mathvariant":4609},[1071,13021,3799],{},[1086,13023,13024,13026],{},[1071,13025,11120],{"mathvariant":4609},[1068,13027,13028,13030],{},[1071,13029,1090],{},[1071,13031,1093],{},[1075,13033,3111],{},[1086,13035,13036,13038],{},[1071,13037,12429],{"mathvariant":4609},[1068,13039,13040,13042,13044],{},[1071,13041,3799],{},[1075,13043,3988],{},[1121,13045,1123],{},[1086,13047,13048,13050],{},[1071,13049,11120],{"mathvariant":4609},[1068,13051,13052,13054],{},[1071,13053,10971],{},[1071,13055,1093],{},[1075,13057,3111],{},[1086,13059,13060,13062],{},[1071,13061,10727],{"mathvariant":4609},[1071,13063,1093],{},[1075,13065,1107],{"stretchy":1080},[1075,13067,1097],{"separator":1096},[1128,13069,13070],{"encoding":1130},"\\mathbf{I}_t = \\sigma(\\mathbf{X}_t \\mathbf{W}_{xi} + \\mathbf{H}_{t-1} \\mathbf{W}_{hi} + \\mathbf{b}_i),",[1052,13072,13074,13130,13238,13348],{"className":13073,"ariaHidden":1096},[1135],[1052,13075,13077,13081,13121,13124,13127],{"className":13076},[1139],[1052,13078],{"className":13079,"style":13080},[1143],"height:0.8361em;vertical-align:-0.15em;",[1052,13082,13084,13087],{"className":13083},[1148],[1052,13085,12586],{"className":13086},[1148,4785],[1052,13088,13090],{"className":13089},[1184],[1052,13091,13093,13113],{"className":13092},[1188,1189],[1052,13094,13096,13110],{"className":13095},[1193],[1052,13097,13099],{"className":13098,"style":11602},[1197],[1052,13100,13101,13104],{"style":1201},[1052,13102],{"className":13103,"style":1206},[1205],[1052,13105,13107],{"className":13106},[1210,1211,1212,1213],[1052,13108,3799],{"className":13109},[1148,1149,1213],[1052,13111,1221],{"className":13112},[1220],[1052,13114,13116],{"className":13115},[1193],[1052,13117,13119],{"className":13118,"style":1228},[1197],[1052,13120],{},[1052,13122],{"className":13123,"style":1155},[1154],[1052,13125,1077],{"className":13126},[1159],[1052,13128],{"className":13129,"style":1155},[1154],[1052,13131,13133,13136,13139,13142,13182,13229,13232,13235],{"className":13132},[1139],[1052,13134],{"className":13135,"style":1597},[1143],[1052,13137,13013],{"className":13138,"style":1245},[1148,1149],[1052,13140,1084],{"className":13141},[1173],[1052,13143,13145,13148],{"className":13144},[1148],[1052,13146,4571],{"className":13147},[1148,4785],[1052,13149,13151],{"className":13150},[1184],[1052,13152,13154,13174],{"className":13153},[1188,1189],[1052,13155,13157,13171],{"className":13156},[1193],[1052,13158,13160],{"className":13159,"style":11602},[1197],[1052,13161,13162,13165],{"style":1201},[1052,13163],{"className":13164,"style":1206},[1205],[1052,13166,13168],{"className":13167},[1210,1211,1212,1213],[1052,13169,3799],{"className":13170},[1148,1149,1213],[1052,13172,1221],{"className":13173},[1220],[1052,13175,13177],{"className":13176},[1193],[1052,13178,13180],{"className":13179,"style":1228},[1197],[1052,13181],{},[1052,13183,13185,13188],{"className":13184},[1148],[1052,13186,11120],{"className":13187,"style":4786},[1148,4785],[1052,13189,13191],{"className":13190},[1184],[1052,13192,13194,13221],{"className":13193},[1188,1189],[1052,13195,13197,13218],{"className":13196},[1193],[1052,13198,13200],{"className":13199,"style":1198},[1197],[1052,13201,13203,13206],{"style":13202},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[1052,13204],{"className":13205,"style":1206},[1205],[1052,13207,13209],{"className":13208},[1210,1211,1212,1213],[1052,13210,13212,13215],{"className":13211},[1148,1213],[1052,13213,1090],{"className":13214},[1148,1149,1213],[1052,13216,1093],{"className":13217},[1148,1149,1213],[1052,13219,1221],{"className":13220},[1220],[1052,13222,13224],{"className":13223},[1193],[1052,13225,13227],{"className":13226,"style":1228},[1197],[1052,13228],{},[1052,13230],{"className":13231,"style":3378},[1154],[1052,13233,3111],{"className":13234},[2428],[1052,13236],{"className":13237,"style":3378},[1154],[1052,13239,13241,13245,13294,13339,13342,13345],{"className":13240},[1139],[1052,13242],{"className":13243,"style":13244},[1143],"height:0.8944em;vertical-align:-0.2083em;",[1052,13246,13248,13251],{"className":13247},[1148],[1052,13249,12429],{"className":13250},[1148,4785],[1052,13252,13254],{"className":13253},[1184],[1052,13255,13257,13286],{"className":13256},[1188,1189],[1052,13258,13260,13283],{"className":13259},[1193],[1052,13261,13263],{"className":13262,"style":4690},[1197],[1052,13264,13265,13268],{"style":1201},[1052,13266],{"className":13267,"style":1206},[1205],[1052,13269,13271],{"className":13270},[1210,1211,1212,1213],[1052,13272,13274,13277,13280],{"className":13273},[1148,1213],[1052,13275,3799],{"className":13276},[1148,1149,1213],[1052,13278,3988],{"className":13279},[2428,1213],[1052,13281,1123],{"className":13282},[1148,1213],[1052,13284,1221],{"className":13285},[1220],[1052,13287,13289],{"className":13288},[1193],[1052,13290,13292],{"className":13291,"style":11713},[1197],[1052,13293],{},[1052,13295,13297,13300],{"className":13296},[1148],[1052,13298,11120],{"className":13299,"style":4786},[1148,4785],[1052,13301,13303],{"className":13302},[1184],[1052,13304,13306,13331],{"className":13305},[1188,1189],[1052,13307,13309,13328],{"className":13308},[1193],[1052,13310,13313],{"className":13311,"style":13312},[1197],"height:0.3361em;",[1052,13314,13315,13318],{"style":13202},[1052,13316],{"className":13317,"style":1206},[1205],[1052,13319,13321],{"className":13320},[1210,1211,1212,1213],[1052,13322,13324],{"className":13323},[1148,1213],[1052,13325,13327],{"className":13326},[1148,1149,1213],"hi",[1052,13329,1221],{"className":13330},[1220],[1052,13332,13334],{"className":13333},[1193],[1052,13335,13337],{"className":13336,"style":1228},[1197],[1052,13338],{},[1052,13340],{"className":13341,"style":3378},[1154],[1052,13343,3111],{"className":13344},[2428],[1052,13346],{"className":13347,"style":3378},[1154],[1052,13349,13351,13354,13394,13397],{"className":13350},[1139],[1052,13352],{"className":13353,"style":1597},[1143],[1052,13355,13357,13360],{"className":13356},[1148],[1052,13358,10727],{"className":13359},[1148,4785],[1052,13361,13363],{"className":13362},[1184],[1052,13364,13366,13386],{"className":13365},[1188,1189],[1052,13367,13369,13383],{"className":13368},[1193],[1052,13370,13372],{"className":13371,"style":1198},[1197],[1052,13373,13374,13377],{"style":1201},[1052,13375],{"className":13376,"style":1206},[1205],[1052,13378,13380],{"className":13379},[1210,1211,1212,1213],[1052,13381,1093],{"className":13382},[1148,1149,1213],[1052,13384,1221],{"className":13385},[1220],[1052,13387,13389],{"className":13388},[1193],[1052,13390,13392],{"className":13391,"style":1228},[1197],[1052,13393],{},[1052,13395,1107],{"className":13396},[1284],[1052,13398,1097],{"className":13399},[1234],[1052,13401,13403],{"className":13402},[1865],[1052,13404,13406,13482],{"className":13405},[1055],[1052,13407,13409],{"className":13408},[1059],[1061,13410,13411],{"xmlns":1063,"display":1874},[1065,13412,13413,13479],{},[1068,13414,13415,13421,13423,13425,13427,13433,13443,13445,13457,13467,13469,13475,13477],{},[1086,13416,13417,13419],{},[1071,13418,3810],{"mathvariant":4609},[1071,13420,3799],{},[1075,13422,1077],{},[1071,13424,13013],{},[1075,13426,1084],{"stretchy":1080},[1086,13428,13429,13431],{},[1071,13430,4571],{"mathvariant":4609},[1071,13432,3799],{},[1086,13434,13435,13437],{},[1071,13436,11120],{"mathvariant":4609},[1068,13438,13439,13441],{},[1071,13440,1090],{},[1071,13442,1746],{},[1075,13444,3111],{},[1086,13446,13447,13449],{},[1071,13448,12429],{"mathvariant":4609},[1068,13450,13451,13453,13455],{},[1071,13452,3799],{},[1075,13454,3988],{},[1121,13456,1123],{},[1086,13458,13459,13461],{},[1071,13460,11120],{"mathvariant":4609},[1068,13462,13463,13465],{},[1071,13464,10971],{},[1071,13466,1746],{},[1075,13468,3111],{},[1086,13470,13471,13473],{},[1071,13472,10727],{"mathvariant":4609},[1071,13474,1746],{},[1075,13476,1107],{"stretchy":1080},[1075,13478,1097],{"separator":1096},[1128,13480,13481],{"encoding":1130},"\\mathbf{F}_t = \\sigma(\\mathbf{X}_t \\mathbf{W}_{xf} + \\mathbf{H}_{t-1} \\mathbf{W}_{hf} + \\mathbf{b}_f),",[1052,13483,13485,13540,13648,13759],{"className":13484,"ariaHidden":1096},[1135],[1052,13486,13488,13491,13531,13534,13537],{"className":13487},[1139],[1052,13489],{"className":13490,"style":13080},[1143],[1052,13492,13494,13497],{"className":13493},[1148],[1052,13495,3810],{"className":13496},[1148,4785],[1052,13498,13500],{"className":13499},[1184],[1052,13501,13503,13523],{"className":13502},[1188,1189],[1052,13504,13506,13520],{"className":13505},[1193],[1052,13507,13509],{"className":13508,"style":11602},[1197],[1052,13510,13511,13514],{"style":1201},[1052,13512],{"className":13513,"style":1206},[1205],[1052,13515,13517],{"className":13516},[1210,1211,1212,1213],[1052,13518,3799],{"className":13519},[1148,1149,1213],[1052,13521,1221],{"className":13522},[1220],[1052,13524,13526],{"className":13525},[1193],[1052,13527,13529],{"className":13528,"style":1228},[1197],[1052,13530],{},[1052,13532],{"className":13533,"style":1155},[1154],[1052,13535,1077],{"className":13536},[1159],[1052,13538],{"className":13539,"style":1155},[1154],[1052,13541,13543,13547,13550,13553,13593,13639,13642,13645],{"className":13542},[1139],[1052,13544],{"className":13545,"style":13546},[1143],"height:1.0361em;vertical-align:-0.2861em;",[1052,13548,13013],{"className":13549,"style":1245},[1148,1149],[1052,13551,1084],{"className":13552},[1173],[1052,13554,13556,13559],{"className":13555},[1148],[1052,13557,4571],{"className":13558},[1148,4785],[1052,13560,13562],{"className":13561},[1184],[1052,13563,13565,13585],{"className":13564},[1188,1189],[1052,13566,13568,13582],{"className":13567},[1193],[1052,13569,13571],{"className":13570,"style":11602},[1197],[1052,13572,13573,13576],{"style":1201},[1052,13574],{"className":13575,"style":1206},[1205],[1052,13577,13579],{"className":13578},[1210,1211,1212,1213],[1052,13580,3799],{"className":13581},[1148,1149,1213],[1052,13583,1221],{"className":13584},[1220],[1052,13586,13588],{"className":13587},[1193],[1052,13589,13591],{"className":13590,"style":1228},[1197],[1052,13592],{},[1052,13594,13596,13599],{"className":13595},[1148],[1052,13597,11120],{"className":13598,"style":4786},[1148,4785],[1052,13600,13602],{"className":13601},[1184],[1052,13603,13605,13631],{"className":13604},[1188,1189],[1052,13606,13608,13628],{"className":13607},[1193],[1052,13609,13611],{"className":13610,"style":13312},[1197],[1052,13612,13613,13616],{"style":13202},[1052,13614],{"className":13615,"style":1206},[1205],[1052,13617,13619],{"className":13618},[1210,1211,1212,1213],[1052,13620,13622,13625],{"className":13621},[1148,1213],[1052,13623,1090],{"className":13624},[1148,1149,1213],[1052,13626,1746],{"className":13627,"style":1762},[1148,1149,1213],[1052,13629,1221],{"className":13630},[1220],[1052,13632,13634],{"className":13633},[1193],[1052,13635,13637],{"className":13636,"style":1855},[1197],[1052,13638],{},[1052,13640],{"className":13641,"style":3378},[1154],[1052,13643,3111],{"className":13644},[2428],[1052,13646],{"className":13647,"style":3378},[1154],[1052,13649,13651,13655,13704,13750,13753,13756],{"className":13650},[1139],[1052,13652],{"className":13653,"style":13654},[1143],"height:0.9722em;vertical-align:-0.2861em;",[1052,13656,13658,13661],{"className":13657},[1148],[1052,13659,12429],{"className":13660},[1148,4785],[1052,13662,13664],{"className":13663},[1184],[1052,13665,13667,13696],{"className":13666},[1188,1189],[1052,13668,13670,13693],{"className":13669},[1193],[1052,13671,13673],{"className":13672,"style":4690},[1197],[1052,13674,13675,13678],{"style":1201},[1052,13676],{"className":13677,"style":1206},[1205],[1052,13679,13681],{"className":13680},[1210,1211,1212,1213],[1052,13682,13684,13687,13690],{"className":13683},[1148,1213],[1052,13685,3799],{"className":13686},[1148,1149,1213],[1052,13688,3988],{"className":13689},[2428,1213],[1052,13691,1123],{"className":13692},[1148,1213],[1052,13694,1221],{"className":13695},[1220],[1052,13697,13699],{"className":13698},[1193],[1052,13700,13702],{"className":13701,"style":11713},[1197],[1052,13703],{},[1052,13705,13707,13710],{"className":13706},[1148],[1052,13708,11120],{"className":13709,"style":4786},[1148,4785],[1052,13711,13713],{"className":13712},[1184],[1052,13714,13716,13742],{"className":13715},[1188,1189],[1052,13717,13719,13739],{"className":13718},[1193],[1052,13720,13722],{"className":13721,"style":13312},[1197],[1052,13723,13724,13727],{"style":13202},[1052,13725],{"className":13726,"style":1206},[1205],[1052,13728,13730],{"className":13729},[1210,1211,1212,1213],[1052,13731,13733,13736],{"className":13732},[1148,1213],[1052,13734,10971],{"className":13735},[1148,1149,1213],[1052,13737,1746],{"className":13738,"style":1762},[1148,1149,1213],[1052,13740,1221],{"className":13741},[1220],[1052,13743,13745],{"className":13744},[1193],[1052,13746,13748],{"className":13747,"style":1855},[1197],[1052,13749],{},[1052,13751],{"className":13752,"style":3378},[1154],[1052,13754,3111],{"className":13755},[2428],[1052,13757],{"className":13758,"style":3378},[1154],[1052,13760,13762,13765,13805,13808],{"className":13761},[1139],[1052,13763],{"className":13764,"style":13546},[1143],[1052,13766,13768,13771],{"className":13767},[1148],[1052,13769,10727],{"className":13770},[1148,4785],[1052,13772,13774],{"className":13773},[1184],[1052,13775,13777,13797],{"className":13776},[1188,1189],[1052,13778,13780,13794],{"className":13779},[1193],[1052,13781,13783],{"className":13782,"style":13312},[1197],[1052,13784,13785,13788],{"style":1201},[1052,13786],{"className":13787,"style":1206},[1205],[1052,13789,13791],{"className":13790},[1210,1211,1212,1213],[1052,13792,1746],{"className":13793,"style":1762},[1148,1149,1213],[1052,13795,1221],{"className":13796},[1220],[1052,13798,13800],{"className":13799},[1193],[1052,13801,13803],{"className":13802,"style":1855},[1197],[1052,13804],{},[1052,13806,1107],{"className":13807},[1284],[1052,13809,1097],{"className":13810},[1234],[1052,13812,13814],{"className":13813},[1865],[1052,13815,13817,13891],{"className":13816},[1055],[1052,13818,13820],{"className":13819},[1059],[1061,13821,13822],{"xmlns":1063,"display":1874},[1065,13823,13824,13888],{},[1068,13825,13826,13832,13834,13836,13838,13844,13854,13856,13868,13878,13880,13886],{},[1086,13827,13828,13830],{},[1071,13829,12864],{"mathvariant":4609},[1071,13831,3799],{},[1075,13833,1077],{},[1071,13835,13013],{},[1075,13837,1084],{"stretchy":1080},[1086,13839,13840,13842],{},[1071,13841,4571],{"mathvariant":4609},[1071,13843,3799],{},[1086,13845,13846,13848],{},[1071,13847,11120],{"mathvariant":4609},[1068,13849,13850,13852],{},[1071,13851,1090],{},[1071,13853,1907],{},[1075,13855,3111],{},[1086,13857,13858,13860],{},[1071,13859,12429],{"mathvariant":4609},[1068,13861,13862,13864,13866],{},[1071,13863,3799],{},[1075,13865,3988],{},[1121,13867,1123],{},[1086,13869,13870,13872],{},[1071,13871,11120],{"mathvariant":4609},[1068,13873,13874,13876],{},[1071,13875,10971],{},[1071,13877,1907],{},[1075,13879,3111],{},[1086,13881,13882,13884],{},[1071,13883,10727],{"mathvariant":4609},[1071,13885,1907],{},[1075,13887,1107],{"stretchy":1080},[1128,13889,13890],{"encoding":1130},"\\mathbf{O}_t = \\sigma(\\mathbf{X}_t \\mathbf{W}_{xo} + \\mathbf{H}_{t-1} \\mathbf{W}_{ho} + \\mathbf{b}_o)",[1052,13892,13894,13949,14056,14166],{"className":13893,"ariaHidden":1096},[1135],[1052,13895,13897,13900,13940,13943,13946],{"className":13896},[1139],[1052,13898],{"className":13899,"style":13080},[1143],[1052,13901,13903,13906],{"className":13902},[1148],[1052,13904,12864],{"className":13905},[1148,4785],[1052,13907,13909],{"className":13908},[1184],[1052,13910,13912,13932],{"className":13911},[1188,1189],[1052,13913,13915,13929],{"className":13914},[1193],[1052,13916,13918],{"className":13917,"style":11602},[1197],[1052,13919,13920,13923],{"style":1201},[1052,13921],{"className":13922,"style":1206},[1205],[1052,13924,13926],{"className":13925},[1210,1211,1212,1213],[1052,13927,3799],{"className":13928},[1148,1149,1213],[1052,13930,1221],{"className":13931},[1220],[1052,13933,13935],{"className":13934},[1193],[1052,13936,13938],{"className":13937,"style":1228},[1197],[1052,13939],{},[1052,13941],{"className":13942,"style":1155},[1154],[1052,13944,1077],{"className":13945},[1159],[1052,13947],{"className":13948,"style":1155},[1154],[1052,13950,13952,13955,13958,13961,14001,14047,14050,14053],{"className":13951},[1139],[1052,13953],{"className":13954,"style":1597},[1143],[1052,13956,13013],{"className":13957,"style":1245},[1148,1149],[1052,13959,1084],{"className":13960},[1173],[1052,13962,13964,13967],{"className":13963},[1148],[1052,13965,4571],{"className":13966},[1148,4785],[1052,13968,13970],{"className":13969},[1184],[1052,13971,13973,13993],{"className":13972},[1188,1189],[1052,13974,13976,13990],{"className":13975},[1193],[1052,13977,13979],{"className":13978,"style":11602},[1197],[1052,13980,13981,13984],{"style":1201},[1052,13982],{"className":13983,"style":1206},[1205],[1052,13985,13987],{"className":13986},[1210,1211,1212,1213],[1052,13988,3799],{"className":13989},[1148,1149,1213],[1052,13991,1221],{"className":13992},[1220],[1052,13994,13996],{"className":13995},[1193],[1052,13997,13999],{"className":13998,"style":1228},[1197],[1052,14000],{},[1052,14002,14004,14007],{"className":14003},[1148],[1052,14005,11120],{"className":14006,"style":4786},[1148,4785],[1052,14008,14010],{"className":14009},[1184],[1052,14011,14013,14039],{"className":14012},[1188,1189],[1052,14014,14016,14036],{"className":14015},[1193],[1052,14017,14019],{"className":14018,"style":1825},[1197],[1052,14020,14021,14024],{"style":13202},[1052,14022],{"className":14023,"style":1206},[1205],[1052,14025,14027],{"className":14026},[1210,1211,1212,1213],[1052,14028,14030,14033],{"className":14029},[1148,1213],[1052,14031,1090],{"className":14032},[1148,1149,1213],[1052,14034,1907],{"className":14035},[1148,1149,1213],[1052,14037,1221],{"className":14038},[1220],[1052,14040,14042],{"className":14041},[1193],[1052,14043,14045],{"className":14044,"style":1228},[1197],[1052,14046],{},[1052,14048],{"className":14049,"style":3378},[1154],[1052,14051,3111],{"className":14052},[2428],[1052,14054],{"className":14055,"style":3378},[1154],[1052,14057,14059,14062,14111,14157,14160,14163],{"className":14058},[1139],[1052,14060],{"className":14061,"style":13244},[1143],[1052,14063,14065,14068],{"className":14064},[1148],[1052,14066,12429],{"className":14067},[1148,4785],[1052,14069,14071],{"className":14070},[1184],[1052,14072,14074,14103],{"className":14073},[1188,1189],[1052,14075,14077,14100],{"className":14076},[1193],[1052,14078,14080],{"className":14079,"style":4690},[1197],[1052,14081,14082,14085],{"style":1201},[1052,14083],{"className":14084,"style":1206},[1205],[1052,14086,14088],{"className":14087},[1210,1211,1212,1213],[1052,14089,14091,14094,14097],{"className":14090},[1148,1213],[1052,14092,3799],{"className":14093},[1148,1149,1213],[1052,14095,3988],{"className":14096},[2428,1213],[1052,14098,1123],{"className":14099},[1148,1213],[1052,14101,1221],{"className":14102},[1220],[1052,14104,14106],{"className":14105},[1193],[1052,14107,14109],{"className":14108,"style":11713},[1197],[1052,14110],{},[1052,14112,14114,14117],{"className":14113},[1148],[1052,14115,11120],{"className":14116,"style":4786},[1148,4785],[1052,14118,14120],{"className":14119},[1184],[1052,14121,14123,14149],{"className":14122},[1188,1189],[1052,14124,14126,14146],{"className":14125},[1193],[1052,14127,14129],{"className":14128,"style":13312},[1197],[1052,14130,14131,14134],{"style":13202},[1052,14132],{"className":14133,"style":1206},[1205],[1052,14135,14137],{"className":14136},[1210,1211,1212,1213],[1052,14138,14140,14143],{"className":14139},[1148,1213],[1052,14141,10971],{"className":14142},[1148,1149,1213],[1052,14144,1907],{"className":14145},[1148,1149,1213],[1052,14147,1221],{"className":14148},[1220],[1052,14150,14152],{"className":14151},[1193],[1052,14153,14155],{"className":14154,"style":1228},[1197],[1052,14156],{},[1052,14158],{"className":14159,"style":3378},[1154],[1052,14161,3111],{"className":14162},[2428],[1052,14164],{"className":14165,"style":3378},[1154],[1052,14167,14169,14172,14212],{"className":14168},[1139],[1052,14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\\mathbf{W}_{xf}, \\mathbf{W}_{xo} \\in \\mathbb{R}^{d \\times h} \\text{ 和 } \\mathbf{W}_{hi}, \\mathbf{W}_{hf}, \\mathbf{W}_{ho} \\in \\mathbb{R}^{h \\times h}",[1052,14334,14336,14501,14717],{"className":14335,"ariaHidden":1096},[1135],[1052,14337,14339,14342,14388,14391,14394,14440,14443,14446,14492,14495,14498],{"className":14338},[1139],[1052,14340],{"className":14341,"style":13654},[1143],[1052,14343,14345,14348],{"className":14344},[1148],[1052,14346,11120],{"className":14347,"style":4786},[1148,4785],[1052,14349,14351],{"className":14350},[1184],[1052,14352,14354,14380],{"className":14353},[1188,1189],[1052,14355,14357,14377],{"className":14356},[1193],[1052,14358,14360],{"className":14359,"style":1198},[1197],[1052,14361,14362,14365],{"style":13202},[1052,14363],{"className":14364,"style":1206},[1205],[1052,14366,14368],{"className":14367},[1210,1211,1212,1213],[1052,14369,14371,14374],{"className":14370},[1148,1213],[1052,14372,1090],{"className":14373},[1148,1149,1213],[1052,14375,1093],{"className":14376},[1148,1149,1213],[1052,14378,1221],{"className":14379},[1220],[1052,14381,14383],{"className":14382},[1193],[1052,14384,14386],{"className":14385,"style":1228},[1197],[1052,14387],{},[1052,14389,1097],{"className":14390},[1234],[1052,14392],{"className":14393,"style":1238},[1154],[1052,14395,14397,14400],{"className":14396},[1148],[1052,14398,11120],{"className":14399,"style":4786},[1148,4785],[1052,14401,14403],{"className":14402},[1184],[1052,14404,14406,14432],{"className":14405},[1188,1189],[1052,14407,14409,14429],{"className":14408},[1193],[1052,14410,14412],{"className":14411,"style":13312},[1197],[1052,14413,14414,14417],{"style":13202},[1052,14415],{"className":14416,"style":1206},[1205],[1052,14418,14420],{"className":14419},[1210,1211,1212,1213],[1052,14421,14423,14426],{"className":14422},[1148,1213],[1052,14424,1090],{"className":14425},[1148,1149,1213],[1052,14427,1746],{"className":14428,"style":1762},[1148,1149,1213],[1052,14430,1221],{"className":14431},[1220],[1052,14433,14435],{"className":14434},[1193],[1052,14436,14438],{"className":14437,"style":1855},[1197],[1052,14439],{},[1052,14441,1097],{"className":14442},[1234],[1052,14444],{"className":14445,"style":1238},[1154],[1052,14447,14449,14452],{"className":14448},[1148],[1052,14450,11120],{"className":14451,"style":4786},[1148,4785],[1052,14453,14455],{"className":14454},[1184],[1052,14456,14458,14484],{"className":14457},[1188,1189],[1052,14459,14461,14481],{"className":14460},[1193],[1052,14462,14464],{"className":14463,"style":1825},[1197],[1052,14465,14466,14469],{"style":13202},[1052,14467],{"className":14468,"style":1206},[1205],[1052,14470,14472],{"className":14471},[1210,1211,1212,1213],[1052,14473,14475,14478],{"className":14474},[1148,1213],[1052,14476,1090],{"className":14477},[1148,1149,1213],[1052,14479,1907],{"className":14480},[1148,1149,1213],[1052,14482,1221],{"className":14483},[1220],[1052,14485,14487],{"className":14486},[1193],[1052,14488,14490],{"className":14489,"style":1228},[1197],[1052,14491],{},[1052,14493],{"className":14494,"style":1155},[1154],[1052,14496,12288],{"className":14497},[1159],[1052,14499],{"className":14500,"style":1155},[1154],[1052,14502,14504,14508,14547,14561,14604,14607,14610,14656,14659,14662,14708,14711,14714],{"className":14503},[1139],[1052,14505],{"className":14506,"style":14507},[1143],"height:1.1352em;vertical-align:-0.2861em;",[1052,14509,14511,14515],{"className":14510},[1148],[1052,14512,2558],{"className":14513},[1148,14514],"mathbb",[1052,14516,14518],{"className":14517},[1184],[1052,14519,14521],{"className":14520},[1188],[1052,14522,14524],{"className":14523},[1193],[1052,14525,14527],{"className":14526,"style":12370},[1197],[1052,14528,14529,14532],{"style":1327},[1052,14530],{"className":14531,"style":1206},[1205],[1052,14533,14535],{"className":14534},[1210,1211,1212,1213],[1052,14536,14538,14541,14544],{"className":14537},[1148,1213],[1052,14539,2370],{"className":14540},[1148,1149,1213],[1052,14542,2329],{"className":14543},[2428,1213],[1052,14545,10971],{"className":14546},[1148,1149,1213],[1052,14548,14551,14554,14558],{"className":14549},[1148,14550],"text",[1052,14552,10123],{"className":14553},[1148],[1052,14555,2755],{"className":14556},[1148,14557],"cjk_fallback",[1052,14559,10123],{"className":14560},[1148],[1052,14562,14564,14567],{"className":14563},[1148],[1052,14565,11120],{"className":14566,"style":4786},[1148,4785],[1052,14568,14570],{"className":14569},[1184],[1052,14571,14573,14596],{"className":14572},[1188,1189],[1052,14574,14576,14593],{"className":14575},[1193],[1052,14577,14579],{"className":14578,"style":13312},[1197],[1052,14580,14581,14584],{"style":13202},[1052,14582],{"className":14583,"style":1206},[1205],[1052,14585,14587],{"className":14586},[1210,1211,1212,1213],[1052,14588,14590],{"className":14589},[1148,1213],[1052,14591,13327],{"className":14592},[1148,1149,1213],[1052,14594,1221],{"className":14595},[1220],[1052,14597,14599],{"className":14598},[1193],[1052,14600,14602],{"className":14601,"style":1228},[1197],[1052,14603],{},[1052,14605,1097],{"className":14606},[1234],[1052,1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\\mathbf{b}_f, \\mathbf{b}_o \\in \\mathbb{R}^{1 \\times h}",[1052,14814,14816,14963],{"className":14815,"ariaHidden":1096},[1135],[1052,14817,14819,14822,14862,14865,14868,14908,14911,14914,14954,14957,14960],{"className":14818},[1139],[1052,14820],{"className":14821,"style":1806},[1143],[1052,14823,14825,14828],{"className":14824},[1148],[1052,14826,10727],{"className":14827},[1148,4785],[1052,14829,14831],{"className":14830},[1184],[1052,14832,14834,14854],{"className":14833},[1188,1189],[1052,14835,14837,14851],{"className":14836},[1193],[1052,14838,14840],{"className":14839,"style":1198},[1197],[1052,14841,14842,14845],{"style":1201},[1052,14843],{"className":14844,"style":1206},[1205],[1052,14846,14848],{"className":14847},[1210,1211,1212,1213],[1052,14849,1093],{"className":14850},[1148,1149,1213],[1052,14852,1221],{"className":14853},[1220],[1052,14855,14857],{"className":14856},[1193],[1052,14858,14860],{"className":14859,"style":1228},[1197],[1052,14861],{},[1052,14863,1097],{"className":14864},[1234],[1052,14866],{"className":14867,"style":1238},[1154],[1052,14869,14871,14874],{"className":14870},[1148],[1052,14872,10727],{"className":14873},[1148,4785],[1052,14875,14877],{"className":14876},[1184],[1052,14878,14880,14900],{"className":14879},[1188,1189],[1052,14881,14883,14897],{"className":14882},[1193],[1052,14884,14886],{"className":14885,"style":13312},[1197],[1052,14887,14888,14891],{"style":1201},[1052,14889],{"className":14890,"style":1206},[1205],[1052,14892,14894],{"className":14893},[1210,1211,1212,1213],[1052,14895,1746],{"className":14896,"style":1762},[1148,1149,1213],[1052,14898,1221],{"className":14899},[1220],[1052,14901,14903],{"className":14902},[1193],[1052,14904,14906],{"className":14905,"style":1855},[1197],[1052,14907],{},[1052,14909,1097],{"className":14910},[1234],[1052,14912],{"className":14913,"style":1238},[1154],[1052,14915,14917,14920],{"className":14916},[1148],[1052,14918,10727],{"className":14919},[1148,4785],[1052,14921,14923],{"className":14922},[1184],[1052,14924,14926,14946],{"className":14925},[1188,1189],[1052,14927,14929,14943],{"className":14928},[1193],[1052,14930,14932],{"className":14931,"style":1825},[1197],[1052,14933,14934,14937],{"style":1201},[1052,14935],{"className":14936,"style":1206},[1205],[1052,14938,14940],{"className":14939},[1210,1211,1212,1213],[1052,14941,1907],{"className":14942},[1148,1149,1213],[1052,14944,1221],{"className":14945},[1220],[1052,14947,14949],{"className":14948},[1193],[1052,14950,14952],{"className":14951,"style":1228},[1197],[1052,14953],{},[1052,14955],{"className":14956,"style":1155},[1154],[1052,14958,12288],{"className":14959},[1159],[1052,14961],{"className":14962,"style":1155},[1154],[1052,14964,14966,14969],{"className":14965},[1139],[1052,14967],{"className":14968,"style":12370},[1143],[1052,14970,14972,14975],{"className":14971},[1148],[1052,14973,2558],{"className":14974},[1148,14514],[1052,14976,14978],{"className":14977},[1184],[1052,14979,14981],{"className":14980},[1188],[1052,14982,14984],{"className":14983},[1193],[1052,14985,14987],{"className":14986,"style":12370},[1197],[1052,14988,14989,14992],{"style":1327},[1052,14990],{"className":14991,"style":1206},[1205],[1052,14993,14995],{"className":14994},[1210,1211,1212,1213],[1052,14996,14998,15001,15004],{"className":14997},[1148,1213],[1052,14999,1123],{"className":15000},[1148,1213],[1052,15002,2329],{"className":15003},[2428,1213],[1052,15005,10971],{"className":15006},[1148,1149,1213],"是偏置参数。",[4548,15009,15011],{"id":15010},"候选记忆元-candidate-memory-cell","候选记忆元 (candidate memory cell)",[1052,15013,15015],{"className":15014},[1865],[1052,15016,15018,15056],{"className":15017},[1055],[1052,15019,15021],{"className":15020},[1059],[1061,15022,15023],{"xmlns":1063,"display":1874},[1065,15024,15025,15053],{},[1068,15026,15027,15039,15041],{},[1086,15028,15029,15037],{},[15030,15031,15032,15034],"mover",{"accent":1096},[1071,15033,3595],{"mathvariant":4609},[1075,15035,15036],{},"~",[1071,15038,3799],{},[1075,15040,12288],{},[5084,15042,15043,15045],{},[1071,15044,2558],{"mathvariant":14270},[1068,15046,15047,15049,15051],{},[1071,15048,1126],{},[1075,15050,2329],{},[1071,15052,10971],{},[1128,15054,15055],{"encoding":1130},"\\tilde{\\mathbf{C}}_t \\in \\mathbb{R}^{n \\times h}",[1052,15057,15059,15149],{"className":15058,"ariaHidden":1096},[1135],[1052,15060,15062,15066,15140,15143,15146],{"className":15061},[1139],[1052,15063],{"className":15064,"style":15065},[1143],"height:1.073em;vertical-align:-0.15em;",[1052,15067,15069,15106],{"className":15068},[1148],[1052,15070,15073],{"className":15071},[1148,15072],"accent",[1052,15074,15076],{"className":15075},[1188],[1052,15077,15079],{"className":15078},[1193],[1052,15080,15083,15092],{"className":15081,"style":15082},[1197],"height:0.923em;",[1052,15084,15086,15089],{"style":15085},"top:-3em;",[1052,15087],{"className":15088,"style":1976},[1205],[1052,15090,3595],{"className":15091},[1148,4785],[1052,15093,15095,15098],{"style":15094},"top:-3.6051em;",[1052,15096],{"className":15097,"style":1976},[1205],[1052,15099,15103],{"className":15100,"style":15102},[15101],"accent-body","left:-0.25em;",[1052,15104,15036],{"className":15105},[1148],[1052,15107,15109],{"className":15108},[1184],[1052,15110,15112,15132],{"className":15111},[1188,1189],[1052,15113,15115,15129],{"className":15114},[1193],[1052,15116,15118],{"className":15117,"style":11602},[1197],[1052,15119,15120,15123],{"style":1201},[1052,15121],{"className":15122,"style":1206},[1205],[1052,15124,15126],{"className":15125},[1210,1211,1212,1213],[1052,15127,3799],{"className":15128},[1148,1149,1213],[1052,15130,1221],{"className":15131},[1220],[1052,15133,15135],{"className":15134},[1193],[1052,15136,15138],{"className":15137,"style":1228},[1197],[1052,15139],{},[1052,15141],{"className":15142,"style":1155},[1154],[1052,15144,12288],{"className":15145},[1159],[1052,15147],{"className":15148,"style":1155},[1154],[1052,15150,15152,15156],{"className":15151},[1139],[1052,15153],{"className":15154,"style":15155},[1143],"height:0.8991em;",[1052,15157,15159,15162],{"className":15158},[1148],[1052,15160,2558],{"className":15161},[1148,14514],[1052,15163,15165],{"className":15164},[1184],[1052,15166,15168],{"className":15167},[1188],[1052,15169,15171],{"className":15170},[1193],[1052,15172,15174],{"className":15173,"style":15155},[1197],[1052,15175,15176,15179],{"style":5234},[1052,15177],{"className":15178,"style":1206},[1205],[1052,15180,15182],{"className":15181},[1210,1211,1212,1213],[1052,15183,15185,15188,15191],{"className":15184},[1148,1213],[1052,15186,1126],{"className":15187},[1148,1149,1213],[1052,15189,2329],{"className":15190},[2428,1213],[1052,15192,10971],{"className":15193},[1148,1149,1213],[996,15195,15196],{},"它的计算与上面描述的三个门的计算类似，但是使用tanh函数作为激活函数，函数的值范围为(-1,1)。下面导出在时间步t处的方程：",[1052,15198,15200],{"className":15199},[1865],[1052,15201,15203,15285],{"className":15202},[1055],[1052,15204,15206],{"className":15205},[1059],[1061,15207,15208],{"xmlns":1063,"display":1874},[1065,15209,15210,15282],{},[1068,15211,15212,15222,15224,15227,15230,15232,15238,15248,15250,15262,15272,15274,15280],{},[1086,15213,15214,15220],{},[15030,15215,15216,15218],{"accent":1096},[1071,15217,3595],{"mathvariant":4609},[1075,15219,15036],{},[1071,15221,3799],{},[1075,15223,1077],{},[1071,15225,15226],{},"tanh",[1075,15228,15229],{},"⁡",[1075,15231,1084],{"stretchy":1080},[1086,15233,15234,15236],{},[1071,15235,4571],{"mathvariant":4609},[1071,15237,3799],{},[1086,15239,15240,15242],{},[1071,15241,11120],{"mathvariant":4609},[1068,15243,15244,15246],{},[1071,15245,1090],{},[1071,15247,4136],{},[1075,15249,3111],{},[1086,15251,15252,15254],{},[1071,15253,12429],{"mathvariant":4609},[1068,15255,15256,15258,15260],{},[1071,15257,3799],{},[1075,15259,3988],{},[1121,15261,1123],{},[1086,15263,15264,15266],{},[1071,15265,11120],{"mathvariant":4609},[1068,15267,15268,15270],{},[1071,15269,10971],{},[1071,15271,4136],{},[1075,15273,3111],{},[1086,15275,15276,15278],{},[1071,15277,10727],{"mathvariant":4609},[1071,15279,4136],{},[1075,15281,1107],{"stretchy":1080},[1128,15283,15284],{"encoding":1130},"\\tilde{\\mathbf{C}}_t = \\tanh(\\mathbf{X}_t \\mathbf{W}_{xc} + \\mathbf{H}_{t-1} \\mathbf{W}_{hc} + \\mathbf{b}_c)",[1052,15286,15288,15371,15478,15588],{"className":15287,"ariaHidden":1096},[1135],[1052,15289,15291,15294,15362,15365,15368],{"className":15290},[1139],[1052,15292],{"className":15293,"style":15065},[1143],[1052,15295,15297,15328],{"className":15296},[1148],[1052,15298,15300],{"className":15299},[1148,15072],[1052,15301,15303],{"className":15302},[1188],[1052,15304,15306],{"className":15305},[1193],[1052,15307,15309,15317],{"className":15308,"style":15082},[1197],[1052,15310,15311,15314],{"style":15085},[1052,15312],{"className":15313,"style":1976},[1205],[1052,15315,3595],{"className":15316},[1148,4785],[1052,15318,15319,15322],{"style":15094},[1052,15320],{"className":15321,"style":1976},[1205],[1052,15323,15325],{"className":15324,"style":15102},[15101],[1052,15326,15036],{"className":15327},[1148],[1052,15329,15331],{"className":15330},[1184],[1052,15332,15334,15354],{"className":15333},[1188,1189],[1052,15335,15337,15351],{"className":15336},[1193],[1052,15338,15340],{"className":15339,"style":11602},[1197],[1052,15341,15342,15345],{"style":1201},[1052,15343],{"className":15344,"style":1206},[1205],[1052,15346,15348],{"className":15347},[1210,1211,1212,1213],[1052,15349,3799],{"className":15350},[1148,1149,1213],[1052,15352,1221],{"className":15353},[1220],[1052,15355,15357],{"className":15356},[1193],[1052,15358,15360],{"className":15359,"style":1228},[1197],[1052,15361],{},[1052,15363],{"className":15364,"style":1155},[1154],[1052,15366,1077],{"className":15367},[1159],[1052,15369],{"className":15370,"style":1155},[1154],[1052,15372,15374,15377,15380,15383,15423,15469,15472,15475],{"className":15373},[1139],[1052,15375],{"className":15376,"style":1597},[1143],[1052,15378,15226],{"className":15379},[2027],[1052,15381,1084],{"className":15382},[1173],[1052,15384,15386,15389],{"className":15385},[1148],[1052,15387,4571],{"className":15388},[1148,4785],[1052,15390,15392],{"className":15391},[1184],[1052,15393,15395,15415],{"className":15394},[1188,1189],[1052,15396,15398,15412],{"className":15397},[1193],[1052,15399,15401],{"className":15400,"style":11602},[1197],[1052,15402,15403,15406],{"style":1201},[1052,15404],{"className":15405,"style":1206},[1205],[1052,15407,15409],{"className":15408},[1210,1211,1212,1213],[1052,15410,3799],{"className":15411},[1148,1149,1213],[1052,15413,1221],{"className":15414},[1220],[1052,15416,15418],{"className":15417},[1193],[1052,15419,15421],{"className":15420,"style":1228},[1197],[1052,15422],{},[1052,15424,15426,15429],{"className":15425},[1148],[1052,15427,11120],{"className":15428,"style":4786},[1148,4785],[1052,15430,15432],{"className":15431},[1184],[1052,15433,15435,15461],{"className":15434},[1188,1189],[1052,15436,15438,15458],{"className":15437},[1193],[1052,15439,15441],{"className":15440,"style":1825},[1197],[1052,15442,15443,15446],{"style":13202},[1052,15444],{"className":15445,"style":1206},[1205],[1052,15447,15449],{"className":15448},[1210,1211,1212,1213],[1052,15450,15452,15455],{"className":15451},[1148,1213],[1052,15453,1090],{"className":15454},[1148,1149,1213],[1052,15456,4136],{"className":15457},[1148,1149,1213],[1052,15459,1221],{"className":15460},[1220],[1052,15462,15464],{"className":15463},[1193],[1052,15465,15467],{"className":15466,"style":1228},[1197],[1052,15468],{},[1052,15470],{"className":15471,"style":3378},[1154],[1052,15473,3111],{"className":15474},[2428],[1052,15476],{"className":15477,"style":3378},[1154],[1052,15479,15481,15484,15533,15579,15582,15585],{"className":15480},[1139],[1052,15482],{"className":15483,"style":13244},[1143],[1052,15485,15487,15490],{"className":15486},[1148],[1052,15488,12429],{"className":15489},[1148,4785],[1052,15491,15493],{"className":15492},[1184],[1052,15494,15496,15525],{"className":15495},[1188,1189],[1052,15497,15499,15522],{"className":15498},[1193],[1052,15500,15502],{"className":15501,"style":4690},[1197],[1052,15503,15504,15507],{"style":1201},[1052,15505],{"className":15506,"style":1206},[1205],[1052,15508,15510],{"className":15509},[1210,1211,1212,1213],[1052,15511,15513,15516,15519],{"className":15512},[1148,1213],[1052,15514,3799],{"className":15515},[1148,1149,1213],[1052,15517,3988],{"className":15518},[2428,1213],[1052,15520,1123],{"className":15521},[1148,1213],[1052,15523,1221],{"className":15524},[1220],[1052,15526,15528],{"className":15527},[1193],[1052,15529,15531],{"className":15530,"style":11713},[1197],[1052,15532],{},[1052,15534,15536,15539],{"className":15535},[1148],[1052,15537,11120],{"className":15538,"style":4786},[1148,4785],[1052,15540,15542],{"className":15541},[1184],[1052,15543,15545,15571],{"className":15544},[1188,1189],[1052,15546,15548,15568],{"className":15547},[1193],[1052,15549,15551],{"className":15550,"style":13312},[1197],[1052,15552,15553,15556],{"style":13202},[1052,15554],{"className":15555,"style":1206},[1205],[1052,15557,15559],{"className":15558},[1210,1211,1212,1213],[1052,15560,15562,15565],{"className":15561},[1148,1213],[1052,15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\\in \\mathbb{R}^{d \\times h} \\text{ 和 } \\mathbf{W}_{hc} \\in \\mathbb{R}^{h \\times h}",[1052,15705,15707,15768,15880],{"className":15706,"ariaHidden":1096},[1135],[1052,15708,15710,15713,15759,15762,15765],{"className":15709},[1139],[1052,15711],{"className":15712,"style":13080},[1143],[1052,15714,15716,15719],{"className":15715},[1148],[1052,15717,11120],{"className":15718,"style":4786},[1148,4785],[1052,15720,15722],{"className":15721},[1184],[1052,15723,15725,15751],{"className":15724},[1188,1189],[1052,15726,15728,15748],{"className":15727},[1193],[1052,15729,15731],{"className":15730,"style":1825},[1197],[1052,15732,15733,15736],{"style":13202},[1052,15734],{"className":15735,"style":1206},[1205],[1052,15737,15739],{"className":15738},[1210,1211,1212,1213],[1052,15740,15742,15745],{"className":15741},[1148,1213],[1052,15743,1090],{"className":15744},[1148,1149,1213],[1052,15746,4136],{"className":15747},[1148,1149,1213],[1052,15749,1221],{"className":15750},[1220],[1052,15752,15754],{"className":15753},[1193],[1052,15755,15757],{"className":15756,"style":1228},[1197],[1052,15758],{},[1052,15760],{"className":15761,"style":1155},[1154],[1052,15763,12288],{"className":15764},[1159],[1052,15766],{"className":15767,"style":1155},[1154],[1052,15769,15771,15775,15813,15825,15871,15874,15877],{"className":15770},[1139],[1052,15772],{"className":15773,"style":15774},[1143],"height:0.9991em;vertical-align:-0.15em;",[1052,15776,15778,15781],{"className":15777},[1148],[1052,15779,2558],{"className":15780},[1148,14514],[1052,15782,15784],{"className":15783},[1184],[1052,15785,15787],{"className":15786},[1188],[1052,15788,15790],{"className":15789},[1193],[1052,15791,15793],{"className":15792,"style":12370},[1197],[1052,15794,15795,15798],{"style":1327},[1052,15796],{"className":15797,"style":1206},[1205],[1052,15799,15801],{"className":15800},[1210,1211,1212,1213],[1052,15802,15804,15807,15810],{"className":15803},[1148,1213],[1052,15805,2370],{"className":15806},[1148,1149,1213],[1052,15808,2329],{"className":15809},[2428,1213],[1052,15811,10971],{"className":15812},[1148,1149,1213],[1052,15814,15816,15819,15822],{"className":15815},[1148,14550],[1052,15817,10123],{"className":15818},[1148],[1052,15820,2755],{"className":15821},[1148,14557],[1052,15823,10123],{"className":15824},[1148],[1052,15826,15828,15831],{"className":15827},[1148],[1052,15829,11120],{"className":15830,"style":4786},[1148,4785],[1052,15832,15834],{"className":15833},[1184],[1052,15835,15837,15863],{"className":15836},[1188,1189],[1052,15838,15840,15860],{"className":15839},[1193],[1052,15841,15843],{"className":15842,"style":13312},[1197],[1052,15844,15845,15848],{"style":13202},[1052,15846],{"className":15847,"style":1206},[1205],[1052,15849,15851],{"className":15850},[1210,1211,1212,1213],[1052,15852,15854,15857],{"className":15853},[1148,1213],[1052,15855,10971],{"className":15856},[1148,1149,1213],[1052,15858,4136],{"className":15859},[1148,1149,1213],[1052,15861,1221],{"className":15862},[1220],[1052,15864,15866],{"className":15865},[1193],[1052,15867,15869],{"className":15868,"style":1228},[1197],[1052,15870],{},[1052,15872],{"className":15873,"style":1155},[1154],[1052,15875,12288],{"className":15876},[1159],[1052,15878],{"className":15879,"style":1155},[1154],[1052,15881,15883,15886],{"className":15882},[1139],[1052,15884],{"className":15885,"style":12370},[1143],[1052,15887,15889,15892],{"className":15888},[1148],[1052,15890,2558],{"className":15891},[1148,14514],[1052,15893,15895],{"className":15894},[1184],[1052,15896,15898],{"className":15897},[1188],[1052,15899,15901],{"className":15900},[1193],[1052,15902,15904],{"className":15903,"style":12370},[1197],[1052,15905,15906,15909],{"style":1327},[1052,15907],{"className":15908,"style":1206},[1205],[1052,15910,15912],{"className":15911},[1210,1211,1212,1213],[1052,15913,15915,15918,15921],{"className":15914},[1148,1213],[1052,15916,10971],{"className":15917},[1148,1149,1213],[1052,15919,2329],{"className":15920},[2428,1213],[1052,15922,10971],{"className":15923},[1148,1149,1213],[1052,15925,15927,15959],{"className":15926},[1055],[1052,15928,15930],{"className":15929},[1059],[1061,15931,15932],{"xmlns":1063},[1065,15933,15934,15956],{},[1068,15935,15936,15942,15944],{},[1086,15937,15938,15940],{},[1071,15939,10727],{"mathvariant":4609},[1071,15941,4136],{},[1075,15943,12288],{},[5084,15945,15946,15948],{},[1071,15947,2558],{"mathvariant":14270},[1068,15949,15950,15952,15954],{},[1121,15951,1123],{},[1075,15953,2329],{},[1071,15955,10971],{},[1128,15957,15958],{"encoding":1130},"\\mathbf{b}_c \\in \\mathbb{R}^{1 \\times h}",[1052,15960,15962,16017],{"className":15961,"ariaHidden":1096},[1135],[1052,15963,15965,15968,16008,16011,16014],{"className":15964},[1139],[1052,15966],{"className":15967,"style":11871},[1143],[1052,15969,15971,15974],{"className":15970},[1148],[1052,15972,10727],{"className":15973},[1148,4785],[1052,15975,15977],{"className":15976},[1184],[1052,15978,15980,16000],{"className":15979},[1188,1189],[1052,15981,15983,15997],{"className":15982},[1193],[1052,15984,15986],{"className":15985,"style":1825},[1197],[1052,15987,15988,15991],{"style":1201},[1052,15989],{"className":15990,"style":1206},[1205],[1052,15992,15994],{"className":15993},[1210,1211,1212,1213],[1052,15995,4136],{"className":15996},[1148,1149,1213],[1052,15998,1221],{"className":15999},[1220],[1052,16001,16003],{"className":16002},[1193],[1052,16004,16006],{"className":16005,"style":1228},[1197],[1052,16007],{},[1052,16009],{"className":16010,"style":1155},[1154],[1052,16012,12288],{"className":16013},[1159],[1052,16015],{"className":16016,"style":1155},[1154],[1052,16018,16020,16023],{"className":16019},[1139],[1052,16021],{"className":16022,"style":12370},[1143],[1052,16024,16026,16029],{"className":16025},[1148],[1052,16027,2558],{"className":16028},[1148,14514],[1052,16030,16032],{"className":16031},[1184],[1052,16033,16035],{"className":16034},[1188],[1052,16036,16038],{"className":16037},[1193],[1052,16039,16041],{"className":16040,"style":12370},[1197],[1052,16042,16043,16046],{"style":1327},[1052,16044],{"className":16045,"style":1206},[1205],[1052,16047,16049],{"className":16048},[1210,1211,1212,1213],[1052,16050,16052,16055,16058],{"className":16051},[1148,1213],[1052,16053,1123],{"className":16054},[1148,1213],[1052,16056,2329],{"className":16057},[2428,1213],[1052,16059,10971],{"className":16060},[1148,1149,1213],[996,16062,16063],{},[1711,16064],{"alt":1711,"src":16065},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-26.png",[1045,16067,16068],{"id":16068},"记忆控制",[996,16070,16071,16072,16174,16175,16245,16246,16398],{},"在长短期记忆网络中，也有两个门用于这样的目的：输入门控制采用多少来自",[1052,16073,16075,16097],{"className":16074},[1055],[1052,16076,16078],{"className":16077},[1059],[1061,16079,16080],{"xmlns":1063},[1065,16081,16082,16094],{},[1068,16083,16084],{},[1086,16085,16086,16092],{},[15030,16087,16088,16090],{"accent":1096},[1071,16089,3595],{"mathvariant":4609},[1075,16091,15036],{},[1071,16093,3799],{},[1128,16095,16096],{"encoding":1130},"\\tilde{\\mathbf{C}}_t",[1052,16098,16100],{"className":16099,"ariaHidden":1096},[1135],[1052,16101,16103,16106],{"className":16102},[1139],[1052,16104],{"className":16105,"style":15065},[1143],[1052,16107,16109,16140],{"className":16108},[1148],[1052,16110,16112],{"className":16111},[1148,15072],[1052,16113,16115],{"className":16114},[1188],[1052,16116,16118],{"className":16117},[1193],[1052,16119,16121,16129],{"className":16120,"style":15082},[1197],[1052,16122,16123,16126],{"style":15085},[1052,16124],{"className":16125,"style":1976},[1205],[1052,16127,3595],{"className":16128},[1148,4785],[1052,16130,16131,16134],{"style":15094},[1052,16132],{"className":16133,"style":1976},[1205],[1052,16135,16137],{"className":16136,"style":15102},[15101],[1052,16138,15036],{"className":16139},[1148],[1052,16141,16143],{"className":16142},[1184],[1052,16144,16146,16166],{"className":16145},[1188,1189],[1052,16147,16149,16163],{"className":16148},[1193],[1052,16150,16152],{"className":16151,"style":11602},[1197],[1052,16153,16154,16157],{"style":1201},[1052,16155],{"className":16156,"style":1206},[1205],[1052,16158,16160],{"className":16159},[1210,1211,1212,1213],[1052,16161,3799],{"className":16162},[1148,1149,1213],[1052,16164,1221],{"className":16165},[1220],[1052,16167,16169],{"className":16168},[1193],[1052,16170,16172],{"className":16171,"style":1228},[1197],[1052,16173],{},"的新数据，而遗忘门",[1052,16176,16178,16196],{"className":16177},[1055],[1052,16179,16181],{"className":16180},[1059],[1061,16182,16183],{"xmlns":1063},[1065,16184,16185,16193],{},[1068,16186,16187],{},[1086,16188,16189,16191],{},[1071,16190,3810],{"mathvariant":4609},[1071,16192,3799],{},[1128,16194,16195],{"encoding":1130},"\\mathbf{F}_t",[1052,16197,16199],{"className":16198,"ariaHidden":1096},[1135],[1052,16200,16202,16205],{"className":16201},[1139],[1052,16203],{"className":16204,"style":13080},[1143],[1052,16206,16208,16211],{"className":16207},[1148],[1052,16209,3810],{"className":16210},[1148,4785],[1052,16212,16214],{"className":16213},[1184],[1052,16215,16217,16237],{"className":16216},[1188,1189],[1052,16218,16220,16234],{"className":16219},[1193],[1052,16221,16223],{"className":16222,"style":11602},[1197],[1052,16224,16225,16228],{"style":1201},[1052,16226],{"className":16227,"style":1206},[1205],[1052,16229,16231],{"className":16230},[1210,1211,1212,1213],[1052,16232,3799],{"className":16233},[1148,1149,1213],[1052,16235,1221],{"className":16236},[1220],[1052,16238,16240],{"className":16239},[1193],[1052,16241,16243],{"className":16242,"style":1228},[1197],[1052,16244],{},"控制保留多少过去的记忆元",[1052,16247,16249,16287],{"className":16248},[1055],[1052,16250,16252],{"className":16251},[1059],[1061,16253,16254],{"xmlns":1063},[1065,16255,16256,16284],{},[1068,16257,16258,16270,16272],{},[1086,16259,16260,16262],{},[1071,16261,3595],{"mathvariant":4609},[1068,16263,16264,16266,16268],{},[1071,16265,3799],{},[1075,16267,3988],{},[1121,16269,1123],{},[1075,16271,12288],{},[5084,16273,16274,16276],{},[1071,16275,2558],{"mathvariant":14270},[1068,16277,16278,16280,16282],{},[1071,16279,1126],{},[1075,16281,2329],{},[1071,16283,10971],{},[1128,16285,16286],{"encoding":1130},"\\mathbf{C}_{t-1} \\in \\mathbb{R}^{n \\times h}",[1052,16288,16290,16354],{"className":16289,"ariaHidden":1096},[1135],[1052,16291,16293,16296,16345,16348,16351],{"className":16292},[1139],[1052,16294],{"className":16295,"style":13244},[1143],[1052,16297,16299,16302],{"className":16298},[1148],[1052,16300,3595],{"className":16301},[1148,4785],[1052,16303,16305],{"className":16304},[1184],[1052,16306,16308,16337],{"className":16307},[1188,1189],[1052,16309,16311,16334],{"className":16310},[1193],[1052,16312,16314],{"className":16313,"style":4690},[1197],[1052,16315,16316,16319],{"style":1201},[1052,16317],{"className":16318,"style":1206},[1205],[1052,16320,16322],{"className":16321},[1210,1211,1212,1213],[1052,16323,16325,16328,16331],{"className":16324},[1148,1213],[1052,16326,3799],{"className":16327},[1148,1149,1213],[1052,16329,3988],{"className":16330},[2428,1213],[1052,16332,1123],{"className":16333},[1148,1213],[1052,16335,1221],{"className":16336},[1220],[1052,16338,16340],{"className":16339},[1193],[1052,16341,16343],{"className":16342,"style":11713},[1197],[1052,16344],{},[1052,16346],{"className":16347,"style":1155},[1154],[1052,16349,12288],{"className":16350},[1159],[1052,16352],{"className":16353,"style":1155},[1154],[1052,16355,16357,16360],{"className":16356},[1139],[1052,16358],{"className":16359,"style":12370},[1143],[1052,16361,16363,16366],{"className":16362},[1148],[1052,16364,2558],{"className":16365},[1148,14514],[1052,16367,16369],{"className":16368},[1184],[1052,16370,16372],{"className":16371},[1188],[1052,16373,16375],{"className":16374},[1193],[1052,16376,16378],{"className":16377,"style":12370},[1197],[1052,16379,16380,16383],{"style":1327},[1052,16381],{"className":16382,"style":1206},[1205],[1052,16384,16386],{"className":16385},[1210,1211,1212,1213],[1052,16387,16389,16392,16395],{"className":16388},[1148,1213],[1052,16390,1126],{"className":16391},[1148,1149,1213],[1052,16393,2329],{"className":16394},[2428,1213],[1052,16396,10971],{"className":16397},[1148,1149,1213],"的内容。使用按元素乘法，得出：",[1052,16400,16402],{"className":16401},[1865],[1052,16403,16405,16466],{"className":16404},[1055],[1052,16406,16408],{"className":16407},[1059],[1061,16409,16410],{"xmlns":1063,"display":1874},[1065,16411,16412,16463],{},[1068,16413,16414,16420,16422,16428,16431,16443,16445,16451,16453],{},[1086,16415,16416,16418],{},[1071,16417,3595],{"mathvariant":4609},[1071,16419,3799],{},[1075,16421,1077],{},[1086,16423,16424,16426],{},[1071,16425,3810],{"mathvariant":4609},[1071,16427,3799],{},[1075,16429,16430],{},"⊙",[1086,16432,16433,16435],{},[1071,16434,3595],{"mathvariant":4609},[1068,16436,16437,16439,16441],{},[1071,16438,3799],{},[1075,16440,3988],{},[1121,16442,1123],{},[1075,16444,3111],{},[1086,16446,16447,16449],{},[1071,16448,12586],{"mathvariant":4609},[1071,16450,3799],{},[1075,16452,16430],{},[1086,16454,16455,16461],{},[15030,16456,16457,16459],{"accent":1096},[1071,16458,3595],{"mathvariant":4609},[1075,16460,15036],{},[1071,16462,3799],{},[1128,16464,16465],{"encoding":1130},"\\mathbf{C}_t = \\mathbf{F}_t \\odot \\mathbf{C}_{t-1} + \\mathbf{I}_t \\odot \\tilde{\\mathbf{C}}_t",[1052,16467,16469,16524,16579,16643,16698],{"className":16468,"ariaHidden":1096},[1135],[1052,16470,16472,16475,16515,16518,16521],{"className":16471},[1139],[1052,16473],{"className":16474,"style":13080},[1143],[1052,16476,16478,16481],{"className":16477},[1148],[1052,16479,3595],{"className":16480},[1148,4785],[1052,16482,16484],{"className":16483},[1184],[1052,16485,16487,16507],{"className":16486},[1188,1189],[1052,16488,16490,16504],{"className":16489},[1193],[1052,16491,16493],{"className":16492,"style":11602},[1197],[1052,16494,16495,16498],{"style":1201},[1052,16496],{"className":16497,"style":1206},[1205],[1052,16499,16501],{"className":16500},[1210,1211,1212,1213],[1052,16502,3799],{"className":16503},[1148,1149,1213],[1052,16505,1221],{"className":16506},[1220],[1052,16508,16510],{"className":16509},[1193],[1052,16511,16513],{"className":16512,"style":1228},[1197],[1052,16514],{},[1052,16516],{"className":16517,"style":1155},[1154],[1052,16519,1077],{"className":16520},[1159],[1052,16522],{"className":16523,"style":1155},[1154],[1052,16525,16527,16530,16570,16573,16576],{"className":16526},[1139],[1052,16528],{"className":16529,"style":13080},[1143],[1052,16531,16533,16536],{"className":16532},[1148],[1052,16534,3810],{"className":16535},[1148,4785],[1052,16537,16539],{"className":16538},[1184],[1052,16540,16542,16562],{"className":16541},[1188,1189],[1052,16543,16545,16559],{"className":16544},[1193],[1052,16546,16548],{"className":16547,"style":11602},[1197],[1052,16549,16550,16553],{"style":1201},[1052,16551],{"className":16552,"style":1206},[1205],[1052,16554,16556],{"className":16555},[1210,1211,1212,1213],[1052,16557,3799],{"className":16558},[1148,1149,1213],[1052,16560,1221],{"className":16561},[1220],[1052,16563,16565],{"className":16564},[1193],[1052,16566,16568],{"className":16567,"style":1228},[1197],[1052,16569],{},[1052,16571],{"className":16572,"style":3378},[1154],[1052,16574,16430],{"className":16575},[2428],[1052,16577],{"className":16578,"style":3378},[1154],[1052,16580,16582,16585,16634,16637,16640],{"className":16581},[1139],[1052,16583],{"className":16584,"style":13244},[1143],[1052,16586,16588,16591],{"className":16587},[1148],[1052,16589,3595],{"className":16590},[1148,4785],[1052,16592,16594],{"className":16593},[1184],[1052,16595,16597,16626],{"className":16596},[1188,1189],[1052,16598,16600,16623],{"className":16599},[1193],[1052,16601,16603],{"className":16602,"style":4690},[1197],[1052,16604,16605,16608],{"style":1201},[1052,16606],{"className":16607,"style":1206},[1205],[1052,16609,16611],{"className":16610},[1210,1211,1212,1213],[1052,16612,16614,16617,16620],{"className":16613},[1148,1213],[1052,16615,3799],{"className":16616},[1148,1149,1213],[1052,16618,3988],{"className":16619},[2428,1213],[1052,16621,1123],{"className":16622},[1148,1213],[1052,16624,1221],{"className":16625},[1220],[1052,16627,16629],{"className":16628},[1193],[1052,16630,16632],{"className":16631,"style":11713},[1197],[1052,16633],{},[1052,16635],{"className":16636,"style":3378},[1154],[1052,16638,3111],{"className":16639},[2428],[1052,16641],{"className":16642,"style":3378},[1154],[1052,16644,16646,16649,16689,16692,16695],{"className":16645},[1139],[1052,16647],{"className":16648,"style":13080},[1143],[1052,16650,16652,16655],{"className":16651},[1148],[1052,16653,12586],{"className":16654},[1148,4785],[1052,16656,16658],{"className":16657},[1184],[1052,16659,16661,16681],{"className":16660},[1188,1189],[1052,16662,16664,16678],{"className":16663},[1193],[1052,16665,16667],{"className":16666,"style":11602},[1197],[1052,16668,16669,16672],{"style":1201},[1052,16670],{"className":16671,"style":1206},[1205],[1052,16673,16675],{"className":16674},[1210,1211,1212,1213],[1052,16676,3799],{"className":16677},[1148,1149,1213],[1052,16679,1221],{"className":16680},[1220],[1052,16682,16684],{"className":16683},[1193],[1052,16685,16687],{"className":16686,"style":1228},[1197],[1052,16688],{},[1052,16690],{"className":16691,"style":3378},[1154],[1052,16693,16430],{"className":16694},[2428],[1052,16696],{"className":16697,"style":3378},[1154],[1052,16699,16701,16704],{"className":16700},[1139],[1052,16702],{"className":16703,"style":15065},[1143],[1052,16705,16707,16738],{"className":16706},[1148],[1052,16708,16710],{"className":16709},[1148,15072],[1052,16711,16713],{"className":16712},[1188],[1052,16714,16716],{"className":16715},[1193],[1052,16717,16719,16727],{"className":16718,"style":15082},[1197],[1052,16720,16721,16724],{"style":15085},[1052,16722],{"className":16723,"style":1976},[1205],[1052,16725,3595],{"className":16726},[1148,4785],[1052,16728,16729,16732],{"style":15094},[1052,16730],{"className":16731,"style":1976},[1205],[1052,16733,16735],{"className":16734,"style":15102},[15101],[1052,16736,15036],{"className":16737},[1148],[1052,16739,16741],{"className":16740},[1184],[1052,16742,16744,16764],{"className":16743},[1188,1189],[1052,16745,16747,16761],{"className":16746},[1193],[1052,16748,16750],{"className":16749,"style":11602},[1197],[1052,16751,16752,16755],{"style":1201},[1052,16753],{"className":16754,"style":1206},[1205],[1052,16756,16758],{"className":16757},[1210,1211,1212,1213],[1052,16759,3799],{"className":16760},[1148,1149,1213],[1052,16762,1221],{"className":16763},[1220],[1052,16765,16767],{"className":16766},[1193],[1052,16768,16770],{"className":16769,"style":1228},[1197],[1052,16771],{},[996,16773,16774,16775,16860],{},"如果遗忘门始终为1且输入门始终为0，则过去的记忆元",[1052,16776,16778,16802],{"className":16777},[1055],[1052,16779,16781],{"className":16780},[1059],[1061,16782,16783],{"xmlns":1063},[1065,16784,16785,16799],{},[1068,16786,16787],{},[1086,16788,16789,16791],{},[1071,16790,3595],{"mathvariant":4609},[1068,16792,16793,16795,16797],{},[1071,16794,3799],{},[1075,16796,3988],{},[1121,16798,1123],{},[1128,16800,16801],{"encoding":1130},"\\mathbf{C}_{t-1}",[1052,16803,16805],{"className":16804,"ariaHidden":1096},[1135],[1052,16806,16808,16811],{"className":16807},[1139],[1052,16809],{"className":16810,"style":13244},[1143],[1052,16812,16814,16817],{"className":16813},[1148],[1052,16815,3595],{"className":16816},[1148,4785],[1052,16818,16820],{"className":16819},[1184],[1052,16821,16823,16852],{"className":16822},[1188,1189],[1052,16824,16826,16849],{"className":16825},[1193],[1052,16827,16829],{"className":16828,"style":4690},[1197],[1052,16830,16831,16834],{"style":1201},[1052,16832],{"className":16833,"style":1206},[1205],[1052,16835,16837],{"className":16836},[1210,1211,1212,1213],[1052,16838,16840,16843,16846],{"className":16839},[1148,1213],[1052,16841,3799],{"className":16842},[1148,1149,1213],[1052,16844,3988],{"className":16845},[2428,1213],[1052,16847,1123],{"className":16848},[1148,1213],[1052,16850,1221],{"className":16851},[1220],[1052,16853,16855],{"className":16854},[1193],[1052,16856,16858],{"className":16857,"style":11713},[1197],[1052,16859],{},"将随时间被保存并传递到当前时间步。引入这种设计是为了缓解梯度消失问题，并更好地捕获序列中的长距离依赖关系。",[1045,16862,16863],{"id":16863},"隐状态计算",[996,16865,16866,16867,17004,17005,17075],{},"定义如何计算隐状态",[1052,16868,16870,16902],{"className":16869},[1055],[1052,16871,16873],{"className":16872},[1059],[1061,16874,16875],{"xmlns":1063},[1065,16876,16877,16899],{},[1068,16878,16879,16885,16887],{},[1086,16880,16881,16883],{},[1071,16882,12429],{"mathvariant":4609},[1071,16884,3799],{},[1075,16886,12288],{},[5084,16888,16889,16891],{},[1071,16890,2558],{"mathvariant":14270},[1068,16892,16893,16895,16897],{},[1071,16894,1126],{},[1075,16896,2329],{},[1071,16898,10971],{},[1128,16900,16901],{"encoding":1130},"\\mathbf{H}_t \\in \\mathbb{R}^{n \\times h}",[1052,16903,16905,16960],{"className":16904,"ariaHidden":1096},[1135],[1052,16906,16908,16911,16951,16954,16957],{"className":16907},[1139],[1052,16909],{"className":16910,"style":13080},[1143],[1052,16912,16914,16917],{"className":16913},[1148],[1052,16915,12429],{"className":16916},[1148,4785],[1052,16918,16920],{"className":16919},[1184],[1052,16921,16923,16943],{"className":16922},[1188,1189],[1052,16924,16926,16940],{"className":16925},[1193],[1052,16927,16929],{"className":16928,"style":11602},[1197],[1052,16930,16931,16934],{"style":1201},[1052,16932],{"className":16933,"style":1206},[1205],[1052,16935,16937],{"className":16936},[1210,1211,1212,1213],[1052,16938,3799],{"className":16939},[1148,1149,1213],[1052,16941,1221],{"className":16942},[1220],[1052,16944,16946],{"className":16945},[1193],[1052,16947,16949],{"className":16948,"style":1228},[1197],[1052,16950],{},[1052,16952],{"className":16953,"style":1155},[1154],[1052,16955,12288],{"className":16956},[1159],[1052,16958],{"className":16959,"style":1155},[1154],[1052,16961,16963,16966],{"className":16962},[1139],[1052,16964],{"className":16965,"style":12370},[1143],[1052,16967,16969,16972],{"className":16968},[1148],[1052,16970,2558],{"className":16971},[1148,14514],[1052,16973,16975],{"className":16974},[1184],[1052,16976,16978],{"className":16977},[1188],[1052,16979,16981],{"className":16980},[1193],[1052,16982,16984],{"className":16983,"style":12370},[1197],[1052,16985,16986,16989],{"style":1327},[1052,16987],{"className":16988,"style":1206},[1205],[1052,16990,16992],{"className":16991},[1210,1211,1212,1213],[1052,16993,16995,16998,17001],{"className":16994},[1148,1213],[1052,16996,1126],{"className":16997},[1148,1149,1213],[1052,16999,2329],{"className":17000},[2428,1213],[1052,17002,10971],{"className":17003},[1148,1149,1213],"这就是输出门发挥作用的地方。在长短期记忆网络中，它仅仅是记忆元的tanh的门控版本。这就确保了",[1052,17006,17008,17026],{"className":17007},[1055],[1052,17009,17011],{"className":17010},[1059],[1061,17012,17013],{"xmlns":1063},[1065,17014,17015,17023],{},[1068,17016,17017],{},[1086,17018,17019,17021],{},[1071,17020,12429],{"mathvariant":4609},[1071,17022,3799],{},[1128,17024,17025],{"encoding":1130},"\\mathbf{H}_t",[1052,17027,17029],{"className":17028,"ariaHidden":1096},[1135],[1052,17030,17032,17035],{"className":17031},[1139],[1052,17033],{"className":17034,"style":13080},[1143],[1052,17036,17038,17041],{"className":17037},[1148],[1052,17039,12429],{"className":17040},[1148,4785],[1052,17042,17044],{"className":17043},[1184],[1052,17045,17047,17067],{"className":17046},[1188,1189],[1052,17048,17050,17064],{"className":17049},[1193],[1052,17051,17053],{"className":17052,"style":11602},[1197],[1052,17054,17055,17058],{"style":1201},[1052,17056],{"className":17057,"style":1206},[1205],[1052,17059,17061],{"className":17060},[1210,1211,1212,1213],[1052,17062,3799],{"className":17063},[1148,1149,1213],[1052,17065,1221],{"className":17066},[1220],[1052,17068,17070],{"className":17069},[1193],[1052,17071,17073],{"className":17072,"style":1228},[1197],[1052,17074],{},"的值始终在区间(-1,1)内：",[1052,17077,17079],{"className":17078},[1865],[1052,17080,17082,17135],{"className":17081},[1055],[1052,17083,17085],{"className":17084},[1059],[1061,17086,17087],{"xmlns":1063,"display":1874},[1065,17088,17089,17132],{},[1068,17090,17091,17097,17099,17105,17107,17109,17111,17113,17119,17121,17123,17125,17127,17130],{},[1086,17092,17093,17095],{},[1071,17094,12429],{"mathvariant":4609},[1071,17096,3799],{},[1075,17098,1077],{},[1086,17100,17101,17103],{},[1071,17102,12864],{"mathvariant":4609},[1071,17104,3799],{},[1075,17106,16430],{},[1071,17108,15226],{},[1075,17110,15229],{},[1075,17112,1084],{"stretchy":1080},[1086,17114,17115,17117],{},[1071,17116,3595],{"mathvariant":4609},[1071,17118,3799],{},[1075,17120,1107],{"stretchy":1080},[1071,17122,1576],{"mathvariant":1575},[1154,17124],{"width":4606},[1075,17126,1084],{"stretchy":1080},[1121,17128,17129],{},"9.2.4",[1075,17131,1107],{"stretchy":1080},[1128,17133,17134],{"encoding":1130},"\\mathbf{H}_t = \\mathbf{O}_t \\odot \\tanh(\\mathbf{C}_t). \\quad (9.2.4)",[1052,17136,17138,17193,17248],{"className":17137,"ariaHidden":1096},[1135],[1052,17139,17141,17144,17184,17187,17190],{"className":17140},[1139],[1052,17142],{"className":17143,"style":13080},[1143],[1052,17145,17147,17150],{"className":17146},[1148],[1052,17148,12429],{"className":17149},[1148,4785],[1052,17151,17153],{"className":17152},[1184],[1052,17154,17156,17176],{"className":17155},[1188,1189],[1052,17157,17159,17173],{"className":17158},[1193],[1052,17160,17162],{"className":17161,"style":11602},[1197],[1052,17163,17164,17167],{"style":1201},[1052,17165],{"className":17166,"style":1206},[1205],[1052,17168,17170],{"className":17169},[1210,1211,1212,1213],[1052,17171,3799],{"className":17172},[1148,1149,1213],[1052,17174,1221],{"className":17175},[1220],[1052,17177,17179],{"className":17178},[1193],[1052,17180,17182],{"className":17181,"style":1228},[1197],[1052,17183],{},[1052,17185],{"className":17186,"style":1155},[1154],[1052,17188,1077],{"className":17189},[1159],[1052,17191],{"className":17192,"style":1155},[1154],[1052,17194,17196,17199,17239,17242,17245],{"className":17195},[1139],[1052,17197],{"className":17198,"style":13080},[1143],[1052,17200,17202,17205],{"className":17201},[1148],[1052,17203,12864],{"className":17204},[1148,4785],[1052,17206,17208],{"className":17207},[1184],[1052,17209,17211,17231],{"className":17210},[1188,1189],[1052,17212,17214,17228],{"className":17213},[1193],[1052,17215,17217],{"className":17216,"style":11602},[1197],[1052,17218,17219,17222],{"style":1201},[1052,17220],{"className":17221,"style":1206},[1205],[1052,17223,17225],{"className":17224},[1210,1211,1212,1213],[1052,17226,3799],{"className":17227},[1148,1149,1213],[1052,17229,1221],{"className":17230},[1220],[1052,17232,17234],{"className":17233},[1193],[1052,17235,17237],{"className":17236,"style":1228},[1197],[1052,17238],{},[1052,17240],{"className":17241,"style":3378},[1154],[1052,17243,16430],{"className":17244},[2428],[1052,17246],{"className":17247,"style":3378},[1154],[1052,17249,17251,17254,17257,17260,17300,17303,17306,17309,17312,17315],{"className":17250},[1139],[1052,17252],{"className":17253,"style":1597},[1143],[1052,17255,15226],{"className":17256},[2027],[1052,17258,1084],{"className":17259},[1173],[1052,17261,17263,17266],{"className":17262},[1148],[1052,17264,3595],{"className":17265},[1148,4785],[1052,17267,17269],{"className":17268},[1184],[1052,17270,17272,17292],{"className":17271},[1188,1189],[1052,17273,17275,17289],{"className":17274},[1193],[1052,17276,17278],{"className":17277,"style":11602},[1197],[1052,17279,17280,17283],{"style":1201},[1052,17281],{"className":17282,"style":1206},[1205],[1052,17284,17286],{"className":17285},[1210,1211,1212,1213],[1052,17287,3799],{"className":17288},[1148,1149,1213],[1052,17290,1221],{"className":17291},[1220],[1052,17293,17295],{"className":17294},[1193],[1052,17296,17298],{"className":17297,"style":1228},[1197],[1052,17299],{},[1052,17301,1107],{"className":17302},[1284],[1052,17304,1576],{"className":17305},[1148],[1052,17307],{"className":17308,"style":4778},[1154],[1052,17310,1084],{"className":17311},[1173],[1052,17313,17129],{"className":17314},[1148],[1052,17316,1107],{"className":17317},[1284],[996,17319,17320],{},"只要输出门接近1，我们就能够有效地将所有记忆信息传递给预测部分，而对于输出门接近0，我们只保留记忆元内的所有信息，而不需要更新隐状态。",[1042,17322,17324],{"id":17323},"_5门控循环单元gru","5、门控循环单元（GRU）",[993,17326,17327],{},[996,17328,17329,17330,17333,17334,17337,17338,17341,17346,17349],{},"门控循环单元与普通的循环神经网络之间的",[1020,17331,17332],{},"关键区别","在于： ",[1532,17335,17336],{},"前者支持隐状态的门控","。 这意味着模型有专门的机制来",[1532,17339,17340],{},"确定应该何时更新隐状态， 以及应该",[1532,17342,17343],{},[1020,17344,17345],{},"何时",[1532,17347,17348],{},"重置隐状态","。 这些机制是可学习的",[996,17351,17352],{},[1711,17353],{"alt":1711,"src":17354},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-27.png",[1045,17356,17357],{"id":17357},"重置门和更新门",[4548,17359,17361,17362],{"id":17360},"重置门-reset-gate-控制遗忘-rtrnhmathbfr_t-in-mathbbrn-times-hrtrnh","重置门 (Reset Gate) - 控制\"遗忘\"        ",[1052,17363,17365,17397],{"className":17364},[1055],[1052,17366,17368],{"className":17367},[1059],[1061,17369,17370],{"xmlns":1063},[1065,17371,17372,17394],{},[1068,17373,17374,17380,17382],{},[1086,17375,17376,17378],{},[1071,17377,2558],{"mathvariant":4609},[1071,17379,3799],{},[1075,17381,12288],{},[5084,17383,17384,17386],{},[1071,17385,2558],{"mathvariant":14270},[1068,17387,17388,17390,17392],{},[1071,17389,1126],{},[1075,17391,2329],{},[1071,17393,10971],{},[1128,17395,17396],{"encoding":1130},"\\mathbf{R}_t \\in \\mathbb{R}^{n \\times h}",[1052,17398,17400,17455],{"className":17399,"ariaHidden":1096},[1135],[1052,17401,17403,17406,17446,17449,17452],{"className":17402},[1139],[1052,17404],{"className":17405,"style":13080},[1143],[1052,17407,17409,17412],{"className":17408},[1148],[1052,17410,2558],{"className":17411},[1148,4785],[1052,17413,17415],{"className":17414},[1184],[1052,17416,17418,17438],{"className":17417},[1188,1189],[1052,17419,17421,17435],{"className":17420},[1193],[1052,17422,17424],{"className":17423,"style":11602},[1197],[1052,17425,17426,17429],{"style":1201},[1052,17427],{"className":17428,"style":1206},[1205],[1052,17430,17432],{"className":17431},[1210,1211,1212,1213],[1052,17433,3799],{"className":17434},[1148,1149,1213],[1052,17436,1221],{"className":17437},[1220],[1052,17439,17441],{"className":17440},[1193],[1052,17442,17444],{"className":17443,"style":1228},[1197],[1052,17445],{},[1052,17447],{"className":17448,"style":1155},[1154],[1052,17450,12288],{"className":17451},[1159],[1052,17453],{"className":17454,"style":1155},[1154],[1052,17456,17458,17461],{"className":17457},[1139],[1052,17459],{"className":17460,"style":12370},[1143],[1052,17462,17464,17467],{"className":17463},[1148],[1052,17465,2558],{"className":17466},[1148,14514],[1052,17468,17470],{"className":17469},[1184],[1052,17471,17473],{"className":17472},[1188],[1052,17474,17476],{"className":17475},[1193],[1052,17477,17479],{"className":17478,"style":12370},[1197],[1052,17480,17481,17484],{"style":1327},[1052,17482],{"className":17483,"style":1206},[1205],[1052,17485,17487],{"className":17486},[1210,1211,1212,1213],[1052,17488,17490,17493,17496],{"className":17489},[1148,1213],[1052,17491,1126],{"className":17492},[1148,1149,1213],[1052,17494,2329],{"className":17495},[2428,1213],[1052,17497,10971],{"className":17498},[1148,1149,1213],[978,17500,17501,17507],{},[983,17502,17503,17506],{},[1020,17504,17505],{},"重置门 ≈ 0","：完全忘记过去信息，候选隐状态只依赖当前输入（像全新开始）",[983,17508,17509,17512],{},[1020,17510,17511],{},"重置门 ≈ 1","：完全保留过去信息，候选隐状态同时考虑当前输入和历史状态",[4548,17514,17516,17517],{"id":17515},"更新门-update-gate-控制更新-ztrnhmathbfz_t-in-mathbbrn-times-hztrnh","更新门 (Update Gate) - 控制\"更新\"      ",[1052,17518,17520,17553],{"className":17519},[1055],[1052,17521,17523],{"className":17522},[1059],[1061,17524,17525],{"xmlns":1063},[1065,17526,17527,17550],{},[1068,17528,17529,17536,17538],{},[1086,17530,17531,17534],{},[1071,17532,17533],{"mathvariant":4609},"Z",[1071,17535,3799],{},[1075,17537,12288],{},[5084,17539,17540,17542],{},[1071,17541,2558],{"mathvariant":14270},[1068,17543,17544,17546,17548],{},[1071,17545,1126],{},[1075,17547,2329],{},[1071,17549,10971],{},[1128,17551,17552],{"encoding":1130},"\\mathbf{Z}_t \\in \\mathbb{R}^{n \\times h}",[1052,17554,17556,17611],{"className":17555,"ariaHidden":1096},[1135],[1052,17557,17559,17562,17602,17605,17608],{"className":17558},[1139],[1052,17560],{"className":17561,"style":13080},[1143],[1052,17563,17565,17568],{"className":17564},[1148],[1052,17566,17533],{"className":17567},[1148,4785],[1052,17569,17571],{"className":17570},[1184],[1052,17572,17574,17594],{"className":17573},[1188,1189],[1052,17575,17577,17591],{"className":17576},[1193],[1052,17578,17580],{"className":17579,"style":11602},[1197],[1052,17581,17582,17585],{"style":1201},[1052,17583],{"className":17584,"style":1206},[1205],[1052,17586,17588],{"className":17587},[1210,1211,1212,1213],[1052,17589,3799],{"className":17590},[1148,1149,1213],[1052,17592,1221],{"className":17593},[1220],[1052,17595,17597],{"className":17596},[1193],[1052,17598,17600],{"className":17599,"style":1228},[1197],[1052,17601],{},[1052,17603],{"className":17604,"style":1155},[1154],[1052,17606,12288],{"className":17607},[1159],[1052,17609],{"className":17610,"style":1155},[1154],[1052,17612,17614,17617],{"className":17613},[1139],[1052,17615],{"className":17616,"style":12370},[1143],[1052,17618,17620,17623],{"className":17619},[1148],[1052,17621,2558],{"className":17622},[1148,14514],[1052,17624,17626],{"className":17625},[1184],[1052,17627,17629],{"className":17628},[1188],[1052,17630,17632],{"className":17631},[1193],[1052,17633,17635],{"className":17634,"style":12370},[1197],[1052,17636,17637,17640],{"style":1327},[1052,17638],{"className":17639,"style":1206},[1205],[1052,17641,17643],{"className":17642},[1210,1211,1212,1213],[1052,17644,17646,17649,17652],{"className":17645},[1148,1213],[1052,17647,1126],{"className":17648},[1148,1149,1213],[1052,17650,2329],{"className":17651},[2428,1213],[1052,17653,10971],{"className":17654},[1148,1149,1213],[978,17656,17657,17663],{},[983,17658,17659,17662],{},[1020,17660,17661],{},"更新门 ≈ 0","：大量采用新信息（候选隐状态），少量保留旧信息",[983,17664,17665,17668],{},[1020,17666,17667],{},"更新门 ≈ 1","：大量保留旧信息（前一时刻隐状态），少量采用新信息",[996,17670,17671,17672,17809,17810,17962,17963,18099,18100],{},"门控循环单元的数学表达。对于给定的时间步t，假设输入是一个小批量",[1052,17673,17675,17707],{"className":17674},[1055],[1052,17676,17678],{"className":17677},[1059],[1061,17679,17680],{"xmlns":1063},[1065,17681,17682,17704],{},[1068,17683,17684,17690,17692],{},[1086,17685,17686,17688],{},[1071,17687,4571],{"mathvariant":4609},[1071,17689,3799],{},[1075,17691,12288],{},[5084,17693,17694,17696],{},[1071,17695,2558],{"mathvariant":14270},[1068,17697,17698,17700,17702],{},[1071,17699,1126],{},[1075,17701,2329],{},[1071,17703,2370],{},[1128,17705,17706],{"encoding":1130},"\\mathbf{X}_t \\in \\mathbb{R}^{n \\times d}",[1052,17708,17710,17765],{"className":17709,"ariaHidden":1096},[1135],[1052,17711,17713,17716,17756,17759,17762],{"className":17712},[1139],[1052,17714],{"className":17715,"style":13080},[1143],[1052,17717,17719,17722],{"className":17718},[1148],[1052,17720,4571],{"className":17721},[1148,4785],[1052,17723,17725],{"className":17724},[1184],[1052,17726,17728,17748],{"className":17727},[1188,1189],[1052,17729,17731,17745],{"className":17730},[1193],[1052,17732,17734],{"className":17733,"style":11602},[1197],[1052,17735,17736,17739],{"style":1201},[1052,17737],{"className":17738,"style":1206},[1205],[1052,17740,17742],{"className":17741},[1210,1211,1212,1213],[1052,17743,3799],{"className":17744},[1148,1149,1213],[1052,17746,1221],{"className":17747},[1220],[1052,17749,17751],{"className":17750},[1193],[1052,17752,17754],{"className":17753,"style":1228},[1197],[1052,17755],{},[1052,17757],{"className":17758,"style":1155},[1154],[1052,17760,12288],{"className":17761},[1159],[1052,17763],{"className":17764,"style":1155},[1154],[1052,17766,17768,17771],{"className":17767},[1139],[1052,17769],{"className":17770,"style":12370},[1143],[1052,17772,17774,17777],{"className":17773},[1148],[1052,17775,2558],{"className":17776},[1148,14514],[1052,17778,17780],{"className":17779},[1184],[1052,17781,17783],{"className":17782},[1188],[1052,17784,17786],{"className":17785},[1193],[1052,17787,17789],{"className":17788,"style":12370},[1197],[1052,17790,17791,17794],{"style":1327},[1052,17792],{"className":17793,"style":1206},[1205],[1052,17795,17797],{"className":17796},[1210,1211,1212,1213],[1052,17798,17800,17803,17806],{"className":17799},[1148,1213],[1052,17801,1126],{"className":17802},[1148,1149,1213],[1052,17804,2329],{"className":17805},[2428,1213],[1052,17807,2370],{"className":17808},[1148,1149,1213],"（样本个数n，输入个数d），上一个时间步的隐状态是",[1052,17811,17813,17851],{"className":17812},[1055],[1052,17814,17816],{"className":17815},[1059],[1061,17817,17818],{"xmlns":1063},[1065,17819,17820,17848],{},[1068,17821,17822,17834,17836],{},[1086,17823,17824,17826],{},[1071,17825,12429],{"mathvariant":4609},[1068,17827,17828,17830,17832],{},[1071,17829,3799],{},[1075,17831,3988],{},[1121,17833,1123],{},[1075,17835,12288],{},[5084,17837,17838,17840],{},[1071,17839,2558],{"mathvariant":14270},[1068,17841,17842,17844,17846],{},[1071,17843,1126],{},[1075,17845,2329],{},[1071,17847,10971],{},[1128,17849,17850],{"encoding":1130},"\\mathbf{H}_{t-1} \\in \\mathbb{R}^{n \\times h}",[1052,17852,17854,17918],{"className":17853,"ariaHidden":1096},[1135],[1052,17855,17857,17860,17909,17912,17915],{"className":17856},[1139],[1052,17858],{"className":17859,"style":13244},[1143],[1052,17861,17863,17866],{"className":17862},[1148],[1052,17864,12429],{"className":17865},[1148,4785],[1052,17867,17869],{"className":17868},[1184],[1052,17870,17872,17901],{"className":17871},[1188,1189],[1052,17873,17875,17898],{"className":17874},[1193],[1052,17876,17878],{"className":17877,"style":4690},[1197],[1052,17879,17880,17883],{"style":1201},[1052,17881],{"className":17882,"style":1206},[1205],[1052,17884,17886],{"className":17885},[1210,1211,1212,1213],[1052,17887,17889,17892,17895],{"className":17888},[1148,1213],[1052,17890,3799],{"className":17891},[1148,1149,1213],[1052,17893,3988],{"className":17894},[2428,1213],[1052,17896,1123],{"className":17897},[1148,1213],[1052,17899,1221],{"className":17900},[1220],[1052,17902,17904],{"className":17903},[1193],[1052,17905,17907],{"className":17906,"style":11713},[1197],[1052,17908],{},[1052,17910],{"className":17911,"style":1155},[1154],[1052,17913,12288],{"className":17914},[1159],[1052,17916],{"className":17917,"style":1155},[1154],[1052,17919,17921,17924],{"className":17920},[1139],[1052,17922],{"className":17923,"style":12370},[1143],[1052,17925,17927,17930],{"className":17926},[1148],[1052,17928,2558],{"className":17929},[1148,14514],[1052,17931,17933],{"className":17932},[1184],[1052,17934,17936],{"className":17935},[1188],[1052,17937,17939],{"className":17938},[1193],[1052,17940,17942],{"className":17941,"style":12370},[1197],[1052,17943,17944,17947],{"style":1327},[1052,17945],{"className":17946,"style":1206},[1205],[1052,17948,17950],{"className":17949},[1210,1211,1212,1213],[1052,17951,17953,17956,17959],{"className":17952},[1148,1213],[1052,17954,1126],{"className":17955},[1148,1149,1213],[1052,17957,2329],{"className":17958},[2428,1213],[1052,17960,10971],{"className":17961},[1148,1149,1213],"（隐藏单元个数h）。那么，重置门",[1052,17964,17966,17997],{"className":17965},[1055],[1052,17967,17969],{"className":17968},[1059],[1061,17970,17971],{"xmlns":1063},[1065,17972,17973,17995],{},[1068,17974,17975,17981,17983],{},[1086,17976,17977,17979],{},[1071,17978,2558],{"mathvariant":4609},[1071,17980,3799],{},[1075,17982,12288],{},[5084,17984,17985,17987],{},[1071,17986,2558],{"mathvariant":14270},[1068,17988,17989,17991,17993],{},[1071,17990,1126],{},[1075,17992,2329],{},[1071,17994,10971],{},[1128,17996,17396],{"encoding":1130},[1052,17998,18000,18055],{"className":17999,"ariaHidden":1096},[1135],[1052,18001,18003,18006,18046,18049,18052],{"className":18002},[1139],[1052,18004],{"className":18005,"style":13080},[1143],[1052,18007,18009,18012],{"className":18008},[1148],[1052,18010,2558],{"className":18011},[1148,4785],[1052,18013,18015],{"className":18014},[1184],[1052,18016,18018,18038],{"className":18017},[1188,1189],[1052,18019,18021,18035],{"className":18020},[1193],[1052,18022,18024],{"className":18023,"style":11602},[1197],[1052,18025,18026,18029],{"style":1201},[1052,18027],{"className":18028,"style":1206},[1205],[1052,18030,18032],{"className":18031},[1210,1211,1212,1213],[1052,18033,3799],{"className":18034},[1148,1149,1213],[1052,18036,1221],{"className":18037},[1220],[1052,18039,18041],{"className":18040},[1193],[1052,18042,18044],{"className":18043,"style":1228},[1197],[1052,18045],{},[1052,18047],{"className":18048,"style":1155},[1154],[1052,18050,12288],{"className":18051},[1159],[1052,18053],{"className":18054,"style":1155},[1154],[1052,18056,18058,18061],{"className":18057},[1139],[1052,18059],{"className":18060,"style":12370},[1143],[1052,18062,18064,18067],{"className":18063},[1148],[1052,18065,2558],{"className":18066},[1148,14514],[1052,18068,18070],{"className":18069},[1184],[1052,18071,18073],{"className":18072},[1188],[1052,18074,18076],{"className":18075},[1193],[1052,18077,18079],{"className":18078,"style":12370},[1197],[1052,18080,18081,18084],{"style":1327},[1052,18082],{"className":18083,"style":1206},[1205],[1052,18085,18087],{"className":18086},[1210,1211,1212,1213],[1052,18088,18090,18093,18096],{"className":18089},[1148,1213],[1052,18091,1126],{"className":18092},[1148,1149,1213],[1052,18094,2329],{"className":18095},[2428,1213],[1052,18097,10971],{"className":18098},[1148,1149,1213],"和更新门",[1052,18101,18103,18134],{"className":18102},[1055],[1052,18104,18106],{"className":18105},[1059],[1061,18107,18108],{"xmlns":1063},[1065,18109,18110,18132],{},[1068,18111,18112,18118,18120],{},[1086,18113,18114,18116],{},[1071,18115,17533],{"mathvariant":4609},[1071,18117,3799],{},[1075,18119,12288],{},[5084,18121,18122,18124],{},[1071,18123,2558],{"mathvariant":14270},[1068,18125,18126,18128,18130],{},[1071,18127,1126],{},[1075,18129,2329],{},[1071,18131,10971],{},[1128,18133,17552],{"encoding":1130},[1052,18135,18137,18192],{"className":18136,"ariaHidden":1096},[1135],[1052,18138,18140,18143,18183,18186,18189],{"className":18139},[1139],[1052,18141],{"className":18142,"style":13080},[1143],[1052,18144,18146,18149],{"className":18145},[1148],[1052,18147,17533],{"className":18148},[1148,4785],[1052,18150,18152],{"className":18151},[1184],[1052,18153,18155,18175],{"className":18154},[1188,1189],[1052,18156,18158,18172],{"className":18157},[1193],[1052,18159,18161],{"className":18160,"style":11602},[1197],[1052,18162,18163,18166],{"style":1201},[1052,18164],{"className":18165,"style":1206},[1205],[1052,18167,18169],{"className":18168},[1210,1211,1212,1213],[1052,18170,3799],{"className":18171},[1148,1149,1213],[1052,18173,1221],{"className":18174},[1220],[1052,18176,18178],{"className":18177},[1193],[1052,18179,18181],{"className":18180,"style":1228},[1197],[1052,18182],{},[1052,18184],{"className":18185,"style":1155},[1154],[1052,18187,12288],{"className":18188},[1159],[1052,18190],{"className":18191,"style":1155},[1154],[1052,18193,18195,18198],{"className":18194},[1139],[1052,18196],{"className":18197,"style":12370},[1143],[1052,18199,18201,18204],{"className":18200},[1148],[1052,18202,2558],{"className":18203},[1148,14514],[1052,18205,18207],{"className":18206},[1184],[1052,18208,18210],{"className":18209},[1188],[1052,18211,18213],{"className":18212},[1193],[1052,18214,18216],{"className":18215,"style":12370},[1197],[1052,18217,18218,18221],{"style":1327},[1052,18219],{"className":18220,"style":1206},[1205],[1052,18222,18224],{"className":18223},[1210,1211,1212,1213],[1052,18225,18227,18230,18233],{"className":18226},[1148,1213],[1052,18228,1126],{"className":18229},[1148,1149,1213],[1052,18231,2329],{"className":18232},[2428,1213],[1052,18234,10971],{"className":18235},[1148,1149,1213],[996,18237,18238],{},"的计算如下所示：",[1052,18240,18242],{"className":18241},[1865],[1052,18243,18245,18319],{"className":18244},[1055],[1052,18246,18248],{"className":18247},[1059],[1061,18249,18250],{"xmlns":1063,"display":1874},[1065,18251,18252,18316],{},[1068,18253,18254,18260,18262,18264,18266,18272,18282,18284,18296,18306,18308,18314],{},[1086,18255,18256,18258],{},[1071,18257,2558],{"mathvariant":4609},[1071,18259,3799],{},[1075,18261,1077],{},[1071,18263,13013],{},[1075,18265,1084],{"stretchy":1080},[1086,18267,18268,18270],{},[1071,18269,4571],{"mathvariant":4609},[1071,18271,3799],{},[1086,18273,18274,18276],{},[1071,18275,11120],{"mathvariant":4609},[1068,18277,18278,18280],{},[1071,18279,1090],{},[1071,18281,2687],{},[1075,18283,3111],{},[1086,18285,18286,18288],{},[1071,18287,12429],{"mathvariant":4609},[1068,18289,18290,18292,18294],{},[1071,18291,3799],{},[1075,18293,3988],{},[1121,18295,1123],{},[1086,18297,18298,18300],{},[1071,18299,11120],{"mathvariant":4609},[1068,18301,18302,18304],{},[1071,18303,10971],{},[1071,18305,2687],{},[1075,18307,3111],{},[1086,18309,18310,18312],{},[1071,18311,10727],{"mathvariant":4609},[1071,18313,2687],{},[1075,18315,1107],{"stretchy":1080},[1128,18317,18318],{"encoding":1130},"\\mathbf{R}_t = \\sigma(\\mathbf{X}_t \\mathbf{W}_{xr} + \\mathbf{H}_{t-1} \\mathbf{W}_{hr} + \\mathbf{b}_r)",[1052,18320,18322,18377,18484,18594],{"className":18321,"ariaHidden":1096},[1135],[1052,18323,18325,18328,18368,18371,18374],{"className":18324},[1139],[1052,18326],{"className":18327,"style":13080},[1143],[1052,18329,18331,18334],{"className":18330},[1148],[1052,18332,2558],{"className":18333},[1148,4785],[1052,18335,18337],{"className":18336},[1184],[1052,18338,18340,18360],{"className":18339},[1188,1189],[1052,18341,18343,18357],{"className":18342},[1193],[1052,18344,18346],{"className":18345,"style":11602},[1197],[1052,18347,18348,18351],{"style":1201},[1052,18349],{"className":18350,"style":1206},[1205],[1052,18352,18354],{"className":18353},[1210,1211,1212,1213],[1052,18355,3799],{"className":18356},[1148,1149,1213],[1052,18358,1221],{"className":18359},[1220],[1052,18361,18363],{"className":18362},[1193],[1052,18364,18366],{"className":18365,"style":1228},[1197],[1052,18367],{},[1052,18369],{"className":18370,"style":1155},[1154],[1052,18372,1077],{"className":18373},[1159],[1052,18375],{"className":18376,"style":1155},[1154],[1052,18378,18380,18383,18386,18389,18429,18475,18478,18481],{"className":18379},[1139],[1052,18381],{"className":18382,"style":1597},[1143],[1052,18384,13013],{"className":18385,"style":1245},[1148,1149],[1052,18387,1084],{"className":18388},[1173],[1052,18390,18392,18395],{"className":18391},[1148],[1052,18393,4571],{"className":18394},[1148,4785],[1052,18396,18398],{"className":18397},[1184],[1052,18399,18401,18421],{"className":18400},[1188,1189],[1052,18402,18404,18418],{"className":18403},[1193],[1052,18405,18407],{"className":18406,"style":11602},[1197],[1052,18408,18409,18412],{"style":1201},[1052,18410],{"className":18411,"style":1206},[1205],[1052,18413,18415],{"className":18414},[1210,1211,1212,1213],[1052,18416,3799],{"className":18417},[1148,1149,1213],[1052,18419,1221],{"className":18420},[1220],[1052,18422,18424],{"className":18423},[1193],[1052,18425,18427],{"className":18426,"style":1228},[1197],[1052,18428],{},[1052,18430,18432,18435],{"className":18431},[1148],[1052,18433,11120],{"className":18434,"style":4786},[1148,4785],[1052,18436,18438],{"className":18437},[1184],[1052,18439,18441,18467],{"className":18440},[1188,1189],[1052,18442,18444,18464],{"className":18443},[1193],[1052,18445,18447],{"className":18446,"style":1825},[1197],[1052,18448,18449,18452],{"style":13202},[1052,18450],{"className":18451,"style":1206},[1205],[1052,18453,18455],{"className":18454},[1210,1211,1212,1213],[1052,18456,18458,18461],{"className":18457},[1148,1213],[1052,18459,1090],{"className":18460},[1148,1149,1213],[1052,18462,2687],{"className":18463,"style":1150},[1148,1149,1213],[1052,18465,1221],{"className":18466},[1220],[1052,18468,18470],{"className":18469},[1193],[1052,18471,18473],{"className":18472,"style":1228},[1197],[1052,18474],{},[1052,18476],{"className":18477,"style":3378},[1154],[1052,18479,3111],{"className":18480},[2428],[1052,18482],{"className":18483,"style":3378},[1154],[1052,18485,18487,18490,18539,18585,18588,18591],{"className":18486},[1139],[1052,18488],{"className":18489,"style":13244},[1143],[1052,18491,18493,18496],{"className":18492},[1148],[1052,18494,12429],{"className":18495},[1148,4785],[1052,18497,18499],{"className":18498},[1184],[1052,18500,18502,18531],{"className":18501},[1188,1189],[1052,18503,18505,18528],{"className":18504},[1193],[1052,18506,18508],{"className":18507,"style":4690},[1197],[1052,18509,18510,18513],{"style":1201},[1052,18511],{"className":18512,"style":1206},[1205],[1052,18514,18516],{"className":18515},[1210,1211,1212,1213],[1052,18517,18519,18522,18525],{"className":18518},[1148,1213],[1052,18520,3799],{"className":18521},[1148,1149,1213],[1052,18523,3988],{"className":18524},[2428,1213],[1052,18526,1123],{"className":18527},[1148,1213],[1052,18529,1221],{"className":18530},[1220],[1052,18532,18534],{"className":18533},[1193],[1052,18535,18537],{"className":18536,"style":11713},[1197],[1052,18538],{},[1052,18540,18542,18545],{"className":18541},[1148],[1052,18543,11120],{"className":18544,"style":4786},[1148,4785],[1052,18546,18548],{"className":18547},[1184],[1052,18549,18551,18577],{"className":18550},[1188,1189],[1052,18552,18554,18574],{"className":18553},[1193],[1052,18555,18557],{"className":18556,"style":13312},[1197],[1052,18558,18559,18562],{"style":13202},[1052,18560],{"className":18561,"style":1206},[1205],[1052,18563,18565],{"className":18564},[1210,1211,1212,1213],[1052,18566,18568,18571],{"className":18567},[1148,1213],[1052,18569,10971],{"className":18570},[1148,1149,1213],[1052,18572,2687],{"className":18573,"style":1150},[1148,1149,1213],[1052,18575,1221],{"className":18576},[1220],[1052,18578,18580],{"className":18579},[1193],[1052,18581,18583],{"className":18582,"style":1228},[1197],[1052,18584],{},[1052,18586],{"className":18587,"style":3378},[1154],[1052,18589,3111],{"className":18590},[2428],[1052,18592],{"className":18593,"style":3378},[1154],[1052,18595,18597,18600,18640],{"className":18596},[1139],[1052,18598],{"className":18599,"style":1597},[1143],[1052,18601,18603,18606],{"className":18602},[1148],[1052,18604,10727],{"className":18605},[1148,4785],[1052,18607,18609],{"className":18608},[1184],[1052,18610,18612,18632],{"className":18611},[1188,1189],[1052,18613,18615,18629],{"className":18614},[1193],[1052,18616,18618],{"className":18617,"style":1825},[1197],[1052,18619,18620,18623],{"style":1201},[1052,18621],{"className":18622,"style":1206},[1205],[1052,18624,18626],{"className":18625},[1210,1211,1212,1213],[1052,18627,2687],{"className":18628,"style":1150},[1148,1149,1213],[1052,18630,1221],{"className":18631},[1220],[1052,18633,18635],{"className":18634},[1193],[1052,18636,18638],{"className":18637,"style":1228},[1197],[1052,18639],{},[1052,18641,1107],{"className":18642},[1284],[1052,18644,18646],{"className":18645},[1865],[1052,18647,18649,18724],{"className":18648},[1055],[1052,18650,18652],{"className":18651},[1059],[1061,18653,18654],{"xmlns":1063,"display":1874},[1065,18655,18656,18721],{},[1068,18657,18658,18664,18666,18668,18670,18676,18687,18689,18701,18711,18713,18719],{},[1086,18659,18660,18662],{},[1071,18661,17533],{"mathvariant":4609},[1071,18663,3799],{},[1075,18665,1077],{},[1071,18667,13013],{},[1075,18669,1084],{"stretchy":1080},[1086,18671,18672,18674],{},[1071,18673,4571],{"mathvariant":4609},[1071,18675,3799],{},[1086,18677,18678,18680],{},[1071,18679,11120],{"mathvariant":4609},[1068,18681,18682,18684],{},[1071,18683,1090],{},[1071,18685,18686],{},"z",[1075,18688,3111],{},[1086,18690,18691,18693],{},[1071,18692,12429],{"mathvariant":4609},[1068,18694,18695,18697,18699],{},[1071,18696,3799],{},[1075,18698,3988],{},[1121,18700,1123],{},[1086,18702,18703,18705],{},[1071,18704,11120],{"mathvariant":4609},[1068,18706,18707,18709],{},[1071,18708,10971],{},[1071,18710,18686],{},[1075,18712,3111],{},[1086,18714,18715,18717],{},[1071,18716,10727],{"mathvariant":4609},[1071,18718,18686],{},[1075,18720,1107],{"stretchy":1080},[1128,18722,18723],{"encoding":1130},"\\mathbf{Z}_t = \\sigma(\\mathbf{X}_t \\mathbf{W}_{xz} + \\mathbf{H}_{t-1} \\mathbf{W}_{hz} + \\mathbf{b}_z)",[1052,18725,18727,18782,18890,19000],{"className":18726,"ariaHidden":1096},[1135],[1052,18728,18730,18733,18773,18776,18779],{"className":18729},[1139],[1052,18731],{"className":18732,"style":13080},[1143],[1052,18734,18736,18739],{"className":18735},[1148],[1052,18737,17533],{"className":18738},[1148,4785],[1052,18740,18742],{"className":18741},[1184],[1052,18743,18745,18765],{"className":18744},[1188,1189],[1052,18746,18748,18762],{"className":18747},[1193],[1052,18749,18751],{"className":18750,"style":11602},[1197],[1052,18752,18753,18756],{"style":1201},[1052,18754],{"className":18755,"style":1206},[1205],[1052,18757,18759],{"className":18758},[1210,1211,1212,1213],[1052,18760,3799],{"className":18761},[1148,1149,1213],[1052,18763,1221],{"className":18764},[1220],[1052,18766,18768],{"className":18767},[1193],[1052,18769,18771],{"className":18770,"style":1228},[1197],[1052,18772],{},[1052,18774],{"className":18775,"style":1155},[1154],[1052,18777,1077],{"className":18778},[1159],[1052,18780],{"className":18781,"style":1155},[1154],[1052,18783,18785,18788,18791,18794,18834,18881,18884,18887],{"className":18784},[1139],[1052,18786],{"className":18787,"style":1597},[1143],[1052,18789,13013],{"className":18790,"style":1245},[1148,1149],[1052,18792,1084],{"className":18793},[1173],[1052,18795,18797,18800],{"className":18796},[1148],[1052,18798,4571],{"className":18799},[1148,4785],[1052,18801,18803],{"className":18802},[1184],[1052,18804,18806,18826],{"className":18805},[1188,1189],[1052,18807,18809,18823],{"className":18808},[1193],[1052,18810,18812],{"className":18811,"style":11602},[1197],[1052,18813,18814,18817],{"style":1201},[1052,18815],{"className":18816,"style":1206},[1205],[1052,18818,18820],{"className":18819},[1210,1211,1212,1213],[1052,18821,3799],{"className":18822},[1148,1149,1213],[1052,18824,1221],{"className":18825},[1220],[1052,18827,18829],{"className":18828},[1193],[1052,18830,18832],{"className":18831,"style":1228},[1197],[1052,18833],{},[1052,18835,18837,18840],{"className":18836},[1148],[1052,18838,11120],{"className":18839,"style":4786},[1148,4785],[1052,18841,18843],{"className":18842},[1184],[1052,18844,18846,18873],{"className":18845},[1188,1189],[1052,18847,18849,18870],{"className":18848},[1193],[1052,18850,18852],{"className":18851,"style":1825},[1197],[1052,18853,18854,18857],{"style":13202},[1052,18855],{"className":18856,"style":1206},[1205],[1052,18858,18860],{"className":18859},[1210,1211,1212,1213],[1052,18861,18863,18866],{"className":18862},[1148,1213],[1052,18864,1090],{"className":18865},[1148,1149,1213],[1052,18867,18686],{"className":18868,"style":18869},[1148,1149,1213],"margin-right:0.044em;",[1052,18871,1221],{"className":18872},[1220],[1052,18874,18876],{"className":18875},[1193],[1052,18877,18879],{"className":18878,"style":1228},[1197],[1052,18880],{},[1052,18882],{"className":18883,"style":3378},[1154],[1052,18885,3111],{"className":18886},[2428],[1052,18888],{"className":18889,"style":3378},[1154],[1052,18891,18893,18896,18945,18991,18994,18997],{"className":18892},[1139],[1052,18894],{"className":18895,"style":13244},[1143],[1052,18897,18899,18902],{"className":18898},[1148],[1052,18900,12429],{"className":18901},[1148,4785],[1052,18903,18905],{"className":18904},[1184],[1052,18906,18908,18937],{"className":18907},[1188,1189],[1052,18909,18911,18934],{"className":18910},[1193],[1052,18912,18914],{"className":18913,"style":4690},[1197],[1052,18915,18916,18919],{"style":1201},[1052,18917],{"className":18918,"style":1206},[1205],[1052,18920,18922],{"className":18921},[1210,1211,1212,1213],[1052,18923,18925,18928,18931],{"className":18924},[1148,1213],[1052,18926,3799],{"className":18927},[1148,1149,1213],[1052,18929,3988],{"className":18930},[2428,1213],[1052,18932,1123],{"className":18933},[1148,1213],[1052,18935,1221],{"className":18936},[1220],[1052,18938,18940],{"className":18939},[1193],[1052,18941,18943],{"className":18942,"style":11713},[1197],[1052,18944],{},[1052,18946,18948,18951],{"className":18947},[1148],[1052,18949,11120],{"className":18950,"style":4786},[1148,4785],[1052,18952,18954],{"className":18953},[1184],[1052,18955,18957,18983],{"className":18956},[1188,1189],[1052,18958,18960,18980],{"className":18959},[1193],[1052,18961,18963],{"className":18962,"style":13312},[1197],[1052,18964,18965,18968],{"style":13202},[1052,18966],{"className":18967,"style":1206},[1205],[1052,18969,18971],{"className":18970},[1210,1211,1212,1213],[1052,18972,18974,18977],{"className":18973},[1148,1213],[1052,18975,10971],{"className":18976},[1148,1149,1213],[1052,18978,18686],{"className":18979,"style":18869},[1148,1149,1213],[1052,18981,1221],{"className":18982},[1220],[1052,18984,18986],{"className":18985},[1193],[1052,18987,18989],{"className":18988,"style":1228},[1197],[1052,18990],{},[1052,18992],{"className":18993,"style":3378},[1154],[1052,18995,3111],{"className":18996},[2428],[1052,18998],{"className":18999,"style":3378},[1154],[1052,19001,19003,19006,19046],{"className":19002},[1139],[1052,19004],{"className":19005,"style":1597},[1143],[1052,19007,19009,19012],{"className":19008},[1148],[1052,19010,10727],{"className":19011},[1148,4785],[1052,19013,19015],{"className":19014},[1184],[1052,19016,19018,19038],{"className":19017},[1188,1189],[1052,19019,19021,19035],{"className":19020},[1193],[1052,19022,19024],{"className":19023,"style":1825},[1197],[1052,19025,19026,19029],{"style":1201},[1052,19027],{"className":19028,"style":1206},[1205],[1052,19030,19032],{"className":19031},[1210,1211,1212,1213],[1052,19033,18686],{"className":19034,"style":18869},[1148,1149,1213],[1052,19036,1221],{"className":19037},[1220],[1052,19039,19041],{"className":19040},[1193],[1052,19042,19044],{"className":19043,"style":1228},[1197],[1052,19045],{},[1052,19047,1107],{"className":19048},[1284],[996,19050,19051,19052,19122,19123,19126,19127],{},"接下来，让我们将重置门",[1052,19053,19055,19073],{"className":19054},[1055],[1052,19056,19058],{"className":19057},[1059],[1061,19059,19060],{"xmlns":1063},[1065,19061,19062,19070],{},[1068,19063,19064],{},[1086,19065,19066,19068],{},[1071,19067,2558],{"mathvariant":4609},[1071,19069,3799],{},[1128,19071,19072],{"encoding":1130},"\\mathbf{R}_t",[1052,19074,19076],{"className":19075,"ariaHidden":1096},[1135],[1052,19077,19079,19082],{"className":19078},[1139],[1052,19080],{"className":19081,"style":13080},[1143],[1052,19083,19085,19088],{"className":19084},[1148],[1052,19086,2558],{"className":19087},[1148,4785],[1052,19089,19091],{"className":19090},[1184],[1052,19092,19094,19114],{"className":19093},[1188,1189],[1052,19095,19097,19111],{"className":19096},[1193],[1052,19098,19100],{"className":19099,"style":11602},[1197],[1052,19101,19102,19105],{"style":1201},[1052,19103],{"className":19104,"style":1206},[1205],[1052,19106,19108],{"className":19107},[1210,1211,1212,1213],[1052,19109,3799],{"className":19110},[1148,1149,1213],[1052,19112,1221],{"className":19113},[1220],[1052,19115,19117],{"className":19116},[1193],[1052,19118,19120],{"className":19119,"style":1228},[1197],[1052,19121],{},"与",[1020,19124,19125],{},"常规隐状态更新机制集成","，得到在时间步t的候选隐状态 (candidate hidden state)",[1052,19128,19130,19166],{"className":19129},[1055],[1052,19131,19133],{"className":19132},[1059],[1061,19134,19135],{"xmlns":1063},[1065,19136,19137,19163],{},[1068,19138,19139,19149,19151],{},[1086,19140,19141,19147],{},[15030,19142,19143,19145],{"accent":1096},[1071,19144,12429],{"mathvariant":4609},[1075,19146,15036],{},[1071,19148,3799],{},[1075,19150,12288],{},[5084,19152,19153,19155],{},[1071,19154,2558],{"mathvariant":14270},[1068,19156,19157,19159,19161],{},[1071,19158,1126],{},[1075,19160,2329],{},[1071,19162,10971],{},[1128,19164,19165],{"encoding":1130},"\\tilde{\\mathbf{H}}_t \\in \\mathbb{R}^{n \\times 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= \\tanh(\\mathbf{X}_t \\mathbf{W}_{xh} + (\\mathbf{R}_t \\odot \\mathbf{H}_{t-1}) \\mathbf{W}_{hh} + \\mathbf{b}_h), \\quad 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操作，将所有输入序列填充到相同长度",[996,20708,20709],{},[1711,20710],{"alt":1711,"src":20711},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-28.png",[993,20713,20714],{},[996,20715,20716],{},"PAD加到后面——不好；加到前面——好，因为语句越往后越重要，由此也引出了双向循环神经网络",[1045,20718,20720],{"id":20719},"_7双向循环神经网络bidirectional-recurrent-neural-network-birnn","7、双向循环神经网络（Bidirectional Recurrent Neural Network, BiRNN）",[996,20722,20723,20724,20727],{},"双向循环神经网络（Bidirectional Recurrent Neural Network, BiRNN）是一种特殊的循环神经网络结构，它通过在",[1020,20725,20726],{},"时间序列的正向和反向两个方向","上同时进行信息处理，从而更好地捕捉序列数据中的上下文信息",[996,20729,20730],{},[1711,20731],{"alt":1711,"src":20732},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-29.png",[1010,20734,20736],{"id":20735},"六transformer结构","六、Transformer结构",[1042,20738,20740],{"id":20739},"_1注意力机制","1、注意力机制",[1045,20742,1047],{"id":20743},"基本概念-3",[996,20745,20746],{},[1711,20747],{"alt":1711,"src":20748},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-30.png",[996,20750,20751,20752,20885,20886,20956,20957,3028],{},"首先生成每个单词的内嵌向量（包含了单词在句子中位置编码向量信息），记为",[1052,20753,20755,20789],{"className":20754},[1055],[1052,20756,20758],{"className":20757},[1059],[1061,20759,20760],{"xmlns":1063},[1065,20761,20762,20786],{},[1068,20763,20764,20770,20772,20774,20777,20779,20781,20784],{},[1086,20765,20766,20768],{},[1071,20767,10976],{},[1071,20769,1093],{},[1075,20771,1084],{"stretchy":1080},[1121,20773,1123],{},[1075,20775,20776],{},"≤",[1071,20778,1093],{},[1075,20780,20776],{},[1121,20782,20783],{},"4",[1075,20785,1107],{"stretchy":1080},[1128,20787,20788],{"encoding":1130},"w_i(1≤i≤4)",[1052,20790,20792,20854,20873],{"className":20791,"ariaHidden":1096},[1135],[1052,20793,20795,20798,20839,20842,20845,20848,20851],{"className":20794},[1139],[1052,20796],{"className":20797,"style":1597},[1143],[1052,20799,20801,20804],{"className":20800},[1148],[1052,20802,10976],{"className":20803,"style":11006},[1148,1149],[1052,20805,20807],{"className":20806},[1184],[1052,20808,20810,20831],{"className":20809},[1188,1189],[1052,20811,20813,20828],{"className":20812},[1193],[1052,20814,20816],{"className":20815,"style":1198},[1197],[1052,20817,20819,20822],{"style":20818},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[1052,20820],{"className":20821,"style":1206},[1205],[1052,20823,20825],{"className":20824},[1210,1211,1212,1213],[1052,20826,1093],{"className":20827},[1148,1149,1213],[1052,20829,1221],{"className":20830},[1220],[1052,20832,20834],{"className":20833},[1193],[1052,20835,20837],{"className":20836,"style":1228},[1197],[1052,20838],{},[1052,20840,1084],{"className":20841},[1173],[1052,20843,1123],{"className":20844},[1148],[1052,20846],{"className":20847,"style":1155},[1154],[1052,20849,20776],{"className":20850},[1159],[1052,20852],{"className":20853,"style":1155},[1154],[1052,20855,20857,20861,20864,20867,20870],{"className":20856},[1139],[1052,20858],{"className":20859,"style":20860},[1143],"height:0.7955em;vertical-align:-0.136em;",[1052,20862,1093],{"className":20863},[1148,1149],[1052,20865],{"className":20866,"style":1155},[1154],[1052,20868,20776],{"className":20869},[1159],[1052,20871],{"className":20872,"style":1155},[1154],[1052,20874,20876,20879,20882],{"className":20875},[1139],[1052,20877],{"className":20878,"style":1597},[1143],[1052,20880,20783],{"className":20881},[1148],[1052,20883,1107],{"className":20884},[1284],",如下计算每个单词",[1052,20887,20889,20907],{"className":20888},[1055],[1052,20890,20892],{"className":20891},[1059],[1061,20893,20894],{"xmlns":1063},[1065,20895,20896,20904],{},[1068,20897,20898],{},[1086,20899,20900,20902],{},[1071,20901,10976],{},[1071,20903,1093],{},[1128,20905,20906],{"encoding":1130},"w_i",[1052,20908,20910],{"className":20909,"ariaHidden":1096},[1135],[1052,20911,20913,20916],{"className":20912},[1139],[1052,20914],{"className":20915,"style":1380},[1143],[1052,20917,20919,20922],{"className":20918},[1148],[1052,20920,10976],{"className":20921,"style":11006},[1148,1149],[1052,20923,20925],{"className":20924},[1184],[1052,20926,20928,20948],{"className":20927},[1188,1189],[1052,20929,20931,20945],{"className":20930},[1193],[1052,20932,20934],{"className":20933,"style":1198},[1197],[1052,20935,20936,20939],{"style":20818},[1052,20937],{"className":20938,"style":1206},[1205],[1052,20940,20942],{"className":20941},[1210,1211,1212,1213],[1052,20943,1093],{"className":20944},[1148,1149,1213],[1052,20946,1221],{"className":20947},[1220],[1052,20949,20951],{"className":20950},[1193],[1052,20952,20954],{"className":20953,"style":1228},[1197],[1052,20955],{},"的",[1532,20958,20959],{},"查询向量（query）、键向量(key)和值向量(value)",[1015,20961,20962,21151,21343],{},[983,20963,20964,20965],{},"查询向量：",[1052,20966,20968,21003],{"className":20967},[1055],[1052,20969,20971],{"className":20970},[1059],[1061,20972,20973],{"xmlns":1063},[1065,20974,20975,21000],{},[1068,20976,20977,20984,20986,20992,20994],{},[1086,20978,20979,20982],{},[1071,20980,20981],{},"q",[1071,20983,1093],{},[1075,20985,1077],{},[5084,20987,20988,20990],{},[1071,20989,11120],{},[1071,20991,20981],{},[1075,20993,2329],{},[1086,20995,20996,20998],{},[1071,20997,10976],{},[1071,20999,1093],{},[1128,21001,21002],{"encoding":1130},"q_i=W^q×w_i",[1052,21004,21006,21061,21105],{"className":21005,"ariaHidden":1096},[1135],[1052,21007,21009,21012,21052,21055,21058],{"className":21008},[1139],[1052,21010],{"className":21011,"style":1452},[1143],[1052,21013,21015,21018],{"className":21014},[1148],[1052,21016,20981],{"className":21017,"style":1245},[1148,1149],[1052,21019,21021],{"className":21020},[1184],[1052,21022,21024,21044],{"className":21023},[1188,1189],[1052,21025,21027,21041],{"className":21026},[1193],[1052,21028,21030],{"className":21029,"style":1198},[1197],[1052,21031,21032,21035],{"style":1260},[1052,21033],{"className":21034,"style":1206},[1205],[1052,21036,21038],{"className":21037},[1210,1211,1212,1213],[1052,21039,1093],{"className":21040},[1148,1149,1213],[1052,21042,1221],{"className":21043},[1220],[1052,21045,21047],{"className":21046},[1193],[1052,21048,21050],{"className":21049,"style":1228},[1197],[1052,21051],{},[1052,21053],{"className":21054,"style":1155},[1154],[1052,21056,1077],{"className":21057},[1159],[1052,21059],{"className":21060,"style":1155},[1154],[1052,21062,21064,21067,21096,21099,21102],{"className":21063},[1139],[1052,21065],{"className":21066,"style":11137},[1143],[1052,21068,21070,21073],{"className":21069},[1148],[1052,21071,11120],{"className":21072,"style":2283},[1148,1149],[1052,21074,21076],{"className":21075},[1184],[1052,21077,21079],{"className":21078},[1188],[1052,21080,21082],{"className":21081},[1193],[1052,21083,21085],{"className":21084,"style":1303},[1197],[1052,21086,21087,21090],{"style":1327},[1052,21088],{"className":21089,"style":1206},[1205],[1052,21091,21093],{"className":21092},[1210,1211,1212,1213],[1052,21094,20981],{"className":21095,"style":1245},[1148,1149,1213],[1052,21097],{"className":21098,"style":3378},[1154],[1052,21100,2329],{"className":21101},[2428],[1052,21103],{"className":21104,"style":3378},[1154],[1052,21106,21108,21111],{"className":21107},[1139],[1052,21109],{"className":21110,"style":1380},[1143],[1052,21112,21114,21117],{"className":21113},[1148],[1052,21115,10976],{"className":21116,"style":11006},[1148,1149],[1052,21118,21120],{"className":21119},[1184],[1052,21121,21123,21143],{"className":21122},[1188,1189],[1052,21124,21126,21140],{"className":21125},[1193],[1052,21127,21129],{"className":21128,"style":1198},[1197],[1052,21130,21131,21134],{"style":20818},[1052,21132],{"className":21133,"style":1206},[1205],[1052,21135,21137],{"className":21136},[1210,1211,1212,1213],[1052,21138,1093],{"className":21139},[1148,1149,1213],[1052,21141,1221],{"className":21142},[1220],[1052,21144,21146],{"className":21145},[1193],[1052,21147,21149],{"className":21148,"style":1228},[1197],[1052,21150],{},[983,21152,21153,21154],{},"健向量：",[1052,21155,21157,21192],{"className":21156},[1055],[1052,21158,21160],{"className":21159},[1059],[1061,21161,21162],{"xmlns":1063},[1065,21163,21164,21189],{},[1068,21165,21166,21173,21175,21181,21183],{},[1086,21167,21168,21171],{},[1071,21169,21170],{},"k",[1071,21172,1093],{},[1075,21174,1077],{},[5084,21176,21177,21179],{},[1071,21178,11120],{},[1071,21180,21170],{},[1075,21182,2329],{},[1086,21184,21185,21187],{},[1071,21186,10976],{},[1071,21188,1093],{},[1128,21190,21191],{"encoding":1130},"k_i=W^k×w_i",[1052,21193,21195,21252,21297],{"className":21194,"ariaHidden":1096},[1135],[1052,21196,21198,21201,21243,21246,21249],{"className":21197},[1139],[1052,21199],{"className":21200,"style":11871},[1143],[1052,21202,21204,21208],{"className":21203},[1148],[1052,21205,21170],{"className":21206,"style":21207},[1148,1149],"margin-right:0.0315em;",[1052,21209,21211],{"className":21210},[1184],[1052,21212,21214,21235],{"className":21213},[1188,1189],[1052,21215,21217,21232],{"className":21216},[1193],[1052,21218,21220],{"className":21219,"style":1198},[1197],[1052,21221,21223,21226],{"style":21222},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[1052,21224],{"className":21225,"style":1206},[1205],[1052,21227,21229],{"className":21228},[1210,1211,1212,1213],[1052,21230,1093],{"className":21231},[1148,1149,1213],[1052,21233,1221],{"className":21234},[1220],[1052,21236,21238],{"className":21237},[1193],[1052,21239,21241],{"className":21240,"style":1228},[1197],[1052,21242],{},[1052,21244],{"className":21245,"style":1155},[1154],[1052,21247,1077],{"className":21248},[1159],[1052,21250],{"className":21251,"style":1155},[1154],[1052,21253,21255,21259,21288,21291,21294],{"className":21254},[1139],[1052,21256],{"className":21257,"style":21258},[1143],"height:0.9324em;vertical-align:-0.0833em;",[1052,21260,21262,21265],{"className":21261},[1148],[1052,21263,11120],{"className":21264,"style":2283},[1148,1149],[1052,21266,21268],{"className":21267},[1184],[1052,21269,21271],{"className":21270},[1188],[1052,21272,21274],{"className":21273},[1193],[1052,21275,21277],{"className":21276,"style":12370},[1197],[1052,21278,21279,21282],{"style":1327},[1052,21280],{"className":21281,"style":1206},[1205],[1052,21283,21285],{"className":21284},[1210,1211,1212,1213],[1052,21286,21170],{"className":21287,"style":21207},[1148,1149,1213],[1052,21289],{"className":21290,"style":3378},[1154],[1052,21292,2329],{"className":21293},[2428],[1052,21295],{"className":21296,"style":3378},[1154],[1052,21298,21300,21303],{"className":21299},[1139],[1052,21301],{"className":21302,"style":1380},[1143],[1052,21304,21306,21309],{"className":21305},[1148],[1052,21307,10976],{"className":21308,"style":11006},[1148,1149],[1052,21310,21312],{"className":21311},[1184],[1052,21313,21315,21335],{"className":21314},[1188,1189],[1052,21316,21318,21332],{"className":21317},[1193],[1052,21319,21321],{"className":21320,"style":1198},[1197],[1052,21322,21323,21326],{"style":20818},[1052,21324],{"className":21325,"style":1206},[1205],[1052,21327,21329],{"className":21328},[1210,1211,1212,1213],[1052,21330,1093],{"className":21331},[1148,1149,1213],[1052,21333,1221],{"className":21334},[1220],[1052,21336,21338],{"className":21337},[1193],[1052,21339,21341],{"className":21340,"style":1228},[1197],[1052,21342],{},[983,21344,21345,21346],{},"值向量：",[1052,21347,21349,21384],{"className":21348},[1055],[1052,21350,21352],{"className":21351},[1059],[1061,21353,21354],{"xmlns":1063},[1065,21355,21356,21381],{},[1068,21357,21358,21365,21367,21373,21375],{},[1086,21359,21360,21363],{},[1071,21361,21362],{},"v",[1071,21364,1093],{},[1075,21366,1077],{},[5084,21368,21369,21371],{},[1071,21370,11120],{},[1071,21372,21362],{},[1075,21374,2329],{},[1086,21376,21377,21379],{},[1071,21378,10976],{},[1071,21380,1093],{},[1128,21382,21383],{"encoding":1130},"v_i=W^v×w_i",[1052,21385,21387,21442,21486],{"className":21386,"ariaHidden":1096},[1135],[1052,21388,21390,21393,21433,21436,21439],{"className":21389},[1139],[1052,21391],{"className":21392,"style":1380},[1143],[1052,21394,21396,21399],{"className":21395},[1148],[1052,21397,21362],{"className":21398,"style":1245},[1148,1149],[1052,21400,21402],{"className":21401},[1184],[1052,21403,21405,21425],{"className":21404},[1188,1189],[1052,21406,21408,21422],{"className":21407},[1193],[1052,21409,21411],{"className":21410,"style":1198},[1197],[1052,21412,21413,21416],{"style":1260},[1052,21414],{"className":21415,"style":1206},[1205],[1052,21417,21419],{"className":21418},[1210,1211,1212,1213],[1052,21420,1093],{"className":21421},[1148,1149,1213],[1052,21423,1221],{"className":21424},[1220],[1052,21426,21428],{"className":21427},[1193],[1052,21429,21431],{"className":21430,"style":1228},[1197],[1052,21432],{},[1052,21434],{"className":21435,"style":1155},[1154],[1052,21437,1077],{"className":21438},[1159],[1052,21440],{"className":21441,"style":1155},[1154],[1052,21443,21445,21448,21477,21480,21483],{"className":21444},[1139],[1052,21446],{"className":21447,"style":11137},[1143],[1052,21449,21451,21454],{"className":21450},[1148],[1052,21452,11120],{"className":21453,"style":2283},[1148,1149],[1052,21455,21457],{"className":21456},[1184],[1052,21458,21460],{"className":21459},[1188],[1052,21461,21463],{"className":21462},[1193],[1052,21464,21466],{"className":21465,"style":1303},[1197],[1052,21467,21468,21471],{"style":1327},[1052,21469],{"className":21470,"style":1206},[1205],[1052,21472,21474],{"className":21473},[1210,1211,1212,1213],[1052,21475,21362],{"className":21476,"style":1245},[1148,1149,1213],[1052,21478],{"className":21479,"style":3378},[1154],[1052,21481,2329],{"className":21482},[2428],[1052,21484],{"className":21485,"style":3378},[1154],[1052,21487,21489,21492],{"className":21488},[1139],[1052,21490],{"className":21491,"style":1380},[1143],[1052,21493,21495,21498],{"className":21494},[1148],[1052,21496,10976],{"className":21497,"style":11006},[1148,1149],[1052,21499,21501],{"className":21500},[1184],[1052,21502,21504,21524],{"className":21503},[1188,1189],[1052,21505,21507,21521],{"className":21506},[1193],[1052,21508,21510],{"className":21509,"style":1198},[1197],[1052,21511,21512,21515],{"style":20818},[1052,21513],{"className":21514,"style":1206},[1205],[1052,21516,21518],{"className":21517},[1210,1211,1212,1213],[1052,21519,1093],{"className":21520},[1148,1149,1213],[1052,21522,1221],{"className":21523},[1220],[1052,21525,21527],{"className":21526},[1193],[1052,21528,21530],{"className":21529,"style":1228},[1197],[1052,21531],{},[993,21533,21534],{},[996,21535,21536],{},[1532,21537,21538],{},[1020,21539,21540],{},"自注意力模型就是要挖掘单词Wi与其他单词在句子中因为上下文（context）关联而具有的自注意力取值大小",[996,21542,21543,21757,21758,1766,21828,21831],{},[1052,21544,21546,21586],{"className":21545},[1055],[1052,21547,21549],{"className":21548},[1059],[1061,21550,21551],{"xmlns":1063},[1065,21552,21553,21583],{},[1068,21554,21555,21561,21575,21577],{},[1086,21556,21557,21559],{},[1075,21558,1891],{},[1071,21560,1093],{},[1109,21562,21563,21565,21572],{},[1071,21564,3796],{},[1068,21566,21567,21570],{},[1121,21568,21569],{},"3",[1071,21571,1093],{},[10121,21573,21574],{},"′",[1075,21576,2329],{},[1086,21578,21579,21581],{},[1071,21580,21362],{},[1071,21582,1093],{},[1128,21584,21585],{"encoding":1130},"∑_ia_{3i}^′×v_i",[1052,21587,21589,21711],{"className":21588,"ariaHidden":1096},[1135],[1052,21590,21592,21596,21641,21644,21702,21705,21708],{"className":21591},[1139],[1052,21593],{"className":21594,"style":21595},[1143],"height:1.0516em;vertical-align:-0.2997em;",[1052,21597,21599,21604],{"className":21598},[2027],[1052,21600,1891],{"className":21601,"style":21603},[2027,2072,21602],"small-op","position:relative;top:0em;",[1052,21605,21607],{"className":21606},[1184],[1052,21608,21610,21632],{"className":21609},[1188,1189],[1052,21611,21613,21629],{"className":21612},[1193],[1052,21614,21617],{"className":21615,"style":21616},[1197],"height:0.162em;",[1052,21618,21620,21623],{"style":21619},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[1052,21621],{"className":21622,"style":1206},[1205],[1052,21624,21626],{"className":21625},[1210,1211,1212,1213],[1052,21627,1093],{"className":21628},[1148,1149,1213],[1052,21630,1221],{"className":21631},[1220],[1052,21633,21635],{"className":21634},[1193],[1052,21636,21639],{"className":21637,"style":21638},[1197],"height:0.2997em;",[1052,21640],{},[1052,21642],{"className":21643,"style":1238},[1154],[1052,21645,21647,21650],{"className":21646},[1148],[1052,21648,3796],{"className":21649},[1148,1149],[1052,21651,21653],{"className":21652},[1184],[1052,21654,21656,21694],{"className":21655},[1188,1189],[1052,21657,21659,21691],{"className":21658},[1193],[1052,21660,21663,21680],{"className":21661,"style":21662},[1197],"height:0.7519em;",[1052,21664,21665,21668],{"style":1306},[1052,21666],{"className":21667,"style":1206},[1205],[1052,21669,21671],{"className":21670},[1210,1211,1212,1213],[1052,21672,21674,21677],{"className":21673},[1148,1213],[1052,21675,21569],{"className":21676},[1148,1213],[1052,21678,1093],{"className":21679},[1148,1149,1213],[1052,21681,21682,21685],{"style":1327},[1052,21683],{"className":21684,"style":1206},[1205],[1052,21686,21688],{"className":21687},[1210,1211,1212,1213],[1052,21689,21574],{"className":21690},[1148,1213],[1052,21692,1221],{"className":21693},[1220],[1052,21695,21697],{"className":21696},[1193],[1052,21698,21700],{"className":21699,"style":1346},[1197],[1052,21701],{},[1052,21703],{"className":21704,"style":3378},[1154],[1052,21706,2329],{"className":21707},[2428],[1052,21709],{"className":21710,"style":3378},[1154],[1052,21712,21714,21717],{"className":21713},[1139],[1052,21715],{"className":21716,"style":1380},[1143],[1052,21718,21720,21723],{"className":21719},[1148],[1052,21721,21362],{"className":21722,"style":1245},[1148,1149],[1052,21724,21726],{"className":21725},[1184],[1052,21727,21729,21749],{"className":21728},[1188,1189],[1052,21730,21732,21746],{"className":21731},[1193],[1052,21733,21735],{"className":21734,"style":1198},[1197],[1052,21736,21737,21740],{"style":1260},[1052,21738],{"className":21739,"style":1206},[1205],[1052,21741,21743],{"className":21742},[1210,1211,1212,1213],[1052,21744,1093],{"className":21745},[1148,1149,1213],[1052,21747,1221],{"className":21748},[1220],[1052,21750,21752],{"className":21751},[1193],[1052,21753,21755],{"className":21754,"style":1228},[1197],[1052,21756],{},", 这个结果作为在当前句子语境下单词",[1052,21759,21761,21779],{"className":21760},[1055],[1052,21762,21764],{"className":21763},[1059],[1061,21765,21766],{"xmlns":1063},[1065,21767,21768,21776],{},[1068,21769,21770],{},[1086,21771,21772,21774],{},[1071,21773,10976],{},[1121,21775,21569],{},[1128,21777,21778],{"encoding":1130},"w_3",[1052,21780,21782],{"className":21781,"ariaHidden":1096},[1135],[1052,21783,21785,21788],{"className":21784},[1139],[1052,21786],{"className":21787,"style":1380},[1143],[1052,21789,21791,21794],{"className":21790},[1148],[1052,21792,10976],{"className":21793,"style":11006},[1148,1149],[1052,21795,21797],{"className":21796},[1184],[1052,21798,21800,21820],{"className":21799},[1188,1189],[1052,21801,21803,21817],{"className":21802},[1193],[1052,21804,21806],{"className":21805,"style":4690},[1197],[1052,21807,21808,21811],{"style":20818},[1052,21809],{"className":21810,"style":1206},[1205],[1052,21812,21814],{"className":21813},[1210,1211,1212,1213],[1052,21815,21569],{"className":21816},[1148,1213],[1052,21818,1221],{"className":21819},[1220],[1052,21821,21823],{"className":21822},[1193],[1052,21824,21826],{"className":21825,"style":1228},[1197],[1052,21827],{},[1020,21829,21830],{},"“注意”到","与其他单词的关联程度（self-attention）",[1045,21833,21835],{"id":21834},"多头注意力机制找多重关系","多头注意力机制——找多重关系",[993,21837,21838],{},[996,21839,21840,21841,21844,21845],{},"可引入",[1020,21842,21843],{},"“多头”注意力（multi-headed attention）","机制从更多角度来挖掘某个单词与其他单词之间概率关联，",[1020,21846,21847],{},"每个单词自注意力关联可以并行计算",[996,21849,21850],{},[1711,21851],{"alt":1711,"src":21852},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-31.png",[996,21854,21855],{},"多头注意⼒机制是由于两个token之间可能存在【多种不同类型的关系】（如语法关系、语义关系等）",[1042,21857,21859],{"id":21858},"_2正则化与批归一化","2、正则化与批归一化",[1045,21861,3037],{"id":3037},[996,21863,21864],{},[1711,21865],{"alt":1711,"src":21866},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-32.png",[1045,21868,21869],{"id":21869},"批归一化",[996,21871,21872,21873,21878],{},"通过规范化手段，把神经网络每层中任意神经元",[1532,21874,21875],{},[1020,21876,21877],{},"的输入值分布改变成均值为0、方差为1的标准正态分布","，把偏移较大的分布强制映射为标准正态分布。经过批归一化处理，激活函数的输入值被映射到非线性函数梯度较大的区域，使得梯度变大从而克服梯度消失问题，进而加快收敛速度",[2930,21880,21882],{"style":21881},"width: 100%; border: 0; border-collapse: collapse; border-spacing: 0;",[2953,21883,21884],{},[2936,21885,21887,21888,21887,21896],{"style":21886},"border: none;","\n  ",[2958,21889,21891,21887],{"style":21890},"width: 53.0%; text-align: center; border: none; padding: 0.3em; margin: 0;",[996,21892,21893],{},[1711,21894],{"alt":1711,"src":21895},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-33.png",[2958,21897,21899,21887],{"style":21898},"width: 46.0%; text-align: center; border: none; padding: 0.3em; margin: 0;",[996,21900,21901],{},[1711,21902],{"alt":1711,"src":21903},"\u002Fimg\u002FAIFrontier\u002FAIFrontier-34.png",[996,21905,21906,21907],{},"实际上就是",[1020,21908,21909],{},"概统里面的正态分布估计——大数定律",[1042,21911,21913],{"id":21912},"_3transformer","3、Transformer",[996,21915,21916,21917],{},"不同于传统卷积神经网络使用固定的卷积核（感受野），Transformer 模型使用自注意力来",[1020,21918,21919],{},"自适应计算感受野",[1045,21921,21922],{"id":21922},"基本组件",[978,21924,21925,22334,22731,23122,23459,23601],{},[983,21926,21927,21930,21931,22028,22029,22112],{},[1020,21928,21929],{},"Image",": ",[1052,21932,21934,21962],{"className":21933},[1055],[1052,21935,21937],{"className":21936},[1059],[1061,21938,21939],{"xmlns":1063},[1065,21940,21941,21959],{},[1068,21942,21943,21945,21947],{},[1071,21944,12586],{},[1075,21946,12288],{},[5084,21948,21949,21951],{},[1071,21950,2558],{"mathvariant":14270},[1068,21952,21953,21955,21957],{},[1071,21954,3595],{},[1075,21956,2329],{},[1071,21958,3615],{},[1128,21960,21961],{"encoding":1130},"I \\in \\mathbb{R}^{C \\times N}",[1052,21963,21965,21984],{"className":21964,"ariaHidden":1096},[1135],[1052,21966,21968,21972,21975,21978,21981],{"className":21967},[1139],[1052,21969],{"className":21970,"style":21971},[1143],"height:0.7224em;vertical-align:-0.0391em;",[1052,21973,12586],{"className":21974,"style":4653},[1148,1149],[1052,21976],{"className":21977,"style":1155},[1154],[1052,21979,12288],{"className":21980},[1159],[1052,21982],{"className":21983,"style":1155},[1154],[1052,21985,21987,21990],{"className":21986},[1139],[1052,21988],{"className":21989,"style":7013},[1143],[1052,21991,21993,21996],{"className":21992},[1148],[1052,21994,2558],{"className":21995},[1148,14514],[1052,21997,21999],{"className":21998},[1184],[1052,22000,22002],{"className":22001},[1188],[1052,22003,22005],{"className":22004},[1193],[1052,22006,22008],{"className":22007,"style":7013},[1197],[1052,22009,22010,22013],{"style":1327},[1052,22011],{"className":22012,"style":1206},[1205],[1052,22014,22016],{"className":22015},[1210,1211,1212,1213],[1052,22017,22019,22022,22025],{"className":22018},[1148,1213],[1052,22020,3595],{"className":22021,"style":3642},[1148,1149,1213],[1052,22023,2329],{"className":22024},[2428,1213],[1052,22026,3615],{"className":22027,"style":3704},[1148,1149,1213],"  ",[1052,22030,22032,22058],{"className":22031},[1055],[1052,22033,22035],{"className":22034},[1059],[1061,22036,22037],{"xmlns":1063},[1065,22038,22039,22055],{},[1068,22040,22041,22043,22045,22047,22049,22051,22053],{},[1075,22042,1084],{"stretchy":1080},[1071,22044,3615],{},[1075,22046,1077],{},[1071,22048,12429],{},[1075,22050,2329],{},[1071,22052,11120],{},[1075,22054,1107],{"stretchy":1080},[1128,22056,22057],{"encoding":1130},"(N = H \\times W)",[1052,22059,22061,22082,22100],{"className":22060,"ariaHidden":1096},[1135],[1052,22062,22064,22067,22070,22073,22076,22079],{"className":22063},[1139],[1052,22065],{"className":22066,"style":1597},[1143],[1052,22068,1084],{"className":22069},[1173],[1052,22071,3615],{"className":22072,"style":3704},[1148,1149],[1052,22074],{"className":22075,"style":1155},[1154],[1052,22077,1077],{"className":22078},[1159],[1052,22080],{"className":22081,"style":1155},[1154],[1052,22083,22085,22088,22091,22094,22097],{"className":22084},[1139],[1052,22086],{"className":22087,"style":11137},[1143],[1052,22089,12429],{"className":22090,"style":12471},[1148,1149],[1052,22092],{"className":22093,"style":3378},[1154],[1052,22095,2329],{"className":22096},[2428],[1052,22098],{"className":22099,"style":3378},[1154],[1052,22101,22103,22106,22109],{"className":22102},[1139],[1052,22104],{"className":22105,"style":1597},[1143],[1052,22107,11120],{"className":22108,"style":2283},[1148,1149],[1052,22110,1107],{"className":22111},[1284],[978,22113,22114],{},[983,22115,22116,22117],{},"绝对位置编码：",[1052,22118,22120,22181],{"className":22119},[1055],[1052,22121,22123],{"className":22122},[1059],[1061,22124,22125],{"xmlns":1063},[1065,22126,22127,22178],{},[1068,22128,22129,22145,22147,22163,22165,22168,22170,22172,22174,22176],{},[5084,22130,22131,22133],{},[1071,22132,12586],{},[1068,22134,22135,22137,22139,22141,22143],{},[1075,22136,1084],{"stretchy":1080},[1071,22138,1090],{},[1075,22140,1097],{"separator":1096},[1071,22142,1102],{},[1075,22144,1107],{"stretchy":1080},[1075,22146,1077],{},[5084,22148,22149,22151],{},[1071,22150,3810],{"mathvariant":4609},[1068,22152,22153,22155,22157,22159,22161],{},[1075,22154,1084],{"stretchy":1080},[1071,22156,1090],{},[1075,22158,1097],{"separator":1096},[1071,22160,1102],{},[1075,22162,1107],{"stretchy":1080},[1075,22164,3111],{},[10121,22166,22167],{},"Emb",[1075,22169,1084],{"stretchy":1080},[1071,22171,1090],{},[1075,22173,1097],{"separator":1096},[1071,22175,1102],{},[1075,22177,1107],{"stretchy":1080},[1128,22179,22180],{"encoding":1130},"I^{(x,y)} = \\mathbf{F}^{(x,y)} + \\text{Emb}(x, y)",[1052,22182,22184,22244,22304],{"className":22183,"ariaHidden":1096},[1135],[1052,22185,22187,22191,22235,22238,22241],{"className":22186},[1139],[1052,22188],{"className":22189,"style":22190},[1143],"height:0.888em;",[1052,22192,22194,22197],{"className":22193},[1148],[1052,22195,12586],{"className":22196,"style":4653},[1148,1149],[1052,22198,22200],{"className":22199},[1184],[1052,22201,22203],{"className":22202},[1188],[1052,22204,22206],{"className":22205},[1193],[1052,22207,22209],{"className":22208,"style":22190},[1197],[1052,22210,22211,22214],{"style":1327},[1052,22212],{"className":22213,"style":1206},[1205],[1052,22215,22217],{"className":22216},[1210,1211,1212,1213],[1052,22218,22220,22223,22226,22229,22232],{"className":22219},[1148,1213],[1052,22221,1084],{"className":22222},[1173,1213],[1052,22224,1090],{"className":22225},[1148,1149,1213],[1052,22227,1097],{"className":22228},[1234,1213],[1052,22230,1102],{"className":22231,"style":1245},[1148,1149,1213],[1052,22233,1107],{"className":22234},[1284,1213],[1052,22236],{"className":22237,"style":1155},[1154],[1052,22239,1077],{"className":22240},[1159],[1052,22242],{"className":22243,"style":1155},[1154],[1052,22245,22247,22251,22295,22298,22301],{"className":22246},[1139],[1052,22248],{"className":22249,"style":22250},[1143],"height:0.9713em;vertical-align:-0.0833em;",[1052,22252,22254,22257],{"className":22253},[1148],[1052,22255,3810],{"className":22256},[1148,4785],[1052,22258,22260],{"className":22259},[1184],[1052,22261,22263],{"className":22262},[1188],[1052,22264,22266],{"className":22265},[1193],[1052,22267,22269],{"className":22268,"style":22190},[1197],[1052,22270,22271,22274],{"style":1327},[1052,22272],{"className":22273,"style":1206},[1205],[1052,22275,22277],{"className":22276},[1210,1211,1212,1213],[1052,22278,22280,22283,22286,22289,22292],{"className":22279},[1148,1213],[1052,22281,1084],{"className":22282},[1173,1213],[1052,22284,1090],{"className":22285},[1148,1149,1213],[1052,22287,1097],{"className":22288},[1234,1213],[1052,22290,1102],{"className":22291,"style":1245},[1148,1149,1213],[1052,22293,1107],{"className":22294},[1284,1213],[1052,22296],{"className":22297,"style":3378},[1154],[1052,22299,3111],{"className":22300},[2428],[1052,22302],{"className":22303,"style":3378},[1154],[1052,22305,22307,22310,22316,22319,22322,22325,22328,22331],{"className":22306},[1139],[1052,22308],{"className":22309,"style":1597},[1143],[1052,22311,22313],{"className":22312},[1148,14550],[1052,22314,22167],{"className":22315},[1148],[1052,22317,1084],{"className":22318},[1173],[1052,22320,1090],{"className":22321},[1148,1149],[1052,22323,1097],{"className":22324},[1234],[1052,22326],{"className":22327,"style":1238},[1154],[1052,22329,1102],{"className":22330,"style":1245},[1148,1149],[1052,22332,1107],{"className":22333},[1284],[983,22335,22336,21930,22339,22560,22561],{},[1020,22337,22338],{},"Query",[1052,22340,22342,22388],{"className":22341},[1055],[1052,22343,22345],{"className":22344},[1059],[1061,22346,22347],{"xmlns":1063},[1065,22348,22349,22385],{},[1068,22350,22351,22354,22356,22362,22364,22366,22368],{},[1071,22352,22353],{"mathvariant":4609},"Q",[1075,22355,1077],{},[1086,22357,22358,22360],{},[1071,22359,11120],{"mathvariant":4609},[1071,22361,20981],{},[1075,22363,9761],{},[1071,22365,12586],{"mathvariant":4609},[1075,22367,12288],{},[5084,22369,22370,22372],{},[1071,22371,2558],{"mathvariant":14270},[1068,22373,22374,22381,22383],{},[5084,22375,22376,22378],{},[1071,22377,3595],{},[1075,22379,21574],{"mathvariant":1575,"lspace":22380,"rspace":22380},"0em",[1075,22382,2329],{},[1071,22384,3615],{},[1128,22386,22387],{"encoding":1130},"\\mathbf{Q} = \\mathbf{W}_q \\cdot \\mathbf{I} \\in \\mathbb{R}^{C' \\times N}",[1052,22389,22391,22410,22465,22484],{"className":22390,"ariaHidden":1096},[1135],[1052,22392,22394,22398,22401,22404,22407],{"className":22393},[1139],[1052,22395],{"className":22396,"style":22397},[1143],"height:0.8805em;vertical-align:-0.1944em;",[1052,22399,22353],{"className":22400},[1148,4785],[1052,22402],{"className":22403,"style":1155},[1154],[1052,22405,1077],{"className":22406},[1159],[1052,22408],{"className":22409,"style":1155},[1154],[1052,22411,22413,22416,22456,22459,22462],{"className":22412},[1139],[1052,22414],{"className":22415,"style":13654},[1143],[1052,22417,22419,22422],{"className":22418},[1148],[1052,22420,11120],{"className":22421,"style":4786},[1148,4785],[1052,22423,22425],{"className":22424},[1184],[1052,22426,22428,22448],{"className":22427},[1188,1189],[1052,22429,22431,22445],{"className":22430},[1193],[1052,22432,22434],{"className":22433,"style":1825},[1197],[1052,22435,22436,22439],{"style":13202},[1052,22437],{"className":22438,"style":1206},[1205],[1052,22440,22442],{"className":22441},[1210,1211,1212,1213],[1052,22443,20981],{"className":22444,"style":1245},[1148,1149,1213],[1052,22446,1221],{"className":22447},[1220],[1052,22449,22451],{"className":22450},[1193],[1052,22452,22454],{"className":22453,"style":1855},[1197],[1052,22455],{},[1052,22457],{"className":22458,"style":3378},[1154],[1052,22460,9761],{"className":22461},[2428],[1052,22463],{"className":22464,"style":3378},[1154],[1052,22466,22468,22472,22475,22478,22481],{"className":22467},[1139],[1052,22469],{"className":22470,"style":22471},[1143],"height:0.7252em;vertical-align:-0.0391em;",[1052,22473,12586],{"className":22474},[1148,4785],[1052,22476],{"className":22477,"style":1155},[1154],[1052,22479,12288],{"className":22480},[1159],[1052,22482],{"className":22483,"style":1155},[1154],[1052,22485,22487,22491],{"className":22486},[1139],[1052,22488],{"className":22489,"style":22490},[1143],"height:0.9425em;",[1052,22492,22494,22497],{"className":22493},[1148],[1052,22495,2558],{"className":22496},[1148,14514],[1052,22498,22500],{"className":22499},[1184],[1052,22501,22503],{"className":22502},[1188],[1052,22504,22506],{"className":22505},[1193],[1052,22507,22509],{"className":22508,"style":22490},[1197],[1052,22510,22511,22514],{"style":1327},[1052,22512],{"className":22513,"style":1206},[1205],[1052,22515,22517],{"className":22516},[1210,1211,1212,1213],[1052,22518,22520,22554,22557],{"className":22519},[1148,1213],[1052,22521,22523,22526],{"className":22522},[1148,1213],[1052,22524,3595],{"className":22525,"style":3642},[1148,1149,1213],[1052,22527,22529],{"className":22528},[1184],[1052,22530,22532],{"className":22531},[1188],[1052,22533,22535],{"className":22534},[1193],[1052,22536,22539],{"className":22537,"style":22538},[1197],"height:0.8278em;",[1052,22540,22542,22545],{"style":22541},"top:-2.931em;margin-right:0.0714em;",[1052,22543],{"className":22544,"style":10371},[1205],[1052,22546,22548],{"className":22547},[1210,10375,10376,1213],[1052,22549,22551],{"className":22550},[1148,1213],[1052,22552,21574],{"className":22553},[1148,1213],[1052,22555,2329],{"className":22556},[2428,1213],[1052,22558,3615],{"className":22559,"style":3704},[1148,1149,1213],", ",[1052,22562,22564,22600],{"className":22563},[1055],[1052,22565,22567],{"className":22566},[1059],[1061,22568,22569],{"xmlns":1063},[1065,22570,22571,22597],{},[1068,22572,22573,22579,22581],{},[1086,22574,22575,22577],{},[1071,22576,11120],{"mathvariant":4609},[1071,22578,20981],{},[1075,22580,12288],{},[5084,22582,22583,22585],{},[1071,22584,2558],{"mathvariant":14270},[1068,22586,22587,22593,22595],{},[5084,22588,22589,22591],{},[1071,22590,3595],{},[1075,22592,21574],{"mathvariant":1575,"lspace":22380,"rspace":22380},[1075,22594,2329],{},[1071,22596,3595],{},[1128,22598,22599],{"encoding":1130},"\\mathbf{W}_q \\in \\mathbb{R}^{C' \\times 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= \\mathbf{W}_k \\cdot \\mathbf{I} \\in \\mathbb{R}^{C' \\times 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之间的相关性，",[1052,24433,24435,24453],{"className":24434},[1055],[1052,24436,24438],{"className":24437},[1059],[1061,24439,24440],{"xmlns":1063},[1065,24441,24442,24450],{},[1068,24443,24444],{},[1086,24445,24446,24448],{},[1071,24447,23620],{"mathvariant":4609},[1071,24449,10727],{},[1128,24451,24452],{"encoding":1130},"\\mathbf{V}_b",[1052,24454,24456],{"className":24455,"ariaHidden":1096},[1135],[1052,24457,24459,24462],{"className":24458},[1139],[1052,24460],{"className":24461,"style":13080},[1143],[1052,24463,24465,24468],{"className":24464},[1148],[1052,24466,23620],{"className":24467,"style":4786},[1148,4785],[1052,24469,24471],{"className":24470},[1184],[1052,24472,24474,24494],{"className":24473},[1188,1189],[1052,24475,24477,24491],{"className":24476},[1193],[1052,24478,24480],{"className":24479,"style":13312},[1197],[1052,24481,24482,24485],{"style":13202},[1052,24483],{"className":24484,"style":1206},[1205],[1052,24486,24488],{"className":24487},[1210,1211,1212,1213],[1052,24489,10727],{"className":24490},[1148,1149,1213],[1052,24492,1221],{"className":24493},[1220],[1052,24495,24497],{"className":24496},[1193],[1052,24498,24500],{"className":24499,"style":1228},[1197],[1052,24501],{}," 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