{"id":5100,"date":"2024-06-12T10:23:14","date_gmt":"2024-06-12T01:23:14","guid":{"rendered":"https:\/\/blog.since2020.jp\/?p=5100"},"modified":"2024-06-12T10:23:14","modified_gmt":"2024-06-12T01:23:14","slug":"supportvector_theoretical","status":"publish","type":"post","link":"https:\/\/since2020.jp\/media\/supportvector_theoretical\/","title":{"rendered":"\u6a5f\u68b0\u5b66\u7fd2\u3011\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u89e3\u8aac\uff5e\u6570\u7406\u7684\u89e3\u8aac\u2460\uff5e"},"content":{"rendered":"\n<p>\u4eca\u56de\u306f\u6a5f\u68b0\u5b66\u7fd2\u624b\u6cd5\u306e\u4e00\u3064\u3067\u3042\u308b\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u306b\u3064\u3044\u3066\u6570\u7406\u7684\u306a\u89e3\u8aac\u3092\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n<h2>\u524d\u56de\u306e\u89e3\u8aac<\/h2>\n<p><blockquote class=\"wp-embedded-content\" data-secret=\"zn17Ti6op4\"><a href=\"https:\/\/blog.since2020.jp\/ai\/machinelearning_svm_part1\/\">\u3010\u6a5f\u68b0\u5b66\u7fd2\u3011\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u89e3\u8aac\u3000part1\u76f4\u89b3\u7684\u7406\u89e3<\/a><\/blockquote><iframe class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" style=\"position: absolute; clip: rect(1px, 1px, 1px, 1px);\" title=\"&#8220;\u3010\u6a5f\u68b0\u5b66\u7fd2\u3011\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u89e3\u8aac\u3000part1\u76f4\u89b3\u7684\u7406\u89e3&#8221; &#8212; Data Driven Knowledgebase\" src=\"https:\/\/blog.since2020.jp\/ai\/machinelearning_svm_part1\/embed\/#?secret=VXZPvqQshw#?secret=zn17Ti6op4\" data-secret=\"zn17Ti6op4\" width=\"500\" height=\"282\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\"><\/iframe><\/p>\n\n<h2>\u306f\u3058\u3081\u306b<\/h2>\n<p>\u4eca\u56de\u306f\u524d\u56de\u6271\u3063\u305f\u6a5f\u68b0\u5b66\u7fd2\u624b\u6cd5\u306e\u4e00\u3064\u3067\u3042\u308b\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u306e\u6570\u7406\u7684\u306a\u89e3\u8aac\u3092\u3057\u3066\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u524d\u56de\u306f\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u306e\u809d\u3067\u3042\u308b\u300c\u30de\u30fc\u30b8\u30f3\u306e\u6700\u5927\u5316\u300d\u306b\u3064\u3044\u3066\u76f4\u89b3\u7684\u306b\u7406\u89e3\u3067\u304d\u308b\u3088\u3046\u306b\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/p>\r\n<p>\u4eca\u56de\u306f\u76f4\u89b3\u7684\u306a\u8868\u73fe\u3092\u6570\u7406\u7684\u306a\u8868\u73fe\u306b\u843d\u3068\u3057\u8fbc\u3093\u3067\u6570\u5b66\u7684\u306b\u53d6\u308a\u6271\u3048\u308b\u5f62\u306b\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\r\n<p>\u307e\u305f\u3001\u524d\u56de\u3001\u7dda\u5f62\u306e\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u306e\u307f\u306b\u3064\u3044\u3066\u6271\u3063\u305f\u306e\u3067\u4eca\u56de\u306e\u89e3\u8aac\u3082\u7dda\u5f62\u306b\u7d5e\u308a\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n<h2>\u30c7\u30fc\u30bf\u306e\u8868\u73fe<\/h2>\n<p>\u307e\u305a\u306f\u3001\u30c7\u30fc\u30bf\u8868\u73fe\u306e\u6574\u7406\u304b\u3089\u884c\u3063\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\r\n<p>\\(\\mathbf{x_i} :\\) i\u756a\u76ee\u306e\u30c7\u30fc\u30bf\u70b9\u3002\u4e00\u500b\u4e00\u500b\u306e\u30c7\u30fc\u30bf\u306e\u3053\u3068<\/p>\r\n<p>\\(y_i :\\) i\u756a\u76ee\u306e\u30c7\u30fc\u30bf\u70b9\u306e\u30af\u30e9\u30b9\u30e9\u30d9\u30eb\u3002\u4eca\u56de\u306f\u4e8c\u5024\u5206\u985e\u3092\u6271\u3046\u306e\u3067 \\(y_i \\in (-1,1)\\) \u3068\u306a\u308b\u3002<\/p>\r\n<p><strong>\u8d85\u5e73\u9762<\/strong><\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x} + b = 0 $$<\/p>\r\n<p>\\(\\mathbf{w}\\)\u306f\u91cd\u307f\u30d9\u30af\u30c8\u30eb\u3001b\u306f\u30d0\u30a4\u30a2\u30b9\u9805\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u3053\u3067\u8d85\u5e73\u9762\u3068\u306f\u4e8c\u3064\u306e\u30af\u30e9\u30b9\u3092\u5206\u3051\u308b\u3046\u3048\u3067\u306e\u5883\u754c\u5e73\u9762\u306e\u3053\u3068\u3067\u3059\u3002\u91cd\u307f\u30d9\u30af\u30c8\u30eb\u3068\u30d0\u30a4\u30a2\u30b9\u9805\u304c\u6c7a\u5b9a\u3059\u308b\u3068\u8d85\u5e73\u9762\u304c\u305f\u3060\u4e00\u3064\u306b\u6c7a\u307e\u308b\u306e\u3067\u5b66\u7fd2\u306e\u969b\u306b\u306f\u91cd\u307f\u30d9\u30af\u30c8\u30eb\u3068\u30d0\u30a4\u30a2\u30b9\u9805\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\r\n<p><strong>\u6c7a\u5b9a\u5883\u754c<\/strong><\/p>\r\n<p>$$ f(\\mathbf{x}) = \\mathbf{w} \\cdot \\mathbf{x} + b $$<\/p>\r\n<p>\u3053\u306e\u3088\u3046\u306b\u8868\u73fe\u3059\u308b\u3053\u3068\u3067\u65b0\u305f\u306a\u30c7\u30fc\u30bf\u304c\u5165\u529b\u3055\u308c\u305f\u3068\u304d\u305d\u306e\u95a2\u6570\u306e\u7b26\u53f7\u3092\u898b\u308b\u3053\u3068\u3067\u30af\u30e9\u30b9\u304c\u6c7a\u5b9a\u3055\u308c\u307e\u3059\u3002<\/p>\r\n<b>\u30de\u30fc\u30b8\u30f3\u306e\u5b9a\u7fa9<\/b>\r\n<p>\u30de\u30fc\u30b8\u30f3\u3068\u306f\u8d85\u5e73\u9762\u306b\u6700\u3082\u8fd1\u3044\u30c7\u30fc\u30bf\u70b9\u306e\u3053\u3068\u3067\u3057\u305f\u3002\u4eca\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u304c\u6b21\u306e\u3044\u305a\u308c\u304b\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3068\u3057\u307e\u3059\u3002<\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x} + {b} = 1 $$<\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x} + b = -1 $$<\/p>\r\n<p>\u3053\u308c\u306f\u5404\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u304c\u6c7a\u5b9a\u5883\u754c\u306e\u6b63\u306e\u5074\u306b\u3044\u308b\u304b\u3001\u8ca0\u306e\u5074\u306b\u3044\u308b\u304b\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\r\n<p>\u3053\u306e\u6642\u3001\u30de\u30fc\u30b8\u30f3\u306f\u3053\u308c\u3089\u4e8c\u3064\u306e\u5e73\u9762\u9593\u306e\u8ddd\u96e2\u3067\u3042\u308b\u306e\u3067<\/p>\r\n<p>\u6b63\u306e\u5074\u306e\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u3068\u8d85\u5e73\u9762\u3068\u306e\u8ddd\u96e2<\/p>\r\n<p>$$ d_+ = \\frac{|\\mathbf{w} \\cdot \\mathbf{x}_+ + b &#8211; 1|}{\\|\\mathbf{w}\\|} = \\frac{1}{\\|\\mathbf{w}\\|} $$<\/p>\r\n<p>\u8ca0\u306e\u5074\u306e\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u3068\u8d85\u5e73\u9762\u3068\u306e\u8ddd\u96e2<\/p>\r\n<p>$$ d_- = \\frac{|\\mathbf{w} \\cdot \\mathbf{x}_- + b + 1|}{\\|\\mathbf{w}\\|} = \\frac{1}{\\|\\mathbf{w}\\|} $$<\/p>\r\n<p>\u3067\u3042\u308b\u306e\u3067\u305d\u306e\u548c\u3092\u53d6\u308b\u3053\u3068\u3067\u30de\u30fc\u30b8\u30f3\u306f<\/p>\r\n<p>$$ \u204d $$<\/p>\r\n<p>\u3068\u8868\u73fe\u3055\u308c\u307e\u3059\u3002<\/p>\r\n<b>\u6570\u7406\u6700\u9069\u5316\u3078<\/b>\r\n<p>\u3053\u3053\u3067\u30de\u30fc\u30b8\u30f3\u3092\u6700\u5927\u5316\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u304c\u3001\u30de\u30fc\u30b8\u30f3\u3092\u6700\u5927\u5316\u3059\u308b\u306b\u306f$\\|\\mathbf{w}\\|$\u306e\u6700\u5c0f\u5316\u3068\u540c\u5024\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n<p>\u307e\u305f\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u306f<\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x} + b = 1 $$<\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x} + b = -1 $$<\/p>\r\n<p>\u306e\u3044\u305a\u308c\u304b\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3068\u3057\u3001\u6700\u3082\u8d85\u5e73\u9762\u306b\u8fd1\u3044\u30c7\u30fc\u30bf\u304c\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u3067\u3042\u308b\u306e\u3067\u5404\u30c7\u30fc\u30bf\u306b\u3064\u3044\u3066<\/p>\r\n<p>\u6b63\u306e\u5074\u306b\u3042\u308b\u30c7\u30fc\u30bf\u306f<\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x}_i + b \\geq 1 $$<\/p>\r\n<p>\u8ca0\u306e\u5074\u306b\u3042\u308b\u30c7\u30fc\u30bf\u306f<\/p>\r\n<p>$$ \\mathbf{w} \\cdot \\mathbf{x}_i + b \\leq -1 $$<\/p>\r\n<p>\u3092\u6e80\u305f\u3059\u5fc5\u8981\u304c\u3042\u308a\u3001\u3053\u308c\u3089\u3092\u307e\u3068\u3081\u308b\u3068<\/p>\r\n<p>$$ y_i (\\mathbf{w} \\cdot \\mathbf{x}_i + b) \\geq 1, \\forall i $$<\/p>\r\n<p>\u3068\u3057\u3066\u8868\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\r\n<p>\u3057\u305f\u304c\u3063\u3066\u3053\u306e\u5236\u7d04\u6761\u4ef6\u306e\u4e0b\u3067\u30de\u30fc\u30b8\u30f3\u3092\u6700\u5927\u306b\u3059\u308b\u3053\u3068\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\r\n<p>\u306a\u304a\u76ee\u7684\u95a2\u6570\u306f\u6b21\u56de\u4ee5\u964d\u306e\u8a08\u7b97\u4e0a\u6271\u3044\u3084\u3059\u3044\u3088\u3046\u306b\u4e8c\u4e57\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\r\n<p>$$ \\min_{\\mathbf{w}, b} \\frac{1}{2} \\|\\mathbf{w}\\|^2 $$<\/p>\r\n<p>$$ \\text{subject to } y_i (\\mathbf{w} \\cdot \\mathbf{x}_i + b) \\geq 1, \\forall i $$<\/p>\r\n<p>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u3092\u6570\u5f0f\u3068\u3057\u3066\u8868\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\r\n<p><!-- notionvc: 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--><\/p>\n\n<h2>\u7d42\u308f\u308a\u306b<\/h2>\n<p>\u4eca\u56de\u306f\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u306e\u7dda\u5f62\u5206\u985e\u306b\u304a\u3051\u308b\u6570\u7406\u7684\u7406\u89e3\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/p>\r\n<p>\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u3092\u6570\u7406\u7684\u306b\u8868\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u304c\u5b9f\u969b\u3001\u3069\u306e\u3088\u3046\u306b\u9069\u5207\u306a\u8d85\u5e73\u9762\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u6b21\u56de\u306e\u30d6\u30ed\u30b0\u3067\u306f\u3069\u306e\u3088\u3046\u306a\u51e6\u7406\u306b\u3088\u308a\u6700\u9069\u5316\u554f\u984c\u3092\u89e3\u3051\u3070\u3088\u3044\u306e\u304b\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>","protected":false},"excerpt":{"rendered":"<p>\u4eca\u56de\u306f\u6a5f\u68b0\u5b66\u7fd2\u624b\u6cd5\u306e\u4e00\u3064\u3067\u3042\u308b\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u306b\u3064\u3044\u3066\u6570\u7406\u7684\u306a\u89e3\u8aac\u3092\u3057\u3066\u3044\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002 \u524d\u56de\u306e\u89e3\u8aac \u3010\u6a5f\u68b0\u5b66\u7fd2\u3011\u30b5\u30dd\u30fc\u30c8\u30d9\u30af\u30bf\u30fc\u30de\u30b7\u30f3\u89e3\u8aac\u3000part1\u76f4\u89b3\u7684\u7406\u89e3 \u306f\u3058\u3081\u306b \u4eca\u56de\u306f\u524d\u56de\u6271\u3063\u305f\u6a5f\u68b0\u5b66\u7fd2\u624b\u6cd5\u306e\u4e00\u3064\u3067 [&hellip;]<\/p>\n","protected":false},"author":44,"featured_media":2525,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"content-type":"","swell_btn_cv_data":"","footnotes":"","_wp_rev_ctl_limit":""},"categories":[1249],"tags":[240,422,419],"class_list":["post-5100","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-knowledge","tag-240","tag-422","tag-419"],"_links":{"self":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/5100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/users\/44"}],"replies":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/comments?post=5100"}],"version-history":[{"count":0,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/posts\/5100\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/media\/2525"}],"wp:attachment":[{"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/media?parent=5100"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/categories?post=5100"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/since2020.jp\/media\/wp-json\/wp\/v2\/tags?post=5100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}