{"id":3194,"date":"2022-02-27T13:09:48","date_gmt":"2022-02-27T05:09:48","guid":{"rendered":"https:\/\/egonlin.com\/?p=3194"},"modified":"2022-02-27T13:09:48","modified_gmt":"2022-02-27T05:09:48","slug":"%e7%ac%ac%e4%b8%89%e8%8a%82%ef%bc%9a%e4%b8%bb%e6%88%90%e5%88%86%e5%88%86%e6%9e%90pca","status":"publish","type":"post","link":"https:\/\/egonlin.com\/?p=3194","title":{"rendered":"\u7b2c\u4e09\u8282\uff1a\u4e3b\u6210\u5206\u5206\u6790(PCA)"},"content":{"rendered":"

\u4e3b\u6210\u5206\u5206\u6790(PCA)<\/h1>\n

\u7ef4\u6570\u707e\u96be\u548c\u964d\u7ef4<\/h1>\n

  \u5728KNN\u7b97\u6cd5\u4e2d\u66fe\u8bb2\u5230\uff0c\u5bf9\u4e8e\u9ad8\u7ef4\u6570\u636e\uff0c\u4f1a\u51fa\u73b0\u6570\u636e\u6837\u672c\u7a00\u758f\u3001\u8ddd\u79bb\u8ba1\u7b97\u56f0\u96be\u7b49\u95ee\u9898\u3002\u4f46\u662f\u8fd9\u79cd\u95ee\u9898\u5e76\u4e0d\u662f\u4ec5\u4ec5\u9488\u5bf9KNN\u7b97\u6cd5\uff0c\u53ea\u662f\u5728KNN\u7b97\u6cd5\u4e2d\u8fd9\u79cd\u95ee\u9898\u4f1a\u88ab\u653e\u5927\uff0c\u800c\u5176\u4ed6\u7684\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u4e5f\u4f1a\u56e0\u4e3a\u9ad8\u7ef4\u6570\u636e\u5bf9\u8bad\u7ec3\u6a21\u578b\u9020\u6210\u6781\u5927\u7684\u969c\u788d\uff0c\u8fd9\u79cd\u95ee\u9898\u4e00\u822c\u88ab\u79f0\u4e3a\u7ef4\u6570\u707e\u96be(curse of dimensionality)\u3002<\/p>\n

  \u89e3\u51b3\u7ef4\u6570\u707e\u96be\u6700\u5e38\u7528\u7684\u65b9\u6cd5\u662f\u964d\u7ef4(dimension reduction)\uff0c\u5373\u901a\u8fc7\u67d0\u79cd\u6570\u5b66\u53d8\u6362\u5c06\u539f\u59cb\u9ad8\u7ef4\u7279\u5f81\u7a7a\u95f4\u8f6c\u53d8\u4e3a\u4e00\u4e2a\u4f4e\u7ef4\u5b50\u7a7a\u95f4\uff0c\u5728\u8fd9\u4e2a\u5b50\u7a7a\u95f4\u4e2d\u6837\u672c\u5bc6\u5ea6\u5927\u5e45\u63d0\u9ad8\uff0c\u8ddd\u79bb\u8ba1\u7b97\u4e5f\u53d8\u5f97\u66f4\u5bb9\u6613\u3002<\/p>\n

# \u7ef4\u6570\u707e\u96be\u548c\u964d\u7ef4\u56fe\u4f8b\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom matplotlib.font_manager import FontProperties\nfrom sklearn.decomposition import PCA\n%matplotlib inline\nfont = FontProperties(fname='\/Library\/Fonts\/Heiti.ttc')\n\nnp.random.seed(0)\nX = np.empty((100, 2))\nX[:, 0] = np.random.uniform(0, 100, size=100)\nX[:, 1] = 0.75 * X[:, 0] + 3. + np.random.normal(0, 10, size=100)\npca = PCA(n_components=1)\nX_reduction = pca.fit_transform(X)\nX_restore = pca.inverse_transform(X_reduction)\n\nplt.scatter(X[:, 0], X[:, 1], color='g', label='\u539f\u59cb\u6570\u636e')\nplt.scatter(X_restore[:, 0], X_restore[:, 1],\n            color='r', label='\u964d\u7ef4\u540e\u7684\u6570\u636e')\nplt.annotate(s='',xytext=(40,60),xy=(65,30),arrowprops=dict(arrowstyle='-',color='b',linewidth=5))\nplt.legend(prop=font)\nplt.show()<\/code><\/pre>\n
\"\"<\/div><\/p>\n
  \u5982\u4e0a\u56fe\u6240\u793a\uff0c\u7eff\u70b9\u5373\u539f\u59cb\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u6837\u672c\u70b9\uff0c\u7ea2\u70b9\u5373\u6211\u4eec\u964d\u7ef4\u540e\u7684\u6837\u672c\u70b9\u3002\u7531\u4e8e\u56fe\u4e2d\u7684\u9ad8\u7ef4\u662f\u4e8c\u7ef4\uff0c\u4f4e\u7ef4\u662f\u4e00\u7ef4\uff0c\u6240\u4ee5\u6837\u672c\u5728\u4f4e\u7ef4\u7a7a\u95f4\u662f\u4e00\u6761\u76f4\u7ebf\u3002<\/p>\n
  \u63a5\u4e0b\u6765\u6211\u4eec\u7684\u76ee\u6807\u5c31\u662f\u804a\u4e00\u804a\u5982\u4f55\u505a\u5230\u628a\u9ad8\u7ef4\u7a7a\u95f4\u6837\u672c\u70b9\u6620\u5c04\u5230\u4f4e\u7ef4\u7a7a\u95f4\uff0c\u5373\u5404\u79cd\u964d\u7ef4\u7b97\u6cd5\u3002<\/p>\n
\u4e3b\u6210\u5206\u5206\u6790\u5b66\u4e60\u76ee\u6807<\/h1>\n
    \n
  1. \u7ef4\u6570\u707e\u96be\u548c\u964d\u7ef4<\/li>\n
  2. \u4e3b\u6210\u5206\u5206\u6790\u4e24\u4e2a\u6761\u4ef6<\/li>\n
  3. \u57fa\u4e8e\u4e3b\u6210\u5206\u5206\u6790\u4e24\u4e2a\u6761\u4ef6\u63a8\u5bfc\u4e3b\u6210\u5206\u5206\u6790<\/li>\n
  4. \u6838\u4e3b\u6210\u5206\u5206\u6790<\/li>\n
  5. \u4e3b\u6210\u5206\u5206\u6790\u4f18\u7f3a\u70b9<\/li>\n<\/ol>\n
    \u4e3b\u6210\u5206\u5206\u6790\u8be6\u89e3<\/h1>\n
      \u4e3b\u6210\u5206\u5206\u6790(principal component analysis\uff0cPCA)\u662f\u6700\u5e38\u7528\u7684\u4e00\u79cd\u964d\u7ef4\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5229\u7528\u201c\u7ef4\u6570\u707e\u96be\u548c\u964d\u7ef4\u56fe\u4f8b\u201d\u89e3\u91ca\u4e86\u964d\u7ef4\u7684\u8fc7\u7a0b\uff0cPCA\u7684\u964d\u7ef4\u8fc7\u7a0b\u5219\u662f\u5c3d\u53ef\u80fd\u7684\u4f7f\u7528\u6570\u636e\u6700\u4e3b\u8981\u7684\u7279\u5f81\u6765\u4ee3\u8868\u6570\u636e\u539f\u6709\u7684\u6240\u6709\u7279\u5f81\u3002\u4f46\u662f\u6709\u6ca1\u6709\u540c\u5b66\u60f3\u8fc7\u4e3a\u4ec0\u4e48\u4f7f\u7528PCA\u964d\u7ef4\u662f\u4e0a\u56fe\u7684\u7ea2\u70b9\u7ec4\u6210\u7684\u7ebf\u800c\u4e0d\u662f\u84dd\u7ebf\u5462\uff1f\u8fd9\u91cc\u5c31\u9700\u8981\u8bf4\u5230\u6211\u4eecPCA\u7684\u4e24\u4e2a\u6761\u4ef6\u4e86\u3002<\/p>\n
    \u4e3b\u6210\u5206\u5206\u6790\u4e24\u4e2a\u6761\u4ef6<\/h2>\n
      \u5bf9\u4e8e\u201c\u7ef4\u6570\u707e\u96be\u548c\u964d\u7ef4\u56fe\u4f8b\u201d\u4e2d\u7684\u7ea2\u7ebf\u548c\u84dd\u7ebf\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u770b\u6210\u4e00\u4e2a\u8d85\u5e73\u9762$S$\uff0c\u7406\u8bba\u4e0a\u7ea2\u7ebf\u548c\u84dd\u7ebf\u6784\u6210\u7684\u8d85\u5e73\u9762\u90fd\u53ef\u4ee5\u505a\u5230\u5bf9\u6837\u672c\u7279\u5f81\u7684\u964d\u7ef4\uff0c\u4f46\u662f\u4e00\u822c\u6211\u4eec\u5e0c\u671b\u8fd9\u79cd\u80fd\u591f\u505a\u5230\u964d\u7ef4\u7684\u8d85\u5e73\u9762\u6ee1\u8db3\u4ee5\u4e0b\u4e24\u4e2a\u6761\u4ef6<\/p>\n
      \n
    1. \u6700\u8fd1\u91cd\u6784\u6027\uff1a\u6837\u672c\u70b9\u5230\u8fd9\u4e2a\u8d85\u5e73\u9762\u7684\u8ddd\u79bb\u90fd\u8db3\u591f\u8fd1<\/li>\n
    2. \u6700\u5927\u53ef\u5206\u6027\uff1a\u6837\u672c\u70b9\u5230\u8fd9\u4e2a\u8d85\u5e73\u9762\u4e0a\u7684\u6295\u5f71\u5c3d\u53ef\u80fd\u5206\u5f00<\/li>\n<\/ol>\n
        \u57fa\u4e8e\u6700\u8fd1\u91cd\u6784\u6027\u548c\u6700\u5927\u53ef\u5206\u6027\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u4e3b\u6210\u5206\u5206\u6790\u7684\u4e24\u79cd\u7b49\u4ef7\u63a8\u5bfc\u3002<\/p>\n
      \u57fa\u4e8e\u6700\u8fd1\u91cd\u6784\u6027\u63a8\u5bfcPCA<\/h2>\n
      \u4e3b\u6210\u5206\u5206\u6790\u76ee\u6807\u51fd\u6570<\/h3>\n
        \u6211\u4eec\u9996\u5148\u4ece\u6700\u8fd1\u91cd\u6784\u6027\u63a8\u5bfcPCA\uff0c\u5373\u6837\u672c\u70b9\u5230\u8fd9\u4e2a\u8d85\u5e73\u9762\u7684\u8ddd\u79bb\u8db3\u591f\u8fd1\u3002<\/p>\n
        \u5047\u8bbe$m$\u4e2a$n$\u7ef4\u6570\u636e$(x^{(1)},x^{(2)},\\cdots,x^{(m)})$\u90fd\u5df2\u7ecf\u8fdb\u884c\u4e86\u4e2d\u5fc3\u5316\uff0c\u5373$\\sum_{i=1}^mx^{(i)}=0$\uff1b\u5728\u5047\u8bbe\u6295\u5f71\u53d8\u6362\u540e\u5f97\u5230\u7684\u65b0\u5750\u6807\u7cfb\u4e3a${w_1,w_2,\\cdots,w_n}$\uff0c\u5176\u4e2d$w_i$\u662f\u6807\u51c6\u6b63\u4ea4\u57fa\u5411\u91cf\uff0c\u5373$||w_i||=1\uff0cw_i^Tw_j=0$\uff0c\u5176\u4e2d$i\\neq{j}$\u3002<\/p>\n
        \u5982\u679c\u628a\u6570\u636e\u4ece$n$\u7ef4\u964d\u5230$n’$\u7ef4\uff0c\u5373\u4e22\u5f03\u65b0\u5750\u6807\u7cfb\u4e2d\u7684\u90e8\u5206\u5750\u6807\uff0c\u5219\u65b0\u7684\u5750\u6807\u7cfb\u4e3a${w_1,w2,\\cdots,w<\/em>{n’}}$\uff0c\u5219\u6837\u672c\u70b9$x^{(i)}$\u5728$n’$\u7ef4\u5750\u6807\u7cfb\u4e2d\u7684\u6295\u5f71\u4e3a
      \n$$
      \nz{i} = (z<\/em>{i1},z{i2},\\cdots,z<\/em>{id’})^T
      \n$$
      \n\u5176\u4e2d$z_{ij}=w_j^Tx_i$\uff0c\u662f$x_i$\u5728\u4f4e\u7ef4\u5750\u6807\u7cfb\u4e0b\u7b2c$j$\u7ef4\u7684\u5750\u6807\u3002<\/p>\n
        \u5982\u679c\u6211\u4eec\u7528$z^{(i)}$\u91cd\u6784$x^{(i)}$\uff0c\u5219\u53ef\u4ee5\u6062\u590d\u7684\u539f\u59cb\u6570\u636e\u4e3a
      \n$$
      \n\\hat{xi}=\\sum<\/em>{j=1}^{d’}z_{ij}w_j
      \n$$
      \n  \u73b0\u5728\u8003\u8651\u6574\u4e2a\u6837\u672c\u96c6\uff0c\u65e2\u53ef\u4ee5\u83b7\u5f97\u539f\u6837\u672c\u70b9$x_i$\u5230\u57fa\u4e8e\u6295\u5f71\u91cd\u6784\u7684\u6837\u672c\u70b9$\\hat{xi}$\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e3a
      \n$$
      \n\\begin{align}
      \n\\sum<\/em>{i=1}^m{||\\hat{x_i}-xi||}^2 & = \\sum<\/em>{i=1}^m{||Wz_i-xi||}^2 \\
      \n& = \\sum<\/em>{i=1}^m(Wz_i)^T(Wzi)-2\\sum<\/em>{i=1}^m(Wz_i)^Txi+\\sum<\/em>{i=1}^mx_i^Txi \\
      \n& = \\sum<\/em>{i=1}^mz_i^Tzi – 2\\sum<\/em>{i=1}^mz_i^TW^Txi+\\sum<\/em>{i=1}^mx_i^Txi \\
      \n& = \\sum<\/em>{i=1}^mz_i^Tzi-2\\sum<\/em>{i=1}^mz_i^Tzi+\\sum<\/em>{i=1}^mx_i^Txi \\
      \n& = -\\sum<\/em>{i=1}^mz_i^Tzi + \\sum<\/em>{i=1}^mx_i^Txi \\
      \n& = -tr(W^T(\\sum<\/em>{i=1}^mx_ixi^T)W)+\\sum<\/em>{i=1}^mx_i^Txi \\
      \n& = -tr(W^TXX^TW)+\\sum<\/em>{i=1}^mx_i^Tx_i
      \n\\end{align}
      \n$$
      \n\u7531\u4e8e\u6d89\u53ca\u8fc7\u591a\u77e9\u9635\u63a8\u5bfc\uff0c\u6b64\u5904\u4e0d\u591a\u8d58\u8ff0\uff0c\u770b\u4e0d\u61c2\u7684\u53ef\u4ee5\u8df3\u8fc7\u3002<\/p>\n
      \u5176\u4e2d$W=(w_1,w_2,\\cdots,wd)$\uff0c\u5176\u4e2d$\\sum<\/em>{i=1}^mx_i^Tx_i$\u662f\u6570\u636e\u96c6\u7684\u534f\u65b9\u5dee\u77e9\u9635\uff0c$W$\u7684\u6bcf\u4e00\u4e2a\u5411\u91cf$wj$\u662f\u6807\u51c6\u6b63\u4ea4\u57fa\uff0c\u800c$\\sum<\/em>{i=1}^mx_i^Tx_i$\u662f\u4e00\u4e2a\u5e38\u91cf\uff0c\u6700\u5c0f\u5316\u4e0a\u5f0f\u7b49\u4ef7\u4e8e
      \n$$
      \n\\begin{align}
      \n& \\underbrace{min}_W\\,-tr(W^TXX^TW) \\
      \n& s.t.\\,W^TW=I
      \n\\end{align}
      \n$$<\/p>\n
      \u4e3b\u6210\u5206\u5206\u6790\u76ee\u6807\u51fd\u6570\u4f18\u5316<\/h3>\n
        \u4e3b\u6210\u5206\u5206\u6790\u76ee\u6807\u51fd\u6570\u4e3a
      \n$$
      \n\\begin{align}
      \n& \\underbrace{min}_W\\,-tr(W^TXX^TW) \\
      \n& s.t.\\,W^TW=I
      \n\\end{align}
      \n$$
      \n  \u6700\u5c0f\u5316\u8be5\u76ee\u6807\u51fd\u6570\u5176\u5b9e\u5e76\u4e0d\u96be\uff0c\u53ef\u4ee5\u53d1\u73b0\u6700\u5c0f\u5316\u76ee\u6807\u51fd\u6570\u5bf9\u5e94\u7684$W$\u7531\u534f\u65b9\u5dee\u77e9\u9635$XX^T$\u6700\u5927\u7684$n’$\u4e2a\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u7ec4\u6210\uff0c\u5229\u7528\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5\u53ef\u5f97
      \n$$
      \nJ(W)=-tr(W^TXX^TW+\\lambda_i(W^TW-I))
      \n$$
      \n  \u5bf9$W$\u6c42\u5bfc\u7b49\u4e8e0\u5373\u53ef\u5f97
      \n$$
      \n\\begin{align}
      \n& -XX^TW+\\lambda{W}=0 \\
      \n& XX^TW = \\lambda{W}
      \n\\end{align}
      \n$$
      \n  \u4ece\u4e0a\u5f0f\u53ef\u4ee5\u770b\u51fa\uff0c$W$\u662f$XX^T$\u7684$n’$\u4e2a\u7279\u5f81\u5411\u91cf\u7ec4\u6210\u7684\u77e9\u9635\uff0c\u800c$\\lambda$\u6709\u82e5\u5e72\u4e2a\u7279\u5f81\u503c\u7ec4\u6210\u7684\u77e9\u9635\uff0c\u7279\u5f81\u503c\u5728\u5bf9\u89d2\u7ebf\u4e0a\uff0c\u5176\u4f59\u4f4d\u7f6e\u4e3a0\u3002\u5f53\u6211\u4eec\u5c06\u6570\u636e\u96c6\u4ece$n$\u7ef4\u964d\u5230$n’$\u7ef4\u65f6\uff0c\u9700\u8981\u627e\u5230\u6700\u5927\u7684$n’$\u4e2a\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u8fd9\u4e2a$n’$\u4e2a\u7279\u5f81\u5411\u91cf\u7ec4\u6210\u7684\u77e9\u9635$W$\u5373\u6211\u4eec\u9700\u8981\u7684\u77e9\u9635\u3002\u5bf9\u4e8e\u539f\u59cb\u6570\u636e\u96c6\uff0c\u6211\u4eec\u53ea\u9700\u8981\u7528$z_i=W^Tx_i$\uff0c\u5c31\u53ef\u4ee5\u628a\u539f\u59cb\u6570\u636e\u96c6\u964d\u5230\u6700\u5c0f\u6295\u5f71\u8ddd\u79bb\u7684$n’$\u7ef4\u6570\u636e\u96c6\u3002<\/p>\n
      \u57fa\u4e8e\u6700\u5927\u53ef\u5206\u6027\u63a8\u5bfcPCA<\/h2>\n
        \u4ece\u6700\u5927\u53ef\u5206\u6027\u51fa\u53d1\uff0c\u6837\u672c\u70b9$x_i$\u5728\u65b0\u7a7a\u95f4\u4e2d\u8d85\u5e73\u9762\u7684\u6295\u5f71\u662f$W^Tx_i$\uff0c\u5982\u679c\u6240\u6709\u6837\u672c\u70b9\u7684\u6295\u5f71\u5c3d\u53ef\u80fd\u5206\u5f00\uff0c\u5219\u5e94\u8be5\u4f7f\u6295\u5f71\u540e\u6837\u672c\u70b9\u7684\u65b9\u5dee\u6700\u5927\u5316\u3002<\/p>\n
        \u6295\u5f71\u540e\u6837\u672c\u70b9\u7684\u65b9\u5dee\u662f$\\sum_{i=1}^mW^Tx_ix_i^TW$\uff0c\u56e0\u6b64\u76ee\u6807\u51fd\u6570\u53ef\u4ee5\u5199\u6210
      \n$$
      \n\\begin{align}
      \n& \\underbrace{max}_W\\,-tr(W^TXX^TW) \\
      \n& s.t.\\,W^TW=I
      \n\\end{align}
      \n$$
      \n\u4e0a\u5f0f\u5176\u5b9e\u548c\u57fa\u4e8e\u6700\u8fd1\u91cd\u6784\u6027\u63a8\u5bfcPCA\u7684\u76ee\u6807\u51fd\u6570\u5176\u5b9e\u5dee\u4e0d\u591a\uff0c\u5176\u4e2d\u4e00\u4e2a\u662f\u52a0\u8d1f\u53f7\u7684\u6700\u5c0f\u5316\uff0c\u4e00\u4e2a\u662f\u6700\u5927\u5316\u3002
      \n  \u5bf9\u57fa\u4e8e\u6700\u5927\u53ef\u5206\u6027\u63a8\u5bfc\u5f97\u5230\u7684\u76ee\u6807\u51fd\u6570\u6700\u5927\u5316\uff0c\u5229\u7528\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5\u53ef\u4ee5\u5f97\u5230
      \n$$
      \nXX^TW = -\\lambda{W}
      \n$$<\/p>\n
      \u6838\u4e3b\u6210\u5206\u5206\u6790(KPCA)<\/h2>\n
        PCA\u4e2d\uff0c\u6211\u4eec\u5047\u8bbe\u5b58\u5728\u4e00\u4e2a\u7ebf\u6027\u7684\u8d85\u5e73\u9762\uff0c\u53ef\u4ee5\u5bf9\u6570\u636e\u6295\u5f71\uff0c\u4f46\u5de5\u4e1a\u4e0a\u5927\u591a\u6570\u65f6\u5019\u6570\u636e\u90fd\u662f\u7ebf\u6027\u4e0d\u53ef\u5206\u7684\uff0c\u8fd9\u91cc\u5c31\u9700\u8981\u7528\u5230\u548c\u6838SVM\u4e00\u6837\u7684\u601d\u60f3\uff0c\u5373\u6838\u4e3b\u6210\u5206\u5206\u6790(kernelized PCA\uff0cKPCA)\uff0c\u662f\u57fa\u4e8e\u6838\u6280\u5de7\u5bf9\u975e\u7ebf\u6027\u53ef\u5206\u6570\u636e\u8fdb\u884c\u964d\u7ef4\u3002<\/p>\n
        KPCA\u9996\u5148\u4f1a\u628a\u6570\u636e\u4ece$n$\u7ef4\u6620\u5c04\u5230\u66f4\u9ad8\u7684$N$\u7ef4\uff0c\u8ba9\u6570\u636e\u7ebf\u6027\u53ef\u5206\u540e\u53c8\u4f1a\u628a\u6570\u636e\u6620\u5c04\u56de\u4f4e\u7ef4$n’$\uff0c\u5373$n'<n<N$\u3002<\/p>\n
        \u5047\u8bbe\u6211\u4eec\u5c06\u5728\u9ad8\u7ef4\u7279\u5f81\u7a7a\u95f4\u628a\u6570\u636e\u6295\u5f71\u5230\u7531$W=(w_1,w_2,\\cdots,wd)$\u786e\u5b9a\u7684\u8d85\u5e73\u9762\u4e0a\uff0c\u5219$W$\u4e3a
      \n$$
      \nZZ^TW = (\\sum<\/em>{i=1}^mz_iz_i^T)W=\\lambda{W}
      \n$$
      \n\u5176\u4e2d$zi$\u662f\u6837\u672c\u70b9\u518d\u9ad8\u7ef4\u7279\u5f81\u7a7a\u95f4\u4e2d\u7684\u50cf\uff0c\u5373\u7279\u5f81\u5206\u89e3\u95ee\u9898\u53d8\u4e3a
      \n$$
      \n\\begin{align}
      \nW & = {\\frac{1}{\\lambda}}(\\sum<\/em>{i=1}^mz_izi^T)W \\
      \n& = \\sum<\/em>{i=1}^mz_i{\\frac{zi^TW}{\\lambda}} \\
      \n& = \\sum<\/em>{i=1}^mz_i\\alpha_i^j
      \n\\end{align}
      \n$$
      \n\u5176\u4e2d$a_i^j={\\frac{1}{\\lambda}}z_i^TW$\u662f$\\alpha_i$\u7684\u7b2c$j$\u4e2a\u5206\u91cf\u3002<\/p>\n
        \u5047\u8bbe$z_i$\u662f\u7531\u539f\u59cb\u6837\u672c\u70b9$x_i$\u901a\u8fc7\u6620\u5c04$\\phi$\u4ea7\u751f\uff0c\u5373$z_i=\\phi(xi)$\uff0c\u5219\u7279\u5f81\u5206\u89e3\u95ee\u9898\u53d8\u4e3a
      \n$$
      \n(\\sum<\/em>{i=1}^m\\phi(x_i)\\phi(xi)^T)W = \\lambda{W}
      \n$$
      \n  $W$\u53d8\u4e3a
      \n$$
      \nW=\\sum<\/em>{i=1}^m\\phi(x_i)\\alpha_i^j
      \n$$
      \n  \u7531\u4e8e\u6211\u4eec\u5e76\u4e0d\u77e5\u9053$\\phi$\u662f\u4ec0\u4e48\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b$\\phi$\u4e0d\u9700\u8981\u663e\u793a\u8ba1\u7b97\uff0c\u901a\u8fc7\u6838\u51fd\u6570\u8f6c\u6362\u5373\u53ef\u3002\u56e0\u6b64\u5f15\u5165\u6838\u51fd\u6570
      \n$$
      \nk(x_i,x_j)=\\phi(x_i)^T\\phi(x_j)
      \n$$
      \n  \u5c06\u6838\u51fd\u6570\u548c$w_j$\u4ee3\u5165\u7279\u5f81\u5206\u89e3\u95ee\u9898\uff0c\u53ef\u5f97
      \n$$
      \nK\\alpha^j=\\lambda\\alpha^j
      \n$$
      \n\u5176\u4e2d$K$\u4e3a$k$\u5bf9\u5e94\u7684\u6838\u77e9\u9635\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u7279\u5f81\u503c\u5206\u89e3\u95ee\u9898\uff0c\u53bb$K$\u6700\u5927\u7684$d’$\u4e2a\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u5373\u53ef\u3002<\/p>\n
        \u5bf9\u4e8e\u65b0\u6837\u672c$x$\uff0c\u4ed6\u6295\u5f71\u540e\u7684\u7b2c$j\\quad(j=1,2,\\cdots,d’)$\u7ef4\u5750\u6807\u4e3a
      \n$$
      \n\\begin{align}
      \nzj & = W^T\\phi(x) \\
      \n& = \\sum<\/em>{i=1}^m\\alpha_i^j\\phi(xi)^T\\phi(x) \\
      \n& = \\sum<\/em>{i=1}^m\\alpha_i^jk(x_i,x)
      \n\\end{align}
      \n$$
      \n  \u4ece\u4e0a\u8ff0\u7279\u5f81\u5206\u89e3\u53ef\u4ee5\u770b\u51fa\uff0cKPCA\u9700\u8981\u5bf9\u6240\u6709\u6837\u672c\u6c42\u548c\uff0c\u56e0\u6b64\u5b83\u7684\u8ba1\u7b97\u5f00\u9500\u8f83\u5927\u3002<\/p>\n
      \u4e3b\u6210\u5206\u5206\u6790\u6d41\u7a0b<\/h1>\n
      \u8f93\u5165<\/h2>\n
        \u6837\u672c\u96c6$D={x_1,x_2,\\cdots,x_n}$\uff1b\u4f4e\u7ef4\u7a7a\u95f4\u7ef4\u6570$n’$\u3002<\/p>\n
      \u8f93\u51fa<\/h2>\n
        \u964d\u7ef4\u540e\u7684\u6837\u672c\u96c6$D’$\u3002<\/p>\n
      \u6d41\u7a0b<\/h2>\n
        \n
      1. \u5bf9\u6240\u6709\u6837\u672c\u8fdb\u884c\u4e2d\u5fc3\u5316\uff1a$x_i\\leftarrow{xi}-{\\frac{1}{m}}\\sum<\/em>{i=1}^m{x_i}$<\/li>\n
      2. \u8ba1\u7b97\u6837\u672c\u7684\u534f\u65b9\u5dee\u77e9\u9635$XX^T$<\/li>\n
      3. \u5bf9\u534f\u65b9\u5dee\u77e9\u9635$XX^T$\u505a\u7279\u5f81\u503c\u5206\u89e3<\/li>\n
      4. \u53d6\u6700\u5927\u7684$n’$\u4e2a\u7279\u5f81\u503c\u6240\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf$(w_1,w2,\\cdots,w<\/em>{n’})$\uff0c\u5c06\u6240\u6709\u7684\u7279\u5f81\u5411\u91cf\u6807\u51c6\u5316\u540e\uff0c\u7ec4\u6210\u7279\u5f81\u5411\u91cf\u77e9\u9635$W$<\/li>\n
      5. \u5bf9\u6837\u672c\u96c6\u4e2d\u7684\u6bcf\u4e00\u4e2a\u6837\u672c$x^{(i)}$\uff0c\u8f6c\u5316\u4e3a\u65b0\u7684\u6837\u672c$z^{(i)}=W^Tx^{(i)}$<\/li>\n
      6. \u5f97\u5230\u8f93\u51fa\u6837\u672c\u96c6$n’=(z^{(1)},z^{(2)},\\cdots,z^{(m)})$<\/li>\n<\/ol>\n
          \u964d\u7ef4\u540e\u4f4e\u7ef4\u7a7a\u95f4\u7684\u7ef4\u6570$n’$\u901a\u5e38\u662f\u7528\u6237\u4e8b\u5148\u6307\u5b9a\u7684\uff0c\u4e00\u822c\u9009\u62e9\u4f7f\u7528\u4ea4\u53c9\u9a8c\u8bc1\u7684\u65b9\u6cd5\u9009\u62e9\u6700\u597d\u7684$n’$\u503c\u3002\u5bf9\u4e8ePCA\uff0c\u6709\u65f6\u5019\u4e5f\u4f1a\u4ece\u91cd\u6784\u7684\u89d2\u5ea6\u6307\u5b9a\u4e00\u4e2a\u964d\u7ef4\u5230\u7684\u4e3b\u6210\u5206\u6bd4\u91cd\u9608\u503c$t$\uff0c\u8fd9\u4e2a\u9608\u503c\u7684\u8303\u56f4\u4e00\u822c\u662f$(0,1]$\uff0c\u7136\u540e\u9009\u53d6\u4f7f\u4e0b\u5f0f\u6210\u7acb\u7684\u6700\u5c0f$n’$\u503c
        \n$$
        \n{\\frac{\\sum_{i=1}^{n’}\\lambdai}{\\sum<\/em>{i=1}^{n}\\lambda_i}}\\geq{t}
        \n$$<\/p>\n
        \u4e3b\u6210\u5206\u5206\u6790\u4f18\u7f3a\u70b9<\/h1>\n
        \u4f18\u70b9<\/h2>\n
          \n
        1. \u53ea\u9700\u8981\u4ee5\u65b9\u5dee\u8861\u91cf\u4fe1\u606f\u91cf\uff0c\u4e0d\u53d7\u6570\u636e\u96c6\u4ee5\u5916\u7684\u56e0\u7d20\u5f71\u54cd<\/li>\n
        2. \u4e3b\u8981\u8ba1\u7b97\u662f\u7279\u5f81\u503c\u5206\u89e3\uff0c\u8ba1\u7b97\u7b80\u5355\uff0c\u6613\u4e8e\u5b9e\u73b0<\/li>\n<\/ol>\n
          \u7f3a\u70b9<\/h2>\n
            \n
          1. \u4e3b\u6210\u5206\u7531\u4e8e\u662f\u964d\u7ef4\u5f97\u5230\uff0c\u6ca1\u6709\u539f\u59cb\u6837\u672c\u90a3\u6837\u8f83\u5f3a\u7684\u89e3\u91ca\u6027<\/li>\n
          2. \u7531\u4e8ePCA\u964d\u7ef4\u4f1a\u4e22\u6389\u4e0d\u5c11\u7684\u4fe1\u606f\uff0c\u53ef\u80fd\u5bf9\u540e\u7eed\u7684\u6570\u636e\u5904\u7406\u6709\u5f71\u54cd<\/li>\n<\/ol>\n
            \u5c0f\u7ed3<\/h1>\n
              PCA\u4f5c\u4e3a\u4e00\u4e2a\u65e0\u76d1\u7763\u5b66\u4e60\u7684\u964d\u7ef4\u65b9\u6cd5\uff0c\u53ea\u9700\u8981\u5bf9\u7279\u5f81\u503c\u5206\u89e3\uff0c\u5c31\u53ef\u4ee5\u538b\u7f29\u6570\u636e\uff0c\u5bf9\u6570\u636e\u53bb\u566a\u58f0\u3002\u4f46\u662fPCA\u8fd8\u662f\u6709\u4e0d\u5c11\u7f3a\u70b9\u7684\uff0c\u9488\u5bf9PCA\u7684\u7f3a\u70b9\uff0c\u4e5f\u51fa\u73b0\u4e86\u5f88\u591aPCA\u7684\u53d8\u79cd\uff0c\u5982\u89e3\u51b3\u975e\u7ebf\u6027\u6570\u636e\u964d\u7ef4\u7684KPCA\uff1b\u89e3\u51b3\u5185\u5b58\u9650\u5236\u7684\u589e\u91cf\u7684Incremental PCA\uff1b\u89e3\u51b3\u7a00\u758f\u6570\u636e\u964d\u7ef4\u7684Sparse PCA\u7b49\u3002<\/p>\n
              \u7531\u4e8ePCA\u6d89\u53ca\u8fc7\u591a\u7684\u6570\u5b66\u516c\u5f0f\uff0c\u4ee5\u53ca\u6709\u7740\u8f83\u5f3a\u903b\u8f91\u548c\u7a7a\u95f4\u5904\u7406\u3002\u5982\u679c\u4e0d\u662f\u5f88\u61c2\u53ef\u4ee5\u7ed3\u5408\u4ee3\u7801\u7136\u540e\u591a\u770b\u51e0\u904d\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
            \u4e3b\u6210\u5206\u5206\u6790(PCA) \u7ef4\u6570\u707e\u96be\u548c\u964d\u7ef4   \u5728KNN\u7b97\u6cd5\u4e2d\u66fe\u8bb2\u5230\uff0c\u5bf9\u4e8e\u9ad8\u7ef4\u6570\u636e\uff0c\u4f1a\u51fa\u73b0\u6570\u636e\u6837 […]<\/p>\n","protected":false},"author":3,"featured_media":3189,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[276,299],"tags":[],"_links":{"self":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/posts\/3194"}],"collection":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3194"}],"version-history":[{"count":0,"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/posts\/3194\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=\/wp\/v2\/media\/3189"}],"wp:attachment":[{"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3194"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3194"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/egonlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}