{"id":8733,"date":"2026-03-25T23:27:26","date_gmt":"2026-03-25T14:27:26","guid":{"rendered":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=8733"},"modified":"2026-04-21T00:58:45","modified_gmt":"2026-04-20T15:58:45","slug":"%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/","title":{"rendered":"\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\u3068\u5236\u9650\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3"},"content":{"rendered":"\r\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 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href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%83%9C%E3%83%AB%E3%83%84%E3%83%9E%E3%83%B3%E3%83%9E%E3%82%B7%E3%83%B3%E3%81%AE%E5%AE%9A%E7%BE%A9\" >\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\u306e\u5b9a\u7fa9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#KL%E6%83%85%E5%A0%B1%E9%87%8F%E3%81%AE%E6%9C%80%E5%B0%8F%E5%8C%96\" >KL\u60c5\u5831\u91cf\u306e\u6700\u5c0f\u5316<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E6%9B%B4%E6%96%B0%E5%BC%8F%E3%81%AE%E5%B0%8E%E5%87%BA\" >\u66f4\u65b0\u5f0f\u306e\u5c0e\u51fa<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%83%91%E3%83%A9%E3%83%A1%E3%83%BC%E3%82%BF%E6%9B%B4%E6%96%B0\" >\u30d1\u30e9\u30e1\u30fc\u30bf\u66f4\u65b0<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%83%90%E3%82%A4%E3%82%A2%E3%82%B9%E9%A0%85%E3%81%AE%E5%BE%AE%E5%88%86\" >\u30d0\u30a4\u30a2\u30b9\u9805\u306e\u5fae\u5206<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E9%87%8D%E3%81%BF%E9%A0%85%E3%81%AE%E5%BE%AE%E5%88%86\" >\u91cd\u307f\u9805\u306e\u5fae\u5206<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E5%8B%BE%E9%85%8D%E5%BC%8F%E3%81%B8%E3%81%AE%E4%BB%A3%E5%85%A5\" >\u52fe\u914d\u5f0f\u3078\u306e\u4ee3\u5165<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97\" >\u671f\u5f85\u5024\u306e\u8a08\u7b97<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E5%85%B7%E4%BD%93%E7%9A%84%E3%81%AA%E8%A8%88%E7%AE%97%E6%96%B9%E6%B3%95%EF%BC%88%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0%EF%BC%89\" >\u5177\u4f53\u7684\u306a\u8a08\u7b97\u65b9\u6cd5\uff08\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff09<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%83%87%E3%83%BC%E3%82%BF%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97\" >\u30c7\u30fc\u30bf\u671f\u5f85\u5024\u306e\u8a08\u7b97<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%83%A2%E3%83%87%E3%83%AB%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97\" >\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%82%AE%E3%83%96%E3%82%B9%E3%82%B5%E3%83%B3%E3%83%97%E3%83%AA%E3%83%B3%E3%82%B0%E3%81%AE%E5%85%B7%E4%BD%93%E7%9A%84%E3%81%AA%E6%89%8B%E9%A0%86\" 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href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%82%B3%E3%83%B3%E3%83%88%E3%83%A9%E3%82%B9%E3%83%86%E3%82%A3%E3%83%96%E3%83%BB%E3%83%80%E3%82%A4%E3%83%90%E3%83%BC%E3%82%B8%E3%82%A7%E3%83%B3%E3%82%B9%EF%BC%88CD%E6%B3%95%EF%BC%89\" >\u30b3\u30f3\u30c8\u30e9\u30b9\u30c6\u30a3\u30d6\u30fb\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\uff08CD\u6cd5\uff09<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E3%82%B3%E3%83%B3%E3%83%88%E3%83%A9%E3%82%B9%E3%83%86%E3%82%A3%E3%83%96%E3%83%BB%E3%83%80%E3%82%A4%E3%83%90%E3%83%BC%E3%82%B8%E3%82%A7%E3%83%B3%E3%82%B9%E3%81%AE%E5%9F%BA%E6%9C%AC%E7%9A%84%E3%81%AA%E8%80%83%E3%81%88%E6%96%B9\" >\u30b3\u30f3\u30c8\u30e9\u30b9\u30c6\u30a3\u30d6\u30fb\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\u306e\u57fa\u672c\u7684\u306a\u8003\u3048\u65b9<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#CD-1_%E3%81%AE%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0\" >CD-1 \u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE\" >\u53c2\u8003\u6587\u732e<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2026\/03\/25\/%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3%e3%81%a8%e5%88%b6%e9%99%90%e3%83%9c%e3%83%ab%e3%83%84%e3%83%9e%e3%83%b3%e3%83%9e%e3%82%b7%e3%83%b3\/#%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E4%BD%9C%E6%88%90%E8%80%85\" >\u672c\u8a18\u4e8b\u306e\u4f5c\u6210\u8005<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%A6%82%E8%A6%81\"><\/span>\u6982\u8981<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u672c\u7a3f\u3067\u306f\u3001\u307e\u305a\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08BM\uff09\u306e\u5b66\u7fd2\u65b9\u6cd5\u306b\u3064\u3044\u3066\u3001\u306a\u308b\u3079\u304f\u5f0f\u5909\u5f62\u3092\u7701\u7565\u305b\u305a\u306b\u89e3\u8aac\u3057\u307e\u3059\u3002\u7d50\u8ad6\u304b\u3089\u8ff0\u3079\u308b\u3068\u3001BM\u3067\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u66f4\u65b0\u306e\u969b\u306b\u5fc5\u8981\u3068\u306a\u308b\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97\u91cf\u304c\u975e\u5e38\u306b\u5927\u304d\u304f\u3001\u5b9f\u7528\u4e0a\u306e\u5927\u304d\u306a\u8ab2\u984c\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u63d0\u6848\u3055\u308c\u305f\u306e\u304c\u3001\u5236\u9650\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08RBM\uff09\u3067\u3059\u3002RBM\u3067\u306f\u3001\u30e6\u30cb\u30c3\u30c8\u306e\u63a5\u7d9a\u65b9\u6cd5\u3092\u5de5\u592b\u3059\u308b\u3053\u3068\u3067\u4e26\u5217\u8a08\u7b97\u3092\u53ef\u80fd\u306b\u3057\u3001\u8a08\u7b97\u91cf\u3092\u5927\u5e45\u306b\u6291\u3048\u308b\u3053\u3068\u306b\u6210\u529f\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u30e2\u30c7\u30eb\u306e\u69cb\u9020\u3084\u904e\u7a0b\u3092\u5de5\u592b\u3057\u3066\u8a08\u7b97\u3092\u5bb9\u6613\u306b\u3059\u308b\u3068\u3044\u3046\u767a\u60f3\u306f\u3001\u751f\u6210\u30e2\u30c7\u30eb\u306e\u7814\u7a76\u306b\u304a\u3044\u3066\u91cd\u8981\u306a\u8003\u3048\u65b9\u306e\u4e00\u3064\u3067\u3059\u3002<\/p>\r\n<div class=\"bm-rbm-article\">\r\n<h2><span class=\"ez-toc-section\" id=\"%E3%83%9C%E3%83%AB%E3%83%84%E3%83%9E%E3%83%B3%E3%83%9E%E3%82%B7%E3%83%B3%E3%81%AE%E5%AE%9A%E7%BE%A9\"><\/span>\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\u306e\u5b9a\u7fa9<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08Boltzmann Machine; BM\uff09\u3067\u306f\u3001\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u3092 <span class=\"math-inline\">\\( \\vec{x}=(x_1,\\dots,x_N) \\)<\/span> \u3068\u3057\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ E_\\theta(\\vec{x}) = -\\sum_i b_i x_i -\\sum_{i&lt;j} w_{ij} x_i x_j \\tag{1} $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001<span class=\"math-inline\">\\(x_i \\in \\{0,1\\}\\)<\/span>\u3001<span class=\"math-inline\">\\(b_i\\)<\/span> \u306f\u30d0\u30a4\u30a2\u30b9\u3001<span class=\"math-inline\">\\(w_{ij}\\)<\/span> \u306f\u7d50\u5408\u91cd\u307f\u3092\u8868\u3057\u307e\u3059\u3002\u3053\u308c\u4ee5\u964d\u306f\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u307e\u3068\u3081\u3066 <span class=\"math-inline\">\\(\\theta=\\{\\vec{b}, \\vec{w}\\}\\)<\/span> \u3067\u8868\u8a18\u3057\u307e\u3059\u3002\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u306b\u57fa\u3065\u304d\u3001\u78ba\u7387\u5206\u5e03\uff08\u30dc\u30eb\u30c4\u30de\u30f3\u5206\u5e03\uff09\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\r\n<p>$$ P_\\theta(\\vec{x}) = \\frac{1}{Z_\\theta}\\exp(-E_\\theta(\\vec{x})) \\tag{2} $$<\/p>\r\n<p>$$ Z_\\theta = \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) \\tag{3} $$<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"KL%E6%83%85%E5%A0%B1%E9%87%8F%E3%81%AE%E6%9C%80%E5%B0%8F%E5%8C%96\"><\/span>KL\u60c5\u5831\u91cf\u306e\u6700\u5c0f\u5316<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u30e2\u30c7\u30eb\u5206\u5e03\u3092\u30c7\u30fc\u30bf\u5206\u5e03\u306b\u8fd1\u3065\u3051\u308b\u305f\u3081\u3001KL\u60c5\u5831\u91cf\u3092\u5229\u7528\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} D_{KL}(P_{\\text{data}}||P_\\theta) &amp;= \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x}) \\log \\frac{P_{\\text{data}}(\\vec{x})}{P_\\theta(\\vec{x})} \\\\ &amp;= \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x}) \\left( \\log P_{\\text{data}}(\\vec{x}) &#8211; \\log P_\\theta(\\vec{x}) \\right) \\\\ &amp;= \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x})\\log P_{\\text{data}}(\\vec{x}) &#8211; \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x})\\log P_\\theta(\\vec{x}) \\quad (\\text{\u2235 } \\log \\tfrac{a}{b}=\\log a-\\log b) \\end{aligned} $$<\/p>\r\n<p>\u7b2c\u4e00\u9805\u306f\u30e2\u30c7\u30eb\u30d1\u30e9\u30e1\u30fc\u30bf <span class=\"math-inline\">\\(\\theta\\)<\/span> \u306b\u4f9d\u5b58\u3057\u306a\u3044\u5b9a\u6570\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001KL\u60c5\u5831\u91cf\u306e\u6700\u5c0f\u5316\u306f\u4ee5\u4e0b\u306e\u91cf\u306e\u6700\u5927\u5316\u3068\u7b49\u4fa1\u3067\u3059\u3002<\/p>\r\n<p>$$ \\max_{\\theta} \\quad f(\\theta) := \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x})\\log P_\\theta(\\vec{x}) \\tag{6} $$<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E6%9B%B4%E6%96%B0%E5%BC%8F%E3%81%AE%E5%B0%8E%E5%87%BA\"><\/span>\u66f4\u65b0\u5f0f\u306e\u5c0e\u51fa<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p><span class=\"math-inline\">\\(f(\\theta)\\)<\/span> \u3092\u6700\u5927\u5316\u3059\u308b\u305f\u3081\u3001\u52fe\u914d\u4e0a\u6607\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4ee5\u4e0b\u306e\u52fe\u914d\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\nabla f(\\theta) = \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x})\\nabla \\log P_\\theta(\\vec{x}) \\tag{7} $$<\/p>\r\n<p>\u307e\u305a\u3001<span class=\"math-inline\">\\(\\nabla \\log P_\\theta(\\vec{x})\\)<\/span> \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\nabla \\log P_\\theta(\\vec{x}) &amp;= \\nabla \\log \\frac{\\exp(-E_\\theta(\\vec{x}))} {\\sum_{\\vec{x}}\\exp(-E_\\theta(\\vec{x}))} \\\\ &amp;= \\nabla \\left( \\log \\exp(-E_\\theta(\\vec{x})) &#8211; \\log \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) \\right) \\quad (\\text{\u2235 } \\log \\tfrac{a}{b}=\\log a-\\log b) \\\\ &amp;= -\\nabla E_\\theta(\\vec{x}) &#8211; \\nabla \\log \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) \\quad (\\text{\u2235 } \\log e^a=a) \\end{aligned} \\tag{8} $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001\u7b2c2\u9805\u3092\u5f0f\u5909\u5f62\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\nabla \\log \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) &amp;= \\frac{ \\nabla \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) } { \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) } \\quad (\\text{\u2235 } \\nabla\\log g=\\tfrac{\\nabla g}{g}) \\\\ &amp;= \\frac{ \\sum_{\\vec{x}} \\nabla \\exp(-E_\\theta(\\vec{x})) } { \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) } \\\\ &amp;= \\frac{ \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) (-\\nabla E_\\theta(\\vec{x})) } { \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) } \\quad (\\text{\u2235 } \\nabla e^f=e^f\\nabla f) \\\\ &amp;= &#8211; \\frac{ \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) \\nabla E_\\theta(\\vec{x}) } { \\sum_{\\vec{x}} \\exp(-E_\\theta(\\vec{x})) } \\\\ &amp;= &#8211; \\sum_{\\vec{x}} P_\\theta(\\vec{x}) \\nabla E_\\theta(\\vec{x}) \\end{aligned} $$<\/p>\r\n<p>\u3053\u308c\u3088\u308a\u3001\u5f0f(8)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>$$ \\nabla \\log P_\\theta(\\vec{x}) = -\\nabla E_\\theta(\\vec{x}) + \\sum_{\\vec{x}^{\\,\\prime}} P_\\theta(\\vec{x}^{\\,\\prime}) \\nabla E_\\theta(\\vec{x}^{\\,\\prime}) $$<\/p>\r\n<p>\u3053\u308c\u3092\u5f0f(7)\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001\u6b21\u306e\u52fe\u914d\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\nabla f(\\theta) &amp;= \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x}) \\left( -\\nabla E_\\theta(\\vec{x}) + \\sum_{\\vec{x}^{\\,\\prime}} P_\\theta(\\vec{x}^{\\,\\prime}) \\nabla E_\\theta(\\vec{x}^{\\,\\prime}) \\right) \\\\ &amp;= &#8211; \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x}) \\nabla E_\\theta(\\vec{x}) + \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x}) \\sum_{\\vec{x}^{\\,\\prime}} P_\\theta(\\vec{x}^{\\,\\prime}) \\nabla E_\\theta(\\vec{x}^{\\,\\prime}) \\\\ &amp;= &#8211; \\sum_{\\vec{x}} P_{\\text{data}}(\\vec{x}) \\nabla E_\\theta(\\vec{x}) + \\sum_{\\vec{x}^{\\,\\prime}} P_\\theta(\\vec{x}^{\\,\\prime}) \\nabla E_\\theta(\\vec{x}^{\\,\\prime}) \\quad (\\text{\u2235 } \\sum_{\\vec{x}}P_{\\text{data}}(\\vec{x})=1) \\\\ &amp;= &#8211; \\mathbb{E}_{P_{\\text{data}}} [\\nabla E_\\theta(\\vec{X})] + \\mathbb{E}_{P_\\theta} [\\nabla E_\\theta(\\vec{X})] \\end{aligned} $$<\/p>\r\n<p>\u3053\u308c\u3067\u3001\u52fe\u914d\u306e\u5f62\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u3053\u3053\u304b\u3089\u306f\u3001<span class=\"math-inline\">\\(\\nabla E_\\theta(\\vec{X})\\)<\/span> \u306b\u3064\u3044\u3066\u3001\u5177\u4f53\u7684\u306a\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u5fae\u5206\u8a08\u7b97\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E3%83%91%E3%83%A9%E3%83%A1%E3%83%BC%E3%82%BF%E6%9B%B4%E6%96%B0\"><\/span>\u30d1\u30e9\u30e1\u30fc\u30bf\u66f4\u65b0<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<h3><span class=\"ez-toc-section\" id=\"%E3%83%90%E3%82%A4%E3%82%A2%E3%82%B9%E9%A0%85%E3%81%AE%E5%BE%AE%E5%88%86\"><\/span>\u30d0\u30a4\u30a2\u30b9\u9805\u306e\u5fae\u5206<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u307e\u305a\u3001\u30d0\u30a4\u30a2\u30b9 <span class=\"math-inline\">\\(b_i\\)<\/span> \u306b\u95a2\u3059\u308b\u5fae\u5206\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\frac{\\partial E_\\theta(\\vec{x})}{\\partial b_i} &amp;= \\frac{\\partial}{\\partial b_i} \\left( -\\sum_k b_k x_k &#8211; \\sum_{k&lt;\\ell} w_{k\\ell}x_kx_\\ell \\right) \\\\ &amp;= -x_i \\end{aligned} $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E9%87%8D%E3%81%BF%E9%A0%85%E3%81%AE%E5%BE%AE%E5%88%86\"><\/span>\u91cd\u307f\u9805\u306e\u5fae\u5206<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u540c\u69d8\u306b\u3001\u91cd\u307f <span class=\"math-inline\">\\(w_{ij}\\)<\/span> \u306b\u95a2\u3059\u308b\u5fae\u5206\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\frac{\\partial E_\\theta(\\vec{x})}{\\partial w_{ij}} &amp;= \\frac{\\partial}{\\partial w_{ij}} \\left( -\\sum_k b_k x_k &#8211; \\sum_{k&lt;\\ell} w_{k\\ell}x_kx_\\ell \\right) \\\\ &amp;= -x_ix_j \\end{aligned} $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E5%8B%BE%E9%85%8D%E5%BC%8F%E3%81%B8%E3%81%AE%E4%BB%A3%E5%85%A5\"><\/span>\u52fe\u914d\u5f0f\u3078\u306e\u4ee3\u5165<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u3053\u308c\u3089\u3092\u52fe\u914d\u5f0f\u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\frac{\\partial f}{\\partial b_i} &amp;= &#8211; \\mathbb{E}_{\\text{data}} \\left[ \\frac{\\partial E_\\theta(\\vec{x})}{\\partial b_i} \\right] + \\mathbb{E}_\\theta \\left[ \\frac{\\partial E_\\theta(\\vec{x})}{\\partial b_i} \\right] \\\\ &amp;= \\mathbb{E}_{\\text{data}}[x_i] &#8211; \\mathbb{E}_\\theta[x_i] \\end{aligned} $$<\/p>\r\n<p>$$ \\begin{aligned} \\frac{\\partial f}{\\partial w_{ij}} &amp;= &#8211; \\mathbb{E}_{\\text{data}} \\left[ \\frac{\\partial E_\\theta(\\vec{x})}{\\partial w_{ij}} \\right] + \\mathbb{E}_\\theta \\left[ \\frac{\\partial E_\\theta(\\vec{x})}{\\partial w_{ij}} \\right] \\\\ &amp;= \\mathbb{E}_{\\text{data}}[x_ix_j] &#8211; \\mathbb{E}_\\theta[x_ix_j] \\end{aligned} $$<\/p>\r\n<\/div>\r\n<div class=\"bm-rbm-article\">\r\n<p>\u7e70\u308a\u8fd4\u3057\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u5bfe\u6570\u5c24\u5ea6 <span class=\"math-inline\">\\(f(\\theta)\\)<\/span> \u3092\u6700\u5927\u5316\u3059\u308b\u305f\u3081\u52fe\u914d\u4e0a\u6607\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002<\/p>\r\n<p>$$ \\theta \\leftarrow \\theta + \\eta \\nabla f(\\theta) \\tag{9} $$<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001\u30d0\u30a4\u30a2\u30b9\u3068\u91cd\u307f\u306e\u66f4\u65b0\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>$$ b_i \\leftarrow b_i + \\eta \\left( \\mathbb{E}_{\\text{data}}[x_i] &#8211; \\mathbb{E}_\\theta[x_i] \\right) \\tag{10} $$<\/p>\r\n<p>$$ w_{ij} \\leftarrow w_{ij} + \\eta \\left( \\mathbb{E}_{\\text{data}}[x_ix_j] &#8211; \\mathbb{E}_\\theta[x_ix_j] \\right) \\tag{11} $$<\/p>\r\n<p>\u3053\u308c\u3067\u3001\u66f4\u65b0\u5f0f\u307e\u3067\u5c0e\u51fa\u3067\u304d\u307e\u3057\u305f\u3002BM\u306e\u5b66\u7fd2\u3067\u306f\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u306f\u30c7\u30fc\u30bf\u306e\u671f\u5f85\u5024\u3068\u30e2\u30c7\u30eb\u306e\u671f\u5f85\u5024\u304c\u4e00\u81f4\u3059\u308b\u3088\u3046\u306b\u66f4\u65b0\u3055\u308c\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002\u6b21\u306e\u7bc0\u304b\u3089\u306f\u3001\u3053\u308c\u3089\u306e\u66f4\u65b0\u5f0f\u3092\u5b9f\u969b\u306b\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u306e\u304b\u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97\"><\/span>\u671f\u5f85\u5024\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u66f4\u65b0\u5f0f\u306b\u304a\u3044\u3066\u3001\u30c7\u30fc\u30bf\u5206\u5e03\u306b\u95a2\u3059\u308b\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_{\\text{data}}\\)<\/span> \u306f\u6a19\u672c\u5e73\u5747\u3068\u3057\u3066\u5bb9\u6613\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u4e00\u65b9\u3001\u30e2\u30c7\u30eb\u5206\u5e03\u306b\u95a2\u3059\u308b\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_\\theta\\)<\/span> \u3092\u53b3\u5bc6\u306b\u8a08\u7b97\u3059\u308b\u306b\u306f\u3001\u5206\u914d\u95a2\u6570\u306e\u8a08\u7b97\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u72b6\u614b\u6570 <span class=\"math-inline\">\\(2^N\\)<\/span> \u306e\u7dcf\u548c\u3092\u542b\u3080\u305f\u3081\u3001\u73fe\u5b9f\u7684\u306b\u306f\u8a08\u7b97\u304c\u56f0\u96e3\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u305d\u3053\u3067\u3001\u30e2\u30c7\u30eb\u5206\u5e03 <span class=\"math-inline\">\\(P_\\theta(\\vec{x})\\)<\/span> \u304b\u3089\u30b5\u30f3\u30d7\u30eb\u3092\u751f\u6210\u3057\u3001\u30b5\u30f3\u30d7\u30eb\u5e73\u5747\u306b\u3088\u3063\u3066\u671f\u5f85\u5024\u3092\u8fd1\u4f3c\u7684\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u3053\u3053\u3067\u4e00\u3064\u91cd\u8981\u306a\u554f\u984c\u304c\u3042\u308a\u307e\u3059\u3002\u30e2\u30c7\u30eb\u5206\u5e03<\/p>\r\n<p>$$ P_\\theta(\\vec{x}) = \\frac{1}{Z_\\theta}\\exp(-E_\\theta(\\vec{x})) $$<\/p>\r\n<p>\u304b\u3089\u76f4\u63a5\u30b5\u30f3\u30d7\u30eb\u3057\u3088\u3046\u3068\u3057\u3066\u3082\u3001\u5206\u914d\u95a2\u6570 <span class=\"math-inline\">\\(Z_\\theta\\)<\/span> \u3092\u8a08\u7b97\u3067\u304d\u306a\u3044\u305f\u3081\u3001\u73fe\u5b9f\u7684\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3064\u307e\u308a\u3001\u300c\u30e2\u30c7\u30eb\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30eb\u3057\u305f\u3044\u300d\u3068\u3044\u3046\u8981\u8acb\u306f\u660e\u78ba\u3067\u3042\u308b\u4e00\u65b9\u3067\u3001\u300c\u305d\u306e\u307e\u307e\u3067\u306f\u30b5\u30f3\u30d7\u30eb\u3067\u304d\u306a\u3044\u300d\u3068\u3044\u3046\u56f0\u96e3\u304c\u3042\u308b\u308f\u3051\u3067\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u554f\u984c\u306b\u5bfe\u3057\u3066\u3001\u4e00\u822c\u306b\u306f\u30de\u30eb\u30b3\u30d5\u9023\u9396\u30e2\u30f3\u30c6\u30ab\u30eb\u30ed\uff08Markov Chain Monte Carlo; MCMC\uff09\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002MCMC\u3067\u306f\u3001\u76ee\u7684\u3068\u3059\u308b\u5206\u5e03\u3092\u5b9a\u5e38\u5206\u5e03\u3068\u3057\u3066\u6301\u3064\u30de\u30eb\u30b3\u30d5\u9023\u9396\u3092\u69cb\u6210\u3057\u3001\u305d\u306e\u9023\u9396\u3092\u5341\u5206\u306b\u9577\u304f\u56de\u3059\u3053\u3068\u3067\u3001\u76ee\u7684\u5206\u5e03\u304b\u3089\u306e\u30b5\u30f3\u30d7\u30eb\u3092\u5f97\u307e\u3059\u3002<\/p>\r\n<p>\u3067\u306f\u3001BM\u3067\u306f\u3069\u306e\u3088\u3046\u306b\u30de\u30eb\u30b3\u30d5\u9023\u9396\u3092\u69cb\u6210\u3059\u308c\u3070\u3088\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u3053\u3053\u3067\u91cd\u8981\u306b\u306a\u308b\u306e\u304c\u3001\u540c\u6642\u5206\u5e03 <span class=\"math-inline\">\\(P_\\theta(\\vec{x})\\)<\/span> \u306f\u76f4\u63a5\u6271\u3044\u306b\u304f\u3044\u4e00\u65b9\u3067\u3001\u5404\u5909\u6570\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387<\/p>\r\n<p>$$ P(x_i \\mid \\vec{x}_{-i}) $$<\/p>\r\n<p>\u306f\u5206\u914d\u95a2\u6570\u3092\u542b\u307e\u306a\u3044\u5f62\u3067\u8a08\u7b97\u3067\u304d\u308b\u3068\u3044\u3046\u70b9\u3067\u3059\u3002\u3053\u306e\u6027\u8cea\u3092\u5229\u7528\u3059\u308b\u3068\u3001\u4ed6\u306e\u5909\u6570\u3092\u56fa\u5b9a\u3057\u305f\u6761\u4ef6\u3067\u3001\u3042\u308b\u4e00\u3064\u306e\u5909\u6570\u3060\u3051\u3092\u9806\u756a\u306b\u66f4\u65b0\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3046\u3057\u3066\u5404\u5909\u6570\u3092\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306b\u5f93\u3063\u3066\u9806\u756a\u306b\u66f4\u65b0\u3057\u3066\u3044\u304f\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u6cd5\u304c\u3001<strong>\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\uff08Gibbs sampling\uff09<\/strong> \u3067\u3059\u3002<\/p>\r\n<p>\u5177\u4f53\u7684\u306b\u306f\u3001\u73fe\u5728\u306e\u72b6\u614b <span class=\"math-inline\">\\(\\vec{x}\\)<\/span> \u306b\u5bfe\u3057\u3066\u3001<\/p>\r\n<p>$$ x_1 \\sim P(x_1 \\mid x_2,\\dots,x_N) $$<\/p>\r\n<p>$$ x_2 \\sim P(x_2 \\mid x_1,x_3,\\dots,x_N) $$<\/p>\r\n<p>$$ \\vdots $$<\/p>\r\n<p>$$ x_N \\sim P(x_N \\mid x_1,\\dots,x_{N-1}) $$<\/p>\r\n<p>\u306e\u3088\u3046\u306b\u3001\u4e00\u5909\u6570\u305a\u3064\u66f4\u65b0\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u3053\u306e\u64cd\u4f5c\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u3067\u3001\u72b6\u614b\u5217<\/p>\r\n<p>$$ \\vec{x}^{(1)},\\vec{x}^{(2)},\\vec{x}^{(3)},\\dots $$<\/p>\r\n<p>\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u7406\u8ad6\u7684\u306b\u306f\u3001\u3053\u306e\u30de\u30eb\u30b3\u30d5\u9023\u9396\u306e\u5b9a\u5e38\u5206\u5e03\u306f <span class=\"math-inline\">\\(P_\\theta(\\vec{x})\\)<\/span> \u306b\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u5341\u5206\u306b\u9577\u304f\u9023\u9396\u3092\u56de\u3057\u305f\u5f8c\u306e\u30b5\u30f3\u30d7\u30eb\u3092\u7528\u3044\u308b\u3053\u3068\u3067\u3001<\/p>\r\n<p>$$ \\mathbb{E}_{P_\\theta}[x_i], \\qquad \\mathbb{E}_{P_\\theta}[x_ix_j] $$<\/p>\r\n<p>\u306e\u3088\u3046\u306a\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u3092\u30b5\u30f3\u30d7\u30eb\u5e73\u5747\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u4ee5\u4e0b\u3067\u306f\u3001\u305d\u306e\u305f\u3081\u306b\u5fc5\u8981\u3068\u306a\u308b\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3092\u5177\u4f53\u7684\u306b\u5c0e\u51fa\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u307e\u305a\u3001\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u5b9a\u7fa9\u304b\u3089\u59cb\u3081\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} P(X_i=1\\mid \\vec{x}_{-i}) &amp;= \\frac{P(x_i=1,\\vec{x}_{-i})}{P(\\vec{x}_{-i})} \\\\ &amp;= \\frac{P(x_i=1,\\vec{x}_{-i})} {P(x_i=0,\\vec{x}_{-i})+P(x_i=1,\\vec{x}_{-i})} \\quad (\\text{\u2235 } x_i \\in \\{0,1\\}) \\end{aligned} $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001<span class=\"math-inline\">\\(\\vec{x}_{-i}\\)<\/span> \u306f\u6210\u5206 <span class=\"math-inline\">\\(x_i\\)<\/span> \u3092\u9664\u3044\u305f\u6b8b\u308a\u306e\u6210\u5206\u5168\u4f53\u3092\u8868\u3059\u8a18\u6cd5\u3067\u3059\u3002\u4f8b\u3048\u3070\u3001<\/p>\r\n<p>$$ \\vec{x} = (x_1,\\dots,x_{i-1},x_i,x_{i+1},\\dots,x_N) $$<\/p>\r\n<p>\u3068\u3059\u308b\u3068\u3001<span class=\"math-inline\">\\(\\vec{x}_{-i}\\)<\/span> \u306f\u6b21\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ \\vec{x}_{-i} = (x_1,\\dots,x_{i-1},x_{i+1},\\dots,x_N) $$<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<span class=\"math-inline\">\\((\\vec{x}_{-i},1)\\)<\/span> \u306f\u300c<span class=\"math-inline\">\\(i\\)<\/span> \u756a\u76ee\u306e\u6210\u5206\u3060\u3051\u3092 1 \u306b\u56fa\u5b9a\u3057\u3001\u4ed6\u306e\u6210\u5206\u306f <span class=\"math-inline\">\\(\\vec{x}_{-i}\\)<\/span> \u306e\u307e\u307e\u306b\u3057\u305f\u72b6\u614b\u300d\u3092\u8868\u3057\u307e\u3059\u3002\u540c\u69d8\u306b\u3001<span class=\"math-inline\">\\((\\vec{x}_{-i},0)\\)<\/span> \u306f\u300c<span class=\"math-inline\">\\(i\\)<\/span> \u756a\u76ee\u306e\u6210\u5206\u3060\u3051\u3092 0 \u306b\u56fa\u5b9a\u3057\u305f\u72b6\u614b\u300d\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u3053\u3067\u3001\u30dc\u30eb\u30c4\u30de\u30f3\u5206\u5e03\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5206\u914d\u95a2\u6570\u3092\u6d88\u53bb\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} P(X_i=1\\mid \\vec{x}_{-i}) &amp;= \\frac{ \\frac{1}{Z_\\theta}\\exp(-E_\\theta(\\vec{x}_{-i},1)) }{ \\frac{1}{Z_\\theta}\\exp(-E_\\theta(\\vec{x}_{-i},0)) + \\frac{1}{Z_\\theta}\\exp(-E_\\theta(\\vec{x}_{-i},1)) } \\\\ &amp;= \\frac{\\exp(-E_\\theta(\\vec{x}_{-i},1))} {\\exp(-E_\\theta(\\vec{x}_{-i},0))+\\exp(-E_\\theta(\\vec{x}_{-i},1))} \\quad (\\text{\u2235 } \\tfrac{1}{Z_\\theta} \\text{ \u304c\u5206\u5b50\u5206\u6bcd\u3067\u5171\u901a}) \\\\ &amp;= \\frac{1} {\\exp\\left(-E_\\theta(\\vec{x}_{-i},0)+E_\\theta(\\vec{x}_{-i},1)\\right)+1} \\\\ &amp;= \\frac{1} {1+\\exp\\left(E_\\theta(\\vec{x}_{-i},1)-E_\\theta(\\vec{x}_{-i},0)\\right)} \\end{aligned} $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001\u5206\u6bcd\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5dee\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u306e\u3046\u3061 <span class=\"math-inline\">\\(x_i\\)<\/span> \u306b\u4f9d\u5b58\u3059\u308b\u90e8\u5206\u306e\u307f\u3092\u53d6\u308a\u51fa\u3059\u3068\u3001<\/p>\r\n<p>$$ \\begin{aligned} E_\\theta(\\vec{x}) &amp;= &#8211; b_i x_i &#8211; \\sum_{j\\neq i} w_{ij}x_ix_j + C(\\vec{x}_{-i}) \\\\ &amp;= &#8211; x_i \\left( b_i+\\sum_{j\\neq i} w_{ij}x_j \\right) + C(\\vec{x}_{-i}) \\end{aligned} $$<\/p>\r\n<p>\u3068\u66f8\u3051\u307e\u3059\u3002\u3053\u3053\u3067 <span class=\"math-inline\">\\(C(\\vec{x}_{-i})\\)<\/span> \u306f <span class=\"math-inline\">\\(x_i\\)<\/span> \u306b\u4f9d\u5b58\u3057\u306a\u3044\u9805\u3067\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} E_\\theta(\\vec{x}_{-i},1) &amp;= &#8211; \\left( b_i+\\sum_{j\\neq i} w_{ij}x_j \\right) + C(\\vec{x}_{-i}) \\\\ E_\\theta(\\vec{x}_{-i},0) &amp;= C(\\vec{x}_{-i}) \\end{aligned} $$<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\r\n<p>$$ E_\\theta(\\vec{x}_{-i},1)-E_\\theta(\\vec{x}_{-i},0) = &#8211; \\left( b_i+\\sum_{j\\neq i} w_{ij}x_j \\right) $$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3092\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001\u6b21\u5f0f\u3092\u5f97\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} P(X_i=1\\mid \\vec{x}_{-i}) &amp;= \\frac{1} {1+\\exp\\left( -\\left( b_i+\\sum_{j\\neq i} w_{ij}x_j \\right) \\right)} \\\\ &amp;= \\sigma \\left( b_i+\\sum_{j\\neq i} w_{ij}x_j \\right) \\quad (\\text{\u2235 } \\sigma(z)=\\tfrac{1}{1+e^{-z}}) \\end{aligned} $$<\/p>\r\n<p>\u3053\u308c\u3067\u3001\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u4ee5\u964d\u306f\u3001\u3053\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3092\u7528\u3044\u3066\u5404\u5909\u6570\u3092\u9806\u756a\u306b\u66f4\u65b0\u3057\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u3092\u8fd1\u4f3c\u7684\u306b\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\r\n<\/div>\r\n<div class=\"bm-rbm-article\">\r\n<h2><span class=\"ez-toc-section\" id=\"%E5%85%B7%E4%BD%93%E7%9A%84%E3%81%AA%E8%A8%88%E7%AE%97%E6%96%B9%E6%B3%95%EF%BC%88%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0%EF%BC%89\"><\/span>\u5177\u4f53\u7684\u306a\u8a08\u7b97\u65b9\u6cd5\uff08\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff09<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u3053\u3053\u307e\u3067\u3067\u3001BM \u306e\u5b66\u7fd2\u306b\u5fc5\u8981\u306a\u66f4\u65b0\u5f0f<\/p>\r\n<p>$$ b_i \\leftarrow b_i + \\eta \\left( \\mathbb{E}_{\\text{data}}[x_i] &#8211; \\mathbb{E}_\\theta[x_i] \\right), \\qquad w_{ij} \\leftarrow w_{ij} + \\eta \\left( \\mathbb{E}_{\\text{data}}[x_ix_j] &#8211; \\mathbb{E}_\\theta[x_ix_j] \\right) $$<\/p>\r\n<p>\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u6b21\u306b\u554f\u984c\u3068\u306a\u308b\u306e\u306f\u3001\u3053\u308c\u3089\u306e\u671f\u5f85\u5024\u3092\u5b9f\u969b\u306b\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u304b\u3067\u3059\u3002\u3053\u306e\u7bc0\u3067\u306f\u3001\u305d\u306e\u5177\u4f53\u7684\u306a\u8a08\u7b97\u65b9\u6cd5\u3001\u3059\u306a\u308f\u3061\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E3%83%87%E3%83%BC%E3%82%BF%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97\"><\/span>\u30c7\u30fc\u30bf\u671f\u5f85\u5024\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u307e\u305a\u3001\u30c7\u30fc\u30bf\u5206\u5e03\u306b\u95a2\u3059\u308b\u671f\u5f85\u5024\u306f\u6a19\u672c\u5e73\u5747\u3068\u3057\u3066\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u5b66\u7fd2\u30c7\u30fc\u30bf\u304c<\/p>\r\n<p>$$ \\vec{x}^{(1)}, \\vec{x}^{(2)}, \\dots, \\vec{x}^{(M)} $$<\/p>\r\n<p>\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30c7\u30fc\u30bf\u671f\u5f85\u5024\u306f\u6b21\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<p>$$ \\mathbb{E}_{\\text{data}}[x_i] \\approx \\frac{1}{M} \\sum_{m=1}^{M} x_i^{(m)} $$<\/p>\r\n<p>$$ \\mathbb{E}_{\\text{data}}[x_ix_j] \\approx \\frac{1}{M} \\sum_{m=1}^{M} x_i^{(m)}x_j^{(m)} $$<\/p>\r\n<p>\u3059\u306a\u308f\u3061\u3001\u30c7\u30fc\u30bf\u671f\u5f85\u5024\u306f\u300c\u5404\u30c7\u30fc\u30bf\u306b\u5bfe\u3057\u3066\u5024\u3092\u8a08\u7b97\u3057\u3001\u305d\u306e\u5e73\u5747\u3092\u53d6\u308b\u300d\u3053\u3068\u3067\u6c42\u307e\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u5358\u306a\u308b\u5e73\u5747\u8a08\u7b97\u306a\u306e\u3067\u3001\u6bd4\u8f03\u7684\u5bb9\u6613\u306b\u5b9f\u884c\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E3%83%A2%E3%83%87%E3%83%AB%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97\"><\/span>\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u4e00\u65b9\u3067\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97\u306f\u96e3\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u4f8b\u3048\u3070\u3001<\/p>\r\n<p>$$ \\mathbb{E}_\\theta[x_i] = \\sum_{\\vec{x}} P_\\theta(\\vec{x})x_i, \\qquad \\mathbb{E}_\\theta[x_ix_j] = \\sum_{\\vec{x}} P_\\theta(\\vec{x})x_ix_j $$<\/p>\r\n<p>\u3067\u3059\u304c\u3001\u3053\u3053\u3067\u548c\u3092\u53d6\u308b\u3079\u304d\u72b6\u614b\u6570\u306f <span class=\"math-inline\">\\(2^N\\)<\/span> \u500b\u3042\u308a\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u53b3\u5bc6\u306a\u8a08\u7b97\u306f\u4e8b\u5b9f\u4e0a\u4e0d\u53ef\u80fd\u3067\u3059\u3002<\/p>\r\n<p>\u305d\u3053\u3067\u3001\u524d\u7bc0\u3067\u5c0e\u3044\u305f\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092\u7528\u3044\u3066\u30e2\u30c7\u30eb\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30eb\u5217\u3092\u751f\u6210\u3057\u3001\u305d\u306e\u5e73\u5747\u306b\u3088\u3063\u3066\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u3092\u8fd1\u4f3c\u3057\u307e\u3059\u3002\u30b5\u30f3\u30d7\u30eb\u5217\u3092<\/p>\r\n<p>$$ \\vec{x}^{[1]}, \\vec{x}^{[2]}, \\dots, \\vec{x}^{[K]} $$<\/p>\r\n<p>\u3068\u3059\u308b\u3068\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306f\u6b21\u306e\u3088\u3046\u306b\u8fd1\u4f3c\u3055\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ \\mathbb{E}_\\theta[x_i] \\approx \\frac{1}{K} \\sum_{k=1}^{K} x_i^{[k]} $$<\/p>\r\n<p>$$ \\mathbb{E}_\\theta[x_ix_j] \\approx \\frac{1}{K} \\sum_{k=1}^{K} x_i^{[k]}x_j^{[k]} $$<\/p>\r\n<p>\u3053\u3053\u3067\u91cd\u8981\u306a\u306e\u306f\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306f\u300c\u30e2\u30c7\u30eb\u5206\u5e03\u304b\u3089\u5f97\u3089\u308c\u305f\u30b5\u30f3\u30d7\u30eb\u5217\u306e\u5e73\u5747\u300d\u3068\u3057\u3066\u8a08\u7b97\u3059\u308b\u3001\u3068\u3044\u3046\u70b9\u3067\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E3%82%AE%E3%83%96%E3%82%B9%E3%82%B5%E3%83%B3%E3%83%97%E3%83%AA%E3%83%B3%E3%82%B0%E3%81%AE%E5%85%B7%E4%BD%93%E7%9A%84%E3%81%AA%E6%89%8B%E9%A0%86\"><\/span>\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u306e\u5177\u4f53\u7684\u306a\u624b\u9806<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3067\u306f\u3001\u73fe\u5728\u306e\u72b6\u614b\u30d9\u30af\u30c8\u30eb<\/p>\r\n<p>$$ \\vec{x}=(x_1,\\dots,x_N) $$<\/p>\r\n<p>\u3092\u7528\u610f\u3057\u3001\u5404\u6210\u5206\u3092 1 \u3064\u305a\u3064\u66f4\u65b0\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u30e6\u30cb\u30c3\u30c8 <span class=\"math-inline\">\\(i\\)<\/span> \u306b\u3064\u3044\u3066<\/p>\r\n<p>$$ P(X_i=1 \\mid \\vec{x}_{-i}) = \\sigma \\left( b_i+\\sum_{j\\neq i}w_{ij}x_j \\right) $$<\/p>\r\n<p>\u3092\u8a08\u7b97\u3057\u3001\u3053\u306e\u78ba\u7387\u306b\u5f93\u3063\u3066 <span class=\"math-inline\">\\(x_i\\)<\/span> \u3092 0 \u307e\u305f\u306f 1 \u306b\u66f4\u65b0\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u4f8b\u3048\u3070\u3001\u78ba\u7387\u304c<\/p>\r\n<p>$$ P(X_i=1 \\mid \\vec{x}_{-i}) = 0.7 $$<\/p>\r\n<p>\u3067\u3042\u308c\u3070\u300170% \u306e\u78ba\u7387\u3067 <span class=\"math-inline\">\\(x_i=1\\)<\/span>\u300130% \u306e\u78ba\u7387\u3067 <span class=\"math-inline\">\\(x_i=0\\)<\/span> \u3092\u9078\u3073\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u64cd\u4f5c\u3092 <span class=\"math-inline\">\\(i=1,\\dots,N\\)<\/span> \u306b\u3064\u3044\u3066\u9806\u756a\u306b\u884c\u3044\u307e\u3059\u30021 \u56de\u3059\u3079\u3066\u306e\u30e6\u30cb\u30c3\u30c8\u3092\u66f4\u65b0\u3059\u308b\u64cd\u4f5c\u3092 <strong>1 sweep<\/strong> \u3068\u547c\u3073\u307e\u3059\u3002\u3053\u308c\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u3067\u3001<\/p>\r\n<p>$$ \\vec{x}^{[1]}, \\vec{x}^{[2]}, \\vec{x}^{[3]}, \\dots $$<\/p>\r\n<p>\u3068\u3044\u3046\u30b5\u30f3\u30d7\u30eb\u5217\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"BM_%E3%81%AE%E5%AD%A6%E7%BF%92%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0\"><\/span>BM \u306e\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u4ee5\u4e0a\u3092\u307e\u3068\u3081\u308b\u3068\u3001BM \u306e\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<ol>\r\n<li>\u5b66\u7fd2\u30c7\u30fc\u30bf\u304b\u3089\u30c7\u30fc\u30bf\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_{\\text{data}}[x_i]\\)<\/span>, <span class=\"math-inline\">\\(\\mathbb{E}_{\\text{data}}[x_ix_j]\\)<\/span> \u3092\u6a19\u672c\u5e73\u5747\u3068\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u73fe\u5728\u306e\u30d1\u30e9\u30e1\u30fc\u30bf <span class=\"math-inline\">\\(\\theta\\)<\/span> \u306e\u3082\u3068\u3067\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092\u5b9f\u884c\u3057\u3001\u30b5\u30f3\u30d7\u30eb\u5217 <span class=\"math-inline\">\\(\\vec{x}^{[1]},\\dots,\\vec{x}^{[K]}\\)<\/span> \u3092\u751f\u6210\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u751f\u6210\u3055\u308c\u305f\u30b5\u30f3\u30d7\u30eb\u5217\u304b\u3089\u30e2\u30c7\u30eb\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_\\theta[x_i]\\)<\/span>, <span class=\"math-inline\">\\(\\mathbb{E}_\\theta[x_ix_j]\\)<\/span> \u3092\u30b5\u30f3\u30d7\u30eb\u5e73\u5747\u3068\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u305d\u308c\u3089\u3092\u66f4\u65b0\u5f0f\u306b\u4ee3\u5165\u3057\u3001 <span class=\"math-inline\">\\(b_i\\)<\/span>, <span class=\"math-inline\">\\(w_{ij}\\)<\/span> \u3092\u66f4\u65b0\u3057\u307e\u3059\u3002<\/li>\r\n<\/ol>\r\n<h3><span class=\"ez-toc-section\" id=\"%E8%A8%88%E7%AE%97%E9%87%8F%E4%B8%8A%E3%81%AE%E5%95%8F%E9%A1%8C\"><\/span>\u8a08\u7b97\u91cf\u4e0a\u306e\u554f\u984c<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u3053\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u7406\u8ad6\u7684\u306b\u306f\u6b63\u3057\u3044\u306e\u3067\u3059\u304c\u3001\u5b9f\u969b\u306b\u306f\u975e\u5e38\u306b\u6642\u9593\u304c\u304b\u304b\u308a\u307e\u3059\u3002\u6700\u5927\u306e\u7406\u7531\u306f\u3001\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3067\u306f <strong>1 \u5909\u6570\u305a\u3064\u3057\u304b\u66f4\u65b0\u3067\u304d\u306a\u3044<\/strong> \u304b\u3089\u3067\u3059\u3002<\/p>\r\n<p>\u3059\u306a\u308f\u3061\u3001\u72b6\u614b\u30d9\u30af\u30c8\u30eb\u306b <span class=\"math-inline\">\\(N\\)<\/span> \u500b\u306e\u5909\u6570\u304c\u3042\u308b\u5834\u5408\u30011 sweep \u3060\u3051\u3067\u3082 <span class=\"math-inline\">\\(N\\)<\/span> \u56de\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u8a08\u7b97\u3068\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u30e2\u30c7\u30eb\u5206\u5e03\u306b\u5341\u5206\u8fd1\u3065\u304f\u307e\u3067\u306b\u306f\u591a\u304f\u306e sweep \u3092\u7e70\u308a\u8fd4\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001BM \u3067\u306f<\/p>\r\n<ul>\r\n<li>1 \u56de\u306e\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u306b\u6642\u9593\u304c\u304b\u304b\u308b<\/li>\r\n<li>\u591a\u304f\u306e\u30b5\u30f3\u30d7\u30eb\u3092\u96c6\u3081\u308b\u5fc5\u8981\u304c\u3042\u308b<\/li>\r\n<li>\u7d50\u679c\u3068\u3057\u3066\u5b66\u7fd2\u5168\u4f53\u304c\u975e\u5e38\u306b\u9045\u304f\u306a\u308b<\/li>\r\n<\/ul>\r\n<p>\u3068\u3044\u3046\u554f\u984c\u304c\u751f\u3058\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u300c\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u304c\u9045\u3044\u300d\u3068\u3044\u3046\u70b9\u304c\u3001BM \u3092\u5b9f\u7528\u4e0a\u6271\u3044\u306b\u304f\u304f\u3057\u3066\u3044\u308b\u672c\u8cea\u7684\u306a\u7406\u7531\u3067\u3059\u3002\u6b21\u306e\u7bc0\u3067\u306f\u3001\u3053\u306e\u554f\u984c\u3092\u69cb\u9020\u7684\u306a\u5de5\u592b\u306b\u3088\u3063\u3066\u6539\u5584\u3057\u305f\u30e2\u30c7\u30eb\u3067\u3042\u308b RBM \u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\r\n<\/div>\r\n<div class=\"bm-rbm-article\">\r\n<h2><span class=\"ez-toc-section\" id=\"%E5%88%B6%E9%99%90%E3%83%9C%E3%83%AB%E3%83%84%E3%83%9E%E3%83%B3%E3%83%9E%E3%82%B7%E3%83%B3\"><\/span>\u5236\u9650\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u524d\u7bc0\u3067\u306f\u3001\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08BM\uff09\u306e\u5b66\u7fd2\u5247\u306b\u73fe\u308c\u308b\u30e2\u30c7\u30eb\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_\\theta[x_i x_j]\\)<\/span> \u304c\u3001\u72b6\u614b\u6570 <span class=\"math-inline\">\\(2^N\\)<\/span> \u306b\u308f\u305f\u308b\u7dcf\u548c\u3092\u542b\u3080\u305f\u3081\u3001\u76f4\u63a5\u8a08\u7b97\u304c\u56f0\u96e3\u3067\u3042\u308b\u3053\u3068\u3092\u898b\u307e\u3057\u305f\u3002<\/p>\r\n<p>\u5236\u9650\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08Restricted Boltzmann Machine; RBM\uff09\u306f\u3001\u3053\u306e\u8a08\u7b97\u3092\u6271\u3044\u3084\u3059\u304f\u3059\u308b\u3053\u3068\u3092\u76ee\u7684\u3068\u3057\u3066\u5c0e\u5165\u3055\u308c\u308b\u30e2\u30c7\u30eb\u3067\u3059\u3002RBM \u3067\u306f\u30e6\u30cb\u30c3\u30c8\u3092\u53ef\u8996\u5909\u6570 <span class=\"math-inline\">\\(\\vec{v}=(v_1,\\dots,v_D)\\)<\/span> \u3068\u96a0\u308c\u5909\u6570 <span class=\"math-inline\">\\(\\vec{h}=(h_1,\\dots,h_F)\\)<\/span> \u306e 2 \u5c64\u306b\u5206\u3051\u3001\u5c64\u5185\u7d50\u5408\u3092\u6301\u305f\u306a\u3044\u4e8c\u90e8\u30b0\u30e9\u30d5\u69cb\u9020\u3092\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u5177\u4f53\u7684\u306a\u69cb\u9020\u5236\u7d04\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\r\n<ul>\r\n<li>\u53ef\u8996\u5909\u6570\u540c\u58eb\u306e\u7d50\u5408\u3092\u6301\u3061\u307e\u305b\u3093\u3002<\/li>\r\n<li>\u96a0\u308c\u5909\u6570\u540c\u58eb\u306e\u7d50\u5408\u3092\u6301\u3061\u307e\u305b\u3093\u3002<\/li>\r\n<li>\u53ef\u8996\u5909\u6570\u3068\u96a0\u308c\u5909\u6570\u306e\u9593\u306b\u306e\u307f\u7d50\u5408\u3092\u6301\u3061\u307e\u3059\u3002<\/li>\r\n<\/ul>\r\n<p>BM \u3068\u540c\u69d8\u306b\u3001RBM \u3067\u3082\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u3092\u5c0e\u5165\u3057\u3001\u305d\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306b\u57fa\u3065\u3044\u3066\u78ba\u7387\u5206\u5e03\u3092\u5b9a\u7fa9\u3057\u3001\u5bfe\u6570\u5c24\u5ea6\u3092\u6700\u5927\u5316\u3059\u308b\u3088\u3046\u306b\u5b66\u7fd2\u3057\u307e\u3059\u3002\u4e00\u65b9\u3067\u3001RBM \u3067\u306f\u3053\u306e\u69cb\u9020\u5236\u7d04\u306b\u3088\u3063\u3066\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u6027\u304c\u6210\u7acb\u3059\u308b\u305f\u3081\u3001BM \u306b\u6bd4\u3079\u3066\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3084\u671f\u5f85\u5024\u8a08\u7b97\u304c\u5927\u5e45\u306b\u7c21\u5358\u306b\u306a\u308a\u307e\u3059\u3002\u4ee5\u5f8c\u3001RBM \u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306f <span class=\"math-inline\">\\(\\theta=(\\vec{a},\\vec{b},W)\\)<\/span> \u3068\u307e\u3068\u3081\u3066\u8868\u3057\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"RBM%E3%81%AE%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC%E9%96%A2%E6%95%B0%E3%81%A8%E7%A2%BA%E7%8E%87%E5%88%86%E5%B8%83\"><\/span>RBM\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u3068\u78ba\u7387\u5206\u5e03<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>RBM \u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u306f\u4ee5\u4e0b\u3067\u4e0e\u3048\u307e\u3059\u3002<\/p>\r\n<p>$$ E_\\theta(\\vec{v},\\vec{h}) = -\\sum_{i=1}^{D} a_i v_i -\\sum_{j=1}^{F} b_j h_j -\\sum_{i=1}^{D}\\sum_{j=1}^{F} v_i W_{ij} h_j \\tag{13} $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001<span class=\"math-inline\">\\(v_i \\in \\{0,1\\}\\)<\/span> \u306f\u53ef\u8996\u30e6\u30cb\u30c3\u30c8\u3001<span class=\"math-inline\">\\(h_j \\in \\{0,1\\}\\)<\/span> \u306f\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u3001<span class=\"math-inline\">\\(a_i\\)<\/span> \u306f\u53ef\u8996\u30e6\u30cb\u30c3\u30c8\u306e\u30d0\u30a4\u30a2\u30b9\u3001<span class=\"math-inline\">\\(b_j\\)<\/span> \u306f\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u306e\u30d0\u30a4\u30a2\u30b9\u3001<span class=\"math-inline\">\\(W_{ij}\\)<\/span> \u306f\u53ef\u8996\u30e6\u30cb\u30c3\u30c8 <span class=\"math-inline\">\\(i\\)<\/span> \u3068\u96a0\u308c\u30e6\u30cb\u30c3\u30c8 <span class=\"math-inline\">\\(j\\)<\/span> \u306e\u7d50\u5408\u91cd\u307f\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\r\n<p>BM \u3068\u540c\u69d8\u306b\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u306b\u57fa\u3065\u3044\u3066\u540c\u6642\u5206\u5e03\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\r\n<p>$$ P_\\theta(\\vec{v},\\vec{h}) = \\frac{1}{Z_\\theta}\\exp(-E_\\theta(\\vec{v},\\vec{h})) \\tag{14} $$<\/p>\r\n<p>\u3053\u3053\u3067\u5206\u914d\u95a2\u6570 <span class=\"math-inline\">\\(Z_\\theta\\)<\/span> \u306f\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ Z_\\theta = \\sum_{\\vec{v}} \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) \\tag{15} $$<\/p>\r\n<p>\u3053\u3053\u307e\u3067\u306f BM \u3068\u540c\u69d8\u3067\u3042\u308a\u3001\u300c\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u3092\u5b9a\u3081\u3001\u305d\u306e\u6307\u6570\u95a2\u6570\u304b\u3089\u78ba\u7387\u5206\u5e03\u3092\u69cb\u6210\u3059\u308b\u300d\u3068\u3044\u3046\u8003\u3048\u65b9\u306f\u5171\u901a\u3067\u3059\u3002\u4e00\u65b9\u3067 RBM \u3067\u306f\u3001\u5b66\u7fd2\u30c7\u30fc\u30bf\u3068\u3057\u3066\u76f4\u63a5\u89b3\u6e2c\u3055\u308c\u308b\u306e\u306f\u53ef\u8996\u5909\u6570 <span class=\"math-inline\">\\(\\vec{v}\\)<\/span> \u306e\u307f\u3067\u3042\u308a\u3001\u96a0\u308c\u5909\u6570 <span class=\"math-inline\">\\(\\vec{h}\\)<\/span> \u306f\u89b3\u6e2c\u3055\u308c\u307e\u305b\u3093\u3002\u305d\u306e\u305f\u3081\u3001\u53ef\u8996\u5909\u6570\u306e\u5468\u8fba\u5206\u5e03 <span class=\"math-inline\">\\(P_\\theta(\\vec{v})\\)<\/span> \u3092\u76f4\u63a5\u6271\u3046\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u53ef\u8996\u5909\u6570\u306e\u5468\u8fba\u5206\u5e03\u306f\u4ee5\u4e0b\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ P_\\theta(\\vec{v}) = \\sum_{\\vec{h}} P_\\theta(\\vec{v},\\vec{h}) \\tag{16} $$<\/p>\r\n<p>\u5f0f(14) \u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\u6b21\u5f0f\u3092\u5f97\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} P_\\theta(\\vec{v}) &amp;= \\frac{1}{Z_\\theta} \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) \\quad\u00a0 \\end{aligned} \\tag{17} $$<\/p>\r\n<p>BM \u3067\u306f\u5404\u72b6\u614b <span class=\"math-inline\">\\(\\vec{x}\\)<\/span> \u306e\u78ba\u7387\u3092\u305d\u306e\u307e\u307e\u6271\u3048\u3070\u5341\u5206\u3067\u3057\u305f\u304c\u3001RBM \u3067\u306f\u96a0\u308c\u5909\u6570\u3092\u5468\u8fba\u5316\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u70b9\u304c\u6700\u521d\u306e\u5927\u304d\u306a\u9055\u3044\u3067\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E5%AD%A6%E7%BF%92%E7%9B%AE%E6%A8%99\"><\/span>\u5b66\u7fd2\u76ee\u6a19<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>RBM \u3067\u3082\u3001\u30e2\u30c7\u30eb\u5206\u5e03\u3092\u30c7\u30fc\u30bf\u5206\u5e03\u306b\u8fd1\u3065\u3051\u308b\u3053\u3068\u3092\u76ee\u7684\u3068\u3057\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u3053\u3053\u3067\u76f4\u63a5\u6271\u3046\u306e\u306f\u53ef\u8996\u5909\u6570\u306e\u5206\u5e03\u3067\u3059\u3002\u3053\u306e\u70b9\u306f BM \u306b\u304a\u3051\u308b\u300c\u89b3\u6e2c\u3055\u308c\u308b\u5909\u6570\u306e\u5206\u5e03\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u300d\u3068\u3044\u3046\u8003\u3048\u65b9\u3068\u540c\u3058\u3067\u3059\u304c\u3001RBM \u3067\u306f\u89b3\u6e2c\u5909\u6570\u304c\u53ef\u8996\u5c64\u3060\u3051\u306b\u9650\u5b9a\u3055\u308c\u308b\u70b9\u304c\u7570\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001\u5bfe\u6570\u5c24\u5ea6\u306e\u671f\u5f85\u5024\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5c0e\u5165\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ f(\\theta) = \\mathbb{E}_{P_{\\text{data}}} [\\log P_\\theta(\\vec{v})] $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001\u5f0f(17)\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001<\/p>\r\n<p>$$ \\begin{aligned} \\log P_\\theta(\\vec{v}) &amp;= \\log \\left( \\frac{1}{Z_\\theta} \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) \\right) \\\\ &amp;= \\log \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) &#8211; \\log Z_\\theta \\quad (\\text{\u2235 } \\log \\tfrac{a}{b}=\\log a-\\log b) \\end{aligned} \\tag{18} $$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002BM \u3067\u306f <span class=\"math-inline\">\\(\\log P_\\theta(\\vec{x})\\)<\/span> \u304c\u30a8\u30cd\u30eb\u30ae\u30fc\u9805\u3068\u5206\u914d\u95a2\u6570\u9805\u3060\u3051\u3067\u76f4\u63a5\u8868\u3055\u308c\u307e\u3057\u305f\u304c\u3001RBM \u3067\u306f\u96a0\u308c\u5909\u6570 <span class=\"math-inline\">\\(\\vec{h}\\)<\/span> \u3092\u5468\u8fba\u5316\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u3001\u5f0f(18) \u306e\u3088\u3046\u306b <span class=\"math-inline\">\\(\\log\\sum_{\\vec{h}}\\exp(-E_\\theta(\\vec{v},\\vec{h}))\\)<\/span> \u304c\u73fe\u308c\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E6%9B%B4%E6%96%B0%E5%BC%8F%E3%81%AE%E5%B0%8E%E5%87%BA-2\"><\/span>\u66f4\u65b0\u5f0f\u306e\u5c0e\u51fa<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>BM \u3068\u540c\u69d8\u306b\u3001\u5b66\u7fd2\u76ee\u6a19\u3067\u5c0e\u5165\u3057\u305f\u76ee\u7684\u95a2\u6570 <span class=\"math-inline\">\\(f(\\theta)=\\mathbb{E}_{P_{\\text{data}}}[\\log P_\\theta(\\vec{v})]\\)<\/span> \u3092\u6700\u5927\u5316\u3059\u308b\u305f\u3081\u3001\u52fe\u914d\u4e0a\u6607\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4ee5\u4e0b\u306e\u52fe\u914d\u304c\u8a08\u7b97\u3067\u304d\u308c\u3070\u5341\u5206\u3067\u3059\u3002<\/p>\r\n<p>$$ \\nabla f(\\theta) = \\sum_{\\vec{v}} P_{\\text{data}}(\\vec{v}) \\nabla\\log P_\\theta(\\vec{v}) \\tag{19} $$<\/p>\r\n<p>\u307e\u305a\u3001<span class=\"math-inline\">\\(\\nabla\\log P_\\theta(\\vec{v})\\)<\/span> \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\nabla \\log P_\\theta(\\vec{v}) = \\nabla \\log \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) &#8211; \\nabla \\log Z_\\theta $$<\/p>\r\n<p>\u3053\u3053\u3067\u3082 BM \u3068\u540c\u69d8\u306b\u3001\u300c\u5bfe\u6570\u78ba\u7387\u306e\u52fe\u914d\u3092\u4e8c\u3064\u306e\u9805\u306b\u5206\u3051\u3066\u8a08\u7b97\u3059\u308b\u300d\u3068\u3044\u3046\u6d41\u308c\u306f\u540c\u3058\u3067\u3059\u3002\u305f\u3060\u3057\u3001RBM \u3067\u306f\u7b2c1\u9805\u306b\u96a0\u308c\u5909\u6570\u306b\u95a2\u3059\u308b\u7dcf\u548c\u304c\u5165\u308b\u305f\u3081\u3001\u305d\u306e\u6271\u3044\u304c\u5c11\u3057\u5909\u308f\u308a\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E7%AC%AC%E4%B8%80%E9%A0%85%E3%81%AE%E8%A8%88%E7%AE%97\"><\/span>\u7b2c\u4e00\u9805\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u307e\u305a\u3001<\/p>\r\n<p>$$ \\log \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) $$<\/p>\r\n<p>\u3092\u5fae\u5206\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\begin{aligned} \\nabla \\log \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) &amp;= \\frac{ \\nabla \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) }{ \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) } \\quad (\\text{\u2235 } \\nabla \\log g = \\tfrac{\\nabla g}{g}) \\\\ &amp;= \\frac{ \\sum_{\\vec{h}} \\nabla \\exp(-E_\\theta(\\vec{v},\\vec{h})) }{ \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) } \\\\ &amp;= \\frac{ \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) (-\\nabla E_\\theta(\\vec{v},\\vec{h})) }{ \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) } \\quad (\\text{\u2235 } \\nabla e^f=e^f\\nabla f) \\\\ &amp;= &#8211; \\sum_{\\vec{h}} \\frac{ \\exp(-E_\\theta(\\vec{v},\\vec{h})) }{ \\sum_{\\vec{h}^{\\,\\prime}} \\exp(-E_\\theta(\\vec{v},\\vec{h}^{\\,\\prime})) } \\nabla E_\\theta(\\vec{v},\\vec{h}) \\end{aligned} $$<\/p>\r\n<p>\u3053\u3053\u3067\u3001\u6761\u4ef6\u4ed8\u304d\u5206\u5e03<\/p>\r\n<p>$$ P_\\theta(\\vec{h}\\mid \\vec{v}) = \\frac{\\exp(-E_\\theta(\\vec{v},\\vec{h}))} {\\sum_{\\vec{h}^{\\,\\prime}} \\exp(-E_\\theta(\\vec{v},\\vec{h}^{\\,\\prime}))} $$<\/p>\r\n<p>\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u308c\u3092\u7528\u3044\u308b\u3068\u3001<\/p>\r\n<p>$$ \\begin{aligned} \\nabla \\log \\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) &amp;= &#8211; \\sum_{\\vec{h}} P_\\theta(\\vec{h}\\mid \\vec{v}) \\nabla E_\\theta(\\vec{v},\\vec{h}) \\\\ &amp;= &#8211; \\mathbb{E}_{P_\\theta(\\vec{H}\\mid \\vec{V}=\\vec{v})} [\\nabla E_\\theta(\\vec{v},\\vec{h})] \\end{aligned} $$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002BM \u3067\u306f\u7b2c1\u9805\u306f\u5358\u7d14\u306b <span class=\"math-inline\">\\(-\\nabla E_\\theta(\\vec{x})\\)<\/span> \u3067\u3057\u305f\u304c\u3001RBM \u3067\u306f\u96a0\u308c\u5909\u6570\u3092\u5468\u8fba\u5316\u3057\u3066\u3044\u308b\u305f\u3081\u3001\u6761\u4ef6\u4ed8\u304d\u5206\u5e03 <span class=\"math-inline\">\\(P_\\theta(\\vec{h}\\mid\\vec{v})\\)<\/span> \u306b\u3088\u308b\u671f\u5f85\u5024\u306b\u5909\u308f\u308a\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E7%AC%AC%E4%BA%8C%E9%A0%85%E3%81%AE%E8%A8%88%E7%AE%97\"><\/span>\u7b2c\u4e8c\u9805\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u6b21\u306b\u3001<span class=\"math-inline\">\\(\\nabla \\log Z_\\theta\\)<\/span> \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u5206\u914d\u95a2\u6570\u306f<\/p>\r\n<p>$$ Z_\\theta = \\sum_{\\vec{v}}\\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) $$<\/p>\r\n<p>\u3067\u3042\u308b\u305f\u3081\u3001<\/p>\r\n<p>$$ \\begin{aligned} \\nabla Z_\\theta &amp;= \\sum_{\\vec{v}}\\sum_{\\vec{h}} \\nabla \\exp(-E_\\theta(\\vec{v},\\vec{h})) \\\\ &amp;= \\sum_{\\vec{v}}\\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) (-\\nabla E_\\theta(\\vec{v},\\vec{h})) \\quad (\\text{\u2235 } \\nabla e^f=e^f\\nabla f) \\end{aligned} $$<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\r\n<p>$$ \\begin{aligned} \\nabla \\log Z_\\theta &amp;= \\frac{1}{Z_\\theta}\\nabla Z_\\theta \\\\ &amp;= \\frac{1}{Z_\\theta} \\sum_{\\vec{v}}\\sum_{\\vec{h}} \\exp(-E_\\theta(\\vec{v},\\vec{h})) (-\\nabla E_\\theta(\\vec{v},\\vec{h})) \\\\ &amp;= &#8211; \\sum_{\\vec{v}}\\sum_{\\vec{h}} \\frac{\\exp(-E_\\theta(\\vec{v},\\vec{h}))}{Z_\\theta} \\nabla E_\\theta(\\vec{v},\\vec{h}) \\\\ &amp;= &#8211; \\sum_{\\vec{v}}\\sum_{\\vec{h}} P_\\theta(\\vec{v},\\vec{h}) \\nabla E_\\theta(\\vec{v},\\vec{h}) \\quad (\\text{\u2235 } P_\\theta(\\vec{v},\\vec{h})=\\tfrac{\\exp(-E_\\theta(\\vec{v},\\vec{h}))}{Z_\\theta}) \\\\ &amp;= &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})} [\\nabla E_\\theta(\\vec{v},\\vec{h})] \\end{aligned} $$<\/p>\r\n<p>\u3053\u306e\u90e8\u5206\u306f\u3001BM \u306b\u304a\u3051\u308b\u5206\u914d\u95a2\u6570\u306e\u5fae\u5206\u3068\u5168\u304f\u540c\u3058\u8003\u3048\u65b9\u3067\u3059\u3002\u9055\u3044\u306f\u3001\u72b6\u614b\u7a7a\u9593\u304c <span class=\"math-inline\">\\(\\vec{x}\\)<\/span> \u3067\u306f\u306a\u304f <span class=\"math-inline\">\\((\\vec{v},\\vec{h})\\)<\/span> \u306b\u5e83\u304c\u3063\u3066\u3044\u308b\u3053\u3068\u3060\u3051\u3067\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E5%8B%BE%E9%85%8D%E3%81%AE%E6%95%B4%E7%90%86\"><\/span>\u52fe\u914d\u306e\u6574\u7406<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u4ee5\u4e0a\u3092\u307e\u3068\u3081\u308b\u3068\u3001<\/p>\r\n<p>$$ \\nabla \\log P_\\theta(\\vec{v}) = &#8211; \\mathbb{E}_{P_\\theta(\\vec{H}\\mid \\vec{V}=\\vec{v})} [\\nabla E_\\theta(\\vec{v},\\vec{h})] + \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})} [\\nabla E_\\theta(\\vec{v},\\vec{h})] $$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AB%E3%82%88%E3%82%8B%E8%A1%A8%E7%8F%BE\"><\/span>\u671f\u5f85\u5024\u306b\u3088\u308b\u8868\u73fe<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u4ee5\u4e0a\u306e\u7d50\u679c\u3092\u5f0f(19) \u306b\u4ee3\u5165\u3059\u308b\u3068\u3001\u76ee\u7684\u95a2\u6570\u306e\u52fe\u914d\u306f\u6b21\u306e\u3088\u3046\u306b\u671f\u5f85\u5024\u3067\u8868\u305b\u307e\u3059\u3002<\/p>\r\n<p>$$ \\nabla f(\\theta) = &#8211; \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})} [\\nabla E_\\theta(\\vec{v},\\vec{h})] + \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})} [\\nabla E_\\theta(\\vec{v},\\vec{h})] \\tag{20} $$<\/p>\r\n<p>\u3053\u306e\u5f0f\u306f\u3001\u30c7\u30fc\u30bf\u5206\u5e03\u306b\u304a\u3051\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u52fe\u914d\u3068\u3001\u30e2\u30c7\u30eb\u5206\u5e03\u306b\u304a\u3051\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u52fe\u914d\u306e\u5dee\u3068\u3057\u3066\u52fe\u914d\u304c\u8868\u3055\u308c\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u5f0f(20) \u306f\u3001BM \u306b\u304a\u3051\u308b\u52fe\u914d\u5f0f<\/p>\r\n<p>$$ &#8211; \\mathbb{E}_{P_{\\text{data}}}[\\nabla E_\\theta(\\vec{X})] + \\mathbb{E}_{P_\\theta}[\\nabla E_\\theta(\\vec{X})] $$<\/p>\r\n<p>\u306b\u5bfe\u5fdc\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E3%83%91%E3%83%A9%E3%83%A1%E3%83%BC%E3%82%BF%E6%9B%B4%E6%96%B0-2\"><\/span>\u30d1\u30e9\u30e1\u30fc\u30bf\u66f4\u65b0<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u6b21\u306b\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570\u3092\u5404\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u5fae\u5206\u3057\u3001\u5f0f(20) \u306b\u4ee3\u5165\u3057\u3066\u66f4\u65b0\u5f0f\u3092\u5f97\u307e\u3059\u3002<\/p>\r\n<p>$$ E_\\theta(\\vec{v},\\vec{h}) = -\\sum_i a_i v_i -\\sum_j b_j h_j -\\sum_{i,j} v_iW_{ij}h_j $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E5%8F%AF%E8%A6%96%E3%83%90%E3%82%A4%E3%82%A2%E3%82%B9%E3%81%AE%E5%BE%AE%E5%88%86\"><\/span>\u53ef\u8996\u30d0\u30a4\u30a2\u30b9\u306e\u5fae\u5206<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>$$ \\begin{aligned} \\frac{\\partial E_\\theta(\\vec{v},\\vec{h})}{\\partial a_i} &amp;= \\frac{\\partial}{\\partial a_i} \\left( -\\sum_k a_k v_k -\\sum_j b_j h_j -\\sum_{k,j} v_k W_{kj} h_j \\right) \\\\ &amp;= -v_i \\end{aligned} $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E9%9A%A0%E3%82%8C%E3%83%90%E3%82%A4%E3%82%A2%E3%82%B9%E3%81%AE%E5%BE%AE%E5%88%86\"><\/span>\u96a0\u308c\u30d0\u30a4\u30a2\u30b9\u306e\u5fae\u5206<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>$$ \\begin{aligned} \\frac{\\partial E_\\theta(\\vec{v},\\vec{h})}{\\partial b_j} &amp;= \\frac{\\partial}{\\partial b_j} \\left( -\\sum_i a_i v_i -\\sum_\\ell b_\\ell h_\\ell -\\sum_{i,\\ell} v_i W_{i\\ell} h_\\ell \\right) \\\\\u00a0 &amp;= -h_j \\end{aligned} $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E9%87%8D%E3%81%BF%E3%81%AE%E5%BE%AE%E5%88%86\"><\/span>\u91cd\u307f\u306e\u5fae\u5206<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>$$ \\begin{aligned} \\frac{\\partial E_\\theta(\\vec{v},\\vec{h})}{\\partial W_{ij}} &amp;= \\frac{\\partial}{\\partial W_{ij}} \\left( -\\sum_k a_k v_k -\\sum_\\ell b_\\ell h_\\ell -\\sum_{k,\\ell} v_k W_{k\\ell} h_\\ell \\right) \\\\\u00a0 &amp;= -v_i h_j \\end{aligned} $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E5%8B%BE%E9%85%8D%E5%BC%8F%E3%81%B8%E3%81%AE%E4%BB%A3%E5%85%A5-2\"><\/span>\u52fe\u914d\u5f0f\u3078\u306e\u4ee3\u5165<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u91cd\u307f\u306b\u95a2\u3059\u308b\u52fe\u914d\u306f<\/p>\r\n<p>$$ \\begin{aligned} \\frac{\\partial f}{\\partial W_{ij}} &amp;= &#8211; \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})} \\left[ \\frac{\\partial E_\\theta(\\vec{v},\\vec{h})}{\\partial W_{ij}} \\right] + \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})} \\left[ \\frac{\\partial E_\\theta(\\vec{v},\\vec{h})}{\\partial W_{ij}} \\right] \\\\ &amp;= &#8211; \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})} [-v_i h_j] + \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})} [-v_i h_j] \\quad (\\text{\u2235 } \\tfrac{\\partial E_\\theta(\\vec{v},\\vec{h})}{\\partial W_{ij}}=-v_i h_j) \\\\ &amp;= \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})} [v_i h_j] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})} [v_i h_j] \\end{aligned} $$<\/p>\r\n<p>\u540c\u69d8\u306b\u3001\u53ef\u8996\u30d0\u30a4\u30a2\u30b9\u306b\u3064\u3044\u3066\u306f<\/p>\r\n<p>$$ \\frac{\\partial f}{\\partial a_i} = \\mathbb{E}_{P_{\\text{data}}}[v_i] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i] $$<\/p>\r\n<p>\u96a0\u308c\u30d0\u30a4\u30a2\u30b9\u306b\u3064\u3044\u3066\u306f<\/p>\r\n<p>$$ \\frac{\\partial f}{\\partial b_j} = \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[h_j] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[h_j] $$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E6%9B%B4%E6%96%B0%E5%BC%8F\"><\/span>\u66f4\u65b0\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u3053\u3053\u3067\u3082\u5bfe\u6570\u5c24\u5ea6\u3092\u6700\u5927\u5316\u3059\u308b\u305f\u3081\u3001BM \u3068\u540c\u69d8\u306b\u52fe\u914d\u4e0a\u6607\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002<\/p>\r\n<p>$$ \\theta \\leftarrow \\theta + \\eta \\nabla f(\\theta) $$<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001\u91cd\u307f\u306e\u66f4\u65b0\u5f0f\u306f\u4ee5\u4e0b\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ W_{ij} \\leftarrow W_{ij} + \\eta \\left( \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[v_i h_j] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i h_j] \\right) $$<\/p>\r\n<p>\u540c\u69d8\u306b\u3001\u53ef\u8996\u30d0\u30a4\u30a2\u30b9\u3068\u96a0\u308c\u30d0\u30a4\u30a2\u30b9\u306e\u66f4\u65b0\u306f\u4ee5\u4e0b\u3067\u3059\u3002<\/p>\r\n<p>$$ a_i \\leftarrow a_i + \\eta \\left( \\mathbb{E}_{P_{\\text{data}}}[v_i] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i] \\right) $$<\/p>\r\n<p>$$ b_j \\leftarrow b_j + \\eta \\left( \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[h_j] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[h_j] \\right) $$<\/p>\r\n<p>BM \u306b\u304a\u3044\u3066\u66f4\u65b0\u5f0f\u304c<\/p>\r\n<p>$$ \\mathbb{E}_{\\text{data}}[x_i x_j] &#8211; \\mathbb{E}_\\theta[x_i x_j] $$<\/p>\r\n<p>\u306e\u5f62\u3092\u3068\u3063\u305f\u306e\u306b\u5bfe\u3057\u3001RBM \u3067\u306f<\/p>\r\n<p>$$ \\mathbb{E}_{\\text{data}}[v_i h_j] &#8211; \\mathbb{E}_\\theta[v_i h_j] $$<\/p>\r\n<p>\u306e\u5f62\u306b\u7f6e\u304d\u63db\u308f\u308a\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001BM \u306b\u304a\u3051\u308b\u30e6\u30cb\u30c3\u30c8\u9593\u76f8\u95a2\u304c\u3001RBM \u3067\u306f\u53ef\u8996\u30fb\u96a0\u308c\u9593\u76f8\u95a2\u3068\u3057\u3066\u8868\u308c\u307e\u3059\u3002\u5f0f\u306e\u5f62\u306f\u975e\u5e38\u306b\u3088\u304f\u4f3c\u3066\u3044\u307e\u3059\u304c\u3001RBM \u3067\u306f\u89b3\u6e2c\u5909\u6570\u3068\u6f5c\u5728\u5909\u6570\u306e\u76f8\u95a2\u3092\u5b66\u7fd2\u3057\u3066\u3044\u308b\u70b9\u304c\u7279\u5fb4\u3067\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E6%9D%A1%E4%BB%B6%E4%BB%98%E3%81%8D%E7%8B%AC%E7%AB%8B%E6%80%A7\"><\/span>\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u6027<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>RBM \u304c BM \u3088\u308a\u3082\u6271\u3044\u3084\u3059\u3044\u6700\u5927\u306e\u7406\u7531\u306f\u3001\u69cb\u9020\u5236\u7d04\u306b\u3088\u308a\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u6027\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3067\u3059\u3002BM \u3067\u306f\u4e00\u822c\u306b\u5168\u3066\u306e\u30e6\u30cb\u30c3\u30c8\u304c\u76f8\u4e92\u4f5c\u7528\u3059\u308b\u305f\u3081\u3001\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3067\u3082 1 \u30e6\u30cb\u30c3\u30c8\u305a\u3064\u9806\u756a\u306b\u66f4\u65b0\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3057\u305f\u3002\u4e00\u65b9\u3067 RBM \u3067\u306f\u3001\u53ef\u8996\u5c64\u3068\u96a0\u308c\u5c64\u306e\u4e8c\u90e8\u69cb\u9020\u306b\u3088\u3063\u3066\u3001\u540c\u4e00\u5c64\u5185\u306e\u30e6\u30cb\u30c3\u30c8\u304c\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E9%9A%A0%E3%82%8C%E5%A4%89%E6%95%B0%E3%81%AE%E6%9D%A1%E4%BB%B6%E4%BB%98%E3%81%8D%E5%88%86%E5%B8%83\"><\/span>\u96a0\u308c\u5909\u6570\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u53ef\u8996\u5909\u6570 \\(\\vec v\\) \u3092\u56fa\u5b9a\u3057\u305f\u3068\u304d\u306e\u96a0\u308c\u5909\u6570\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03\u3092\u8003\u3048\u307e\u3059\u3002\u6761\u4ef6\u4ed8\u304d\u5206\u5e03\u306e\u5b9a\u7fa9\u3088\u308a\u3001<\/p>\r\n<p>$$<br \/>P_\\theta(\\vec h \\mid \\vec v)<br \/>= \\frac{P_\\theta(\\vec v,\\vec h)}{P_\\theta(\\vec v)}<br \/>\\propto P_\\theta(\\vec v,\\vec h)<br \/>\\propto \\exp\\bigl(-E_\\theta(\\vec v,\\vec h)\\bigr)<br \/>$$<\/p>\r\n<p>\u3067\u3059\u3002\u3053\u3053\u3067\u3001RBM \u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u95a2\u6570<br \/>$$<br \/>E_\\theta(\\vec v,\\vec h)<br \/>= -\\sum_i a_i v_i &#8211; \\sum_j b_j h_j &#8211; \\sum_{i,j} W_{ij} v_i h_j<br \/>$$<br \/>\u3092\u3001\\(\\vec v\\) \u306b\u306e\u307f\u4f9d\u5b58\u3059\u308b\u9805\u3068 \\(\\vec h\\) \u3092\u542b\u3080\u9805\u306b\u5206\u3051\u308b\u3068\u3001<\/p>\r\n<p>$$<br \/>E_\\theta(\\vec v,\\vec h)<br \/>=<br \/>\\underbrace{-\\sum_i a_i v_i}_{=:C(\\vec v)}<br \/>-\\sum_j h_j \\left( b_j + \\sum_i W_{ij} v_i \\right)<br \/>$$<\/p>\r\n<p>\u3068\u66f8\u3051\u308b\u306e\u3067\u3001<\/p>\r\n<p>$$<br \/>\\begin{aligned}<br \/>P_\\theta(\\vec h \\mid \\vec v)<br \/>&amp;\\propto \\exp\\bigl(-E_\\theta(\\vec v,\\vec h)\\bigr) \\\\<br \/>&amp;= \\exp\\left(-C(\\vec v) + \\sum_j h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right) \\\\<br \/>&amp;= \\exp\\bigl(-C(\\vec v)\\bigr)<br \/>\\prod_j \\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right) \\\\<br \/>&amp;\\propto \\prod_j \\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right).<br \/>\\end{aligned}<br \/>$$<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3001\u6b63\u898f\u5316\u3092\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u305b\u307e\u3059\u3002<\/p>\r\n<p>$$<br \/>\\begin{aligned}<br \/>P_\\theta(\\vec h \\mid \\vec v)<br \/>&amp;=<br \/>\\frac{<br \/>\\prod_j \\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right)<br \/>}{<br \/>\\sum_{\\vec h}<br \/>\\prod_j \\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right)<br \/>} \\\\<br \/>&amp;=<br \/>\\frac{<br \/>\\prod_j \\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right)<br \/>}{<br \/>\\prod_j \\sum_{h_j\\in\\{0,1\\}}<br \/>\\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right)<br \/>} \\\\<br \/>&amp;=<br \/>\\prod_j<br \/>\\frac{<br \/>\\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right)<br \/>}{<br \/>\\sum_{h_j\\in\\{0,1\\}}<br \/>\\exp\\left(h_j\\left(b_j+\\sum_i W_{ij}v_i\\right)\\right)<br \/>}.<br \/>\\end{aligned}<br \/>$$<\/p>\r\n<p>\u3088\u3063\u3066\u3001<\/p>\r\n<p>$$<br \/>P_\\theta(\\vec h \\mid \\vec v)=\\prod_j P_\\theta(h_j\\mid \\vec v) \\tag{21}<br \/>$$<\/p>\r\n<p>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001\\(\\vec v\\) \u3092\u56fa\u5b9a\u3059\u308b\u3068\u3001\u5404\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u306f\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u3001RBM \u3067\u306f\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u540c\u58eb\u306e\u76f8\u4e92\u4f5c\u7528\u9805\u304c\u5b58\u5728\u3057\u306a\u3044\u305f\u3081\u3067\u3059\u3002<\/p>\r\n<p>\u3055\u3089\u306b\u3001\u5404 \\(j\\) \u306b\u3064\u3044\u3066<br \/>$$<br \/>\\lambda_j := b_j+\\sum_i W_{ij}v_i<br \/>$$<br \/>\u3068\u304a\u304f\u3068\u3001\u4e0a\u5f0f\u3088\u308a<\/p>\r\n<p>$$<br \/>P_\\theta(h_j\\mid \\vec v)<br \/>=<br \/>\\frac{\\exp(h_j\\lambda_j)}{\\sum_{h_j\\in\\{0,1\\}}\\exp(h_j\\lambda_j)}<br \/>=<br \/>\\frac{\\exp(h_j\\lambda_j)}{1+\\exp(\\lambda_j)}<br \/>$$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\\(h_j=1\\) \u306e\u5834\u5408\u306b\u306f<\/p>\r\n<p>$$<br \/>\\begin{aligned}<br \/>P_\\theta(H_j=1\\mid \\vec v)<br \/>&amp;=<br \/>\\frac{\\exp(\\lambda_j)}{1+\\exp(\\lambda_j)} \\\\<br \/>&amp;=<br \/>\\frac{1}{1+\\exp(-\\lambda_j)} \\\\<br \/>&amp;=<br \/>\\sigma(\\lambda_j).<br \/>\\end{aligned}<br \/>$$<\/p>\r\n<p>\u3086\u3048\u306b\u3001<\/p>\r\n<p>$$<br \/>P_\\theta(H_j=1\\mid \\vec v)<br \/>=<br \/>\\sigma\\left(b_j+\\sum_i W_{ij}v_i\\right) \\tag{22}<br \/>$$<\/p>\r\n<p>\u3092\u5f97\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E5%8F%AF%E8%A6%96%E5%A4%89%E6%95%B0%E3%81%AE%E6%9D%A1%E4%BB%B6%E4%BB%98%E3%81%8D%E5%88%86%E5%B8%83\"><\/span>\u53ef\u8996\u5909\u6570\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u540c\u69d8\u306b\u3001\u96a0\u308c\u5909\u6570 <span class=\"math-inline\">\\(\\vec{h}\\)<\/span> \u3092\u56fa\u5b9a\u3057\u305f\u3068\u304d\u3001\u53ef\u8996\u5909\u6570\u306e\u6761\u4ef6\u4ed8\u304d\u5206\u5e03\u3082\u5206\u89e3\u3055\u308c\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u6b21\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\r\n<p>$$ P_\\theta(\\vec{v}\\mid \\vec{h}) = \\prod_i P_\\theta(v_i\\mid \\vec{h}) \\tag{23} $$<\/p>\r\n<p>\u5404\u53ef\u8996\u30e6\u30cb\u30c3\u30c8\u306e\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306f\u6b21\u5f0f\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ P_\\theta(V_i=1\\mid \\vec{h}) = \\sigma \\left( a_i+\\sum_j W_{ij}h_j \\right) \\tag{24} $$<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"BM%E3%81%A8%E3%81%AE%E6%AF%94%E8%BC%83\"><\/span>BM\u3068\u306e\u6bd4\u8f03<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>BM \u3067\u306f\u3001\u4e00\u822c\u306b <span class=\"math-inline\">\\(P(X_i\\mid \\vec{X}_{-i})\\)<\/span> \u3092 1 \u30e6\u30cb\u30c3\u30c8\u305a\u3064\u66f4\u65b0\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3057\u305f\u3002\u4e00\u65b9 RBM \u3067\u306f\u3001\u5f0f(21) \u304a\u3088\u3073\u5f0f(23) \u304c\u6210\u7acb\u3059\u308b\u305f\u3081\u3001\u540c\u4e00\u5c64\u5185\u306e\u30e6\u30cb\u30c3\u30c8\u3092\u4e26\u5217\u306b\u66f4\u65b0\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u9055\u3044\u306f\u8a08\u7b97\u91cf\u306b\u76f4\u7d50\u3057\u307e\u3059\u3002BM \u3067\u306f 1 \u30e6\u30cb\u30c3\u30c8\u305a\u3064\u9806\u756a\u306b\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3059\u308b\u305f\u3081\u6642\u9593\u304c\u304b\u304b\u308a\u307e\u3057\u305f\u304c\u3001RBM \u3067\u306f\u53ef\u8996\u5c64\u307e\u305f\u306f\u96a0\u308c\u5c64\u3092\u307e\u3068\u3081\u3066\u66f4\u65b0\u3067\u304d\u308b\u305f\u3081\u3001\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u304c\u5927\u5e45\u306b\u9ad8\u901f\u5316\u3055\u308c\u307e\u3059\u3002\u3053\u3053\u304c\u3001RBM \u304c BM \u3088\u308a\u3082\u5b9f\u7528\u7684\u3067\u3042\u308b\u6700\u5927\u306e\u7406\u7531\u3067\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"RBM%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8B%E5%85%B7%E4%BD%93%E7%9A%84%E3%81%AA%E8%A8%88%E7%AE%97%E6%96%B9%E6%B3%95%EF%BC%88%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0%EF%BC%89\"><\/span>RBM\u306b\u304a\u3051\u308b\u5177\u4f53\u7684\u306a\u8a08\u7b97\u65b9\u6cd5\uff08\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff09<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u3053\u3053\u307e\u3067\u3067\u3001RBM \u306e\u5b66\u7fd2\u306b\u5fc5\u8981\u306a\u66f4\u65b0\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u6b21\u306b\u3001BM \u306e\u5834\u5408\u3068\u540c\u69d8\u306b\u3001\u3053\u308c\u3089\u306e\u671f\u5f85\u5024\u3092\u5b9f\u969b\u306b\u3069\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u306e\u304b\u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E3%83%87%E3%83%BC%E3%82%BF%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97-2\"><\/span>\u30c7\u30fc\u30bf\u671f\u5f85\u5024\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u30c7\u30fc\u30bf\u5074\u306e\u671f\u5f85\u5024\u306f\u3001BM \u306e\u5834\u5408\u3068\u540c\u69d8\u306b\u6a19\u672c\u5e73\u5747\u3068\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u5b66\u7fd2\u30c7\u30fc\u30bf\u304c<\/p>\r\n<p>$$ \\vec{v}^{(1)}, \\vec{v}^{(2)}, \\dots, \\vec{v}^{(M)} $$<\/p>\r\n<p>\u3068\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3068\u3059\u308b\u3068\u3001\u4f8b\u3048\u3070\u53ef\u8996\u30d0\u30a4\u30a2\u30b9\u306e\u30c7\u30fc\u30bf\u671f\u5f85\u5024\u306f<\/p>\r\n<p>$$ \\mathbb{E}_{\\text{data}}[v_i] \\approx \\frac{1}{M} \\sum_{m=1}^{M} v_i^{(m)} $$<\/p>\r\n<p>\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<p>\u4e00\u65b9\u3001\u91cd\u307f\u306e\u66f4\u65b0\u306b\u73fe\u308c\u308b<\/p>\r\n<p>$$ \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[v_i h_j] $$<\/p>\r\n<p>\u306f\u3001\u30c7\u30fc\u30bf <span class=\"math-inline\">\\(\\vec{v}^{(m)}\\)<\/span> \u3092\u4e0e\u3048\u305f\u3068\u304d\u306e\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u306e\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001<\/p>\r\n<p>$$ \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[v_i h_j] \\approx \\frac{1}{M} \\sum_{m=1}^{M} v_i^{(m)} P_\\theta(H_j=1\\mid \\vec{v}^{(m)}) $$<\/p>\r\n<p>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u304c BM \u3068\u7570\u306a\u308b\u70b9\u3067\u3042\u308a\u3001RBM \u3067\u306f\u96a0\u308c\u5909\u6570\u304c\u89b3\u6e2c\u3055\u308c\u306a\u3044\u305f\u3081\u3001\u30c7\u30fc\u30bf\u304b\u3089\u76f4\u63a5\u76f8\u95a2\u3092\u6570\u3048\u308b\u306e\u3067\u306f\u306a\u304f\u3001\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u3092\u4ecb\u3057\u3066\u88dc\u3044\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"%E3%83%A2%E3%83%87%E3%83%AB%E6%9C%9F%E5%BE%85%E5%80%A4%E3%81%AE%E8%A8%88%E7%AE%97-2\"><\/span>\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u4e00\u65b9\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024<\/p>\r\n<p>$$ \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i h_j] $$<\/p>\r\n<p>\u306f\u3001BM \u3068\u540c\u69d8\u306b\u53b3\u5bc6\u8a08\u7b97\u304c\u56f0\u96e3\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u30e2\u30c7\u30eb\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30eb\u3092\u751f\u6210\u3057\u3001\u30b5\u30f3\u30d7\u30eb\u5e73\u5747\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u3053\u3067 RBM \u306e\u5229\u70b9\u304c\u73fe\u308c\u307e\u3059\u3002BM \u3067\u306f 1 \u30e6\u30cb\u30c3\u30c8\u305a\u3064\u3057\u304b\u66f4\u65b0\u3067\u304d\u307e\u305b\u3093\u3067\u3057\u305f\u304c\u3001RBM \u3067\u306f\u5f0f(21) \u3068\u5f0f(23) \u306b\u3088\u308a\u3001\u96a0\u308c\u5c64\u306f\u53ef\u8996\u5c64\u3092\u56fa\u5b9a\u3059\u308b\u3068\u4e26\u5217\u66f4\u65b0\u3067\u304d\u3001\u53ef\u8996\u5c64\u306f\u96a0\u308c\u5c64\u3092\u56fa\u5b9a\u3059\u308b\u3068\u4e26\u5217\u66f4\u65b0\u3067\u304d\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u306f<\/p>\r\n<ol>\r\n<li>\u53ef\u8996\u30d9\u30af\u30c8\u30eb <span class=\"math-inline\">\\(\\vec{v}\\)<\/span> \u3092\u56fa\u5b9a\u3057\u3001\u5404\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u3092\u4e26\u5217\u306b\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3059\u308b<\/li>\r\n<li>\u5f97\u3089\u308c\u305f\u96a0\u308c\u30d9\u30af\u30c8\u30eb <span class=\"math-inline\">\\(\\vec{h}\\)<\/span> \u3092\u56fa\u5b9a\u3057\u3001\u5404\u53ef\u8996\u30e6\u30cb\u30c3\u30c8\u3092\u4e26\u5217\u306b\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3059\u308b<\/li>\r\n<\/ol>\r\n<p>\u3068\u3044\u3046 2 \u6bb5\u968e\u3067\u5b9f\u884c\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u5f97\u3089\u308c\u305f\u30b5\u30f3\u30d7\u30eb\u5217<\/p>\r\n<p>$$ (\\vec{v}^{[1]},\\vec{h}^{[1]}), (\\vec{v}^{[2]},\\vec{h}^{[2]}), \\dots, (\\vec{v}^{[K]},\\vec{h}^{[K]}) $$<\/p>\r\n<p>\u3092\u7528\u3044\u308b\u3068\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306f<\/p>\r\n<p>$$ \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i h_j] \\approx \\frac{1}{K} \\sum_{k=1}^{K} v_i^{[k]} h_j^{[k]} $$<\/p>\r\n<p>\u3068\u3057\u3066\u8fd1\u4f3c\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<h3><span class=\"ez-toc-section\" id=\"RBM_%E3%81%AE%E5%AD%A6%E7%BF%92%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0\"><\/span>RBM \u306e\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<span class=\"ez-toc-section-end\"><\/span><\/h3>\r\n<p>\u4ee5\u4e0a\u3092\u307e\u3068\u3081\u308b\u3068\u3001RBM \u306e\u5b66\u7fd2\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<ol>\r\n<li>\u5b66\u7fd2\u30c7\u30fc\u30bf\u304b\u3089\u3001\u53ef\u8996\u30e6\u30cb\u30c3\u30c8\u306e\u5e73\u5747 <span class=\"math-inline\">\\(\\mathbb{E}_{\\text{data}}[v_i]\\)<\/span> \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u5404\u30c7\u30fc\u30bf <span class=\"math-inline\">\\(\\vec{v}^{(m)}\\)<\/span> \u306b\u5bfe\u3057\u3066 <span class=\"math-inline\">\\(P_\\theta(H_j=1\\mid \\vec{v}^{(m)})\\)<\/span> \u3092\u8a08\u7b97\u3057\u3001\u30c7\u30fc\u30bf\u5074\u306e\u76f8\u95a2 <span class=\"math-inline\">\\(\\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[v_i h_j]\\)<\/span> \u3092\u6c42\u3081\u307e\u3059\u3002<\/li>\r\n<li>\u73fe\u5728\u306e\u30d1\u30e9\u30e1\u30fc\u30bf <span class=\"math-inline\">\\(\\theta\\)<\/span> \u306e\u3082\u3068\u3067\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092\u5b9f\u884c\u3057\u3001\u30e2\u30c7\u30eb\u30b5\u30f3\u30d7\u30eb <span class=\"math-inline\">\\((\\vec{v}^{[k]},\\vec{h}^{[k]})\\)<\/span> \u3092\u751f\u6210\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u305d\u306e\u30b5\u30f3\u30d7\u30eb\u5217\u304b\u3089\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i h_j]\\)<\/span> \u3092\u30b5\u30f3\u30d7\u30eb\u5e73\u5747\u3068\u3057\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u66f4\u65b0\u5f0f\u306b\u4ee3\u5165\u3057\u3001 <span class=\"math-inline\">\\(W_{ij}\\)<\/span>, <span class=\"math-inline\">\\(a_i\\)<\/span>, <span class=\"math-inline\">\\(b_j\\)<\/span> \u3092\u66f4\u65b0\u3057\u307e\u3059\u3002<\/li>\r\n<\/ol>\r\n<h2><span class=\"ez-toc-section\" id=\"%E3%81%BE%E3%81%A8%E3%82%81\"><\/span>\u307e\u3068\u3081<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>RBM \u3067\u306f\u3001\u53ef\u8996\u5909\u6570\u3068\u96a0\u308c\u5909\u6570\u306e\u9593\u3060\u3051\u306b\u7d50\u5408\u3092\u8a31\u3059\u3053\u3068\u3067\u3001BM \u3088\u308a\u3082\u5358\u7d14\u306a\u69cb\u9020\u3092\u5f97\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u69cb\u9020\u5236\u7d04\u306b\u3088\u308a\u3001\u5b66\u7fd2\u306e\u52fe\u914d\u306f\u6b21\u306e\u5f62\u306b\u6574\u7406\u3055\u308c\u307e\u3059\u3002<\/p>\r\n<p>$$ \\mathbb{E}_{\\text{data}}[v_i h_j] &#8211; \\mathbb{E}_{\\text{model}}[v_i h_j] $$<\/p>\r\n<p>\u3055\u3089\u306b\u3001\u6761\u4ef6\u4ed8\u304d\u72ec\u7acb\u6027\u306b\u3088\u308a\u5f0f(22) \u304a\u3088\u3073\u5f0f(24) \u304c\u6210\u308a\u7acb\u3064\u305f\u3081\u3001BM \u306b\u6bd4\u3079\u3066\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3068\u671f\u5f85\u5024\u8a08\u7b97\u304c\u306f\u308b\u304b\u306b\u6271\u3044\u3084\u3059\u304f\u306a\u308a\u307e\u3059\u3002\u7279\u306b\u3001BM \u3067\u306f 1 \u30e6\u30cb\u30c3\u30c8\u305a\u3064\u3057\u304b\u66f4\u65b0\u3067\u304d\u306a\u304b\u3063\u305f\u306e\u306b\u5bfe\u3057\u3001RBM \u3067\u306f\u53ef\u8996\u5c64\u30fb\u96a0\u308c\u5c64\u3092\u307e\u3068\u3081\u3066\u4e26\u5217\u66f4\u65b0\u3067\u304d\u308b\u70b9\u304c\u5927\u304d\u306a\u5229\u70b9\u3067\u3059\u3002<\/p>\r\n<p>\u3082\u3063\u3068\u3082\u3001RBM \u306b\u304a\u3044\u3066\u3082\u30e2\u30c7\u30eb\u671f\u5f85\u5024 <span class=\"math-inline\">\\(\\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i h_j]\\)<\/span> \u3092\u53b3\u5bc6\u306b\u6c42\u3081\u308b\u3053\u3068\u306f\u96e3\u3057\u304f\u3001\u4f9d\u7136\u3068\u3057\u3066\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u304c\u5fc5\u8981\u3067\u3059\u3002\u6b21\u306e\u7bc0\u3067\u306f\u3001\u3053\u306e\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u3092\u3055\u3089\u306b\u52b9\u7387\u3088\u304f\u8fd1\u4f3c\u3059\u308b\u305f\u3081\u306b\u63d0\u6848\u3055\u308c\u305f\u30b3\u30f3\u30c8\u30e9\u30b9\u30c6\u30a3\u30d6\u30fb\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\uff08CD\u6cd5\uff09\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E3%82%B3%E3%83%B3%E3%83%88%E3%83%A9%E3%82%B9%E3%83%86%E3%82%A3%E3%83%96%E3%83%BB%E3%83%80%E3%82%A4%E3%83%90%E3%83%BC%E3%82%B8%E3%82%A7%E3%83%B3%E3%82%B9%EF%BC%88CD%E6%B3%95%EF%BC%89\"><\/span>\u30b3\u30f3\u30c8\u30e9\u30b9\u30c6\u30a3\u30d6\u30fb\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\uff08CD\u6cd5\uff09<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u524d\u7bc0\u3067\u306f\u3001RBM \u306e\u5b66\u7fd2\u52fe\u914d\u304c<\/p>\r\n<p>$$ \\frac{\\partial f}{\\partial W_{ij}} = \\mathbb{E}_{P_{\\text{data}}(\\vec{v})P_\\theta(\\vec{h}\\mid \\vec{v})}[v_i h_j] &#8211; \\mathbb{E}_{P_\\theta(\\vec{V},\\vec{H})}[v_i h_j] $$<\/p>\r\n<p>\u3068\u306a\u308b\u3053\u3068\u3092\u5c0e\u304d\u307e\u3057\u305f\u3002\u3053\u3053\u3067\u554f\u984c\u3068\u306a\u308b\u306e\u304c\u7b2c\u4e8c\u9805\u3067\u3042\u308a\u3001BM\u540c\u69d8\u3001\u76f4\u63a5\u8a08\u7b97\u306f\u4e0d\u53ef\u80fd\u3067\u3059\u3002<\/p>\r\n<p>\u305d\u3053\u3067\u3001\u30e2\u30c7\u30eb\u5206\u5e03\u304b\u3089\u30b5\u30f3\u30d7\u30eb\u3092\u751f\u6210\u3057\u3001\u30b5\u30f3\u30d7\u30eb\u5e73\u5747\u306b\u3088\u3063\u3066\u671f\u5f85\u5024\u3092\u8fd1\u4f3c\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u305f\u3081\u306b\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092\u7528\u3044\u307e\u3059\u3002<\/p>\r\n<p>\u3057\u304b\u3057\u3001\u3053\u3053\u3067\u3082\u65b0\u305f\u306a\u554f\u984c\u304c\u751f\u3058\u307e\u3059\u3002\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u306f\u7406\u8ad6\u7684\u306b\u306f\u30e2\u30c7\u30eb\u5206\u5e03\u306b\u53ce\u675f\u3057\u307e\u3059\u304c\u3001\u305d\u306e\u305f\u3081\u306b\u306f\u975e\u5e38\u306b\u591a\u304f\u306e\u30b9\u30c6\u30c3\u30d7\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u3064\u307e\u308a\u3001<\/p>\r\n<ul>\r\n<li>\u5404\u66f4\u65b0\u3067\u9577\u3044\u30de\u30eb\u30b3\u30d5\u9023\u9396\u3092\u56de\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/li>\r\n<li>\u5b66\u7fd2\u5168\u4f53\u304c\u975e\u5e38\u306b\u9045\u304f\u306a\u308a\u307e\u3059\u3002<\/li>\r\n<\/ul>\r\n<p>\u3068\u3044\u3046\u554f\u984c\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u554f\u984c\u3092\u7de9\u548c\u3059\u308b\u305f\u3081\u306b\u63d0\u6848\u3055\u308c\u305f\u306e\u304c <strong>\u30b3\u30f3\u30c8\u30e9\u30b9\u30c6\u30a3\u30d6\u30fb\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\uff08Contrastive Divergence; CD\uff09<\/strong> \u3067\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"%E3%82%B3%E3%83%B3%E3%83%88%E3%83%A9%E3%82%B9%E3%83%86%E3%82%A3%E3%83%96%E3%83%BB%E3%83%80%E3%82%A4%E3%83%90%E3%83%BC%E3%82%B8%E3%82%A7%E3%83%B3%E3%82%B9%E3%81%AE%E5%9F%BA%E6%9C%AC%E7%9A%84%E3%81%AA%E8%80%83%E3%81%88%E6%96%B9\"><\/span>\u30b3\u30f3\u30c8\u30e9\u30b9\u30c6\u30a3\u30d6\u30fb\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\u306e\u57fa\u672c\u7684\u306a\u8003\u3048\u65b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>CD \u6cd5\u306e\u57fa\u672c\u7684\u306a\u30a2\u30a4\u30c7\u30a2\u306f\u6b21\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\r\n<p>\u901a\u5e38\u306f\u30e2\u30c7\u30eb\u5206\u5e03\u306b\u5341\u5206\u8fd1\u3065\u304f\u307e\u3067\u9577\u3044\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092\u884c\u3046\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u308c\u306b\u5bfe\u3057\u3066 CD \u6cd5\u3067\u306f\u3001<\/p>\r\n<ul>\r\n<li>\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3092 <strong>\u30c7\u30fc\u30bf\u70b9\u304b\u3089\u958b\u59cb\u3057\u307e\u3059\u3002<\/strong><\/li>\r\n<li>\u6570\u30b9\u30c6\u30c3\u30d7\u306e\u30ae\u30d6\u30b9\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3060\u3051\u884c\u3044\u307e\u3059\u3002<\/li>\r\n<li>\u305d\u306e\u7d50\u679c\u3092\u30e2\u30c7\u30eb\u30b5\u30f3\u30d7\u30eb\u3068\u3057\u3066\u7528\u3044\u307e\u3059\u3002<\/li>\r\n<\/ul>\r\n<p>\u3068\u3044\u3046\u8fd1\u4f3c\u3092\u884c\u3044\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u3068\u304d\u3001\u30ae\u30d6\u30b9\u66f4\u65b0\u3092 <span class=\"math-inline\">\\(k\\)<\/span> \u56de\u884c\u3046 CD \u6cd5\u3092 <strong>CD-\\(k\\)<\/strong> \u3068\u547c\u3073\u307e\u3059\u3002\u5b9f\u969b\u306b\u306f <span class=\"math-inline\">\\(k=1\\)<\/span> \u306e CD-1 \u304c\u3088\u304f\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/p>\r\n<h2><span class=\"ez-toc-section\" id=\"CD-1_%E3%81%AE%E3%82%A2%E3%83%AB%E3%82%B4%E3%83%AA%E3%82%BA%E3%83%A0\"><\/span>CD-1 \u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>CD-1 \u306e\u5b66\u7fd2\u624b\u9806\u306f\u6b21\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\r\n<ol>\r\n<li>\u30c7\u30fc\u30bf <span class=\"math-inline\">\\(\\vec{v}^{(0)}\\)<\/span> \u3092\u5165\u529b\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u3092\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3057\u307e\u3059\u3002<\/li>\r\n<\/ol>\r\n<p>$$ P(H_j=1\\mid \\vec{v}^{(0)}) = \\sigma \\left( b_j+\\sum_i W_{ij}v_i^{(0)} \\right) $$<\/p>\r\n<ol start=\"3\">\r\n<li>\u5f97\u3089\u308c\u305f <span class=\"math-inline\">\\(\\vec{h}^{(0)}\\)<\/span> \u304b\u3089\u53ef\u8996\u30e6\u30cb\u30c3\u30c8\u3092\u518d\u69cb\u6210\u3057\u307e\u3059\u3002<\/li>\r\n<\/ol>\r\n<p>$$ P(V_i=1\\mid \\vec{h}^{(0)}) = \\sigma \\left( a_i+\\sum_j W_{ij}h_j^{(0)} \\right) $$<\/p>\r\n<ol start=\"4\">\r\n<li>\u518d\u69cb\u6210\u3055\u308c\u305f\u53ef\u8996\u30d9\u30af\u30c8\u30eb\u3092 <span class=\"math-inline\">\\(\\vec{v}^{(1)}\\)<\/span> \u3068\u3057\u307e\u3059\u3002<\/li>\r\n<li>\u3082\u3046\u4e00\u5ea6\u96a0\u308c\u30e6\u30cb\u30c3\u30c8\u3092\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3057\u307e\u3059\u3002<\/li>\r\n<\/ol>\r\n<p>$$ P(H_j=1\\mid \\vec{v}^{(1)}) = \\sigma \\left( b_j+\\sum_i W_{ij}v_i^{(1)} \\right) $$<\/p>\r\n<p>\u3053\u306e\u3088\u3046\u306b\u3001CD-1 \u3067\u306f\u300c\u30c7\u30fc\u30bf\u304b\u3089 1 \u30b9\u30c6\u30c3\u30d7\u3060\u3051\u518d\u69cb\u6210\u3057\u305f\u30b5\u30f3\u30d7\u30eb\u300d\u3092\u7528\u3044\u3066\u3001\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u3092\u8fd1\u4f3c\u3057\u307e\u3059\u3002<\/p>\r\n<div class=\"bm-rbm-article\">\r\n<h2><span class=\"ez-toc-section\" id=\"%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE\"><\/span>\u53c2\u8003\u6587\u732e<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<ul>\r\n<li>\u6050\u795e\u8cb4\u884c, \u6570\u7406\u3067\u3072\u3082\u3068\u304fAI\u6280\u8853\u306e\u6df1\u5316\u2015\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\u3068\u305f\u3069\u308b\u6700\u5148\u7aef\u3078\u306e\u9053\u2015, \u30b3\u30ed\u30ca\u793e, 2025.<\/li>\r\n<\/ul>\r\n<p>&nbsp;<\/p>\r\n<\/div>\r\n<h2><span class=\"ez-toc-section\" id=\"%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E4%BD%9C%E6%88%90%E8%80%85\"><\/span>\u672c\u8a18\u4e8b\u306e\u4f5c\u6210\u8005<span class=\"ez-toc-section-end\"><\/span><\/h2>\r\n<p>\u9e7f\u5185\u601c\u592e<\/p>\r\n<\/div>\r\n","protected":false},"excerpt":{"rendered":"<p>\u6982\u8981 \u672c\u7a3f\u3067\u306f\u3001\u307e\u305a\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08BM\uff09\u306e\u5b66\u7fd2\u65b9\u6cd5\u306b\u3064\u3044\u3066\u3001\u306a\u308b\u3079\u304f\u5f0f\u5909\u5f62\u3092\u7701\u7565\u305b\u305a\u306b\u89e3\u8aac\u3057\u307e\u3059\u3002\u7d50\u8ad6\u304b\u3089\u8ff0\u3079\u308b\u3068\u3001BM\u3067\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u66f4\u65b0\u306e\u969b\u306b\u5fc5\u8981\u3068\u306a\u308b\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97\u91cf\u304c\u975e\u5e38\u306b\u5927\u304d\u304f\u3001\u5b9f\u7528\u4e0a\u306e\u5927\u304d\u306a\u8ab2\u984c\u3068\u306a\u308a\u307e [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-8733","post","type-post","status-publish","format-standard","hentry","category-glossary"],"yoast_head":"<!-- This site is optimized with the 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content=\"\u672c\u7a3f\u3067\u306f\u3001\u307e\u305a\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08BM\uff09\u306e\u5b66\u7fd2\u65b9\u6cd5\u306b\u3064\u3044\u3066\u3001\u306a\u308b\u3079\u304f\u5f0f\u5909\u5f62\u3092\u7701\u7565\u305b\u305a\u306b\u89e3\u8aac\u3057\u307e\u3059\u3002\u7d50\u8ad6\u304b\u3089\u8ff0\u3079\u308b\u3068\u3001BM\u3067\u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u66f4\u65b0\u306e\u969b\u306b\u5fc5\u8981\u3068\u306a\u308b\u30e2\u30c7\u30eb\u671f\u5f85\u5024\u306e\u8a08\u7b97\u91cf\u304c\u975e\u5e38\u306b\u5927\u304d\u304f\u3001\u5b9f\u7528\u4e0a\u306e\u5927\u304d\u306a\u8ab2\u984c\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u63d0\u6848\u3055\u308c\u305f\u306e\u304c\u3001\u5236\u9650\u30dc\u30eb\u30c4\u30de\u30f3\u30de\u30b7\u30f3\uff08RBM\uff09\u3067\u3059\u3002RBM\u3067\u306f\u3001\u30e6\u30cb\u30c3\u30c8\u306e\u63a5\u7d9a\u65b9\u6cd5\u3092\u5de5\u592b\u3059\u308b\u3053\u3068\u3067\u4e26\u5217\u8a08\u7b97\u3092\u53ef\u80fd\u306b\u3057\u3001\u8a08\u7b97\u91cf\u3092\u5927\u5e45\u306b\u6291\u3048\u308b\u3053\u3068\u306b\u6210\u529f\u3057\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u30e2\u30c7\u30eb\u306e\u69cb\u9020\u3084\u904e\u7a0b\u3092\u5de5\u592b\u3057\u3066\u8a08\u7b97\u3092\u5bb9\u6613\u306b\u3059\u308b\u3068\u3044\u3046\u767a\u60f3\u306f\u3001\u751f\u6210\u30e2\u30c7\u30eb\u306e\u7814\u7a76\u306b\u304a\u3044\u3066\u91cd\u8981\u306a\u8003\u3048\u65b9\u306e\u4e00\u3064\u3067\u3059\u3002\" 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