{"id":88328,"date":"2026-07-31T22:32:33","date_gmt":"2026-07-31T14:32:33","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/88328.html"},"modified":"2026-07-31T22:32:33","modified_gmt":"2026-07-31T14:32:33","slug":"%e7%8b%ac%e7%83%ad%e7%bc%96%e7%a0%81%e4%b8%8e%e4%bd%99%e5%bc%a6%e7%9b%b8%e4%bc%bc%e5%ba%a6","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/88328.html","title":{"rendered":"\u72ec\u70ed\u7f16\u7801\u4e0e\u4f59\u5f26\u76f8\u4f3c\u5ea6"},"content":{"rendered":"<h3>\u4e00\u3001\u72ec\u70ed\u7f16\u7801&#xff08;One-Hot Encoding&#xff09;<\/h3>\n<h4>1.1 \u4e3a\u4ec0\u4e48\u9700\u8981\u72ec\u70ed\u7f16\u7801&#xff1f;<\/h4>\n<p>\u673a\u5668\u5b66\u4e60\u6a21\u578b\u53ea\u80fd\u5904\u7406\u6570\u503c\u578b\u6570\u636e&#xff0c;\u4f46\u73b0\u5b9e\u4e16\u754c\u5145\u6ee1\u7c7b\u522b\u578b\u6570\u636e&#xff1a;<\/p>\n<ul>\n<li>\u989c\u8272&#xff1a;{\u7ea2, \u7eff, \u84dd}<\/li>\n<li>\u57ce\u5e02&#xff1a;{\u5317\u4eac, \u4e0a\u6d77, \u5e7f\u5dde, \u6df1\u5733}<\/li>\n<li>\u5546\u54c1\u7c7b\u522b&#xff1a;{\u7535\u5b50\u4ea7\u54c1, \u670d\u88c5, \u98df\u54c1, \u5bb6\u5c45}<\/li>\n<\/ul>\n<p>\u76f4\u63a5\u8d4b\u6574\u6570\u503c\u7684\u95ee\u9898&#xff1a;<\/p>\n<table>\n<tr>\u989c\u8272\u6574\u6570\u7f16\u7801\u95ee\u9898<\/tr>\n<tbody>\n<tr>\n<td>\u7ea2<\/td>\n<td>0<\/td>\n<td>\u6a21\u578b\u4f1a\u8ba4\u4e3a \u7eff(1) &gt; \u7ea2(0)&#xff0c;\u4e14 \u84dd(2) &#8211; \u7eff(1) &#061; \u7eff(1) &#8211; \u7ea2(0)<\/td>\n<\/tr>\n<tr>\n<td>\u7eff<\/td>\n<td>1<\/td>\n<td>\u8fd9\u4e9b&#034;\u6570\u503c\u5173\u7cfb&#034;\u5728\u7c7b\u522b\u6570\u636e\u4e2d\u6beb\u65e0\u610f\u4e49&#xff01;<\/td>\n<\/tr>\n<tr>\n<td>\u84dd<\/td>\n<td>2<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u72ec\u70ed\u7f16\u7801\u89e3\u51b3&#xff1a;\u5c06\u7c7b\u522b\u8f6c\u6362\u4e3a\u4e8c\u8fdb\u5236\u5411\u91cf&#xff0c;\u6d88\u9664\u865a\u5047\u6570\u503c\u5173\u7cfb\u3002<\/p>\n<h4>1.2 \u6570\u5b66\u5b9a\u4e49<\/h4>\n<p>\u5bf9\u4e8e\u5177\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">KK<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span> \u4e2a\u7c7b\u522b\u7684\u53d8\u91cf&#xff0c;\u7b2c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">ii<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6595em\"><\/span><span class=\"mord mathnormal\">i<\/span><\/span><\/span><\/span><\/span> \u4e2a\u7c7b\u522b\u7684\u72ec\u70ed\u7f16\u7801\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">OneHot(i)&#061;[0,\u22ef\u2009,0,1\u23df\u7b2c\u00a0i\u00a0\u4f4d,0,\u22ef\u2009,0]\u2208RK<br \/>\n\\\\text{OneHot}(i) &#061; [0, \\\\cdots, 0, \\\\underbrace{1}_{\\\\text{\u7b2c } i \\\\text{ \u4f4d}}, 0, \\\\cdots, 0] \\\\in \\\\mathbb{R}^K<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord text\"><span class=\"mord\">OneHot<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0763em;vertical-align: -1.3263em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord munder\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6444em\"><span class=\"\" style=\"top: -1.6737em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord cjk_fallback mtight\">\u7b2c<\/span><span class=\"mord mtight\">\u00a0<\/span><\/span><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mord text mtight\"><span class=\"mord mtight\">\u00a0<\/span><span class=\"mord cjk_fallback mtight\">\u4f4d<\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord munder\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6444em\"><span class=\"svg-align\" style=\"top: -2.352em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"stretchy\" style=\"height: 0.548em;min-width: 1.6em\"><span class=\"brace-left\" style=\"height: 0.548em\"><\/span><span class=\"brace-center\" style=\"height: 0.548em\"><\/span><span class=\"brace-right\" style=\"height: 0.548em\"><\/span><\/span><\/span><span class=\"\" style=\"top: -3em\"><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: 0.648em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3263em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mclose\">]<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8913em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8913em\"><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 mathnormal mtight\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u793a\u4f8b&#xff1a;\u989c\u8272 {\u7ea2, \u7eff, \u84dd}&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">K&#061;3K&#061;3<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/span><\/p>\n<table>\n<tr>\u989c\u8272\u72ec\u70ed\u7f16\u7801<\/tr>\n<tbody>\n<tr>\n<td>\u7ea2<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">[1,0,0][1, 0, 0]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u7eff<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">[0,1,0][0, 1, 0]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u84dd<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">[0,0,1][0, 0, 1]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>1.3 \u4ee3\u7801\u5b9e\u73b0<\/h4>\n<p><span class=\"token keyword\">import<\/span> numpy <span class=\"token keyword\">as<\/span> np<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>preprocessing <span class=\"token keyword\">import<\/span> OneHotEncoder<\/p>\n<p><span class=\"token comment\"># \u65b9\u6cd51&#xff1a;\u624b\u52a8\u5b9e\u73b0<\/span><br \/>\n<span class=\"token keyword\">def<\/span> <span class=\"token function\">one_hot_manual<\/span><span class=\"token punctuation\">(<\/span>labels<span class=\"token punctuation\">,<\/span> num_classes<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    <span class=\"token triple-quoted-string string\">&#034;&#034;&#034;<br \/>\n    \u624b\u52a8\u5b9e\u73b0\u72ec\u70ed\u7f16\u7801<\/p>\n<p>    :param labels: \u7c7b\u522b\u6807\u7b7e\u5217\u8868&#xff0c;\u5982 [0, 2, 1, 0]<br \/>\n    :param num_classes: \u7c7b\u522b\u603b\u6570 K<br \/>\n    :return: \u72ec\u70ed\u7f16\u7801\u77e9\u9635<br \/>\n    &#034;&#034;&#034;<\/span><br \/>\n    n <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>labels<span class=\"token punctuation\">)<\/span><br \/>\n    one_hot <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>zeros<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">,<\/span> num_classes<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    one_hot<span class=\"token punctuation\">[<\/span>np<span class=\"token punctuation\">.<\/span>arange<span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> labels<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> one_hot<\/p>\n<p><span class=\"token comment\"># \u793a\u4f8b<\/span><br \/>\nlabels <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span>  <span class=\"token comment\"># \u7ea2, \u84dd, \u7eff, \u7ea2<\/span><br \/>\nresult <span class=\"token operator\">&#061;<\/span> one_hot_manual<span class=\"token punctuation\">(<\/span>labels<span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>result<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># [[1. 0. 0.]<\/span><br \/>\n<span class=\"token comment\">#  [0. 0. 1.]<\/span><br \/>\n<span class=\"token comment\">#  [0. 1. 0.]<\/span><br \/>\n<span class=\"token comment\">#  [1. 0. 0.]]<\/span><\/p>\n<p><span class=\"token comment\"># \u65b9\u6cd52&#xff1a;sklearn<\/span><br \/>\nencoder <span class=\"token operator\">&#061;<\/span> OneHotEncoder<span class=\"token punctuation\">(<\/span>sparse_output<span class=\"token operator\">&#061;<\/span><span class=\"token boolean\">False<\/span><span class=\"token punctuation\">)<\/span><br \/>\ncolors <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span><span class=\"token string\">&#039;\u7ea2&#039;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token string\">&#039;\u7eff&#039;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token string\">&#039;\u84dd&#039;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token string\">&#039;\u7ea2&#039;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">]<\/span><br \/>\nencoded <span class=\"token operator\">&#061;<\/span> encoder<span class=\"token punctuation\">.<\/span>fit_transform<span class=\"token punctuation\">(<\/span>colors<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>encoded<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># [[1. 0. 0.]<\/span><br \/>\n<span class=\"token comment\">#  [0. 1. 0.]<\/span><br \/>\n<span class=\"token comment\">#  [0. 0. 1.]<\/span><br \/>\n<span class=\"token comment\">#  [1. 0. 0.]]<\/span><\/p>\n<h4>1.4 \u72ec\u70ed\u7f16\u7801\u7684\u7279\u6027<\/h4>\n<table>\n<tr>\u7279\u6027\u8bf4\u660e\u5f71\u54cd<\/tr>\n<tbody>\n<tr>\n<td>\u6b63\u4ea4\u6027<\/td>\n<td>\u4efb\u610f\u4e24\u4e2a\u4e0d\u540c\u7c7b\u522b\u7684\u7f16\u7801\u5185\u79ef\u4e3a 0<\/td>\n<td>\u7c7b\u522b\u95f4\u65e0\u504f\u5e8f\u5173\u7cfb&#xff0c;\u8ddd\u79bb\u76f8\u7b49<\/td>\n<\/tr>\n<tr>\n<td>\u7b49\u8ddd\u6027<\/td>\n<td>\u4efb\u610f\u4e24\u4e2a\u4e0d\u540c\u7c7b\u522b\u7684\u6b27\u6c0f\u8ddd\u79bb\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">2\\\\sqrt{2}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.04em;vertical-align: -0.1328em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9072em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -2.8672em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1328em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u4e0d\u5f15\u5165\u865a\u5047\u7684\u76f8\u4f3c\u5ea6<\/td>\n<\/tr>\n<tr>\n<td>\u7a00\u758f\u6027<\/td>\n<td>\u5411\u91cf\u4e2d\u53ea\u6709\u4e00\u4e2a 1&#xff0c;\u5176\u4f59\u4e3a 0<\/td>\n<td>\u5b58\u50a8\u6548\u7387\u9ad8&#xff0c;\u4f46\u7ef4\u5ea6\u81a8\u80c0<\/td>\n<\/tr>\n<tr>\n<td>\u7ef4\u5ea6\u707e\u96be<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">KK<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span> \u4e2a\u7c7b\u522b\u9700\u8981 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">KK<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span> \u7ef4\u5411\u91cf<\/td>\n<td>\u9ad8\u57fa\u6570\u7c7b\u522b&#xff08;\u5982\u7528\u6237ID\u3001\u5546\u54c1ID&#xff09;\u4e0d\u9002\u7528<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>1.5 \u4f18\u7f3a\u70b9\u4e0e\u66ff\u4ee3\u65b9\u6848<\/h4>\n<p>\u4f18\u70b9&#xff1a;<\/p>\n<ul>\n<li>\u7b80\u5355\u76f4\u89c2&#xff0c;\u5b9e\u73b0\u5bb9\u6613<\/li>\n<li>\u4e0d\u5f15\u5165\u7c7b\u522b\u95f4\u7684\u865a\u5047\u5173\u7cfb<\/li>\n<li>\u4e0e\u795e\u7ecf\u7f51\u7edc\u517c\u5bb9\u6027\u597d<\/li>\n<\/ul>\n<p>\u7f3a\u70b9&#xff1a;<\/p>\n<ul>\n<li>\u7ef4\u5ea6\u7206\u70b8&#xff1a;10000 \u4e2a\u7c7b\u522b \u2192 10000 \u7ef4\u5411\u91cf<\/li>\n<li>\u8bed\u4e49\u7f3a\u5931&#xff1a;\u65e0\u6cd5\u8868\u8fbe\u7c7b\u522b\u95f4\u7684\u5185\u5728\u8054\u7cfb&#xff08;\u5982&#034;\u732b&#034;\u548c&#034;\u864e&#034;\u6bd4&#034;\u732b&#034;\u548c&#034;\u6c7d\u8f66&#034;\u66f4\u76f8\u4f3c&#xff09;<\/li>\n<\/ul>\n<p>\u66ff\u4ee3\u65b9\u6848&#xff1a;<\/p>\n<table>\n<tr>\u65b9\u6cd5\u9002\u7528\u573a\u666f\u6838\u5fc3\u601d\u60f3<\/tr>\n<tbody>\n<tr>\n<td>\u5d4c\u5165&#xff08;Embedding&#xff09;<\/td>\n<td>\u9ad8\u57fa\u6570\u7c7b\u522b\u3001\u8bed\u4e49\u76f8\u4f3c\u5ea6<\/td>\n<td>\u5b66\u4e60\u4f4e\u7ef4\u7a20\u5bc6\u5411\u91cf&#xff0c;\u5982 Word2Vec\u3001DeepWalk<\/td>\n<\/tr>\n<tr>\n<td>\u6807\u7b7e\u7f16\u7801&#xff08;Label Encoding&#xff09;<\/td>\n<td>\u6709\u5e8f\u7c7b\u522b&#xff08;\u5982\u8bc4\u5206 1-5 \u661f&#xff09;<\/td>\n<td>\u4fdd\u7559\u987a\u5e8f\u5173\u7cfb<\/td>\n<\/tr>\n<tr>\n<td>\u76ee\u6807\u7f16\u7801&#xff08;Target Encoding&#xff09;<\/td>\n<td>\u9ad8\u57fa\u6570 &#043; \u6709\u76d1\u7763\u4efb\u52a1<\/td>\n<td>\u7528\u76ee\u6807\u53d8\u91cf\u7edf\u8ba1\u503c\u66ff\u4ee3\u7c7b\u522b<\/td>\n<\/tr>\n<tr>\n<td>\u4e8c\u8fdb\u5236\u7f16\u7801<\/td>\n<td>\u6781\u7aef\u9ad8\u57fa\u6570<\/td>\n<td>\u7528 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">log\u20612K\\\\log_2 K<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;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 class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span> \u4f4d\u4e8c\u8fdb\u5236\u8868\u793a<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u4e8c\u3001\u4f59\u5f26\u76f8\u4f3c\u5ea6&#xff08;Cosine Similarity&#xff09;<\/h3>\n<h4>2.1 \u4e3a\u4ec0\u4e48\u9700\u8981\u4f59\u5f26\u76f8\u4f3c\u5ea6&#xff1f;<\/h4>\n<p>\u5728\u4fe1\u606f\u68c0\u7d22\u3001\u63a8\u8350\u7cfb\u7edf\u3001NLP \u4e2d&#xff0c;\u6838\u5fc3\u95ee\u9898\u662f&#xff1a;\u5982\u4f55\u5ea6\u91cf\u4e24\u4e2a\u5411\u91cf\u7684\u76f8\u4f3c\u7a0b\u5ea6&#xff1f;<\/p>\n<p>\u6b27\u6c0f\u8ddd\u79bb\u7684\u5c40\u9650&#xff1a;<\/p>\n<ul>\n<li>\u53d7\u5411\u91cf\u957f\u5ea6&#xff08;\u6a21&#xff09;\u5f71\u54cd\u5927<\/li>\n<li>\u957f\u6587\u6863\u4e0e\u77ed\u6587\u6863\u5373\u4f7f\u5185\u5bb9\u76f8\u4f3c&#xff0c;\u6b27\u6c0f\u8ddd\u79bb\u4e5f\u53ef\u80fd\u5f88\u5927<\/li>\n<\/ul>\n<p>\u4f59\u5f26\u76f8\u4f3c\u5ea6\u7684\u4f18\u52bf&#xff1a;\u53ea\u5173\u5fc3\u65b9\u5411&#xff0c;\u4e0d\u5173\u5fc3\u957f\u5ea6<\/p>\n<h4>2.2 \u6570\u5b66\u5b9a\u4e49<\/h4>\n<p>\u5bf9\u4e8e\u4e24\u4e2a\u975e\u96f6\u5411\u91cf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">a,b\u2208Rn\\\\mathbf{a}, \\\\mathbf{b} \\\\in \\\\mathbb{R}^n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathbf\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathbf\">b<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6889em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><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 mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">CosineSimilarity(a,b)&#061;a\u22c5b\u2225a\u2225\u2225b\u2225&#061;\u2211i&#061;1naibi\u2211i&#061;1nai2\u22c5\u2211i&#061;1nbi2<br \/>\n\\\\text{CosineSimilarity}(\\\\mathbf{a}, \\\\mathbf{b}) &#061; \\\\frac{\\\\mathbf{a} \\\\cdot \\\\mathbf{b}}{\\\\|\\\\mathbf{a}\\\\| \\\\|\\\\mathbf{b}\\\\|} &#061; \\\\frac{\\\\sum_{i&#061;1}^{n} a_i b_i}{\\\\sqrt{\\\\sum_{i&#061;1}^{n} a_i^2} \\\\cdot \\\\sqrt{\\\\sum_{i&#061;1}^{n} b_i^2}}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord text\"><span class=\"mord\">CosineSimilarity<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathbf\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathbf\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.3074em;vertical-align: -0.936em\"><\/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: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"mord mathbf\">a<\/span><span class=\"mord\">\u2225\u2225<\/span><span class=\"mord mathbf\">b<\/span><span class=\"mord\">\u2225<\/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 mathbf\">a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.624em;vertical-align: -1.13em\"><\/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: 1.494em\"><span class=\"\" style=\"top: -2.1727em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9373em\"><span class=\"svg-align\" style=\"top: -3.2em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;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 mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/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 mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7959em\"><span class=\"\" style=\"top: -2.4231em;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 mathnormal 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 mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2769em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -2.8973em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.28em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3027em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9373em\"><span class=\"svg-align\" style=\"top: -3.2em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;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 mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/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 mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7959em\"><span class=\"\" style=\"top: -2.4231em;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 mathnormal 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 mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2769em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -2.8973em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.28em\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3027em\"><span class=\"\"><\/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: -3.6897em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;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 mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/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 mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><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 mathnormal mtight\">i<\/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=\"mord mathnormal\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3117em\"><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 mathnormal mtight\">i<\/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 class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.13em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u51e0\u4f55\u89e3\u91ca&#xff1a;\u4e24\u4e2a\u5411\u91cf\u5939\u89d2\u7684\u4f59\u5f26\u503c<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">CosineSimilarity&#061;cos\u2061(\u03b8)&#061;a\u22c5b\u2225a\u2225\u2225b\u2225<br \/>\n\\\\text{CosineSimilarity} &#061; \\\\cos(\\\\theta) &#061; \\\\frac{\\\\mathbf{a} \\\\cdot \\\\mathbf{b}}{\\\\|\\\\mathbf{a}\\\\| \\\\|\\\\mathbf{b}\\\\|}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord text\"><span class=\"mord\">CosineSimilarity<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">cos<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.3074em;vertical-align: -0.936em\"><\/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: 1.3714em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"mord mathbf\">a<\/span><span class=\"mord\">\u2225\u2225<\/span><span class=\"mord mathbf\">b<\/span><span class=\"mord\">\u2225<\/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 mathbf\">a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<table>\n<tr>\u5939\u89d2 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u03b8\\\\theta<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><\/span><\/span><\/span>\u4f59\u5f26\u503c\u76f8\u4f3c\u5ea6<\/tr>\n<tbody>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">0\u22180^\\\\circ<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6741em\"><\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6741em\"><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=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;\u540c\u5411&#xff09;<\/td>\n<td>1<\/td>\n<td>\u5b8c\u5168\u76f8\u540c<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">90\u221890^\\\\circ<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6741em\"><\/span><span class=\"mord\">9<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6741em\"><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=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;\u6b63\u4ea4&#xff09;<\/td>\n<td>0<\/td>\n<td>\u65e0\u76f8\u5173\u6027<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">180\u2218180^\\\\circ<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6741em\"><\/span><span class=\"mord\">18<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6741em\"><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=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;\u53cd\u5411&#xff09;<\/td>\n<td>-1<\/td>\n<td>\u5b8c\u5168\u76f8\u53cd<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>2.3 \u4ee3\u7801\u5b9e\u73b0<\/h4>\n<p><span class=\"token keyword\">import<\/span> numpy <span class=\"token keyword\">as<\/span> np<br \/>\n<span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>metrics<span class=\"token punctuation\">.<\/span>pairwise <span class=\"token keyword\">import<\/span> cosine_similarity<\/p>\n<p><span class=\"token comment\"># \u65b9\u6cd51&#xff1a;\u624b\u52a8\u5b9e\u73b0<\/span><br \/>\n<span class=\"token keyword\">def<\/span> <span class=\"token function\">cosine_similarity_manual<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    <span class=\"token triple-quoted-string string\">&#034;&#034;&#034;<br \/>\n    \u624b\u52a8\u5b9e\u73b0\u4f59\u5f26\u76f8\u4f3c\u5ea6<\/p>\n<p>    :param a: \u5411\u91cfa<br \/>\n    :param b: \u5411\u91cfb<br \/>\n    :return: \u4f59\u5f26\u76f8\u4f3c\u5ea6&#xff0c;\u8303\u56f4[-1, 1]<br \/>\n    &#034;&#034;&#034;<\/span><br \/>\n    dot_product <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>dot<span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">)<\/span><br \/>\n    norm_a <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>linalg<span class=\"token punctuation\">.<\/span>norm<span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">)<\/span><br \/>\n    norm_b <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>linalg<span class=\"token punctuation\">.<\/span>norm<span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">)<\/span><\/p>\n<p>    <span class=\"token keyword\">if<\/span> norm_a <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">0<\/span> <span class=\"token keyword\">or<\/span> norm_b <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">:<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> <span class=\"token number\">0.0<\/span>  <span class=\"token comment\"># \u907f\u514d\u9664\u96f6<\/span><\/p>\n<p>    <span class=\"token keyword\">return<\/span> dot_product <span class=\"token operator\">\/<\/span> <span class=\"token punctuation\">(<\/span>norm_a <span class=\"token operator\">*<\/span> norm_b<span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token comment\"># \u793a\u4f8b&#xff1a;\u6587\u6863\u76f8\u4f3c\u5ea6<\/span><br \/>\ndoc1 <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u8bcd\u9891\u5411\u91cf<\/span><br \/>\ndoc2 <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\ndoc3 <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u5b8c\u5168\u4e0d\u540c\u7684\u4e3b\u9898<\/span><\/p>\n<p>sim_12 <span class=\"token operator\">&#061;<\/span> cosine_similarity_manual<span class=\"token punctuation\">(<\/span>doc1<span class=\"token punctuation\">,<\/span> doc2<span class=\"token punctuation\">)<\/span><br \/>\nsim_13 <span class=\"token operator\">&#061;<\/span> cosine_similarity_manual<span class=\"token punctuation\">(<\/span>doc1<span class=\"token punctuation\">,<\/span> doc3<span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string-interpolation\"><span class=\"token string\">f&#034;doc1 vs doc2: <\/span><span class=\"token interpolation\"><span class=\"token punctuation\">{<\/span>sim_12<span class=\"token punctuation\">:<\/span><span class=\"token format-spec\">.4f<\/span><span class=\"token punctuation\">}<\/span><\/span><span class=\"token string\">&#034;<\/span><\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 0.9759&#xff0c;\u9ad8\u5ea6\u76f8\u4f3c<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string-interpolation\"><span class=\"token string\">f&#034;doc1 vs doc3: <\/span><span class=\"token interpolation\"><span class=\"token punctuation\">{<\/span>sim_13<span class=\"token punctuation\">:<\/span><span class=\"token format-spec\">.4f<\/span><span class=\"token punctuation\">}<\/span><\/span><span class=\"token string\">&#034;<\/span><\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 0.0000&#xff0c;\u5b8c\u5168\u4e0d\u76f8\u5173<\/span><\/p>\n<p><span class=\"token comment\"># \u65b9\u6cd52&#xff1a;sklearn<\/span><br \/>\nsimilarity_matrix <span class=\"token operator\">&#061;<\/span> cosine_similarity<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span>doc1<span class=\"token punctuation\">,<\/span> doc2<span class=\"token punctuation\">,<\/span> doc3<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>similarity_matrix<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># [[1.         0.97590007 0.        ]<\/span><br \/>\n<span class=\"token comment\">#  [0.97590007 1.         0.        ]<\/span><br \/>\n<span class=\"token comment\">#  [0.         0.         1.        ]]<\/span><\/p>\n<h4>2.4 \u4f59\u5f26\u76f8\u4f3c\u5ea6 vs \u5176\u4ed6\u5ea6\u91cf<\/h4>\n<table>\n<tr>\u5ea6\u91cf\u65b9\u6cd5\u516c\u5f0f\u7279\u70b9\u9002\u7528\u573a\u666f<\/tr>\n<tbody>\n<tr>\n<td>\u6b27\u6c0f\u8ddd\u79bb<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\u2223a\u2212b\u22232|\\\\mathbf{a} &#8211; \\\\mathbf{b}|_2<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathbf\">a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathbf\">b<\/span><span class=\"mord\"><span class=\"mord\">\u2223<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><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 mtight\">2<\/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><\/td>\n<td>\u53d7\u957f\u5ea6\u5f71\u54cd&#xff0c;\u7edd\u5bf9\u5dee\u5f02<\/td>\n<td>\u805a\u7c7b\u3001KNN<\/td>\n<\/tr>\n<tr>\n<td>\u4f59\u5f26\u76f8\u4f3c\u5ea6<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">a\u22c5b\u2223a\u2223\u2223b\u2223\\\\frac{\\\\mathbf{a} \\\\cdot \\\\mathbf{b}}{|\\\\mathbf{a}| |\\\\mathbf{b}|}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.4001em;vertical-align: -0.52em\"><\/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: 0.8801em\"><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\">\u2223<\/span><span class=\"mord mathbf mtight\">a<\/span><span class=\"mord mtight\">\u2223\u2223<\/span><span class=\"mord mathbf mtight\">b<\/span><span class=\"mord mtight\">\u2223<\/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 mathbf mtight\">a<\/span><span class=\"mbin mtight\">\u22c5<\/span><span class=\"mord mathbf mtight\">b<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.52em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5f52\u4e00\u5316&#xff0c;\u53ea\u5173\u5fc3\u65b9\u5411<\/td>\n<td>\u6587\u672c\u3001\u63a8\u8350\u3001\u68c0\u7d22<\/td>\n<\/tr>\n<tr>\n<td>\u70b9\u79ef<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">a\u22c5b\\\\mathbf{a} \\\\cdot \\\\mathbf{b}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathbf\">a<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathbf\">b<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u672a\u5f52\u4e00\u5316&#xff0c;\u53d7\u957f\u5ea6\u5f71\u54cd<\/td>\n<td>\u5feb\u901f\u8fd1\u4f3c&#xff0c;\u9700\u540e\u7eed\u5f52\u4e00\u5316<\/td>\n<\/tr>\n<tr>\n<td>\u76ae\u5c14\u900a\u76f8\u5173\u7cfb\u6570<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">Cov(a,b)\u03c3a\u03c3b\\\\frac{\\\\text{Cov}(a,b)}{\\\\sigma_a \\\\sigma_b}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.4609em;vertical-align: -0.4509em\"><\/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\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1645em\"><span class=\"\" style=\"top: -2.357em;margin-left: -0.0359em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">a<\/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=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0359em\">\u03c3<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3448em\"><span class=\"\" style=\"top: -2.3488em;margin-left: -0.0359em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1512em\"><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 text mtight\"><span class=\"mord mtight\">Cov<\/span><\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"mpunct mtight\">,<\/span><span class=\"mord mathnormal mtight\">b<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4509em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u4e2d\u5fc3\u5316\u540e\u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6<\/td>\n<td>\u8bc4\u5206\u9884\u6d4b\u3001\u53bb\u8d8b\u52bf\u6570\u636e<\/td>\n<\/tr>\n<tr>\n<td>Jaccard \u76f8\u4f3c\u5ea6<\/td>\n<td>$\\\\frac{<\/td>\n<td>A \\\\cap B<\/td>\n<td>}{<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u5173\u952e\u5bf9\u6bd4&#xff1a;<\/p>\n<p>\u5411\u91cf a &#061; [1, 2, 3]      \u5411\u91cf b &#061; [2, 4, 6] &#061; 2*a  (a\u76842\u500d)<br \/>\n\u5411\u91cf c &#061; [3, 2, 1]      (\u4e0ea\u65b9\u5411\u4e0d\u540c)<\/p>\n<p>\u6b27\u6c0f\u8ddd\u79bb: d(a,b) &#061; ||a &#8211; b|| &#061; ||a &#8211; 2a|| &#061; ||a|| \u2248 3.74  (\u8f83\u5927)<br \/>\n          d(a,c) \u2248 2.83  (\u8f83\u5c0f&#xff0c;\u4f46b\u4e0ea\u5185\u5bb9\u6bd4\u4f8b\u76f8\u540c!)<\/p>\n<p>\u4f59\u5f26\u76f8\u4f3c\u5ea6: sim(a,b) &#061; 1.0  (\u5b8c\u5168\u76f8\u540c\u65b9\u5411&#xff0c;\u5b8c\u7f8e\u76f8\u4f3c)<br \/>\n            sim(a,c) \u2248 0.71  (\u65b9\u5411\u4e0d\u540c&#xff0c;\u76f8\u4f3c\u5ea6\u8f83\u4f4e)<\/p>\n<h4>2.5 \u5e94\u7528\u573a\u666f<\/h4>\n<h5>2.5.1 \u6587\u672c\u76f8\u4f3c\u5ea6&#xff08;NLP&#xff09;<\/h5>\n<p>TF-IDF &#043; \u4f59\u5f26\u76f8\u4f3c\u5ea6 \u662f\u7ecf\u5178\u7684\u4fe1\u606f\u68c0\u7d22\u7ec4\u5408&#xff1a;<\/p>\n<p><span class=\"token keyword\">from<\/span> sklearn<span class=\"token punctuation\">.<\/span>feature_extraction<span class=\"token punctuation\">.<\/span>text <span class=\"token keyword\">import<\/span> TfidfVectorizer<\/p>\n<p>documents <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token string\">&#034;\u673a\u5668\u5b66\u4e60\u662f\u4eba\u5de5\u667a\u80fd\u7684\u4e00\u4e2a\u5206\u652f&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token string\">&#034;\u6df1\u5ea6\u5b66\u4e60\u662f\u673a\u5668\u5b66\u4e60\u7684\u4e00\u79cd\u65b9\u6cd5&#034;<\/span><span class=\"token punctuation\">,<\/span><br \/>\n    <span class=\"token string\">&#034;\u81ea\u7136\u8bed\u8a00\u5904\u7406\u5c5e\u4e8e\u4eba\u5de5\u667a\u80fd\u9886\u57df&#034;<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><\/p>\n<p><span class=\"token comment\"># TF-IDF \u5411\u91cf\u5316<\/span><br \/>\nvectorizer <span class=\"token operator\">&#061;<\/span> TfidfVectorizer<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\ntfidf_matrix <span class=\"token operator\">&#061;<\/span> vectorizer<span class=\"token punctuation\">.<\/span>fit_transform<span class=\"token punctuation\">(<\/span>documents<span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token comment\"># \u8ba1\u7b97\u4f59\u5f26\u76f8\u4f3c\u5ea6<\/span><br \/>\nsimilarities <span class=\"token operator\">&#061;<\/span> cosine_similarity<span class=\"token punctuation\">(<\/span>tfidf_matrix<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>similarities<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># [[1.         0.42&#8230;    0.35&#8230;  ]<\/span><br \/>\n<span class=\"token comment\">#  [0.42&#8230;    1.         0.31&#8230;  ]<\/span><br \/>\n<span class=\"token comment\">#  [0.35&#8230;    0.31&#8230;    1.       ]]<\/span><\/p>\n<h5>2.5.2 \u63a8\u8350\u7cfb\u7edf<\/h5>\n<p>\u7528\u6237-\u7269\u54c1\u5d4c\u5165\u7684\u76f8\u4f3c\u5ea6\u8ba1\u7b97&#xff1a;<\/p>\n<p><span class=\"token comment\"># \u7528\u6237\u5d4c\u5165\u548c\u7269\u54c1\u5d4c\u5165&#xff08;\u6765\u81ea\u77e9\u9635\u5206\u89e3\u6216\u795e\u7ecf\u7f51\u7edc&#xff09;<\/span><br \/>\nuser_embedding <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.3<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.8<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># \u7528\u6237\u5174\u8da3\u5411\u91cf<\/span><br \/>\nitem_embeddings <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">0.6<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.4<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.7<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>   <span class=\"token comment\"># \u7269\u54c1A&#xff1a;\u7535\u5f71<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">0.1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.9<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.2<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>   <span class=\"token comment\"># \u7269\u54c1B&#xff1a;\u97f3\u4e50<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">0.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.3<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.9<\/span><span class=\"token punctuation\">]<\/span>    <span class=\"token comment\"># \u7269\u54c1C&#xff1a;\u7535\u5f71<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token comment\"># \u8ba1\u7b97\u7528\u6237\u4e0e\u6240\u6709\u7269\u54c1\u7684\u76f8\u4f3c\u5ea6<\/span><br \/>\nscores <span class=\"token operator\">&#061;<\/span> cosine_similarity<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span>user_embedding<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> item_embeddings<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string-interpolation\"><span class=\"token string\">f&#034;\u7269\u54c1A\u76f8\u4f3c\u5ea6: <\/span><span class=\"token interpolation\"><span class=\"token punctuation\">{<\/span>scores<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">:<\/span><span class=\"token format-spec\">.4f<\/span><span class=\"token punctuation\">}<\/span><\/span><span class=\"token string\">&#034;<\/span><\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 0.9876&#xff0c;\u6700\u76f8\u5173<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string-interpolation\"><span class=\"token string\">f&#034;\u7269\u54c1B\u76f8\u4f3c\u5ea6: <\/span><span class=\"token interpolation\"><span class=\"token punctuation\">{<\/span>scores<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">:<\/span><span class=\"token format-spec\">.4f<\/span><span class=\"token punctuation\">}<\/span><\/span><span class=\"token string\">&#034;<\/span><\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 0.2341&#xff0c;\u4e0d\u592a\u76f8\u5173<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string-interpolation\"><span class=\"token string\">f&#034;\u7269\u54c1C\u76f8\u4f3c\u5ea6: <\/span><span class=\"token interpolation\"><span class=\"token punctuation\">{<\/span>scores<span class=\"token punctuation\">[<\/span><span class=\"token number\">2<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">:<\/span><span class=\"token format-spec\">.4f<\/span><span class=\"token punctuation\">}<\/span><\/span><span class=\"token string\">&#034;<\/span><\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 0.9567&#xff0c;\u4e5f\u5f88\u76f8\u5173<\/span><\/p>\n<h5>2.5.3 \u56fe\u50cf\u68c0\u7d22&#xff08;CLIP&#xff09;<\/h5>\n<p>\u8de8\u6a21\u6001\u76f8\u4f3c\u5ea6\u8ba1\u7b97&#xff1a;<\/p>\n<p><span class=\"token comment\"># CLIP \u7f16\u7801\u540e\u7684\u56fe\u50cf\u548c\u6587\u672c\u7279\u5f81&#xff08;\u5df2\u5f52\u4e00\u5316&#xff09;<\/span><br \/>\nimage_features <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0.3<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.5<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 512\u7ef4<\/span><br \/>\ntext_features <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0.2<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.6<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token operator\">&#8211;<\/span><span class=\"token number\">0.1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0.3<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">.<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\"># 512\u7ef4<\/span><\/p>\n<p><span class=\"token comment\"># \u4f59\u5f26\u76f8\u4f3c\u5ea6&#xff08;\u70b9\u79ef\u5373\u53ef&#xff0c;\u56e0\u4e3a\u5df2\u5f52\u4e00\u5316&#xff09;<\/span><br \/>\nsimilarity <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>dot<span class=\"token punctuation\">(<\/span>image_features<span class=\"token punctuation\">,<\/span> text_features<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># \u6216\u76f4\u63a5<\/span><br \/>\nsimilarity <span class=\"token operator\">&#061;<\/span> cosine_similarity<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span>image_features<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span>text_features<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><\/p>\n<hr \/>\n<h3>\u4e09\u3001\u72ec\u70ed\u7f16\u7801\u4e0e\u4f59\u5f26\u76f8\u4f3c\u5ea6\u7684\u7ed3\u5408\u5e94\u7528<\/h3>\n<h4>3.1 \u7ecf\u5178\u573a\u666f&#xff1a;\u534f\u540c\u8fc7\u6ee4\u63a8\u8350<\/h4>\n<p>\u7528\u6237-\u7269\u54c1\u8bc4\u5206\u77e9\u9635\u7684\u72ec\u70ed\u8868\u793a &#043; \u4f59\u5f26\u76f8\u4f3c\u5ea6\u627e\u76f8\u4f3c\u7528\u6237&#xff1a;<\/p>\n<p><span class=\"token comment\"># \u7528\u6237-\u7269\u54c1\u4ea4\u4e92\u77e9\u9635&#xff08;\u9690\u5f0f\u53cd\u9988&#xff1a;0\/1&#xff09;<\/span><br \/>\n<span class=\"token comment\"># \u884c&#xff1a;\u7528\u6237&#xff0c;\u5217&#xff1a;\u7269\u54c1&#xff0c;1\u8868\u793a\u4ea4\u4e92\u8fc7<\/span><br \/>\ninteraction_matrix <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>  <span class=\"token comment\"># \u7528\u6237A&#xff1a;\u559c\u6b22\u7269\u54c11,2,4<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>  <span class=\"token comment\"># \u7528\u6237B&#xff1a;\u4e0e\u7528\u6237A\u5b8c\u5168\u76f8\u540c<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>  <span class=\"token comment\"># \u7528\u6237C&#xff1a;\u5b8c\u5168\u4e0d\u540c\u7684\u5174\u8da3<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token comment\"># \u8ba1\u7b97\u7528\u6237\u76f8\u4f3c\u5ea6&#xff08;\u57fa\u4e8e\u72ec\u70ed\u5411\u91cf\u7684\u4f59\u5f26\u76f8\u4f3c\u5ea6&#xff09;<\/span><br \/>\nuser_similarities <span class=\"token operator\">&#061;<\/span> cosine_similarity<span class=\"token punctuation\">(<\/span>interaction_matrix<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>user_similarities<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># [[1.         1.         0.        ]<\/span><br \/>\n<span class=\"token comment\">#  [1.         1.         0.        ]<\/span><br \/>\n<span class=\"token comment\">#  [0.         0.         1.        ]]<\/span><\/p>\n<p><span class=\"token comment\"># \u5e94\u7528&#xff1a;\u7ed9\u7528\u6237A\u63a8\u8350<\/span><br \/>\n<span class=\"token comment\"># \u627e\u5230\u76f8\u4f3c\u7528\u6237B&#xff0c;\u63a8\u8350B\u4ea4\u4e92\u8fc7\u4f46A\u672a\u4ea4\u4e92\u7684\u7269\u54c1<\/span><\/p>\n<h4>3.2 \u591a\u70ed\u7f16\u7801&#xff08;Multi-Hot&#xff09;&#043; \u4f59\u5f26\u76f8\u4f3c\u5ea6<\/h4>\n<p>\u5f53\u7528\u6237\u6709\u591a\u4e2a\u6807\u7b7e\/\u5174\u8da3\u65f6&#xff1a;<\/p>\n<p><span class=\"token comment\"># \u7528\u6237\u5174\u8da3\u6807\u7b7e&#xff08;\u591a\u70ed\u7f16\u7801&#xff09;<\/span><br \/>\n<span class=\"token comment\"># \u6807\u7b7e&#xff1a;\u79d1\u6280, \u4f53\u80b2, \u97f3\u4e50, \u7535\u5f71, \u7f8e\u98df, \u65c5\u6e38<\/span><br \/>\nuser_profiles <span class=\"token operator\">&#061;<\/span> np<span class=\"token punctuation\">.<\/span>array<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>  <span class=\"token comment\"># \u7528\u6237A&#xff1a;\u79d1\u6280&#043;\u97f3\u4e50&#043;\u7535\u5f71<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>  <span class=\"token comment\"># \u7528\u6237B&#xff1a;\u79d1\u6280&#043;\u4f53\u80b2&#043;\u97f3\u4e50&#043;\u65c5\u6e38<\/span><br \/>\n    <span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span>  <span class=\"token comment\"># \u7528\u6237C&#xff1a;\u4f53\u80b2&#043;\u7f8e\u98df&#043;\u65c5\u6e38<\/span><br \/>\n<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>similarities <span class=\"token operator\">&#061;<\/span> cosine_similarity<span class=\"token punctuation\">(<\/span>user_profiles<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>similarities<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># \u7528\u6237A\u548cB\u5171\u4eab\u79d1\u6280\u3001\u97f3\u4e50&#xff0c;\u76f8\u4f3c\u5ea6\u8f83\u9ad8<\/span><br \/>\n<span class=\"token comment\"># \u7528\u6237A\u548cC\u65e0\u5171\u540c\u5174\u8da3&#xff0c;\u76f8\u4f3c\u5ea6\u4e3a0<\/span><\/p>\n<hr \/>\n<h3>\u56db\u3001\u5e38\u89c1\u95ee\u9898\u4e0e\u6ce8\u610f\u4e8b\u9879<\/h3>\n<h4>4.1 \u72ec\u70ed\u7f16\u7801\u7684\u9677\u9631<\/h4>\n<table>\n<tr>\u95ee\u9898\u539f\u56e0\u89e3\u51b3\u65b9\u6848<\/tr>\n<tbody>\n<tr>\n<td>\u7ef4\u5ea6\u707e\u96be<\/td>\n<td>\u7c7b\u522b\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">KK<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0715em\">K<\/span><\/span><\/span><\/span><\/span> \u5f88\u5927<\/td>\n<td>\u4f7f\u7528\u5d4c\u5165\u5c42&#xff08;Embedding&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u8bed\u4e49\u7f3a\u5931<\/td>\n<td>\u65e0\u6cd5\u8868\u8fbe\u7c7b\u522b\u76f8\u4f3c\u6027<\/td>\n<td>\u7528\u9884\u8bad\u7ec3\u5d4c\u5165\u66ff\u4ee3&#xff08;\u5982 Word2Vec&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u7a00\u758f\u5b58\u50a8<\/td>\n<td>\u5927\u91cf\u96f6\u5143\u7d20<\/td>\n<td>\u4f7f\u7528\u7a00\u758f\u77e9\u9635\u683c\u5f0f&#xff08;CSR\/CSC&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u65b0\u7c7b\u522b\u5904\u7406<\/td>\n<td>\u8bad\u7ec3\u65f6\u672a\u89c1\u8fc7\u7684\u7c7b\u522b<\/td>\n<td>\u6dfb\u52a0&#034;\u672a\u77e5&#034;\u7c7b\u522b\u6216\u4f7f\u7528\u54c8\u5e0c\u6280\u5de7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>4.2 \u4f59\u5f26\u76f8\u4f3c\u5ea6\u7684\u5c40\u9650<\/h4>\n<table>\n<tr>\u95ee\u9898\u539f\u56e0\u89e3\u51b3\u65b9\u6848<\/tr>\n<tbody>\n<tr>\n<td>\u5ffd\u7565\u5411\u91cf\u957f\u5ea6<\/td>\n<td>\u4ec5\u5f52\u4e00\u5316\u65b9\u5411<\/td>\n<td>\u7ed3\u5408\u70b9\u79ef\u6216\u6b27\u6c0f\u8ddd\u79bb<\/td>\n<\/tr>\n<tr>\n<td>\u8d1f\u503c\u96be\u4ee5\u89e3\u91ca<\/td>\n<td>\u8303\u56f4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">[\u22121,1][-1, 1]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u6620\u5c04\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">[0,1][0, 1]<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">[<\/span><span class=\"mord\">0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span><\/span>&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">1&#043;cos\u20612\\\\frac{1 &#043; \\\\cos}{2}<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1901em;vertical-align: -0.345em\"><\/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: 0.8451em\"><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\">2<\/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 class=\"mbin mtight\">&#043;<\/span><span class=\"mop mtight\"><span class=\"mtight\">c<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\">s<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u5bf9\u96f6\u5411\u91cf\u672a\u5b9a\u4e49<\/td>\n<td>\u5206\u6bcd\u4e3a\u96f6<\/td>\n<td>\u6dfb\u52a0\u5e73\u6ed1\u9879\u6216\u7279\u6b8a\u5904\u7406<\/td>\n<\/tr>\n<tr>\n<td>\u65e0\u6cd5\u6355\u6349\u975e\u7ebf\u6027\u5173\u7cfb<\/td>\n<td>\u7ebf\u6027\u5ea6\u91cf<\/td>\n<td>\u4f7f\u7528\u6838\u65b9\u6cd5\u6216\u795e\u7ecf\u7f51\u7edc<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>4.3 \u5f52\u4e00\u5316\u7684\u91cd\u8981\u6027<\/h4>\n<p>\u4f59\u5f26\u76f8\u4f3c\u5ea6\u8ba1\u7b97\u524d&#xff0c;\u901a\u5e38\u9700\u8981\u5bf9\u5411\u91cf\u8fdb\u884c L2 \u5f52\u4e00\u5316&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">anormalized&#061;a\u2225a\u22252<br \/>\n\\\\mathbf{a}_{\\\\text{normalized}} &#061; \\\\frac{\\\\mathbf{a}}{\\\\|\\\\mathbf{a}\\\\|_2}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5944em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><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 mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">normalized<\/span><\/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 class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0574em;vertical-align: -0.936em\"><\/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: 1.1214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"mord mathbf\">a<\/span><span class=\"mord\"><span class=\"mord\">\u2225<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><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 mtight\">2<\/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 mathbf\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.936em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5f52\u4e00\u5316\u540e&#xff0c;\u4f59\u5f26\u76f8\u4f3c\u5ea6\u7b49\u4ef7\u4e8e\u70b9\u79ef&#xff1a;<br \/>\n<span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">CosineSimilarity(a,b)&#061;anorm\u22c5bnorm<br \/>\n\\\\text{CosineSimilarity}(\\\\mathbf{a}, \\\\mathbf{b}) &#061; \\\\mathbf{a}_{\\\\text{norm}} \\\\cdot \\\\mathbf{b}_{\\\\text{norm}}<br \/>\n<\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord text\"><span class=\"mord\">CosineSimilarity<\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathbf\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathbf\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5945em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><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 mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">norm<\/span><\/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 class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8444em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathbf\">b<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><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 mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">norm<\/span><\/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><\/span><\/p>\n<p>\u4f18\u52bf&#xff1a;\u8ba1\u7b97\u66f4\u5feb&#xff0c;\u9002\u5408\u5927\u89c4\u6a21\u68c0\u7d22&#xff08;\u5982 FAISS\u3001Milvus \u7b49\u5411\u91cf\u6570\u636e\u5e93&#xff09;\u3002<\/p>\n<hr \/>\n<h3>\u4e94\u3001\u4e00\u53e5\u8bdd\u603b\u7ed3<\/h3>\n<ul>\n<li>\u72ec\u70ed\u7f16\u7801&#xff1a;\u5c06\u79bb\u6563\u7c7b\u522b\u8f6c\u6362\u4e3a\u6b63\u4ea4\u7684\u4e8c\u8fdb\u5236\u5411\u91cf&#xff0c;\u6d88\u9664\u865a\u5047\u6570\u503c\u5173\u7cfb&#xff0c;\u4f46\u9762\u4e34\u7ef4\u5ea6\u707e\u96be\u548c\u8bed\u4e49\u7f3a\u5931\u95ee\u9898&#xff0c;\u9ad8\u57fa\u6570\u573a\u666f\u63a8\u8350\u7528\u5d4c\u5165\u5c42\u66ff\u4ee3\u3002<\/li>\n<li>\u4f59\u5f26\u76f8\u4f3c\u5ea6&#xff1a;\u5ea6\u91cf\u5411\u91cf\u65b9\u5411\u7684\u76f8\u4f3c\u7a0b\u5ea6&#xff0c;\u5ffd\u7565\u957f\u5ea6\u5f71\u54cd&#xff0c;\u662f\u6587\u672c\u3001\u63a8\u8350\u3001\u68c0\u7d22\u7cfb\u7edf\u7684\u6838\u5fc3\u5ea6\u91cf&#xff0c;\u8ba1\u7b97\u9ad8\u6548\u4e14\u53ef\u89e3\u91ca\u6027\u5f3a\u3002<\/li>\n<\/ul>\n<p>\u4e24\u8005\u7ed3\u5408&#xff0c;\u6784\u6210\u4e86\u673a\u5668\u5b66\u4e60\u4e2d\u8868\u793a\u79bb\u6563\u6570\u636e\u5e76\u5ea6\u91cf\u76f8\u4f3c\u6027\u7684\u57fa\u7840\u5de5\u5177\u94fe\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e00\u3001\u72ec\u70ed\u7f16\u7801&#xff08;One-Hot Encoding&#xff09;<br \/>\n1.1 \u4e3a\u4ec0\u4e48\u9700\u8981\u72ec\u70ed\u7f16\u7801&#xff1f;<br \/>\n\u673a\u5668\u5b66\u4e60\u6a21\u578b\u53ea\u80fd\u5904\u7406\u6570\u503c\u578b\u6570\u636e&#xff0c;\u4f46\u73b0\u5b9e\u4e16\u754c\u5145\u6ee1\u7c7b\u522b\u578b\u6570\u636e&#xff1a;<br \/>\n\u989c\u8272&#xff1a;{\u7ea2, \u7eff, \u84dd}\u57ce\u5e02&#xff1a;{\u5317\u4eac, \u4e0a\u6d77, \u5e7f\u5dde, \u6df1\u5733}\u5546\u54c1\u7c7b\u522b&#xff1a;{\u7535\u5b50\u4ea7\u54c1, \u670d\u88c5, \u98df\u54c1, \u5bb6\u5c45}<br \/>\n\u76f4\u63a5\u8d4b\u6574\u6570\u503c\u7684\u95ee\u9898&#xff1a;\u989c\u8272\u6574\u6570\u7f16\u7801\u95ee\u9898\u7ea20\u6a21\u578b\u4f1a\u8ba4\u4e3a \u7eff(1) &gt; \u7ea2(0)&#xff0c;\u4e14 \u84dd(2) 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