{"id":7975,"date":"2024-04-20T09:01:01","date_gmt":"2024-04-20T01:01:01","guid":{"rendered":""},"modified":"2024-04-20T09:01:01","modified_gmt":"2024-04-20T01:01:01","slug":"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES","status":"publish","type":"post","link":"https:\/\/mushiming.com\/7975.html","title":{"rendered":"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES"},"content":{"rendered":"<p><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"display: none;\"> \n <path stroke-linecap=\"round\" d=\"M5,0 0,2.5 5,5z\" id=\"raphael-marker-block\" style=\"-webkit-tap-highlight-color: rgba(0, 0, 0, 0);\"><\/path> \n<\/svg> <\/p>\n<h2>1. \u6982\u8ff0<\/h2>\n<p>\u76f8\u6bd4\u8f83\u4e8e\u57fa\u4e8eCollaborative Filter\u7b97\u6cd5\uff0c\u57fa\u4e8e\u57fa\u7840Graph Embedding\u6a21\u578b\u53ef\u4ee5\u6839\u636e\u7528\u6237\u7684\u884c\u4e3a\u5e8f\u5217\u5b66\u4e60\u51faitem\u7684embedding\uff0c\u5229\u7528item\u5bf9\u5e94\u7684Embedding\u53ef\u4ee5\u65b9\u4fbf\u8ba1\u7b97item\u4e0eitem\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5e76\u5728\u5b9e\u8df5\u4e2d\u88ab\u8bc1\u660e\u662f\u5353\u6709\u6210\u6548\u7684\u65b9\u6cd5\uff0c\u5728\u57fa\u4e8e\u57fa\u7840Graph Embedding\u6a21\u578b\uff0c\u4e3b\u8981\u5305\u62ecitem2vec\uff0cnode2vec\uff0cdeepwalk\u7b49\u7b97\u6cd5\u3002<\/p>\n<p>\u5728\u4f7f\u7528\u57fa\u7840Graph Embedding\u7b97\u6cd5\u7684\u524d\u63d0\u662f\u7528\u6237\u7684\u884c\u4e3a\u5e8f\u5217\uff0c\u4f46\u662f\u5bf9\u4e8e\u4e00\u4e9b\u65b0\u7684item\u6216\u8005\u7528\u6237\u5f88\u5c11\u6709\u884c\u4e3a\u7684item\uff0c\u5373\u51b7\u542f\u52a8\u95ee\u9898\uff0c\u57fa\u7840\u7684Graph Embedding\u7b97\u6cd5\u5f88\u96be\u5b66\u5230\u5bf9\u5e94item\u7684embedding\u8868\u793a\uff0c\u4e3a\u6b64\uff0c\u4e00\u4e9b\u9488\u5bf9item\u51b7\u542f\u52a8\u7684\u65b9\u6cd5\u88ab\u63d0\u51fa\uff0c\u5176\u4e2d\u5c31\u5305\u62ecGES\u548cEGES\u7b97\u6cd5\u3002<\/p>\n<p>GES\u548cEGES\u662f\u963f\u91cc\u57282018\u5e74\u63d0\u51fa\u7684\u4e24\u4e2a\u57fa\u4e8eGraph Embedding\u7684\u7b97\u6cd5\uff0c\u5176\u4e2dGES\u5168\u79f0\u4e3aGraph Embedding with Side Information\uff0cEGES\u5168\u79f0\u4e3aEnhanced Graph Embedding with Side Information\u3002\u4e3a\u4e86\u89e3\u51b3\u51b7\u542f\u52a8\u7684\u95ee\u9898\uff0cGES\u548cEGES\u5728\u8ba1\u7b97item embedding\u7684\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e86side information\u3002<\/p>\n<h2>2. \u7b97\u6cd5\u539f\u7406<\/h2>\n<h3>2.1. side information<\/h3>\n<p>side information\u5728\u63a8\u8350\u7cfb\u7edf\u4e2d\u6709\u7740\u91cd\u8981\u7684\u4f5c\u7528\uff0c\u4e0d\u4ec5\u4ec5\u80fd\u5e94\u7528\u5728\u53ec\u56de\u4e2d\u7528\u4e8e\u5904\u7406\u51b7\u542f\u52a8\u95ee\u9898\uff0c\u540c\u65f6\u5728\u6392\u5e8f\u9636\u6bb5\u4e2d\u4e5f\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\u3002side information\u4e3b\u8981\u6307\u7684\u662f\u4e0eitem\u76f8\u5173\u7684\u4e00\u4e9b\u5148\u9a8c\u4fe1\u606f\uff0c\u5bf9\u4e8e\u5546\u54c1\u800c\u8a00\uff0c\u5148\u9a8c\u4fe1\u606f\u5305\u62ec\uff1a\u7c7b\u522b\uff0c\u5546\u5e97\uff0c\u4ef7\u683c\u7b49\u3002<\/p>\n<h3>2.2. GES\u7b97\u6cd5<\/h3>\n<p>GES\u7b97\u6cd5\u5168\u79f0\u4e3aGraph Embedding with Side Information\uff0c\u5047\u8bbe<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> W \\mathbf{W} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68611em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793aitem\u6216\u8005side information\u7684embedding\u77e9\u9635\uff0c\u5176\u4e2d\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> W v 0 \\mathbf{W}_v^0 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.06111em; vertical-align: -0.247em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.814108em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793aitem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\u7684embedding\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> W v s \\mathbf{W}_v^s <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.93311em; vertical-align: -0.247em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.664392em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7b2c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> s s <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">s<\/span><\/span><\/span><\/span><\/span>\u4e2aside information\uff0citem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\u5171\u6709<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> n n <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">n<\/span><\/span><\/span><\/span><\/span>\u4e2aside information\uff0c\u5219\u5bf9\u4e8eitem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\u5171\u6709<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> n + 1 n+1 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.66666em; vertical-align: -0.08333em;\"><\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.64444em; vertical-align: 0em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>\u4e2a\u5411\u91cf\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> W v 0 \uff0c \u22ef \u2009 , W v n \u2208 R d \\mathbf{W}_v^0\uff0c\\cdots ,\\mathbf{W}_v^n\\in \\mathbb{R}^d <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.11111em; vertical-align: -0.247em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.864108em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord cjk_fallback\">\uff0c<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.714392em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.899108em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.899108em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> d d <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.69444em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">d<\/span><\/span><\/span><\/span><\/span>\u4e3aembedding\u7684\u7ef4\u5ea6\u3002<\/p>\n<p>\u5bf9\u4e8eitem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\uff0c\u4f7f\u7528average-pooling\u5c06\u8fd9<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> n + 1 n+1 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.66666em; vertical-align: -0.08333em;\"><\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.64444em; vertical-align: 0em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>\u4e2a\u5411\u91cf\u805a\u5408\u8d77\u6765\uff0c\u5f97\u5230item <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\u7684\u5411\u91cf\u8868\u793a\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> H v = 1 n + 1 \u2211 s = 0 n W v s \\mathbf{H}_v=\\frac{1}{n+1}\\sum_{s=0}^{n}\\mathbf{W}_v^s <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.83611em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.91851em; vertical-align: -1.26711em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.32144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.76933em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em;\"><span class=\"\" style=\"top: -1.88289em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">s<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05001em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.30001em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.26711em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.714392em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h3>2.3. EGES\u7b97\u6cd5<\/h3>\n<p>EGES\u7b97\u6cd5\u5168\u79f0\u4e3aEnhanced Graph Embedding with Side Information\uff0c\u4ece\u5176\u540d\u5b57\u6765\u770b\u4fbf\u53ef\u4ee5\u77e5\u9053\uff0cEGES\u662fGES\u7684\u589e\u5f3a\u7248\u3002\u5728GES\u4e2d\uff0c\u6bcf\u4e00\u4e2a\u5411\u91cf\uff0c\u5305\u62ec\u4e00\u4e2aitem\u7684\u5411\u91cf\u4ee5\u53ca<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> n n <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">n<\/span><\/span><\/span><\/span><\/span>\u4e2aside information\u7684\u5411\u91cf\uff0c\u8fd9\u4e9b\u5411\u91cf\u7684\u6743\u91cd\u662f\u4e00\u6837\u7684\u3002\u4ece\u5b9e\u9645\u7684\u60c5\u51b5\u6765\u770b\uff0c\u4e0d\u540c\u79cd\u7c7b\u7684side information\u5bf9\u4e8e\u6700\u7ec8\u7684embedding\u7684\u8d21\u732e\u662f\u4e0d\u4e00\u6837\u7684\u3002\u56e0\u6b64EGES\u5bf9GES\u4e2d\u7684\u5411\u91cf\u505a\u4e86\u52a0\u6743\u7684\u64cd\u4f5c\u3002<\/p>\n<p>\u5047\u8bbe\u5bf9\u4e8eitem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\uff0c\u6743\u91cd\u77e9\u9635\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> A \u2208 R \u2223 V \u2223 \u00d7 ( n + 1 ) \\mathbf{A}\\in\\mathbb{R}^{\\left | V \\right |\\times \\left ( n+1 \\right )} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.72521em; vertical-align: -0.0391em;\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">A<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.888em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.888em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">\u2223<\/span><\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.22222em;\">V<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">\u2223<\/span><\/span><\/span><span class=\"mbin mtight\">\u00d7<\/span><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">n<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u5176\u4e2d\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> A i j \\mathbf{A}_{ij} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.972218em; vertical-align: -0.286108em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">A<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.311664em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">i<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286108em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7b2c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> i i <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><\/span><\/span><\/span><\/span>\u4e2aitem\u7684\u7b2c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> j j <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.85396em; vertical-align: -0.19444em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><\/span>\u4e2aside information\u7684\u6743\u91cd\uff0c\u4e3a\u7b80\u5355\uff0c\u8bb0<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> a i j a_i^j <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.21944em; vertical-align: -0.276864em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.942572em;\"><span class=\"\" style=\"top: -2.42314em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i<\/span><\/span><\/span><span class=\"\" style=\"top: -3.18091em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.276864em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> A i j \\mathbf{A}_{ij} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.972218em; vertical-align: -0.286108em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">A<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.311664em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">i<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286108em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> A i 0 \\mathbf{A}_{i0} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.83611em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">A<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.311664em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">i<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u8868\u793a\u7684\u662fitem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> i i <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.65952em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">i<\/span><\/span><\/span><\/span><\/span>\u672c\u8eab\u5411\u91cf\u7684\u6743\u91cd\uff0c\u8bb0\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> a i 0 a_i^0 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.07277em; vertical-align: -0.258664em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.814108em;\"><span class=\"\" style=\"top: -2.44134em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.258664em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u5bf9\u4e8eitem <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><\/span>\uff0c\u52a0\u6743\u5e73\u5747\u540e\u7684\u7ed3\u679c\u4e3a\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> H v = \u2211 j = 0 n e a v j W v j \u2211 j = 0 n e a v j \\mathbf{H}_v=\\frac{\\sum _{j=0}^ne^{a_v^j}\\mathbf{W}_v^j}{\\sum _{j=0}^ne^{a_v^j}} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.83611em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.10344em; vertical-align: -1.28316em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.82028em;\"><span class=\"\" style=\"top: -2.15266em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.804292em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.435818em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.95734em;\"><span class=\"\" style=\"top: -2.989em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.954771em;\"><span class=\"\" style=\"top: -2.34126em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.98369em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.158743em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.82582em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.804292em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.435818em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.99446em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.902086em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.824664em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.28316em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d\uff0c\u4f7f\u7528<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> e a v j e^{a_v^j} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.99446em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.99446em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.902086em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u800c\u4e0d\u662f<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> a v j a_v^j <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.07166em; vertical-align: -0.247em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.824664em;\"><span class=\"\" style=\"top: -2.453em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u662f\u4e3a\u4e86\u4fdd\u8bc1\u6743\u91cd\u5927\u4e8e0\u3002<\/p>\n<h3>2.4. GES\u548cEGES\u7684\u6a21\u578b\u7ed3\u6784<\/h3>\n<p>GES\u548cEGES\u7684\u6a21\u578b\u7ed3\u6784\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/78b81e2f827b40879fbf6ef99b2c9eff.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES (https:\/\/mushiming.com\/)  \u7b2c1\u5f20\" title=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES  \u7b2c1\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> \u5176\u4e2d\uff0cDense Embeddings\u8868\u793a\u7684\u662fitem\u5411\u91cf\u4ee5\u53ca<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> n n <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">n<\/span><\/span><\/span><\/span><\/span>\u4e2aside information\u7684\u5411\u91cf\u3002Hidden Representation\u5373\u4e3a\u5982\u4e0a\u516c\u5f0f\u4e2d\u7684<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> H v \\mathbf{H}_v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.83611em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u4ece\u4e0a\u8ff0\u8fc7\u7a0b\u6765\u770b\uff0cGES\u5373\u4e3aEGES\u6a21\u578b\u7684\u7b80\u5316\u7248\u672c\uff0c\u5373\u6743\u91cd\u90fd\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> 1 n + 1 \\frac{1}{n+1} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.24844em; vertical-align: -0.403331em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.845108em;\"><span class=\"\" style=\"top: -2.655em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.394em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.403331em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h3>2.5. EGES\u4e2ditem\u5411\u91cf\u7684\u6c42\u89e3<\/h3>\n<p>EGES\u7b97\u6cd5\u7684\u6d41\u7a0b\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/39733bf8448a41eaafc9d9bd5718a212.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES (https:\/\/mushiming.com\/)  \u7b2c2\u5f20\" title=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES  \u7b2c2\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> \u4eceEGES\u7b97\u6cd5\u7684\u6d41\u7a0b\u4e2d\uff0c\u7b14\u8005\u53d1\u73b0\uff0c\u5176\u4e0eDeepWalk\u7684\u6d41\u7a0b\u57fa\u672c\u4e00\u81f4\uff0c\u4e0d\u540c\u7684\u4e3b\u8981\u662f\u4e24\u70b9\uff1a1\uff09\u5b66\u4e60\u7684\u53c2\u6570\u4e0d\u540c\uff0c\u5728DeepWalk\u4e2d\u4e3b\u8981\u662fitem\u7684\u5411\u91cf\u8868\u793a\uff0c\u5728EGES\u4e2d\u4e0d\u4ec5\u8981\u5b66\u4e60item\u7684\u5411\u91cf<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> W 0 \\mathbf{W}^0 <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.814108em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.814108em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> n n <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">n<\/span><\/span><\/span><\/span><\/span>\u4e2aside information\u7684\u5411\u91cf<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> W 1 , \u22ef W n \\mathbf{W}^1,\\cdots \\mathbf{W}^n <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.00855em; vertical-align: -0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.814108em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.664392em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u8fd8\u5305\u62ec\u6743\u91cd\u7684\u77e9\u9635<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> A \\mathbf{A} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68611em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">A<\/span><\/span><\/span><\/span><\/span><\/span>\uff1b2\uff09\u5728DeepWalk\u4e2d\u4f7f\u7528\u7684\u662fSkipGram\uff0c\u5728EGES\u4e2d\u4f7f\u7528\u7684\u662fWeightedSkipGram\u3002<\/p>\n<h3>2.6. Weighted Skip-Gram<\/h3>\n<p>Weighted Skip-Gram\u7b97\u6cd5\u7684\u6d41\u7a0b\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/5ef5ca45251c4e4995aa2f5242340001.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES (https:\/\/mushiming.com\/)  \u7b2c3\u5f20\" title=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES  \u7b2c3\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> \u4e3a\u4e86\u80fd\u591f\u66f4\u597d\u7684\u7406\u89e3\u4e0a\u8ff0\u7684\u6d41\u7a0b\uff0c\u6211\u4eec\u9700\u8981\u5148\u4e86\u89e3word2vec\u4e2dSkip-Gram\u6a21\u578b\u7684\u5177\u4f53\u6d41\u7a0b\uff0c\u5728\u8bcd\u5411\u91cf\u7684\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9664\u4e86Skip-Gram\u8fd8\u53ef\u4ee5\u662fCBOW\u6a21\u578b\uff0c\u672c\u6587\u7684\u91cd\u70b9\u662fSkip-Gram\u6a21\u578b\uff0cSkip-Gram\u6a21\u578b\u7684\u7ed3\u6784\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/img.mushiming.top\/app\/mushiming_com\/b62274ac3e1749b9979108cb9ec19d35.png\" =\"\" =\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" alt=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES (https:\/\/mushiming.com\/)  \u7b2c4\u5f20\" title=\"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES  \u7b2c4\u5f20-\u7a46\u4e16\u660e\u535a\u5ba2\" ><br \/> \u4e3a\u8ba8\u8bba\u7684\u65b9\u4fbf\uff0c\u5047\u8bbe\u5728Skip-Gram\u6a21\u578b\u4e2d\uff0c\u6bcf\u4e2a\u8bcd\u7684\u5411\u91cf\u7ef4\u5ea6\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> d d <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.69444em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">d<\/span><\/span><\/span><\/span><\/span>\uff0c\u5728\u8bcd\u5178<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> V V <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68333em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.22222em;\">V<\/span><\/span><\/span><\/span><\/span>\u4e2d\uff0c\u4e2d\u5fc3\u8bcd<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> w c w_c <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u8bcd\u5411\u91cf\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> v c \u2208 R d v_c\\in \\mathbb{R}^d <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6891em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.849108em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.849108em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u80cc\u666f\u8bcd<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> w o w_o <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u8bcd\u5411\u91cf\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> u o \u2208 R d u_o\\in \\mathbb{R}^d <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6891em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.849108em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.849108em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u7ed9\u5b9a\u4e2d\u5fc3\u8bcd\u751f\u6210\u80cc\u666f\u8bcd\u7684\u6761\u4ef6\u6982\u7387\u53ef\u4ee5\u901a\u8fc7\u5bf9\u5411\u91cf\u5185\u79ef\u505asoftmax\u8fd0\u7b97\u800c\u5f97\u5230\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> P ( w o \u2223 w c ) = e x p ( u o T v c ) \u2211 i \u2208 V e x p ( u i T v c ) P\\left ( w_o\\mid w_c \\right )=\\frac{exp\\left ( u_o^Tv_c \\right )}{\\sum _{i\\in V}exp\\left ( u_i^Tv_c \\right )} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.68002em; vertical-align: -1.09001em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.59001em;\"><span class=\"\" style=\"top: -2.26em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.178621em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">i<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.22222em;\">V<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.32708em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mord mathdefault\">p<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.823131em;\"><span class=\"\" style=\"top: -2.42314em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i<\/span><\/span><\/span><span class=\"\" style=\"top: -3.0448em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.276864em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.74001em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mord mathdefault\">p<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.841331em;\"><span class=\"\" style=\"top: -2.453em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.09001em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6b64\u65f6\uff0c\u5bf9\u4e8e\u6574\u4e2a\u6587\u672c\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u7684\u6982\u7387\u5f62\u5f0f\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u220f t = 1 T \u220f \u2212 m \u2a7d j \u2a7d m , j \u2260 0 P ( w ( t + j ) \u2223 w ( t ) ) \\prod_{t=1}^{T}\\prod _{-m\\leqslant j\\leqslant m,j\\neq 0}P\\left ( w^{\\left ( t+j \\right )}\\mid w^{\\left ( t \\right )} \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 3.26656em; vertical-align: -1.43822em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.82834em;\"><span class=\"\" style=\"top: -1.88289em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05001em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.30001em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.26711em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05001em;\"><span class=\"\" style=\"top: -1.84789em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\"><span class=\"mrel mtight\"><span class=\"mord mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.69444em;\"><span class=\"\" style=\"top: -2.7em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"rlap mtight\"><span class=\"strut\" style=\"height: 0.88888em; vertical-align: -0.19444em;\"><\/span><span class=\"inner\"><span class=\"mrel mtight\">\ue020<\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.19444em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel mtight\">=<\/span><\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.43822em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8bed\u8a00\u6a21\u578b\u4e2d\u7684\u76ee\u6807\u662f\u8981\u4f7f\u5f97\u4e0a\u8ff0\u7684\u6982\u7387\u6700\u5927\uff0c\u901a\u8fc7log\u4f3c\u7136\uff0c\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u7684\u635f\u5931\u51fd\u6570\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u2212 \u220f t = 1 T \u220f \u2212 m \u2a7d j \u2a7d m , j \u2260 0 l o g \u2005\u200a P ( w ( t + j ) \u2223 w ( t ) ) -\\prod_{t=1}^{T}\\prod _{-m\\leqslant j\\leqslant m,j\\neq 0}log\\; P\\left ( w^{\\left ( t+j \\right )}\\mid w^{\\left ( t \\right )} \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 3.26656em; vertical-align: -1.43822em;\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.82834em;\"><span class=\"\" style=\"top: -1.88289em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05001em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.30001em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.26711em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05001em;\"><span class=\"\" style=\"top: -1.84789em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\"><span class=\"mrel mtight\"><span class=\"mord mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.69444em;\"><span class=\"\" style=\"top: -2.7em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"rlap mtight\"><span class=\"strut\" style=\"height: 0.88888em; vertical-align: -0.19444em;\"><\/span><span class=\"inner\"><span class=\"mrel mtight\">\ue020<\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.19444em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel mtight\">=<\/span><\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.43822em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5bf9\u4e8e<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> l o g \u2005\u200a P ( w o \u2223 w c ) log\\; P\\left ( w_o\\mid w_c \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u6709\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> l o g \u2005\u200a P ( w o \u2223 w c ) = u o T v c \u2212 l o g ( \u2211 i \u2208 V e x p ( u i T v c ) ) log\\; P\\left ( w_o\\mid w_c \\right )=u_o^Tv_c-log\\left ( \\sum _{i\\in V}exp\\left ( u_i^Tv_c \\right ) \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.13833em; vertical-align: -0.247em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.07171em; vertical-align: -1.32171em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size4\">(<\/span><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05001em;\"><span class=\"\" style=\"top: -1.85566em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">i<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.22222em;\">V<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.32171em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\">e<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mord mathdefault\">p<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">i<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size4\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e3a\u4e86\u80fd\u591f\u5bf9\u5176\u4e2d\u7684\u53c2\u6570\u6c42\u89e3\uff0c\u53ef\u4ee5\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u6cd5\u6c42\u89e3\uff0c\u6b64\u65f6\u9700\u8981\u5bf9\u635f\u5931\u51fd\u6570\u6c42\u5bfc\uff0c\u4ee5<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> \u2202 l o g \u2005\u200a P ( w o \u2223 w c ) \u2202 v c \\frac{\\partial log\\; P\\left ( w_o\\mid w_c \\right )}{\\partial v_c} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.4551em; vertical-align: -0.4451em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.01em;\"><span class=\"\" style=\"top: -2.655em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.164543em;\"><span class=\"\" style=\"top: -2.357em; margin-left: -0.03588em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.485em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault mtight\">o<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace mtight\" style=\"margin-right: 0.325278em;\"><\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.164543em;\"><span class=\"\" style=\"top: -2.357em; margin-left: -0.02691em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel mtight\">\u2223<\/span><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.164543em;\"><span class=\"\" style=\"top: -2.357em; margin-left: -0.02691em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4451em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>\u4e3a\u4f8b\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u2202 l o g \u2005\u200a P ( w o \u2223 w c ) \u2202 v c = u o \u2212 \u2211 j \u2208 V P ( w j \u2223 w c ) \u22c5 u j \\frac{\\partial log\\; P\\left ( w_o\\mid w_c \\right )}{\\partial v_c}=u_o-\\sum _{j\\in V}P\\left ( w_j\\mid w_c \\right )\\cdot u_j <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.263em; vertical-align: -0.836em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.836em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.73333em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.48045em; vertical-align: -1.43044em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05001em;\"><span class=\"\" style=\"top: -1.85566em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.22222em;\">V<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.43044em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.311664em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286108em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.716668em; vertical-align: -0.286108em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.311664em;\"><span class=\"\" style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286108em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4ece\u4e0a\u8ff0\u7684\u516c\u5f0f\u53d1\u73b0\uff0c\u6bcf\u6b21\u7684\u6c42\u5bfc\u6570\u7684\u8fc7\u7a0b\u4e2d\uff0c\u90fd\u9700\u8981\u5bf9\u6574\u4e2a\u8bcd\u5178\u4e2d\u7684\u8bcd\u8ba1\u7b97\uff0c\u5982\u679c\u8bcd\u5178\u8f83\u5927\uff0c\u90a3\u4e48\u6bcf\u6b21\u66f4\u65b0\u65f6\u7684\u8ba1\u7b97\u6210\u672c\u5c31\u6bd4\u8f83\u5927\uff0c\u4e3a\u964d\u4f4e\u8ba1\u7b97\u6210\u672c\uff0c\u8fd1\u4f3c\u7684\u8bad\u7ec3\u65b9\u6cd5\u88ab\u63d0\u51fa\uff0c\u8d1f\u91c7\u6837\uff08Negative Sampling\uff09\u4fbf\u662f\u5176\u4e2d\u7684\u4e00\u79cd\u8fd1\u4f3c\u8ba1\u7b97\u65b9\u6cd5\u3002<\/p>\n<p>\u5bf9\u4e8e\u4e0a\u8ff0\u7ed9\u5b9a\u7684\u4e2d\u5fc3\u8bcd<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> w c w_c <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u80cc\u666f\u7a97\u53e3\uff0c\u5047\u8bbe\u80cc\u666f\u8bcd<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> w o w_o <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u51fa\u73b0\u5728<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> w c w_c <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.58056em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u7684\u80cc\u666f\u7a97\u53e3\u4e2d\u7684\u4e8b\u4ef6\u6982\u7387\u4e3a\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> P ( D = 1 \u2223 w c , w o ) = \u03c3 ( u o T v c ) P\\left ( D=1\\mid w_c,w_o \\right )=\\sigma \\left ( u_o^Tv_c \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.02778em;\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24134em; vertical-align: -0.35001em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">\u03c3<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\">u<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">o<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.03588em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">c<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u957f\u5ea6\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> T T <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68333em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><\/span>\u7684\u6587\u672c\uff0c\u5047\u8bbe\u65f6\u95f4\u6b65<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> t t <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.61508em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><\/span><\/span>\u7684\u8bcd\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> w ( t ) w^{\\left ( t \\right )} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.888em; vertical-align: 0em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.888em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u4e14\u80cc\u666f\u7a97\u53e3\u5927\u5c0f\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> m m <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><\/span><\/span><\/span><\/span>\uff0c\u6b64\u65f6\u8054\u5408\u6982\u7387\u4e3a\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u220f t = 1 T \u220f \u2212 m \u2a7d j \u2a7d m , j \u2260 0 P ( D = 1 \u2223 w ( t ) , w ( t + j ) ) \\prod_{t=1}^{T}\\prod _{-m\\leqslant j\\leqslant m,j\\neq 0}P\\left ( D=1\\mid w^{\\left ( t \\right )},w^{\\left ( t+j \\right )} \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 3.26656em; vertical-align: -1.43822em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.82834em;\"><span class=\"\" style=\"top: -1.88289em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05001em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.30001em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.26711em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05001em;\"><span class=\"\" style=\"top: -1.84789em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\"><span class=\"mrel mtight\"><span class=\"mord mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.69444em;\"><span class=\"\" style=\"top: -2.7em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"rlap mtight\"><span class=\"strut\" style=\"height: 0.88888em; vertical-align: -0.19444em;\"><\/span><span class=\"inner\"><span class=\"mrel mtight\">\ue020<\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.19444em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel mtight\">=<\/span><\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.43822em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.02778em;\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6b64\u65f6\u6a21\u578b\u4e2d\u4ec5\u8003\u8651\u4e86\u6b63\u6837\u672c\uff0c\u901a\u8fc7\u91c7\u6837<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> K K <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68333em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">K<\/span><\/span><\/span><\/span><\/span>\u4e2a\u672a\u51fa\u73b0\u5728\u8be5\u80cc\u666f\u7a97\u53e3\u4e2d\u7684\u8bcd\uff0c\u6b64\u65f6\u7684\u8054\u5408\u6982\u7387\u4e3a\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u220f t = 1 T \u220f \u2212 m \u2a7d j \u2a7d m , j \u2260 0 P ( w ( t + j ) \u2223 w ( t ) ) \\prod_{t=1}^{T}\\prod _{-m\\leqslant j\\leqslant m,j\\neq 0}P\\left ( w^{\\left ( t+j \\right )}\\mid w^{\\left ( t \\right )} \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 3.26656em; vertical-align: -1.43822em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.82834em;\"><span class=\"\" style=\"top: -1.88289em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05001em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.30001em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.26711em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05001em;\"><span class=\"\" style=\"top: -1.84789em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel amsrm mtight\">\u2a7d<\/span><span class=\"mord mathdefault mtight\">m<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\"><span class=\"mrel mtight\"><span class=\"mord mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.69444em;\"><span class=\"\" style=\"top: -2.7em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"rlap mtight\"><span class=\"strut\" style=\"height: 0.88888em; vertical-align: -0.19444em;\"><\/span><span class=\"inner\"><span class=\"mrel mtight\">\ue020<\/span><\/span><span class=\"fix\"><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.19444em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mrel mtight\">=<\/span><\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.43822em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d\uff0c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> P ( w ( t + j ) \u2223 w ( t ) ) P\\left ( w^{\\left ( t+j \\right )}\\mid w^{\\left ( t \\right )} \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.23801em; vertical-align: -0.35001em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.888em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.888em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> P ( w ( t + j ) \u2223 w ( t ) ) = P ( D = 1 \u2223 w ( t ) , w ( t + j ) ) \u22c5 \u220f k = 1 K P ( D = 0 \u2223 w ( t ) , w k ) P\\left ( w^{\\left ( t+j \\right )}\\mid w^{\\left ( t \\right )} \\right )=P\\left ( D=1\\mid w^{\\left ( t \\right )},w^{\\left ( t+j \\right )} \\right )\\cdot \\prod_{k=1}^{K}P\\left ( D=0\\mid w^{\\left ( t \\right )},w_k \\right ) <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.80002em; vertical-align: -0.65002em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.80002em; vertical-align: -0.65002em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.02778em;\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.13045em; vertical-align: -1.30211em;\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.82834em;\"><span class=\"\" style=\"top: -1.84789em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03148em;\">k<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u220f<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em; margin-left: 0em;\"><span class=\"pstrut\" style=\"height: 3.05em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.07153em;\">K<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.30211em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.13889em;\">P<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.02778em;\">D<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.938em;\"><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"minner mtight\"><span class=\"mopen mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">(<\/span><\/span><span class=\"mord mathdefault mtight\">t<\/span><span class=\"mclose mtight delimcenter\" style=\"top: 0em;\"><span class=\"mtight\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right: 0.02691em;\">w<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.336108em;\"><span class=\"\" style=\"top: -2.55em; margin-left: -0.02691em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03148em;\">k<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u53ef\u4ee5\u9a8c\u8bc1\uff0c\u6b64\u65f6\u8ba1\u7b97\u4e0d\u518d\u4e0e\u8bcd\u5178\u5927\u5c0f\u76f8\u5173\uff0c\u800c\u662f\u4e0e\u8d1f\u91c7\u6837\u7684\u53c2\u6570<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> K K <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.68333em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.07153em;\">K<\/span><\/span><\/span><\/span><\/span>\u76f8\u5173\uff0c\u4ee5\u4e0a\u4fbf\u662fSkip-Gram\u6a21\u578b\u4ee5\u53ca\u8d1f\u91c7\u6837\u7684\u76f8\u5173\u5185\u5bb9\u3002<\/p>\n<p>\u5bf9\u4e8e\u91c7\u6837\u5230\u7684\u6837\u672c<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> u u <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.43056em; vertical-align: 0em;\"><\/span><span class=\"mord mathdefault\">u<\/span><\/span><\/span><\/span><\/span>\uff0c\u5176\u5bf9\u5e94\u7684\u5411\u91cf\u4e3a<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"> Z u \\mathbf{Z}_u <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.83611em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\uff0c\u7531\u4e0a\u8ff0\u7684\u7406\u8bba\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> L ( v , u , y ) = \u2212 [ y l o g ( \u03c3 ( H v T Z u ) ) + ( 1 \u2212 y ) l o g ( 1 \u2212 \u03c3 ( H v T Z u ) ) ] \\mathfrak{L}\\left ( v,u,y \\right )=-\\left [ ylog\\left ( \\sigma \\left ( \\mathbf{H}_v^T\\mathbf{Z}_u \\right ) \\right )+\\left ( 1-y \\right )log\\left ( 1-\\sigma \\left ( \\mathbf{H}_v^T\\mathbf{Z}_u \\right ) \\right ) \\right ] <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em; vertical-align: -0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathfrak\">L<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">v<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\">u<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">y<\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24134em; vertical-align: -0.35001em;\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">[<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">y<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">\u03c3<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">y<\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\">)<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.01968em;\">l<\/span><span class=\"mord mathdefault\">o<\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">g<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">\u03c3<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u7684\u5bfc\u6570\uff1a<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u2202 L \u2202 Z u = ( \u03c3 ( H v T Z u ) \u2212 y ) H v \\frac{\\partial \\mathfrak{L}}{\\partial \\mathbf{Z}_u}=\\left ( \\sigma \\left ( \\mathbf{H}_v^T\\mathbf{Z}_u \\right )-y \\right )\\mathbf{H}_v <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.20744em; vertical-align: -0.836em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathfrak\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.836em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24134em; vertical-align: -0.35001em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">\u03c3<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">y<\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u2202 L \u2202 a v s = ( \u03c3 ( H v T Z u ) \u2212 y ) Z u ( \u2211 j = 0 n e a v j ) e a v s W v s \u2212 e a v s \u2211 j = 0 n e a v j W v j ( \u2211 j = 0 n e a v j ) 2 \\frac{\\partial \\mathfrak{L}}{\\partial a_v^s}=\\left ( \\sigma \\left ( \\mathbf{H}_v^T\\mathbf{Z}_u \\right )-y \\right )\\mathbf{Z}_u\\frac{\\left ( \\sum_{j=0}^{n}e^{a_v^j} \\right )e^{a_v^s}\\mathbf{W}_v^s-e^{a_v^s}\\sum_{j=0}^{n}e^{a_v^j}\\mathbf{W}_v^j}{\\left ( \\sum_{j=0}^{n}e^{a_v^j} \\right )^2} <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.30444em; vertical-align: -0.933em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.590392em;\"><span class=\"\" style=\"top: -2.453em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.989em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathfrak\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.933em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 4.08405em; vertical-align: -1.89403em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">\u03c3<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">y<\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 2.19002em;\"><span class=\"\" style=\"top: -2.11em;\"><span class=\"pstrut\" style=\"height: 3.35401em;\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.804292em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.435818em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.95734em;\"><span class=\"\" style=\"top: -2.989em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.954771em;\"><span class=\"\" style=\"top: -2.34126em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.98369em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.158743em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.35401em;\"><span class=\"\" style=\"top: -3.6029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.58401em;\"><span class=\"pstrut\" style=\"height: 3.35401em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -4.39403em;\"><span class=\"pstrut\" style=\"height: 3.35401em;\"><\/span><span class=\"mord\"><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">(<\/span><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.804292em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.435818em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.99446em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.902086em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size2\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.87998em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.738543em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.664392em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.87998em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.738543em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.804292em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.435818em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.99446em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.902086em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.824664em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.89403em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"> \u2202 L \u2202 W v s = e a v s \u2211 j = 0 n e a v j ( \u03c3 ( H v T Z u ) \u2212 y ) Z u \\frac{\\partial \\mathfrak{L}}{\\partial \\mathbf{W}_v^s}=\\frac{e^{a_v^s}}{\\sum_{j=0}^{n}e^{a_v^j}}\\left ( \\sigma \\left ( \\mathbf{H}_v^T\\mathbf{Z}_u \\right )-y \\right )\\mathbf{Z}_u <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.30444em; vertical-align: -0.933em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.37144em;\"><span class=\"\" style=\"top: -2.314em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\" style=\"margin-right: 0.01597em;\">W<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.590392em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.989em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\" style=\"margin-right: 0.05556em;\">\u2202<\/span><span class=\"mord\"><span class=\"mord mathfrak\">L<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.933em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.277778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.84014em; vertical-align: -1.28316em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.55698em;\"><span class=\"\" style=\"top: -2.15266em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative; top: -5e-06em;\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.804292em;\"><span class=\"\" style=\"top: -2.40029em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><span class=\"mrel mtight\">=<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.435818em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.95734em;\"><span class=\"\" style=\"top: -2.989em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.954771em;\"><span class=\"\" style=\"top: -2.34126em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.98369em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.05724em;\">j<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.158743em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em;\"><\/span><\/span><span class=\"\" style=\"top: -3.677em;\"><span class=\"pstrut\" style=\"height: 3em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathdefault\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.87998em;\"><span class=\"\" style=\"top: -3.063em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.738543em;\"><span class=\"\" style=\"top: -2.214em; margin-left: 0em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -2.931em; margin-right: 0.0714286em;\"><span class=\"pstrut\" style=\"height: 2.5em;\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathdefault mtight\">s<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.286em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.28316em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">\u03c3<\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">(<\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">H<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.891331em;\"><span class=\"\" style=\"top: -2.453em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.03588em;\">v<\/span><\/span><\/span><span class=\"\" style=\"top: -3.113em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\" style=\"margin-right: 0.13889em;\">T<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.247em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.222222em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right: 0.03588em;\">y<\/span><span class=\"mclose delimcenter\" style=\"top: 0em;\"><span class=\"delimsizing size1\">)<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.166667em;\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">Z<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.151392em;\"><span class=\"\" style=\"top: -2.55em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">u<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h2>\u53c2\u8003\u6587\u732e<\/h2>\n<ul>\n<li>Wang J, Huang P, Zhao H, et al. Billion-scale commodity embedding for e-commerce recommendation in alibaba[C]\/\/Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery &amp; Data Mining. 2018: 839-848.<\/li>\n<li>[Graph Embedding]\u963f\u91cc\u8d85\u5927\u89c4\u6a21\u5546\u54c1Embedding\u7b56\u7565EGES<\/li>\n<li>Graph Embedding\u5728\u6dd8\u5b9d\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528<\/li>\n<li>NLP\u4e4b\u2014word2vec\u7b97\u6cd5skip-gram\u539f\u7406\u8be6\u89e3<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5e38\u7528\u7b97\u6cd5\u2014\u2014\u57fa\u4e8eGraph Embedding\u7684GES\u548cEGES1.\u6982\u8ff0GES\u548cEGES\u662f\u963f\u91cc\u57282018\u5e74\u63d0\u51fa\u7684\u4e24\u4e2aGraphEmbedding\u7b97\u6cd5\uff0c\u5176\u4e2dGE...","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/7975"}],"collection":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/comments?post=7975"}],"version-history":[{"count":0,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/7975\/revisions"}],"wp:attachment":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/media?parent=7975"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/categories?post=7975"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/tags?post=7975"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}