{"id":397,"date":"2018-07-09T15:00:35","date_gmt":"2018-07-09T06:00:35","guid":{"rendered":"http:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=397"},"modified":"2018-07-09T15:00:35","modified_gmt":"2018-07-09T06:00:35","slug":"%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/","title":{"rendered":"\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 &#8220;Nonnegative\/binary matrix factorization with a D-Wave quantum annealer&#8221; by Daniel O\u2019Malley, et al. (2017)"},"content":{"rendered":"\n<p>\u672c\u8a18\u4e8b\u3067\u7d39\u4ecb\u3059\u308b\u8ad6\u6587\u3067\u306f\u3001D-Wave\u30de\u30b7\u30f3\u3092\u7528\u3044\u305f\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u306b\u3088\u308a\u4e8c\u5024\u5909\u6570\u5236\u7d04\u306e\u3064\u3044\u305f \u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 \u306e\u624b\u6cd5\u3068\u9854\u753b\u50cf\u8a8d\u8b58\u306b\u5fdc\u7528\u3057\u305f\u7d50\u679c\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u901f\u5ea6\u6027\u80fd\u8a55\u4fa1\u306e\u305f\u3081\u306b\u3001\u53e4\u5178\u7684\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7528\u3044\u308bGurobi\u3068qbsolv\uff08\u30bf\u30d6\u30fc\u30b5\u30fc\u30c1\uff09\u3092\u7528\u3044\u305f\u5834\u5408\u3068D-Wave 2X\u3067\u5f97\u3089\u308c\u305f\u6700\u9069\u89e3\u306b\u9039\u3059\u308b\u7d2f\u7a4dTTT (time-to-target)\u3092\u6e2c\u308a\u3001\u305d\u306e\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u56de\u6570\u4f9d\u5b58\u6027\u3092\u6bd4\u8f03\u30fb\u691c\u8a0e\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 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 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\" >\u6587\u732e\u60c5\u5831<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E9%9D%9E%E8%B2%A0%E5%80%A4%E8%A1%8C%E5%88%97%E5%88%86%E8%A7%A3_NMF%E3%81%A8%E3%81%AF\" >\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 (NMF)\u3068\u306f<\/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\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E6%9C%AC%E8%AB%96%E6%96%87%E3%81%A7%E6%89%B1%E3%81%86%E6%89%8B%E6%B3%95_-%E4%BA%8C%E5%80%A4%E5%88%B6%E7%B4%84%E9%9D%9E%E8%B2%A0%E5%80%A4%E8%A1%8C%E5%88%97%E5%88%86%E8%A7%A3_NBNF\" >\u672c\u8ad6\u6587\u3067\u6271\u3046\u624b\u6cd5 -\u4e8c\u5024\u5236\u7d04\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 (NBNF)-<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E9%87%8F%E5%AD%90%E3%82%A2%E3%83%8B%E3%83%BC%E3%83%AA%E3%83%B3%E3%82%B0%E3%82%92%E3%81%A9%E3%81%93%E3%81%AB%E7%94%A8%E3%81%84%E3%82%8B%E3%81%8B\" >\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u3069\u3053\u306b\u7528\u3044\u308b\u304b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#D-Wave%E3%83%9E%E3%82%B7%E3%83%B3%E3%81%AB%E5%AE%9F%E8%A3%85%E3%81%99%E3%82%8B%E3%81%9F%E3%82%81%E3%81%AEQUBO%E8%A1%A8%E7%8F%BE\" >D-Wave\u30de\u30b7\u30f3\u306b\u5b9f\u88c5\u3059\u308b\u305f\u3081\u306eQUBO\u8868\u73fe<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E7%B5%90%E6%9E%9C\" >\u7d50\u679c<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E9%A1%94%E7%94%BB%E5%83%8F%E8%AA%8D%E8%AD%98%E3%81%B8%E3%81%AE%E5%BF%9C%E7%94%A8%E7%B5%90%E6%9E%9C\" >\u9854\u753b\u50cf\u8a8d\u8b58\u3078\u306e\u5fdc\u7528\u7d50\u679c<\/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\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E4%BA%8C%E5%80%A4%E5%88%B6%E7%B4%84%E4%BB%98%E3%81%8D%E3%81%AB%E9%99%90%E5%AE%9A%E3%81%97%E3%81%9F%E5%BD%B1%E9%9F%BF\" >\u4e8c\u5024\u5236\u7d04\u4ed8\u304d\u306b\u9650\u5b9a\u3057\u305f\u5f71\u97ff<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E5%8F%A4%E5%85%B8%E3%82%BD%E3%83%AB%E3%83%90%E3%83%BC%E3%81%A8%E3%81%AE%E9%80%9F%E5%BA%A6%E6%80%A7%E8%83%BD%E6%AF%94%E8%BC%83\" >\u53e4\u5178\u30bd\u30eb\u30d0\u30fc\u3068\u306e\u901f\u5ea6\u6027\u80fd\u6bd4\u8f03<\/a><\/li><\/ul><\/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\/2018\/07\/09\/%e9%9d%9e%e8%b2%a0%e5%80%a4%e8%a1%8c%e5%88%97%e5%88%86%e8%a7%a3-nonnegative-binary-matrix-factorization-with-a-d-wave-quantum-annealer-by-daniel-omalley-et-al-2017\/#%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E6%8B%85%E5%BD%93%E8%80%85\" >\u672c\u8a18\u4e8b\u306e\u62c5\u5f53\u8005<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\"><\/span>\u6587\u732e\u60c5\u5831<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Daniel O\u2019Malley, Velimir V. Vesselinov, Boian S. Alexandrov, Ludmil B. Alexandrov, &#8220;Nonnegative\/binary matrix factorization with a D-Wave quantum annealer&#8221;, <a href=\"https:\/\/arxiv.org\/abs\/1704.01605\">arXiv:1704.01605<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E9%9D%9E%E8%B2%A0%E5%80%A4%E8%A1%8C%E5%88%97%E5%88%86%E8%A7%A3_NMF%E3%81%A8%E3%81%AF\"><\/span>\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 (NMF)\u3068\u306f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 (Nonnegative Matrix Factorization; NMF) \u306f\u3001$n \\times m$\u884c\u5217$V$\u3092\u5168\u3066\u975e\u8ca0\u5024\u306e\u8981\u7d20\u304b\u3089\u306a\u308b$n \\times k$\u884c\u5217$W$\u3068$k \\times m$\u884c\u5217$H$\u306e\u7a4d\u3067\u8868\u3059\u3082\u306e\u3067\u3042\u308b\u3002\u5b8c\u5168\u306a\u5206\u89e3\u304c\u5f97\u3089\u308c\u306a\u3044\u5834\u5408\u3067\u3082\u3001\u4e21\u8fba\u306e\u5dee\u304c\u6700\u3082\u5c0f\u3055\u304f\u306a\u308b\u3088\u3046\u306a\u8fd1\u4f3c\u5024\u3092\u5f97\u308b\u305f\u3081\u306e\u53e4\u5178\u7684\u306a\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u3044\u304f\u3064\u304b\u77e5\u3089\u308c\u3066\u304a\u308a\u3001\u4f8b\u3048\u3070[<a href=\"https:\/\/www.nature.com\/articles\/44565\" rel=\"noopener\" target=\"_blank\">D.D. Lee and H.S. Seung, Nature <strong>401<\/strong> (1999) 788.<\/a>]\u306f\u305d\u306e\u4e00\u4f8b\u3067\u3042\u308a\u3001\u307e\u305f\u9854\u753b\u50cf\u8a8d\u8b58\u3078\u306e\u5fdc\u7528\u304c\u793a\u3055\u308c\u3066\u3044\u308b\u3002\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3\u306f\u3053\u306e\u3088\u3046\u306b\u30d1\u30bf\u30fc\u30f3\u8a8d\u8b58\u3084\u6642\u7cfb\u5217\u89e3\u6790\u306a\u3069\u69d8\u3005\u306a\u5c40\u9762\u3067\u5fdc\u7528\u3055\u308c\u3066\u3044\u308b\u91cd\u8981\u306a\u624b\u6cd5\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%9C%AC%E8%AB%96%E6%96%87%E3%81%A7%E6%89%B1%E3%81%86%E6%89%8B%E6%B3%95_-%E4%BA%8C%E5%80%A4%E5%88%B6%E7%B4%84%E9%9D%9E%E8%B2%A0%E5%80%A4%E8%A1%8C%E5%88%97%E5%88%86%E8%A7%A3_NBNF\"><\/span>\u672c\u8ad6\u6587\u3067\u6271\u3046\u624b\u6cd5 -\u4e8c\u5024\u5236\u7d04\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 (NBNF)-<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u672c\u8a18\u4e8b\u3067\u6271\u3046\u8ad6\u6587\u3067\u306f\u5206\u89e3\u5148\u306e\u4e8c\u3064\u306e\u884c\u5217\u306e\u3046\u3061\u4e00\u3064\u3092\u3001\u305d\u306e\u6210\u5206\u304c0, 1\u306e\u4e8c\u5024\u306e\u307f\u3092\u53d6\u308a\u3046\u308b\u4e8c\u5024\u5236\u7d04\u4ed8\u304d\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3\u3092D-Wave\u30de\u30b7\u30f3\u3092\u7528\u3044\u3066\u884c\u3046\u305f\u3081\u306e\u65b9\u6cd5\u3092\u6271\u3063\u3066\u3044\u308b\u3002\u3059\u306a\u308f\u3061\u3001<\/p>\n\n\n\n<p>$$V \\approx WH$$<\/p>\n\n\n\n<p>\u3053\u3053\u3067$W$\u306e\u3044\u304b\u306a\u308b\u8981\u7d20\u3082\u975e\u8ca0\u5024\u3092\u53d6\u308b\u3053\u3068($W_{ij}&gt;0$)\u3002$H$\u306e\u8981\u7d20\u306f($H_{ij} \\in \\{0, 1\\}$)\u306e\u4e8c\u5024\u3057\u304b\u53d6\u3089\u306a\u3044\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E9%87%8F%E5%AD%90%E3%82%A2%E3%83%8B%E3%83%BC%E3%83%AA%E3%83%B3%E3%82%B0%E3%82%92%E3%81%A9%E3%81%93%E3%81%AB%E7%94%A8%E3%81%84%E3%82%8B%E3%81%8B\"><\/span>\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u3069\u3053\u306b\u7528\u3044\u308b\u304b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u57fa\u672c\u7684\u306bNMF\u306f\u884c\u5217$W$\u3068$H$\u306b\u95a2\u3057\u3066\u3001\u4ea4\u4e92\u306b\u6700\u5c0f\u4e8c\u4e57\u6cd5\u3092\u7528\u3044\u3066\u9010\u6b21\u66f4\u65b0\u3059\u308b\u3053\u3068\u3067\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u306e\u3046\u3061\u4e8c\u5024\u5236\u7d04\u306e\u3042\u308b\u884c\u5217$H$\u306e\u66f4\u65b0\u306b\u306e\u307f\u3001\u4ee5\u4e0b\u306e\u6a21\u5f0f\u56f3\u306e\u3088\u3046\u306bD-Wave\u30de\u30b7\u30f3\u3092\u7528\u3044\u308b\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/05\/795316b92fc766b0181f6fef074f03fa-2.png\" alt=\"\" class=\"wp-image-635\"\/><figcaption>\u4ea4\u4e92\u6700\u5c0f\u4e8c\u4e57\u6cd5\u306b\u3088\u308bNBMF<\/figcaption><\/figure><\/div>\n\n\n\n<p>\u884c\u5217$H$\u306e\u66f4\u65b0\u306b\u969b\u3057\u3066\u306f\u3001<\/p>\n\n\n\n<p>$$<br>H_{i}=\\arg\\min _{X \\in \\{0,1\\}^{k}}\\|V_{i}-(WX)_{i} \\|_{2}<br>$$<\/p>\n\n\n\n<p>\u306e\u3088\u3046\u306b\u884c\u3054\u3068\u306b\u72ec\u7acb\u3057\u3066\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u305f\u3081\u3001\u3053\u308c\u306b\u3088\u308a\u5909\u6570\u3092$k$\u500b\u307e\u3067\u6e1b\u3089\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002D-Wave 2X\u306e\u5099\u3048\u308b\u91cf\u5b50\u30d3\u30c3\u30c8\u306f1024\u3067\u3042\u308a\u3001\u5b9f\u969b\u306b\u7528\u3044\u308b\u3053\u3068\u306e\u3067\u304d\u308b\u5909\u6570\u306e\u6570\u306f\u3053\u308c\u4ee5\u4e0b\u306b\u5236\u9650\u3055\u308c\u308b\u305f\u3081\u3001\u3053\u306e\u3088\u3046\u306b\u5909\u6570\u306e\u6570\u3092\u6e1b\u3089\u3059\u5de5\u592b\u306f\u975e\u5e38\u306b\u5927\u5207\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"D-Wave%E3%83%9E%E3%82%B7%E3%83%B3%E3%81%AB%E5%AE%9F%E8%A3%85%E3%81%99%E3%82%8B%E3%81%9F%E3%82%81%E3%81%AEQUBO%E8%A1%A8%E7%8F%BE\"><\/span>D-Wave\u30de\u30b7\u30f3\u306b\u5b9f\u88c5\u3059\u308b\u305f\u3081\u306eQUBO\u8868\u73fe<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u884c\u5217$H_{i}$\u306e\u66f4\u65b0\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306bQUBO\u8868\u73fe\u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002QUBO\u8868\u73fe\u306e\u4e00\u822c\u5f62\u3092<\/p>\n\n\n\n<p>$$<br>f({\\bf q}) = \\sum^{}_{i} a_{i} q_{i} + \\sum^{}_{i&lt;j}b_{ij} q_{i} q_{j}<br>$$<\/p>\n\n\n\n<p>\u306e\u3088\u3046\u306b\u8868\u3057\u305f\u6642\u3001\u305d\u306e\u6642\u306e\u4fc2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u308b\u3002<\/p>\n\n\n\n<p>$$<br>a_{j} = \\sum^{}_{k}W_{kj} (W_{kj} &#8211; 2V_{ij})<br>$$<\/p>\n\n\n\n<p>$$<br>b_{jk} = 2 \\sum^{}_{l} W_{lj} W_{lk}<br>$$<\/p>\n\n\n\n<p>\u3053\u306e\u554f\u984c\u306b\u304a\u3044\u3066\u306f\u3001D-Wave 2X\u306e\u30c1\u30c3\u30d7\u306e\u6b20\u9665\u3092\u8003\u616e\u3057\u3001\u5909\u6570\u306e\u6570\u309235\u500b\u306b\u5236\u9650\u3057\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%B5%90%E6%9E%9C\"><\/span>\u7d50\u679c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E9%A1%94%E7%94%BB%E5%83%8F%E8%AA%8D%E8%AD%98%E3%81%B8%E3%81%AE%E5%BF%9C%E7%94%A8%E7%B5%90%E6%9E%9C\"><\/span>\u9854\u753b\u50cf\u8a8d\u8b58\u3078\u306e\u5fdc\u7528\u7d50\u679c<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u904e\u53bb\u306b\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3\u3092\u4f7f\u3063\u3066\u9854\u306e\u30d1\u30fc\u30c4\u3092\u5b66\u7fd2\u3059\u308b\u305f\u3081\u306b\u5206\u6790\u3055\u308c\u305f\u3082\u306e\u3068\u540c\u3058\u3067\u3042\u308b2,429\u500b\u306e\u9854\u753b\u50cf\u3092\u5206\u6790\u3057\u305f\u3002\uff08\u53c2\u7167\uff1a<a href=\"https:\/\/www.nature.com\/articles\/44565\">Learning the parts of objects by non-negative matrix factorization<\/a>\uff09 \u4e0b\u56f3\u306f10,000\u56de\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u3092\u5b9f\u884c\u3057\u305f\u3068\u304d\u306b\u5b66\u7fd2\u3055\u308c\u305f\u7279\u5fb4\u3067\u3042\u308b\u3002\u4e00\u898b\u3001\u771f\u3063\u9ed2\u306b\u898b\u3048\u308b\u753b\u50cf\u3082\u3042\u308b\u304c\u3001\u5b9f\u969b\u5fae\u304b\u306a\u7279\u5fb4\u3092\u542b\u3093\u3067\u3044\u308b\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/05\/c8856789ec11ab8b1013037cef6929f9.png\" alt=\"\" class=\"wp-image-647\"\/><figcaption>\u53f3\u306e7\u00d75\u884c\u5217\u304c\u5b66\u7fd2\u3055\u308c\u305f\u7279\u9577\u3002\u5de6\u4e0a\u306e\u753b\u50cf\u304c\u30aa\u30ea\u30b8\u30ca\u30eb\u3002\u5de6\u4e0b\u306e\u753b\u50cf\u306f\u5fa9\u5143\u3055\u308c\u305f\u3082\u306e\u3002\u5fa9\u5143\u753b\u50cf\u306f\u7dd1\u306e\u67a0\u3067\u56f2\u307e\u308c\u305f\u753b\u50cf\u3092\u91cd\u306d\u5408\u308f\u305b\u3066\u5f97\u3089\u308c\u305f\u3002(arXiv:1704.01605)<\/figcaption><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/05\/3a4f695a458cb0ac0aceaa2eb13ac2dd.png\" alt=\"\" class=\"wp-image-648\"\/><figcaption>\u6700\u3082\u6697\u3044\u30d4\u30af\u30bb\u30eb\u3092\u9ed2\u3001\u6700\u3082\u660e\u308b\u3044\u30d4\u30af\u30bb\u30eb\u3092\u767d\u306b\u3057\u3066\u3001\u30b3\u30f3\u30c8\u30e9\u30b9\u30c8\u3092\u6700\u5927\u306b\u3057\u305f (arXiv:1704.01605)<\/figcaption><\/figure><\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E4%BA%8C%E5%80%A4%E5%88%B6%E7%B4%84%E4%BB%98%E3%81%8D%E3%81%AB%E9%99%90%E5%AE%9A%E3%81%97%E3%81%9F%E5%BD%B1%E9%9F%BF\"><\/span>\u4e8c\u5024\u5236\u7d04\u4ed8\u304d\u306b\u9650\u5b9a\u3057\u305f\u5f71\u97ff<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>NBMF\u3092NMF\u306e\u512a\u52a3\u3092\u6b21\u306e\u8868\u306b\u793a\u3059\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th><\/th><th>NMF<\/th><th>NBMF<\/th><th><\/th><\/tr><\/thead><tbody><tr><td>$H$\uff08\u9078\u629e\u3059\u308b\u7279\u5fb4\u306e\u6570\uff09<\/td><td>13% \u30b9\u30d1\u30fc\u30b9<\/td><td>83% \u30b9\u30d1\u30fc\u30b9<\/td><td>NBMF\u304c\u6709\u5229<\/td><\/tr><tr><td>$\\|V-WH\\|_{F}$\uff08\u8aa4\u5dee\uff09<\/td><td>NBMF\u306e\u534a\u5206\u306e\u8aa4\u5dee<\/td><td>NMF\u306e\uff12\u500d\u306e\u8aa4\u5dee<\/td><td>NMF\u304c\u6709\u5229<\/td><\/tr><tr><td>$W$\uff08\u7279\u5fb4\u306e\u8907\u96d1\u3055\uff09<\/td><td>43% \u30b9\u30d1\u30fc\u30b9<\/td><td>\u5bc6<\/td><td>NMF\u304c\u6709\u5229<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>\u3059\u306a\u308f\u3061\u3001\u8aa4\u5dee\u304c2\u500d\u306b\u306a\u308b\u4e00\u65b9\u3001\u5fc5\u8981\u3068\u306a\u308b\u6210\u5206\u306e\u6570\u306f1\/4\u7a0b\u5ea6\u306b\u524a\u6e1b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3044\u308b\u3002\u5fc5\u8981\u3068\u306a\u308b\u30c7\u30fc\u30bf\u6570\u306e\u524a\u6e1b\u306b\u5bfe\u3057\u3066\u3001\u8aa4\u5dee\u306e\u5897\u5927\u306e\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u304c\u5c0f\u3055\u3044\u305f\u3081\u7528\u9014\u306b\u3088\u3063\u3066\u306f\u5341\u5206\u306b\u6d3b\u7528\u3067\u304d\u308b\u5834\u9762\u304c\u5b58\u5728\u3057\u3046\u308b\u3068\u8a18\u4e8b\u62c5\u5f53\u8005\u306f\u8003\u3048\u308b\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%8F%A4%E5%85%B8%E3%82%BD%E3%83%AB%E3%83%90%E3%83%BC%E3%81%A8%E3%81%AE%E9%80%9F%E5%BA%A6%E6%80%A7%E8%83%BD%E6%AF%94%E8%BC%83\"><\/span>\u53e4\u5178\u30bd\u30eb\u30d0\u30fc\u3068\u306e\u901f\u5ea6\u6027\u80fd\u6bd4\u8f03<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>D-Wave\u306e\u30d1\u30d5\u30a9\u30fc\u30de\u30f3\u30b9\u3092\u53e4\u5178\u30bd\u30eb\u30d0\u30fc\u3067\u3042\u308bqbsolv (\u30bf\u30d6\u30fc\u30b5\u30fc\u30c1)\u3068Gurobi\u3067\u6bd4\u8f03\u3059\u308b\u3002D-Wave\u30de\u30b7\u30f3\u306f\u6700\u9069\u89e3\u3092\u5f97\u308b\u305f\u3081\u306b10, 100, 1,000, 10,000\u56de\u306e\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u56de\u6570\u304c\u4e0e\u3048\u3089\u308c\u308b\u3002\u305d\u3053\u3067\u5f97\u3089\u308c\u308b\u6700\u9069\u89e3\u3092\u30bf\u30fc\u30b2\u30c3\u30c8\u5024\u3068\u3057\u3001qbsolv\u3068Gurobi\u304c\u5c11\u306a\u304f\u3068\u3082\u305d\u306e\u5024\u304f\u3089\u3044\u306b\u826f\u3044\u89e3\u3092\u5f97\u308b\u307e\u3067\u306e\u6642\u9593(TTT)\u3092\u8a08\u7b97\u3059\u308b\u3002\u3082\u3057\u300110\u5206\u4ee5\u4e0a\u304b\u304b\u3063\u305f\u5834\u5408\u306f\u8a08\u7b97\u3092\u6253\u3061\u5207\u308a\u3001TTT\u306f10\u5206\u3068\u3057\u305f\u3002\u5b9f\u969b\u306b\u306f\u3001Gurobi\u306f10\u5206\u4ee5\u4e0a\u304b\u304b\u308b\u4e8b\u306f\u7121\u304b\u3063\u305f\u304c\u3001qbsolv\u306f\u3044\u304f\u3064\u304b\u306e\u30b1\u30fc\u30b9\u306b\u304a\u3044\u306610\u5206\u4ee5\u4e0a\u6642\u9593\u3092\u8981\u3057\u305f\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/05\/2b530e80c7d0de90885e285c5d798063.png\" alt=\"\" class=\"wp-image-644\"\/><\/figure><\/div>\n\n\n\n<p>\u8a08\u7b97\u6642\u9593\u306e\u6bd4\u8f03\u7d50\u679c\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/05\/421e72a65abf8d63fdfd2d58a6ce852f.png\" alt=\"\" class=\"wp-image-784\"\/><figcaption>\u7d2f\u7a4dTTT (arXiv:1704.01605)<\/figcaption><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/05\/f4a1b0aed5dc02442c433030ff24c031-1.png\" alt=\"\" class=\"wp-image-798\"\/><figcaption>\u500b\u3005\u306eTTT (arXiv:1704.01605)<\/figcaption><\/figure><\/div>\n\n\n\n<p>Gurobi\u3068qbsolv\u306e\u30d1\u30d5\u30a9\u30fc\u30de\u30f3\u30b9\u306f\u7570\u306a\u308b\u50be\u5411\u3092\u793a\u3057\u305f\u3002qbsolv\u306f\u983b\u7e41\u306b\u4e00\u56de\u4e00\u56de\u306e\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u6642\u9593\u3088\u308a\u3082\u65e9\u304f\u30bf\u30fc\u30b2\u30c3\u30c8\u89e3\u306b\u5230\u9054\u3057\u305f\u304c\u3001\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u6642\u9593\u5185\u306b\u5230\u9054\u3067\u304d\u306a\u304b\u3063\u305f\u3068\u304d\u306f\u6bd4\u8f03\u7684\u306b\u9577\u3044\u6642\u9593\u3092\u8981\u3057\u305f\u3002\u3053\u306e\u305b\u3044\u3067\u3001qbsolv\u306e\u7d2f\u7a4dTTT\u306f\u5927\u304d\u304f\u306a\u3063\u3066\u3044\u308b\u3002D-Wave\u304c\u3088\u308a\u591a\u304f\u306e\u30b5\u30f3\u30d7\u30eb\u3092\u3068\u308b\u306b\u5f93\u3044\u3001\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u6642\u9593\u3092\u8d85\u3048\u308b\u56de\u6570\u304c\u5897\u3048\u305f\u3002Gurobi\u306f\u30bf\u30fc\u30b2\u30c3\u30c8\u89e3\u306b\u5230\u9054\u3059\u308b\u306e\u306b\u9577\u3044\u6642\u9593\u3092\u8981\u3055\u306a\u304b\u3063\u305f\u304c\u3001\u3068\u3066\u3082\u901f\u304f\u89e3\u304f\u3053\u3068\u3082\u6ec5\u591a\u306b\u306a\u304b\u3063\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E6%8B%85%E5%BD%93%E8%80%85\"><\/span>\u672c\u8a18\u4e8b\u306e\u62c5\u5f53\u8005<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u7fbd\u5834\u5ec9\u4e00\u90ce \uff08\u7de8\u96c6\uff1a\u89b3\u5c71\u6b63\u9053\uff09<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u8a18\u4e8b\u3067\u7d39\u4ecb\u3059\u308b\u8ad6\u6587\u3067\u306f\u3001D-Wave\u30de\u30b7\u30f3\u3092\u7528\u3044\u305f\u91cf\u5b50\u30a2\u30cb\u30fc\u30ea\u30f3\u30b0\u306b\u3088\u308a\u4e8c\u5024\u5909\u6570\u5236\u7d04\u306e\u3064\u3044\u305f \u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 \u306e\u624b\u6cd5\u3068\u9854\u753b\u50cf\u8a8d\u8b58\u306b\u5fdc\u7528\u3057\u305f\u7d50\u679c\u3092\u793a\u3057\u3066\u3044\u308b\u3002\u901f\u5ea6\u6027\u80fd\u8a55\u4fa1\u306e\u305f\u3081\u306b\u3001\u53e4\u5178\u7684\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7528\u3044\u308bGurobi\u3068 [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-397","post","type-post","status-publish","format-standard","hentry","category-review"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u975e\u8ca0\u5024\u884c\u5217\u5206\u89e3 &quot;Nonnegative\/binary matrix factorization with a D-Wave quantum annealer&quot; by Daniel O\u2019Malley, et al. 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