{"id":1581,"date":"2020-04-24T22:33:35","date_gmt":"2020-04-24T20:33:35","guid":{"rendered":"https:\/\/myoceane.fr\/?p=1581"},"modified":"2020-04-25T20:05:34","modified_gmt":"2020-04-25T18:05:34","slug":"pca-%e4%b8%bb%e6%88%90%e4%bb%bd%e5%88%86%e6%9e%90","status":"publish","type":"post","link":"https:\/\/myoceane.fr\/index.php\/pca-%e4%b8%bb%e6%88%90%e4%bb%bd%e5%88%86%e6%9e%90\/","title":{"rendered":"[ML] PCA \u4e3b\u6210\u4efd\u5206\u6790"},"content":{"rendered":"
<\/div>\n\n

\u5728\u4e0a\u4e00\u7bc7\u7684\u6587\u7ae0<\/a>\u4e2d\uff0c\u6ab8\u6aac\u7238\u9996\u5148\u4ecb\u7d39\u5982\u4f55\u4f7f\u7528 Python \u8207 Numpy \u51fd\u5f0f\u5eab\u5c07\u60f3\u8981\u5206\u6790\u7684\u5716\u7247\u8f09\u5165\u591a\u7dad\u7684\u7a7a\u9593\u4e2d\uff0c\u63a5\u4e0b\u4f86\u5c31\u662f\u9700\u8981\u958b\u59cb\u5206\u6790\u9019\u4e9b\u5716\u7247\uff0c\u5047\u8a2d\u4e00\u958b\u59cb\u4e26\u4e0d\u77e5\u9053\u9019\u4e9b\u5716\u7247\u7684\u6a19\u7c64\u7684\u6642\u5019\uff0c\u6211\u5011\u6c92\u6709\u8fa6\u6cd5\u57f7\u884c\u5206\u985e\u7684\u8a13\u7df4\u3002\u672c\u7bc7\u60f3\u8981\u4ecb\u7d39\u4e00\u4e0b Principle Component Analysis, PCA \u4e3b\u6210\u4efd\u5206\u6790\u9019\u4e00\u500b\u65b9\u6cd5\u80cc\u5f8c\u7684\u6578\u5b78\u7406\u8ad6\u8207\u7269\u7406\u610f\u7fa9\uff0c\u53c3\u8003\u7684\u662f\u53f0\u5927\u8cc7\u5de5\u7cfb\u6797\u8ed2\u7530\u6559\u6388\u7684\u8b1b\u7fa9<\/a>\uff0c\u5728\u6797\u6559\u6388\u7684\u8b1b\u89e3\u904e\u7a0b\u4e2d\uff0cPCA \u5176\u5be6\u662f Auto-Encoder \u4e2d\u7684\u4e00\u500b\u7dda\u6027\u7279\u4f8b\uff0c\u5982\u679c\u5f9e Auto-Encoder \u7684\u89d2\u5ea6\u4f86\u770b PCA \u7684\u8a71\u53ef\u4ee5\u66f4\u52a0\u4e86\u89e3 PCA \u4e3b\u6210\u4efd\u5206\u6790\u7684\u7269\u7406\u610f\u7fa9\uff01<\/p>\n\n\n\n

\u4e00\u822c\u60c5\u6cc1\u7684 Auto-Encoder<\/h5>\n\n\n\n\n\u53ef\u4ee5\u63cf\u8ff0\u6210\u4ee5\u4e0b\u7684\u6578\u5b78\u5f0f\uff0c\u8f38\u5165\u662f\u7dad\u5ea6\u70ba \"d\" \u7684 \"\textbf{x}\" \u5411\u91cf\uff0c\u4e2d\u9593\u7684\u795e\u7d93\u5143\u6578\u91cf\u70ba \"\tilde{d}\"\uff0c\u8f38\u51fa\u70ba \"h_i(\textbf{x})\" \u7dad\u5ea6\u4e5f\u662f \"d\"\u3002\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\n\n\n

\u7dda\u6027\u7684 Auto-Encoder<\/h5>\n\n\n\n\u7dda\u6027\u7684\u60c5\u6cc1\u5247\u53ef\u4ee5\u63cf\u8ff0\u6210\u4ee5\u4e0b\u7684\u6578\u5b78\u5f0f\uff0c\u8f38\u5165\u662f\u7dad\u5ea6\u70ba \"d\" \u7684 \"\textbf{x}\" \u5411\u91cf\uff0c\u4e2d\u9593\u7684\u795e\u7d93\u5143\u6578\u91cf\u70ba \"\tilde{d}\"\uff0c\u8f38\u51fa\u70ba \"h_i(\textbf{x})\" \u7dad\u5ea6\u4e5f\u662f \"d\"\u3002\n

(2) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\u5047\u8a2d \"w^{(2)}_{ji}=w^{(1)}_{ij}=w_{ij}\" \u4e14\u7528 \"W=[w_{ij}]\" \u7dad\u5ea6\u70ba \"d\times\tilde{d}\" \u7684\u77e9\u9663\u8868\u793a\uff0c\u53e6\u5916\u70ba\u4e86\u5408\u7406\u6027\u4e5f\u5047\u8a2d \"\tilde{d} < d\"\uff0c\u6240\u4ee5\n

(3) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\n\n\n

\u7269\u7406\u610f\u7fa9<\/h5>\n\n\n\n\u9996\u5148\u6211\u5011\u5229\u7528 eigen-decompose \u7684\u6280\u8853\u5c07 \"WW^T\" \u7528\u53e6\u5916\u7684\u65b9\u5f0f\u8868\u793a\uff0c\u5176\u4e2d \"V\" \u662f \"d\times d\" \u7684\u6b63\u4ea4\u77e9\u9663\u4e5f\u5c31\u662f\u8aaa \"VV^T = V^TV = I_d\"\uff0c\u53e6\u5916 \"\Gamma\" \u662f\u5c0d\u89d2\u7dda\u77e9\u9663\uff0c\u800c\u4e14\u7e3d\u5171\u6709 \"\leq\tilde{d}\" \u500b\u503c\u4e0d\u662f\u96f6\uff0c\u4e3b\u8981\u662f\u56e0\u70ba \"W \in R^{d\times\tilde{d}}\" \u7684\u95dc\u4fc2\n

(4) <\/span>   <\/span>\"\begin{equation*} <\/p>\n\n\n\n

\u6240\u4ee5\u7dda\u6027 Auto-Encoder \u7684\u7269\u7406\u610f\u7fa9\u5c31\u662f\uff1a<\/p>\n

    \n
  1. \u5148\u5229\u7528\u4e00\u500b orthonormal \u7684\u57fa\u790e\u5411\u91cf\u96c6\u5408 V \u5c07 x \u5411\u91cf\u8f49\u7f6e\u5230\u53e6\u5916\u4e00\u500b\u5411\u91cf\u7a7a\u9593 (Vector Space)<\/span><\/li>\n
  2. \u5c07\u4e00\u90e8\u5206\u7dad\u5ea6\u653e\u5927\u7e2e\u5c0f\uff0c\u53e6\u5916\u4e00\u90e8\u5206\u7dad\u5ea6\u8a2d\u6210 0<\/span><\/li>\n
  3. \u518d\u5229\u7528\u540c\u4e00\u7d44\u57fa\u790e\u5411\u91cf\u96c6\u5408\u5c07\u8655\u7406\u904e\u5f8c\u7684\u5411\u91cf\u8f49\u56de\u539f\u672c\u7684\u5411\u91cf\u7a7a\u9593<\/span><\/li>\n<\/ol>\n\n\n\n\u5982\u679c \"\Gamma = I\" \u77e9\u9663\u7684\u8a71\uff0c\u5176\u5be6\u6211\u5011\u5c07\u4e00\u7d44\u5411\u91cf\u8f49\u7f6e\u5230\u53e6\u5916\u4e00\u500b\u5411\u91cf\u7a7a\u9593\u7dad\u6301\u500b\u6b63\u4ea4\u7dad\u5ea6\u7684\u9577\u5ea6\u4e0d\u8b8a\u518d\u8f49\u7f6e\u56de\u4f86\u7406\u8ad6\u4e0a\u5c31\u6703\u5f97\u5230\u81ea\u5df1 \n

    (5) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\n\n\n

    Problem Formulation \u554f\u984c\u8868\u8ff0<\/h5>\n\n\n\n\u6709\u4e86\u4ee5\u4e0a\u5c0d Auto-Encoder \u7684\u63cf\u8ff0\uff0c\u6211\u5011\u53ef\u4ee5\u67b6\u69cb\u6211\u5011\u7684\u554f\u984c\u70ba\uff1a\u627e\u51fa\u4e00\u500b\u6700\u4f73\u5316\u7684\u77e9\u9663 \"W\" (\u4e5f\u5c31\u662f\u4e00\u7d44 \"V, \Gamma\") \u4f7f\u5f97 Auto-Encoder \"h(\textbf{x})\" \u7684\u7d50\u679c\u8207\u539f\u59cb\u8cc7\u6599\u7684\u5dee\u8ddd \"E_{\text{in}}(\textbf{h})\" \u6700\u5c0f\u3002\n

    (6) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\n\n\n

    \u6c42\u89e3 W<\/h5>\n

    \u4ee5\u4e0b\u6211\u5011 go through \u4e00\u904d\u8b1b\u7fa9\u4e2d\u7684\u7dda\u6027\u4ee3\u6578\u8b49\u660e\uff0c\u4e5f\u5c31\u662f\u6c42\u89e3\u4ee5\u4e0b\u7684\u6700\u4f73\u5316\u554f\u984c\uff1a<\/p>\n\n\n\n

    (7) <\/span>   <\/span>\"\begin{align*}<\/p>\n\n\n\n

    \u56e0\u70ba\u4f7f\u7528 orthonormal \u7684\u5411\u91cf\u96c6\u5408\u8f49\u7f6e\u5230\u53e6\u5916\u4e00\u500b\u5411\u91cf\u7a7a\u9593\u4e26\u4e0d\u6703\u5f71\u97ff\u9577\u5ea6\uff0c\u6240\u4ee5\u6211\u5011\u5c07\u554f\u984c\u7c21\u5316\uff0c<\/p>\n\n\n\n

    (8) <\/span>   <\/span>\"\begin{align*}<\/p>\n\u53c8\u56e0\u70ba\u6211\u5011\u5e0c\u671b \"(I-\Gamma)\" \u8d8a\u591a\u96f6\u8d8a\u597d\uff0c\u6240\u4ee5\u6700\u4f73\u5316\u7684 \"\Gamma = \begin{bmatrix}\u3002\n\n

    (9) <\/span>   <\/span>\"\begin{align*}<\/p>\n\u5982\u679c \"\tilde{d} = 1\"\uff0c\u90a3\u9ebc\u6700\u4f73\u5316\u7684\u5de5\u4f5c\u5c31\u662f\u627e\u5230\u4e00\u500b \"\textbf{v}\" \u5411\u91cf\u5728\u689d\u4ef6 \"\textbf{v}^T\textbf{v} = 1\" \u7684\u689d\u4ef6\u4e0b\u53ef\u4ee5\u6700\u5927\u5316\n

    (10) <\/span>   <\/span>\"\begin{equation*}<\/p>\n\u56e0\u70ba \"\sum^{N}_{n=1}\textbf{x}_n\textbf{x}^T_n\textbf{v} = X^T X \textbf{v}= \lambda\textbf{v}\" \u6240\u4ee5\u6700\u4f73\u5316\u8aa4\u5dee\u5728 \"\tilde{d}=1\" \u7684\u60c5\u6cc1\u4e0b\u5c31\u662f\u5c0b\u627e \"X^TX\" \u4e2d\u6700\u5927\u7684\u7279\u5fb5\u503c\u5c0d\u61c9\u7684\u7279\u5fb5\u5411\u91cf\u4f5c\u70ba \"\textbf{v}\"\uff0c\u4e00\u822c\u5316\u60c5\u6cc1 \"\tilde{d} \leq d\"\uff0c\"W=\{w_j\}=\{v_j\}^{\tilde{d}}_{j=1}\" \u5c31\u662f\u7531 \"X^TX\" \u4e2d\u6700\u5927\u7684 \"\tilde{d}\" \u500b\u7279\u5fb5\u5411\u91cf\u7d44\u6210\u3002\n\n\n\n

    PCA \u8207 Auto-Encoder \u7684\u5dee\u7570<\/h5>\n

    \u5176\u5be6 PCA \u8207 Auto-Encoder \u5728\u8655\u7406\u7684\u554f\u984c\u9084\u662f\u6709\u4e00\u9ede\u5dee\u7570\u7684\uff0cAuto-Encoder \u662f\u5728\u8655\u7406\u6700\u5927\u5316\u9577\u5ea6\u7684\u554f\u984c\uff0c\u800c PCA \u5247\u662f\u5728\u8655\u7406\u6700\u5927\u5316\u8b8a\u7570\u6578\u7684\u554f\u984c\uff0c\u4f46\u662f\u5176\u5be6\u5169\u500b\u554f\u984c\u7684\u672c\u8cea\u662f\u4e00\u6a23\u7684\u3002<\/p>\n\n\n\n

    (11) <\/span>   <\/span>\"\begin{align*}<\/p>\n\n\n\n

    PCA \u5728 Clustering \u7684\u61c9\u7528\u5be6\u4f8b<\/h5>\n

    \u5728\u4ee5\u4e0b\u9023\u7d50\u4e2d\uff0c\u4f5c\u8005\u4f7f\u7528 scikit-Learn \u4e2d\u7684 PCA \u4e3b\u6210\u4efd\u5206\u6790\u5de5\u5177\u5206\u6790 MNIST \u7684\u5716\u7247\uff1a<\/p>\n\n\n\n

    \u5728 MNIST \u7684\u8cc7\u6599\u4e2d\u5be6\u4f5c PCA, K-Means, LDA, t-SNE \u7b49\u6f14\u7b97\u6cd5<\/a><\/div>\n\n\n\n

    \u539f\u672c\u8cc7\u6599\u7dad\u5ea6\u70ba 28 * 28 = 784 \u7dad\uff0c\u672c\u7bc7\u4f5c\u8005\u5c55\u793a\u5c07\u8cc7\u6599\u964d\u7dad\u6210 40 \u7dad\u4e00\u6a23\u53ef\u4ee5\u4fdd\u6301\u8cc7\u6599\u4e4b\u9593\u7684\u5dee\u7570\u6027\u4fdd\u6301\u4f4f\uff0c\u9019\u908a\u503c\u5f97\u4e00\u63d0\u7684\u4f5c\u8005\u57f7\u884c PCA \u4e4b\u524d\u6709\u7528<\/p>\n

    from sklearn.preprocessing import StandardScaler\nX_std = StandardScaler().fit_transform(X)<\/pre>\n
    \u5148\u5c07\u8cc7\u6599\u53bb\u5e73\u5747\u503c\u8207\u6b63\u898f\u5316\uff0c\u5176\u5be6\u5c31\u662f\u5f0f\u5b50 (11) \u5728\u505a\u7684\u4e8b\u60c5\uff0c\u53d6\u6700\u5927\u8b8a\u7570\u7684\u5169\u500b\u7dad\u5ea6\uff0c\u53ef\u4ee5\u5f97\u5230\u4ee5\u4e0b\u7684\u7d50\u679c\uff0c\u7d50\u679c\u986f\u793a\u5229\u7528 PCA \u964d\u6210\u5169\u500b\u7dad\u5ea6\u5728\u505a\u6a5f\u5668\u5b78\u7fd2\u7684\u6548\u679c\u53ef\u80fd\u90e8\u6703\u592a\u597d\uff0c\u9019\u908a\u53ef\u80fd\u8981\u8003\u616e\u4f7f\u7528\u4e00\u4e9b\u975e\u7dda\u6027\u7684\u5206\u985e\u65b9\u6cd5\uff0c\u4f8b\u5982 t-SNE \u7b49\u7b49\uff01<\/p>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
    \u672c\u7bc7\u60f3\u8981\u4ecb\u7d39\u4e00\u4e0b Principle Component Analysis, PCA \u4e3b\u6210\u4efd\u5206\u6790\u9019\u4e00\u500b\u65b9\u6cd5\u80cc\u5f8c\u7684\u6578\u5b78\u7406\u8ad6\u8207\u7269\u7406\u610f\u7fa9\uff0c\u53c3\u8003\u7684\u662f\u53f0\u5927\u8cc7\u5de5\u7cfb\u6797\u8ed2\u7530\u6559\u6388\u7684\u8b1b\u7fa9\uff0c\u5728\u6797\u6559\u6388\u7684\u8b1b\u89e3\u904e\u7a0b\u4e2d\uff0cPCA \u5176\u5be6\u662f Auto-Encoder \u4e2d\u7684\u4e00\u500b\u7dda\u6027\u7279\u4f8b\uff0c\u5982\u679c\u5f9e Auto-Encoder \u7684\u89d2\u5ea6\u4f86\u770b PCA \u7684\u8a71\u53ef\u4ee5\u66f4\u52a0\u4e86\u89e3 PCA \u4e3b\u6210\u4efd\u5206\u6790\u7684\u7269\u7406\u610f\u7fa9\uff01<\/p>\n","protected":false},"author":1,"featured_media":4088,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,158],"tags":[206,204,207,205],"class_list":["post-1581","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-bigdata-ml","category-probability-and-statistics","tag-mnist","tag-pca","tag-principle-component-analysis","tag-205"],"yoast_head":"\n[ML] PCA \u4e3b\u6210\u4efd\u5206\u6790 - \u60f3\u65b9\u6d89\u6cd5 - \u91cf\u74f6\u5916\u7684\u5929\u7a7a M-Y-Oceane<\/title>\n<meta name=\"description\" content=\"\u672c\u7bc7\u60f3\u8981\u4ecb\u7d39\u4e00\u4e0b Principle Component Analysis PCA \u4e3b\u6210\u4efd\u5206\u6790\u9019\u4e00\u500b\u65b9\u6cd5\u80cc\u5f8c\u7684\u6578\u5b78\u7406\u8ad6\u8207\u7269\u7406\u610f\u7fa9\uff0cPCA \u5176\u5be6\u662f Auto-Encoder \u4e2d\u7684\u4e00\u500b\u7dda\u6027\u7279\u4f8b\uff0c\u5982\u679c\u5f9e Auto-Encoder \u7684\u89d2\u5ea6\u4f86\u770b\u7684\u8a71\u53ef\u4ee5\u66f4\u52a0\u4e86\u89e3 PCA \u4e3b\u6210\u4efd\u5206\u6790\u7684\u7269\u7406\u610f\u7fa9\uff01\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/myoceane.fr\/index.php\/pca-\u4e3b\u6210\u4efd\u5206\u6790\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[ML] PCA \u4e3b\u6210\u4efd\u5206\u6790 - 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