{"id":768,"date":"2024-12-02T21:29:52","date_gmt":"2024-12-02T13:29:52","guid":{"rendered":"https:\/\/blog.sssn.tech\/?p=768"},"modified":"2025-03-19T00:36:37","modified_gmt":"2025-03-18T16:36:37","slug":"2024%e5%b9%b412%e6%9c%88-%e5%8f%88%e8%b0%88svm","status":"publish","type":"post","link":"https:\/\/blog.sssn.tech\/?p=768","title":{"rendered":"2024\u5e7412\u6708, \u65f6\u9694\u4e00\u5e74\u518d\u8c08SVM"},"content":{"rendered":"

24\u5e7412\u6708, \u53c8\u8c08SVM<\/h1>\n

\u4e00\u5e74\u4e4b\u524d\u5199\u8fc7\u4e00\u7bc7SVM\u7684\u535a\u5ba2, \u662f\u5728\u5b66\u5434\u6069\u8fbe\u7684\u65f6\u5019, \u6bd4\u8f83\u6cdb\u6cdb\u800c\u8c08, \u8d28\u91cf\u4e0d\u9ad8. \u8fd9\u5b66\u671f\u521a\u597d\u6709\u673a\u5668\u5b66\u4e60\u8fd9\u95e8\u8bfe, \u7167\u4f8b\u671f\u672b\u62b1\u4f5b\u811a, \u518d\u91cd\u65b0\u5b66\u4e00\u904dSVM, \u8fd9\u6b21\u7565\u5fae\u63a5\u89e6\u4e00\u4e9b\u539f\u7406, \u5e0c\u671b\u80fd\u52a0\u6df1\u7406\u89e3.<\/p>\n

\u53c2\u8003\u4e86\u770b\u4e86\u8fd9\u7bc7\u6587\u7ae0\u4f60\u8fd8\u4e0d\u61c2SVM\u4f60\u5c31\u6765\u6253\u6211<\/a><\/p>\n

\u6982\u8ff0<\/h1>\n

\u652f\u6301\u5411\u91cf\u673a(Support Vector Machine, SVM), \u662f\u4e00\u79cd\u5e38\u89c1\u7684\u76d1\u7763\u5b66\u4e60\u7b97\u6cd5, \u7ecf\u5e38\u88ab\u7528\u4e8e\u5206\u7c7b\u4efb\u52a1, \u5c24\u5176\u64c5\u957f\u5904\u7406\u9ad8\u7ef4\u6570\u636e\u548c\u5c0f\u6837\u672c\u6570\u636e\u96c6.<\/p>\n

\u6838\u5fc3\u601d\u60f3<\/h2>\n

SVM\u7684\u6838\u5fc3\u601d\u60f3\u662f, \u5bf9\u4e8e\u4e00\u7ec4m\u4e2ad\u7ef4\u7684\u6570\u636e\u70b9, \u627e\u51fa\u4e00\u4e2a\u6700\u4f73\u51b3\u7b56\u8fb9\u754c(\u8d85\u5e73\u9762)\u5c06\u6570\u636e\u5206\u4e3a\u4e24\u4e2a\u7c7b\u522b<\/p>\n

    \n
  • \n

    \u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d, svm\u4f1a\u627e\u51fa\u4e00\u6761\u76f4\u7ebf\u6765\u5212\u5206\u5e73\u9762\u4e2d\u7684\u70b9<\/p>\n<\/li>\n

  • \n

    \u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d, svm\u4f1a\u627e\u51fa\u4e00\u4e2a\u5e73\u9762\u6765\u5212\u5206\u7a7a\u95f4\u4e2d\u7684\u70b9<\/p>\n<\/li>\n<\/ul>\n

    SVM\u4f1a\u6700\u5927\u5316\u8d85\u5e73\u9762\u5230\u4e0d\u540c\u7c7b\u7684\u95f4\u9694, \u4ee5\u6b64\u6765\u907f\u514d\u8fc7\u62df\u5408, \u5373, \u753b\u51fa\u5c3d\u53ef\u80fd\u7684\u7c97, \u6216\u8005\u539a\u7684\u8d85\u5e73\u9762<\/p>\n

      \n
    • \u6700\u4f73\u7684\u51b3\u7b56\u8fb9\u754c, \u5373\u8d85\u5e73\u9762, \u4f1a\u4ee4\u8ddd\u79bb\u5e73\u9762\u6700\u8fd1\u7684\u70b9\u7684\u8ddd\u79bb\u6700\u5927<\/li>\n
    • \u8d85\u5e73\u9762\u7684\u4f4d\u7f6e\u5b8c\u5168\u7531\u4e00\u90e8\u5206\u5173\u952e\u6837\u672c\u51b3\u5b9a, \u8fd9\u4e9b\u6837\u672c\u53eb\u505a\u652f\u6301\u5411\u91cf, \u4ed6\u4eec\u662f\u8ddd\u79bb\u8d85\u5e73\u9762\u6700\u8fd1\u7684\u70b9<\/li>\n
    • \u975e\u652f\u6301\u5411\u91cf\u5bf9\u51b3\u7b56\u8fb9\u754c\u6ca1\u6709\u8d21\u732e<\/li>\n<\/ul>\n

      \u7ebf\u6027\u4e0d\u53ef\u5206\u95ee\u9898<\/p>\n

        \n
      • \u5982\u679c\u70b9\u5728\u539f\u59cb\u7a7a\u95f4\u4e2d\u7ebf\u6027\u4e0d\u53ef\u5206, SVM\u53ef\u4ee5\u901a\u8fc7\u6838\u51fd\u6570\u5c06\u6570\u636e\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4, \u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u5bfb\u627e\u8d85\u5e73\u9762<\/li>\n<\/ul>\n

        SVM\u7684\u6570\u5b66\u63cf\u8ff0<\/h1>\n

        \u7ebf\u6027\u53ef\u5206\u7684\u786c\u8fb9\u754cSVM<\/h2>\n

        \u7ebf\u6027\u53ef\u5206\u7684SVM\u662f\u6700\u7b80\u5355\u7684\u4e00\u7c7b\u95ee\u9898, \u4ed6\u4ee3\u8868\u6211\u4eec\u7684\u6837\u672c\u5728\u5f53\u524d\u7a7a\u95f4\u4e2d\u53ef\u4ee5\u76f4\u63a5\u7528\u8d85\u5e73\u9762\u5206\u5f00. <\/p>\n

        \u6b64\u65f6, \u6839\u636e\u662f\u5426\u5141\u8bb8\u5206\u7c7b\u9519\u8bef, \u53ef\u4ee5\u5206\u4e3a\u4e24\u7c7b\u8fb9\u754c<\/p>\n

          \n
        • \u786c\u8fb9\u754c: \u4e0d\u5141\u8bb8\u5206\u7c7b\u9519\u8bef<\/li>\n
        • \u8f6f\u8fb9\u754c: \u5f15\u5165\u60e9\u7f5a\u9879, \u5141\u8bb8\u5206\u7c7b\u9519\u8bef<\/li>\n<\/ul>\n

          \u6211\u4eec\u5148\u8ba8\u8bba\u786c\u8fb9\u754c\u7684\u60c5\u51b5<\/p>\n

          \u5047\u8bbe\u6211\u4eec\u6709m\u7ec4\u6570\u636e:<\/p>\n

          \\{(x_1, y_1), (x_2, y_2), ..., (x_m, y_m)\\}\n<\/code><\/pre>\n
          \u5176\u4e2dx_i \\in \\mathbb{R}^d<\/code>\u662f\u7279\u5f81\u5411\u91cf, y_i \\in \\{1, -1\\}<\/code>\u4ee3\u8868\u6b63\u8d1f\u5206\u7c7b<\/p>\n
          SVM\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u6700\u4f73\u7684\u8d85\u5e73\u9762:<\/p>\n
          w \\cdot x + b = 0<\/code><\/pre>\n
          \u5176\u4e2dw \\in \\mathbb{R}^{d}<\/code>\u662f\u6743\u91cd\u5411\u91cf, b<\/code>\u662f\u504f\u7f6e, \u9700\u8981\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6:<\/p>\n
          y_i(w \\cdot x_i + b) \\ge 1, \\forall i<\/code><\/pre>\n
          \u8fd9\u662f\u56e0\u4e3a<\/p>\n
            \n
          • w \\cdot x_i + b<\/code>\u662f\u6837\u672c\u70b9i<\/code>\u7684\u51b3\u7b56\u51fd\u6570\u503c<\/li>\n
          • \u5982\u679cy_i=+1<\/code>, \u5219\u8981\u6c42w \\cdot x_i + b \\ge 1<\/code>, \u5373\u6b63\u6837\u672c\u5fc5\u987b\u5728\u8d85\u5e73\u9762\u6b63\u4fa7<\/li>\n
          • \u5982\u679cy_i=-1<\/code>, \u5219\u8981\u6c42w \\cdot x_i + b \\le 1<\/code>, \u5373\u8d1f\u6837\u672c\u5fc5\u987b\u5728\u8d85\u5e73\u9762\u8d1f\u4fa7<\/li>\n<\/ul>\n
            \u5176\u4e2d\u652f\u6301\u5411\u91cf\u662f\u4ee4\u7b49\u5f0f\u6210\u7acb\u7684\u70b9, \u5373\u6ee1\u8db3<\/p>\n
            y_i(w \\cdot x_i + b) = 1<\/code><\/pre>\n
            \u5728\u4e0a\u8ff0\u7b49\u5f0f\u4e24\u8fb9\u540c\u65f6\u9664\u4ee5$$||w||$$, \u6839\u636e\u70b9\u5230\u76f4\u7ebf\u7684\u8ddd\u79bb\u516c\u5f0f(\u5e7f\u4e49), \u53ef\u4ee5\u5f97\u5230\u652f\u6301\u5411\u91cf\u5230\u8d85\u5e73\u9762\u7684\u8ddd\u79bb\u662f<\/p>\n
            \\frac{1}{||w||}<\/code><\/pre>\n
            \u4e0d\u96be\u5f97\u5230SVM\u7684\u5206\u7c7b\u95f4\u9694(Margin), \u5373\u8d85\u5e73\u9762\u5230\u4e24\u65b9\u652f\u6301\u5411\u91cf\u7684\u8ddd\u79bb\u548c<\/p>\n
            Margin = \\frac{2}{||w||}<\/code><\/pre>\n
            SVM\u5e0c\u671b\u6700\u5927\u5316\u5206\u7c7b\u95f4\u9694, \u4e5f\u5373\u6700\u5c0f\u5316||w||<\/code><\/p>\n
            \"img\"<\/p>\n
            \u5728\u5b9e\u9645\u60c5\u51b5\u4e2d, \u7531\u4e8e<\/p>\n
            ||w|| = \\sqrt{\\sum w_i^2}<\/code><\/pre>\n
            \u4e0a\u9762\u8fd9\u4e2a\u5f62\u5f0f\u6bd4\u8f83\u590d\u6742, \u6211\u4eec\u6700\u4f18\u5316\u7684\u76ee\u6807\u4f1a\u662f<\/p>\n
            \\underset{w, b}{\\text{min}} \\frac{1}{2} \\|w\\|^2<\/code><\/pre>\n
            \u8fd9\u4e2a\u6700\u4f18\u5316\u662f\u5b8c\u5168\u7b49\u4ef7\u7684,\u5176\u4e2d<\/p>\n
              \n
            • \u8d85\u5e73\u9762\u7684\u89e3\u6790\u5f0f\u662fw \\cdot x + b = 0<\/code><\/li>\n
            • w<\/code>\u662f\u8d85\u5e73\u9762\u7684\u6cd5\u5411\u91cf<\/li>\n
            • b<\/code>\u662f\u8d85\u5e73\u9762\u7684\u504f\u7f6e<\/li>\n
            • ||w||<\/code>\u662f\u8d85\u5e73\u9762\u7684\u8303\u6570, \u4e0e\u95f4\u9694\u5927\u5c0f\u6210\u53cd\u6bd4<\/li>\n<\/ul>\n
              \u7b80\u5355\u603b\u7ed3<\/h2>\n
              \u81f3\u6b64, \u6211\u4eec\u91cd\u65b0\u5ba1\u89c6\u4e0a\u9762\u8bf4\u8fc7\u7684, SVM\u4f1a\u5c3d\u53ef\u80fd\u753b\u51fa\u6700\u7c97\u6216\u8005\u6700\u539a\u7684\u8d85\u5e73\u9762:<\/p>\n
              \u8d85\u5e73\u9762\u7684\u4e2d\u5fc3\u5e73\u9762<\/p>\n
              w \\cdot x + b = 0<\/code><\/pre>\n
              \u8d85\u5e73\u9762\u7684\u4e24\u8fb9\u754c\u5e73\u9762<\/p>\n
              w \\cdot x + b = \\pm1<\/code><\/pre>\n
              \u8fd9\u4e2a\u8fb9\u754c\u662f\u7531\u652f\u6301\u5411\u91cf\u51b3\u5b9a\u7684, \u4ee3\u8868\u652f\u6301\u5411\u91cf\u7684\u70b9\u5728\u8fb9\u754c\u5e73\u9762\u4e0a<\/p>\n
              \u4e0a\u4e0b\u4e24\u8fb9\u754c\u7684\u8ddd\u79bb<\/p>\n
              {Margin} = \\frac{2}{||w||}<\/code><\/pre>\n
              \u662f\u6211\u4eec\u6700\u4f18\u5316(\u6700\u5927\u5316)\u7684\u76ee\u6807, \u4f18\u5316\u7ea6\u675f\u662f<\/p>\n
              y_i(w \\cdot x_i + b) \\ge 1, \\forall i<\/code><\/pre>\n
              \u8fd9\u4e2a\u7ea6\u675f\u4ee3\u8868, \u4e0d\u4ec5\u6240\u6709\u70b9\u90fd\u8981\u6b63\u786e\u9884\u6d4b, \u8fd8\u9700\u8981\u4f4d\u4e8e\u4e0a\u4e0b\u8fb9\u754c\u4e4b\u5916, \u5373<\/p>\n
              dist \\ge \\frac{1}{||w||}<\/code><\/pre>\n
              \u8fd9\u4e2a\u7ea6\u675f\u4e0e\u6700\u4f18\u5316\u6761\u4ef6\u662f\u76f8\u8f85\u76f8\u6210\u7684, \u5728\u5b9e\u9645\u8fd0\u7528\u4e2d, \u6211\u4eec\u4f18\u5316\u7684\u662f\u7b49\u4ef7\u5f62\u5f0f<\/p>\n
              \\underset{w, b}{\\text{min}} \\, \\frac{1}{2} \\|w\\|^2, \\quad \n\\text{s.t.} \\, y_i (w \\cdot x_i + b) \\geq 1, \\, \\forall i<\/code><\/pre>\n
              \u8fd9\u662f\u4e00\u4e2a\u4e25\u683c\u51f8\u4f18\u5316\u95ee\u9898, \u4ed6\u7684\u89e3\u662f\u552f\u4e00\u7684<\/p>\n
              \u6c42\u89e3SVM: \u62c9\u683c\u6717\u65e5\u5bf9\u5076+KKT\u6761\u4ef6<\/h2>\n
              \u4e3a\u4e86\u6c42\u89e3\u4e0b\u9762\u7684\u95ee\u9898<\/p>\n
              \\underset{w, b}{\\text{min}} \\, \\frac{1}{2} \\|w\\|^2, \\quad \n\\text{s.t.} \\, y_i (w \\cdot x_i + b) \\geq 1, \\, \\forall i<\/code><\/pre>\n
              \u6211\u4eec\u6784\u9020\u62c9\u683c\u6717\u65e5\u51fd\u6570<\/p>\n
              L(w, b, \\alpha)=\\frac{1}{2}\\|w\\|^{2}-\\sum_{i=1}^{n} \\alpha_{i}\\left[y_{i}\\left(w \\cdot x_{i}+b\\right)-1\\right]<\/code><\/pre>\n
                \n
              • \u5f15\u5165\u4e86\u62c9\u683c\u6717\u65e5\u4e58\u5b50\\alpha_i \\geq 0<\/code>\u5bf9\u5e94\u6bcf\u4e2a\u7ea6\u675f\u6761\u4ef6\n
                  \n
                • \u5f53\u7ea6\u675f\u6761\u4ef6\u4e0d\u88ab\u6ee1\u8db3\u65f6, \u8fd9\u4e00\u9879\u4f1a\u901a\u8fc7\u52a0\u6cd5\u60e9\u7f5a\u76ee\u6807\u51fd\u6570<\/li>\n<\/ul>\n<\/li>\n
                • \u76ee\u6807\u51fd\u6570\\frac{1}{2} ||w||^2<\/code><\/li>\n
                • \u7ea6\u675fy_i(w \\cdot x_i + b) \\geq 1<\/code><\/li>\n<\/ul>\n
                  \u4e3a\u4e86\u5c06\u539f\u95ee\u9898\u8f6c\u5316\u4e3a\u5bf9\u5076\u95ee\u9898, \u6211\u4eec\u5c06L<\/code>\u5206\u522b\u5bf9w<\/code>\u548cb<\/code>\u6c42\u5bfc, \u5e76\u53e6\u5176\u4e3a0, \u5f97\u5230\u4f18\u5316\u6761\u4ef6<\/p>\n
                  \\frac{\\partial L}{\\partial w}=w-\\sum_{i=1}^{n} \\alpha_{i} y_{i} x_{i}=0 \\quad \\Longrightarrow \\quad w=\\sum_{i=1}^{n} \\alpha_{i} y_{i} x_{i}<\/code><\/pre>\n
                  \\frac{\\partial L}{\\partial b}=-\\sum_{i=1}^{n} \\alpha_{i} y_{i}=0 \\quad \\Longrightarrow \\quad \\sum_{i=1}^{n} \\alpha_{i} y_{i}=0<\/code><\/pre>\n
                  \u5c06\u4e0a\u9762\u4e24\u4e2a\u7ed3\u679c\u5e26\u56de\u62c9\u683c\u6717\u65e5\u51fd\u6570, \u6d88\u53bbw<\/code>\u548cb<\/code>, \u53ef\u4ee5\u5f97\u5230<\/p>\n
                  L(w, b, \\alpha) = \\sum_{i=1}^{n} \\alpha_{i}-\\frac{1}{2} \\sum_{i, j=1}^{n} y_{i} y_{j} \\alpha_{i} \\alpha_{j} \\mathbf{x}_{i}^{} \\mathbf{x}_{j}<\/code><\/pre>\n
                  \u6211\u4eec\u7684\u95ee\u9898\u73b0\u5728\u662f<\/p>\n
                  \\underset{w, b}{\\text{min}} \\, \\underset{\\alpha}{\\text{max}} \\, L(w, b, \\alpha), \\quad \n\\text{s.t.} \\, \\alpha_i \\geq 0, \\, \\forall i, \\quad \n\\sum_{i=1}^n \\alpha_i y_i = 0<\/code><\/pre>\n
                  \u5185\u5c42\u7684max<\/code>\u662f\u56e0\u4e3a<\/p>\n
                    \n
                  • \u6700\u5927\u5316\\alpha<\/code>\u662f\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u7406\u8bba\u7684\u6838\u5fc3\u6b65\u9aa4<\/li>\n
                  • \u4e3a\u4e86\u786e\u4fdd\u7ea6\u675f\u6761\u4ef6\u88ab\u4e25\u683c\u9075\u5b88, \u5373, \u627e\u5230\u7ea6\u675f\u7684\u6700\u5f3a\u5f62\u5f0f<\/li>\n<\/ul>\n
                    \u5916\u5c42\u7684min<\/code>\u662f\u56e0\u4e3a<\/p>\n
                      \n
                    • \u6700\u4f18\u5316\u7684\u6700\u7ec8\u76ee\u6807: minimize: ||w||^2<\/code><\/li>\n<\/ul>\n
                      \u56e0\u4e3aSVM\u7684\u539f\u59cb\u95ee\u9898\u662f\u51f8\u4f18\u5316\u95ee\u9898, \u4e14\u6ee1\u8db3Slater\u6761\u4ef6, \u6240\u4ee5\u4ed6\u662f\u5f3a\u5bf9\u5076\u7684, \u6b64\u65f6<\/p>\n
                      \\underset{w, b}{\\text{min}} \\, \\underset{\\alpha}{\\text{max}} \\, L(w, b, \\alpha) = \\underset{\\alpha}{\\text{max}} \\, \\underset{w, b}{\\text{min}} \\, L(w, b, \\alpha)<\/code><\/pre>\n
                      \n
                      Slater\u6761\u4ef6: \u5728\u51f8\u4f18\u5316\u95ee\u9898\u4e2d, \u5982\u679cSlater\u6761\u4ef6\u6210\u7acb, \u5219\u5f3a\u5bf9\u5076\u6027\u6210\u7acb<\/p>\n
                      KKT\u6761\u4ef6: \u5728\u51f8\u4f18\u5316\u95ee\u9898\u4e2d, KKT\u6761\u4ef6\u662f\u63cf\u8ff0\u4e00\u4e2a\u70b9\u662f\u6700\u4f18\u89e3\u7684\u5145\u8981\u6761\u4ef6<\/p>\n<\/blockquote>\n
                      \u5373\u6c42\u89e3\u6700\u7ec8\u7684\u5bf9\u5076\u95ee\u9898<\/p>\n
                      \\max _{\\alpha} \\,  (\\sum_{i=1}^{n} \\alpha_{i}-\\frac{1}{2} \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\alpha_{i} \\alpha_{j} y_{i} y_{j}\\left(x_{i} \\cdot x_{j}\\right)) \\\\\ns.t. \\, \\alpha_i \\geq 0,  \\forall i \\\\ s.t. \\, \\sum_{i=1}^n \\alpha_i y_i = 0<\/code><\/pre>\n
                      \u8fd9\u4e2a\u6700\u7ec8\u7684\u5f62\u5f0f\u662f\u4e00\u4e2a\u6807\u51c6\u7684\u51f8\u4f18\u5316\u5212\u95ee\u9898, \u6211\u4eec\u53ef\u4ee5\u7528\u51f8\u4f18\u5316\u7684\u6807\u51c6\u5f62\u5f0f\u6c42\u89e3, \u6bd4\u5982QP\u6216\u8005SMO<\/p>\n
                      \u8f6f\u8fb9\u754c<\/h2>\n
                      \u5728\u4e0a\u9762\u7684\u60c5\u51b5\u4e2d, \u6211\u4eec\u90fd\u662f\u7528\u4e86\u7ea6\u675f\u6761\u4ef6<\/p>\n
                      y_i(w \\cdot x_i + b) \\ge 1, \\forall i<\/code><\/pre>\n
                      \u8fd9\u4ee3\u8868\u6240\u6709\u7684\u70b9\u90fd\u5fc5\u987b\u88ab\u6b63\u786e\u5206\u7c7b, \u5e76\u4e14\u4ee4\u8fb9\u754c\u6700\u5927. \u8fd9\u5728\u9762\u5bf9\u4e00\u4e9b\u5b9e\u9645\u60c5\u51b5\u7684\u65f6\u5019\u5e76\u4e0d\u5408\u9002<\/p>\n
                      \"img\"<\/p>\n
                      \u8003\u8651\u4e0a\u56fe\u7684\u60c5\u51b5, \u5982\u679c\u6211\u4eec\u8981\u5728\u5f53\u524d\u4e8c\u7ef4\u7a7a\u95f4\u753b\u51fa\u4e00\u6761\u76f4\u7ebf, \u5219\u4e0d\u53ef\u907f\u514d\u7684\u4f1a\u4ea7\u751f\u4e00\u4e9b\u5206\u7c7b\u9519\u8bef. \u6211\u4eec\u73b0\u5728\u7814\u7a76\u53ef\u4ee5\u5bb9\u5fcd\u9519\u8bef\u7684\u60c5\u51b5, \u5373, \u5141\u8bb8\u6a21\u578b\u5728\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u51fa\u9519(\u4e0d\u6ee1\u8db3\u539f\u7ea6\u675f\u6761\u4ef6). \u8fd9\u79cd\u60c5\u51b5\u6211\u4eec\u79f0\u4f5c\u8f6f\u8fb9\u754c(Soft Margin)<\/p>\n
                      \"img\"<\/p>\n
                      \u5728\u8f6f\u8fb9\u754c\u4e2d, \u6211\u4eec\u6539\u5199\u76ee\u6807\u51fd\u6570\u4e3a<\/p>\n
                      \\min \\frac{1}{2}\\|w\\|^{2}+C \\sum_{i=1}^{n} \\xi_{i}<\/code><\/pre>\n
                      \u7ea6\u675f\u6761\u4ef6<\/p>\n
                      y_{i}\\left(w \\cdot x_{i}+b\\right) \\geq 1-\\xi_{i}, \\quad \\xi_{i} \\geq 0<\/code><\/pre>\n
                        \n
                      • \n
                        w<\/code>\u662f\u8d85\u5e73\u9762\u7684\u6cd5\u5411\u91cf, b<\/code>\u662f\u622a\u8ddd<\/p>\n<\/li>\n
                      • \n
                        \\xi_i<\/code>\u662f\u677e\u5f1b\u53d8\u91cf, \u4ee3\u8868i<\/code>\u53f7\u6837\u672c\u5bf9\u7ea6\u675f\u7684\u8fdd\u53cd\u7a0b\u5ea6<\/p>\n
                          \n
                        • \\xi_i = max(0, 1-y_i(w \\cdot x _ b))<\/code><\/li>\n
                        • \\xi_i = 0<\/code>, \u6837\u672c\u88ab\u6b63\u786e\u5206\u7c7b, \u4e14\u4e0d\u5728\u95f4\u9694\u5185<\/li>\n
                        • 0 < \\xi_i \\le 1<\/code>, \u6837\u672c\u88ab\u6b63\u786e\u5206\u7c7b, \u4f46\u662f\u5728\u95f4\u9694\u5185<\/li>\n
                        • \\xi_i > 1<\/code>, \u6837\u672c\u88ab\u9519\u8bef\u5206\u7c7b<\/li>\n<\/ul>\n<\/li>\n
                        • \n
                          C<\/code>\u662f\u6b63\u5219\u5316\u53c2\u6570, \u4ee3\u8868\u6a21\u578b\u5bf9\u9519\u8bef\u7684\u5bbd\u5bb9\u7a0b\u5ea6<\/p>\n
                            \n
                          • C<\/code>\u8f83\u5927\u65f6, \u6a21\u578b\u4e0d\u5bb9\u5fcd\u9519\u8bef, \u53ef\u80fd\u8fc7\u62df\u5408<\/li>\n
                          • C<\/code>\u8f83\u5c0f\u65f6, \u6a21\u578b\u5141\u8bb8\u66f4\u591a\u9519\u8bef, \u53ef\u80fd\u6b20\u62df\u5408<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
                            \"img\"<\/p>\n
                            \u6838\u65b9\u6cd5<\/h2>\n
                            \u5bf9\u4e8e\u96be\u4ee5\u5728\u5f53\u524d\u7a7a\u95f4\u88ab\u7ebf\u6027\u5206\u5272\u7684\u6570\u636e, SVM\u53ef\u4ee5\u4f7f\u7528\u6838\u65b9\u6cd5\u6765\u628a\u6570\u636e\u4ece\u5f53\u524d\u7a7a\u95f4\u6620\u5c04\u5230\u66f4\u9ad8\u7ef4\u5ea6. \u8fd9\u662f\u7ebf\u6027SVM\u5230\u975e\u7ebf\u5f62SVM\u7684\u63a8\u5e7f<\/p>\n
                            \"img\"<\/p>\n
                            \u6838\u65b9\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f<\/p>\n
                              \n
                            • \u5c06\u5f53\u524d\u7a7a\u95f4\u5185\u7684\u70b9\u6620\u5c04\u5230\u66f4\u9ad8\u7ef4\u5ea6(\u751a\u81f3\u65e0\u7a77\u7ef4)<\/li>\n
                            • \u5728\u6620\u5c04\u540e\u7684\u9ad8\u7ef4\u7a7a\u95f4\u7528\u7ebf\u6027\u65b9\u6cd5\u5b66\u4e60\u8d85\u5e73\u9762<\/li>\n<\/ul>\n
                              \u6838\u51fd\u6570<\/strong><\/p>\n
                              \u8bbe\u51fd\u6570\\phi(x)<\/code>\u53ef\u4ee5\u5c06\u539f\u59cb\u6570\u636e\u4ece\u4f4e\u7ef4\u7a7a\u95f4\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4<\/p>\n
                              x \\in \\mathbb{R}^{n} \\xrightarrow{\\phi} \\phi(x) \\in \\mathbb{R}^{m}, m > n<\/code><\/pre>\n
                              \u4f8b\u5982, \u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u70b9x=[x_1, x_2]<\/code>\u53ef\u4ee5\u6620\u5c04\u5230\u4e09\u7ef4\u7a7a\u95f4<\/p>\n
                              \\phi(x) = [x_1, x_2, x_1^2+x_2^2]<\/code><\/pre>\n
                              \u6838\u51fd\u6570K(x^{(i)}, x^{(j)})<\/code>\u7528\u4ee5\u76f4\u63a5\u8ba1\u7b97\u9ad8\u7ef4\u7a7a\u95f4\u4e2d, \u4e24\u4e2a\u6570\u636e\u70b9\u7684\u5185\u79ef<\/p>\n
                              K(x^{(i)}, y^{(j)}) = \\phi(x^{(i)}) \\cdot \\phi(x^{(j)})<\/code><\/pre>\n
                              \u6838\u51fd\u6570\u76f4\u63a5\u5f97\u5230\u4e86\u4e24\u70b9\u5728\u9ad8\u7ef4\u7a7a\u95f4\u7684\u5185\u79ef, \u7701\u7565\u4e86\u5347\u7ef4\u7684\u5177\u4f53\u8ba1\u7b97, \u663e\u8457\u964d\u4f4e\u4e86\u8ba1\u7b97\u590d\u6742\u6027<\/p>\n
                              \u6838\u51fd\u6570\u5728SVM\u4e2d\u7684\u5e94\u7528<\/strong><\/p>\n
                              SVM\u7684\u4f18\u5316\u4f9d\u8d56\u4e8e\u6837\u672c\u70b9\u4e4b\u95f4\u7684\u5185\u79ef(\u6bd4\u5982\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u5f62\u5f0f\u4e2d). \u91c7\u7528\u6838\u51fd\u6570\u4ee3\u66ff\u5185\u79ef, \u5219\u53ef\u4ee5\u4ee4SVM\u5728\u9ad8\u7ef4\u7a7a\u95f4\u8fdb\u884c\u4f18\u5316, \u5e76\u4e14\u65e0\u9700\u8fdb\u884c\u663e\u5f0f\u7684\u9ad8\u7ef4\u6620\u5c04<\/p>\n
                              \u56de\u987e\u4e4b\u524d\u63d0\u5230\u7684\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u5f62\u5f0f:<\/p>\n
                              \\max _{\\alpha} \\,  (\\sum_{i=1}^{n} \\alpha_{i}-\\frac{1}{2} \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\alpha_{i} \\alpha_{j} y_{i} y_{j}\\left(x_{i} \\cdot x_{j}\\right)) \\\\\ns.t. \\, \\alpha_i \\geq 0,  \\forall i \\\\ s.t. \\, \\sum_{i=1}^n \\alpha_i y_i = 0<\/code><\/pre>\n
                              \u7ecf\u8fc7\u6838\u51fd\u6570\u4fee\u6b63\u540e\u7684\u62c9\u683c\u6717\u65e5\u5bf9\u5076\u5f62\u5f0f: <\/p>\n
                              \\max _{\\alpha} \\sum_{i=1}^{n} \\alpha_{i}-\\frac{1}{2} \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\alpha_{i} \\alpha_{j} y_{i} y_{j} K\\left(x_{i}, x_{j}\\right)<\/code><\/pre>\n
                              -\\alpha_i<\/code>\u662f\u62c9\u683c\u6717\u65e5\u4e58\u5b50<\/p>\n
                                \n
                              • y_i<\/code>\u662f\u6807\u7b7e, \u53d6+1<\/code>\u6216\u8005-1<\/code><\/li>\n
                              • K(x_i,x_j)<\/code>\u662f\u6838\u51fd\u6570, \u8868\u793a\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u8ba1\u7b97 \\phi(x_i)<\/code> \u548c\\phi(x_j)<\/code> \u7684\u5185\u79ef<\/li>\n<\/ul>\n
                                \u5e38\u89c1\u7684\u6838\u51fd\u6570<\/strong><\/p>\n
                                \u7ebf\u6027\u6838<\/strong><\/p>\n
                                K(x_i, x_j) = x_i \\cdot x_j<\/code><\/pre>\n
                                  \n
                                • \u7b49\u4ef7\u4e8e\u4e0d\u5411\u9ad8\u7ef4\u6620\u5c04<\/li>\n<\/ul>\n
                                  \u591a\u9879\u5f0f\u6838<\/strong><\/p>\n
                                  K(x_i, x_j) = (x_i \\cdot x_j + c)^{d}<\/code><\/pre>\n
                                    \n
                                  • \u9002\u7528\u4e8e\u5177\u6709\u591a\u9879\u5f0f\u5173\u7cfb\u7684\u975e\u7ebf\u6027\u6570\u636e<\/li>\n<\/ul>\n
                                    \u9ad8\u65af\u6838<\/strong><\/p>\n
                                    K\\left(x_{i}, x_{j}\\right)=\\exp \\left(-\\frac{\\left\\|x_{i}-x_{j}\\right\\|^{2}}{2 \\sigma^{2}}\\right)<\/code><\/pre>\n
                                      \n
                                    • \n
                                      \u662f\u6700\u5e38\u7528\u7684\u6838\u51fd\u6570, \u9002\u5408\u5904\u7406\u4efb\u610f\u5206\u5e03\u7684\u975e\u7ebf\u6027\u6570\u636e\u3002<\/p>\n<\/li>\n
                                    • \n
                                      \u80fd\u591f\u5c06\u6570\u636e\u6620\u5c04\u5230\u65e0\u9650\u7ef4\u7684\u7279\u5f81\u7a7a\u95f4<\/p>\n<\/li>\n<\/ul>\n
                                      Sigmoid\u6838<\/strong><\/p>\n
                                      K\\left(x_{i}, x_{j}\\right)=\\tanh \\left(\\beta x_{i} \\cdot x_{j}+c\\right)<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
                                      24\u5e7412\u6708, \u53c8\u8c08SVM \u4e00\u5e74\u4e4b\u524d\u5199\u8fc7\u4e00\u7bc7SVM\u7684\u535a\u5ba2, \u662f\u5728\u5b66\u5434\u6069\u8fbe\u7684\u65f6\u5019, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30,18],"tags":[],"class_list":["post-768","post","type-post","status-publish","format-standard","hentry","category-30","category-18"],"_links":{"self":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/768","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=768"}],"version-history":[{"count":14,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/768\/revisions"}],"predecessor-version":[{"id":845,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/768\/revisions\/845"}],"wp:attachment":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=768"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=768"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=768"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}