{"id":553,"date":"2021-12-22T20:26:23","date_gmt":"2021-12-22T12:26:23","guid":{"rendered":"http:\/\/139.9.1.231\/?p=553"},"modified":"2021-12-23T14:30:30","modified_gmt":"2021-12-23T06:30:30","slug":"svd","status":"publish","type":"post","link":"http:\/\/139.9.1.231\/index.php\/2021\/12\/22\/svd\/","title":{"rendered":"SVD\uff08\u5947\u5f02\u503c\u5206\u89e3\uff09\u7b97\u6cd5"},"content":{"rendered":"\n

\u672c\u4ee5\u4e3a\u4ee5\u540e\u518d\u4e5f\u7528\u4e0d\u5230\u5230\u77e9\u9635\u8bba\u7684\u77e5\u8bc6\u4e86\uff0c\u6ca1\u60f3\u5230\u8fd8\u662f\u6ca1\u9003\u8fc7\u53bb\uff0c\u5927\u56db\u5b66\u7684\u77e9\u9635\u8bba\u73b0\u5728\u57fa\u672c\u4e0a\u5fd8\u5149\u4e86\uff0c\u4e8e\u662f\u53c8\u8981\u91cd\u65b0\u62ff\u8d77\u8bfe\u672c\u548c\u7b14\u8bb0\u4e86\uff0c\u6211\u611f\u89c9\u8fd9\u6b21\u5199\u5b8c\u8fc7\u6bb5\u65f6\u95f4\u53c8\u4f1a\u5fd8\u6389\uff0c\u8fd9\u8bb0\u6027….,\u5728\u5b66\u4e60Svd\u7684\u8fc7\u7a0b\u4e2d\uff0c\u771f\u7684\u662f\u6bcf\u4e2a\u6982\u5ff5\u90fd\u8981\u91cd\u65b0\u53bb\u770b\uff0c\u6bd4\u8f83\u9ebb\u70e6\uff0c\u4e0d\u8bf4\u4e86\uff0c\u6478\u9c7c\u53bb\u4e86……<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

\u5947\u5f02\u503c\u5206\u89e3<\/strong>\uff08Singular Value Decompositionm,\u7b80\u79f0SVD\uff09\u662f\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\u5e94\u7528\u8f83\u4e3a\u5e7f\u6cdb\u7684\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4e5f\u662f\u5b66\u4e60\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7ed5\u4e0d\u5f00\u7684\u57fa\u77f3\u4e4b\u4e00\u3002SVD\u7b97\u6cd5\u4e3b\u8981\u7528\u5728\u964d\u7ef4\u7b97\u6cd5\u4e2d\u7684\u7279\u5f81\u5206\u89e3\u3001\u63a8\u8350\u7cfb\u7edf\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406\u8ba1\u7b97\u673a\u89c6\u89c9\u7b49\u9886\u57df\u3002\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u901a\u4fd7\u4e00\u70b9\u8bb2\u5c31\u662f\u5c06\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u5206\u89e3\u4e3a\u4e24\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u4ee3\u8868\u65cb\u8f6c\uff0c\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u4ee3\u8868\u62c9\u4f38\u3002<\/p>\n\n\n\n

$$A = U\\Sigma V^T \\tag{1-1}$$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u6ce8\uff1aSVD\u662f\u5c06\u4e00\u4e2a\u77e9\u9635\u5206\u89e3\u6210\u4e24\u4e2a\u6b63\u4ea4\u77e9\u9635\u548c\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff0c\u6211\u4eec\u77e5\u9053\u6b63\u4ea4\u77e9\u9635\u5bf9\u5e94\u7684\u53d8\u6362\u662f\u65cb\u8f6c\u53d8\u6362\uff0c\u5bf9\u89d2\u77e9\u9635\u5bf9\u5e94\u7684\u53d8\u6362\u662f\u4f38\u7f29\u53d8\u6362\u3002<\/p>

<\/p><\/blockquote>\n\n\n\n

1\u3001SVD\u7b97\u6cd5\u5b9e\u73b0<\/h1>\n\n\n\n

1.1 SVD\u539f\u7406\u7b80\u5355\u56de\u987e<\/h2>\n\n\n\n

\u6709\u4e00\u4e2a\\(m \\times n\\)\u7684\u5b9e\u6570\u77e9\u9635A\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u5206\u89e3\u6210\u5982\u4e0b\u7684\u5f62\u5f0f \\( A = U\\Sigma V^T \\tag{1-1} \\) <\/p>\n\n\n\n

\u5176\u4e2dU\u548cV\u5747\u4e3a\u5355\u4f4d\u6b63\u4ea4\u9635\uff0c\u5373\u6709UU^T=I\u548cVV^T=I\uff0cU\u79f0\u4e3a\u5de6\u5947\u5f02\u77e9\u9635<\/code>\uff0cV\u79f0\u4e3a\u53f3\u5947\u5f02\u77e9\u9635<\/code>\uff0c\\( \\Sigma\\)\u4ec5\u5728\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u6709\u503c\uff0c\u6211\u4eec\u79f0\u5b83\u4e3a\u5947\u5f02\u503c<\/code>\uff0c\u5176\u5b83\u5143\u7d20\u5747\u4e3a0\u3002\u4e0a\u9762\u77e9\u9635\u7684\u7ef4\u5ea6\u5206\u522b\u4e3a $$U \\in \\mathbf{R}^{m\\times m},\\ \\Sigma \\in \\mathbf{R}^{m\\times n},\\ V \\in \\mathbf{R}^{n\\times n}$$\u3002<\/p>\n\n\n\n

\u6b63\u5e38\u6c42\u4e0a\u9762\u7684 \\( U,V,\\Sigma \\) \u4e0d\u4fbf\u4e8e\u6c42\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5982\u4e0b\u6027\u8d28 <\/p>\n\n\n\n\\(AA^T=U\\Sigma V^TV\\Sigma^TU^T=U\\Sigma \\Sigma^TU^T \\tag{1-2}A^TA=V\\Sigma^TU^TU\\Sigma V^T=V\\Sigma^T\\Sigma V^T \\tag{1-3}\\)\n\n\n\n

1.2 SVD\u7b97\u6cd5#<\/a><\/h2>\n\n\n\n

\u636e1.1\u5c0f\u8282\uff0c\u5bf9\u5f0f\uff081-3\uff09\u548c\u5f0f\uff081-4\uff09\u505a\u7279\u5f81\u503c\u5206\u89e3\uff0c\u5373\u53ef\u5f97\u5230\u5947\u5f02\u503c\u5206\u89e3\u7684\u7ed3\u679c\u3002\u4f46\u662f\u6837\u5206\u5f00\u6c42\u5b58\u5728\u4e00\u5b9a\u7684\u95ee\u9898\uff0c\u7531\u4e8e\u505a\u7279\u5f81\u503c\u5206\u89e3\u7684\u65f6\u5019\uff0c\u7279\u5f81\u5411\u91cf\u7684\u6b63\u8d1f\u53f7\u5e76\u4e0d\u5f71\u54cd\u7ed3\u679c\uff0c\u6bd4\u5982\uff0c\u6211\u4eec\u5229\u7528\u5f0f\uff081-3\uff09\u548c\uff081-4\uff09\u505a\u7279\u5f81\u503c\u5206\u89e3 \\( AA^T\\mathbf{u}_i = \\sigma_i \\mathbf{u}_i\\quad \\text{or} \\quad AA^T(-\\mathbf{u}_i) = \\sigma_i (-\\mathbf{u}_i)\\\\ A^TA\\mathbf{v}_i = \\sigma_i \\mathbf{v}_i\\quad \\text{or} \\quad A^TA(-\\mathbf{v}_i) = \\sigma_i (-\\mathbf{v}_i)\\)<\/p>\n\n\n\n

\u5982\u679c\u5728\u8ba1\u7b97\u8fc7\u7a0b\u53d6\uff0c\u53d6\u4e0a\u9762\u7684\\(\\mathbf{u}_i\\)\u7ec4\u6210\u5de6\u5947\u5f02\u77e9\u9635U\uff0c\u53d6 \\( -\\mathbf{v}_i \\) \u7ec4\u6210\u53f3\u5947\u5f02\u77e9\u9635V\uff0c\u6b64\u65f6\\(A\\ne U\\Sigma V^T \\) \u3002\u56e0\u6b64\u6c42\\(\\mathbf{v}_i\\)\u65f6\uff0c\u8981\u6839\u636e\\(\\mathbf{u}_i\\)\u6765\u6c42\uff0c\u8fd9\u6837\u624d\u80fd\u4fdd\u8bc1\\(A= U\\Sigma V^T\\)\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u5982\u4e0b1.1\u8ba1\u7b97SVD\u7684\u7b97\u6cd5\u3002\u5b83\u4e3b\u8981\u662f\u5148\u505a\u7279\u6027\u503c\u5206\u89e3\uff0c\u518d\u6839\u636e\u7279\u5f81\u503c\u5206\u89e3\u5f97\u5230\u7684\u5de6\u5947\u5f02\u77e9\u9635U\u95f4\u63a5\u5730\u6c42\u51fa\u90e8\u5206\u7684\u53f3\u5947\u5f02\u77e9\u9635\\(V’\\in \\mathbf{R}^{m\\times n}\\)\u3002<\/p>\n\n\n\n

\u7b97\u6cd51.1\uff1aSVD<\/a>\u8f93\u5165\uff1a<\/strong>\u6837\u672c\u6570\u636e
\u8f93\u51fa\uff1a<\/strong>\u5de6\u5947\u5f02\u77e9\u9635\uff0c\u5947\u5f02\u503c\u77e9\u9635\uff0c\u53f3\u5947\u5f02\u77e9\u9635<\/p>\n\n\n\n

  1. \u8ba1\u7b97\u7279\u5f81\u503c\uff1a<\/strong>\u00a0\u7279\u5f81\u503c\u5206\u89e3AA^T\uff0c\u5176\u4e2d\\(A \\in \\mathbf{R}^{m\\times n}\\) \u4e3a\u539f\u59cb\u6837\u672c\u6570\u636e\\(AA^T=U\\Sigma \\Sigma^TU^T\\)\u5f97\u5230\u5de6\u5947\u5f02\u77e9\u9635\\(U \\in \\mathbf{R}^{m \\times m}\\) \u548c\u5947\u5f02\u503c\u77e9\u9635\\(\\Sigma’ \\in \\mathbf{R}^{m \\times m}\\)<\/li>
  2. \u95f4\u63a5\u6c42\u90e8\u5206\u53f3\u5947\u5f02\u77e9\u9635\uff1a<\/strong>\u00a0\u6c42\\(V’ \\in \\mathbf{R}^{m \\times n}\\)\u5229\u7528\\(A=U\\Sigma’V’\\)\u53ef\u5f97\\(V’ = (U\\Sigma’)^{-1}A = (\\Sigma’)^{-1}U^TA \\tag{1-4}\\)<\/li>
  3. \u8fd4\u56de\\(U,\\ \\Sigma’,\\ V’\\)\uff0c\u5206\u522b\u4e3a\u5de6\u5947\u5f02\u77e9\u9635\uff0c\u5947\u5f02\u503c\u77e9\u9635\uff0c\u53f3\u5947\u5f02\u77e9\u9635\u3002<\/li><\/ol>\n\n\n\n

    \u6ce8\uff1a<\/strong>\u8fd9\u91cc\u5f97\u5230\u7684\\(\\Sigma’\\)\u548cV’\u4e0e\u5f0f\uff081-2\uff09\u6240\u5f97\u5230\u7684\\(\\Sigma,\\ V\\)\u6709\u533a\u522b\uff0c\u5b83\u4eec\u7684\u7ef4\u5ea6\u4e0d\u4e00\u6837\u3002\\(\\Sigma’\\)\u662f\u53ea\u53d6\u4e86\u524dm\u4e2a\u5947\u5f02\u503c\u5f62\u6210\u7684\u5bf9\u89d2\u65b9\u9635\uff0c\u5373\\(\\Sigma’ \\in \\mathbf{R}^{m \\times m}\uff1bV’\\)\u4e0d\u662f\u4e00\u4e2a\u65b9\u9635\uff0c\u5b83\u53ea\u53d6\u4e86\\(V \\in \\mathbf{R}^{m \\times n}\\)\u7684\u524dm\u884c\uff08\u5047\u8bbem < n\uff09\uff0c\u5373\u6709\\(V’ = V(:m,\\cdot)\\)\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u540c\u6837\u6709\u7c7b\u4f3c\u5f0f\uff081-1\uff09\u7684\u6570\u5b66\u5173\u7cfb\u6210\u7acb\uff0c\u5373\\(A = U\\Sigma’ (V’)^T\\tag{1-5}\\)<\/p>\n\n\n\n

    \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u6b64\u5173\u7cfb\u91cd\u5efa\u539f\u59cb\u6570\u636e\u3002<\/p>\n\n\n\n

    2\u3001SVD\u7684Python\u5b9e\u73b0#<\/a><\/h1>\n\n\n\n

    \u4ee5\u4e0b\u4ee3\u7801\u7684\u8fd0\u884c\u73af\u5883\u4e3apython3.6<\/code>+jupyter5.4<\/code>\u3002<\/p>\n\n\n\n

    2.1 SVD\u5b9e\u73b0\u8fc7\u7a0b#<\/a><\/h2>\n\n\n\n

    \u8bfb\u53d6\u6570\u636e#<\/a><\/h3>\n\n\n\n

    \u8fd9\u91cc\u9762\u7684\u6570\u636e\u96c6\u5927\u5bb6\u968f\u4fbf\u627e\u4e00\u4e2a\u6570\u636e\u5c31\u597d\uff0c\u5982\u679c\u6709\u9700\u8981\u6211\u7684\u6570\u636e\u96c6\uff0c\u53ef\u4ee5\u4e0b\u5728\u9762\u7559\u8a00\u3002<\/p>\n\n\n\n

    import numpy as np\nimport pandas as pd\nfrom scipy.io import loadmat\n\n# \u8bfb\u53d6\u6570\u636e\uff0c\u4f7f\u7528\u81ea\u5df1\u6570\u636e\u96c6\u7684\u8def\u5f84\u3002\ntrain_data_mat = loadmat(\"..\/data\/train_data2.mat\")\ntrain_data = train_data_mat[\"Data\"]\nprint(train_data.shape)\n<\/code><\/pre>\n\n\n\n
    \u7279\u5f81\u503c\u5206\u89e3#<\/a><\/h3>\n\n\n\n
    # \u6570\u636e\u5fc5\u9700\u5148\u8f6c\u4e3a\u6d6e\u70b9\u578b\uff0c\u5426\u5219\u5728\u8ba1\u7b97\u7684\u8fc7\u7a0b\u4e2d\u4f1a\u6ea2\u51fa\uff0c\u5bfc\u81f4\u7ed3\u679c\u4e0d\u51c6\u786e\ntrain_dataFloat = train_data \/ 255.0\n# \u8ba1\u7b97\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\neval_sigma1,evec_u = np.linalg.eigh(train_dataFloat.dot(train_dataFloat.T))\n<\/code><\/pre>\n\n\n\n
    \u8ba1\u7b97\u53f3\u5947\u5f02\u77e9\u9635#<\/a><\/h3>\n\n\n\n
    #\u964d\u5e8f\u6392\u5217\u540e\uff0c\u9006\u5e8f\u8f93\u51fa\neval1_sort_idx = np.argsort(eval_sigma1)[::-1]\n# \u5c06\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u4e5f\u5bf9\u5e94\u6392\u597d\u5e8f\neval_sigma1 = np.sort(eval_sigma1)[::-1]\nevec_u = evec_u[:,eval1_sort_idx]\n# \u8ba1\u7b97\u5947\u5f02\u503c\u77e9\u9635\u7684\u9006\neval_sigma1 = np.sqrt(eval_sigma1)\neval_sigma1_inv = np.linalg.inv(np.diag(eval_sigma1))\n# \u8ba1\u7b97\u53f3\u5947\u5f02\u77e9\u9635\nevec_part_v = eval_sigma1_inv.dot((evec_u.T).dot(train_dataFloat))\n<\/code><\/pre>\n\n\n\n
    \u4e0a\u9762\u7684\u8ba1\u7b97\u51fa\u7684evec_u, eval_sigma1, evec_part_v\u5206\u522b\u4e3a\u5de6\u5947\u5f02\u77e9\u9635\uff0c\u6240\u6709\u5947\u5f02\u503c\uff0c\u53f3\u5947\u5f02\u77e9\u9635\u3002<\/p>\n\n\n\n
    2.2 SVD\u964d\u7ef4\u540e\u91cd\u5efa\u6570\u636e#<\/a><\/h2>\n\n\n\n
    \u53d6\u4e0d\u540c\u4e2a\u6570\u7684\u5947\u5f02\u503c\uff0c\u91cd\u5efa\u56fe\u7247\uff0c\u8ba1\u7b97\u51fa\u5747\u65b9\u8bef\u5dee\uff0c\u5982\u56fe2-1\u6240\u793a\u3002\u4ece\u56fe\u4e2d\u53ef\u4ee5\u770b\u51fa\uff0c\u968f\u7740\u5947\u5f02\u503c\u7684\u589e\u52a0\uff0c\u5747\u65b9\u8bef\u5dee\uff08MSE\uff09\u5728\u51cf\u5c0f\uff0c\u4e14\u5947\u5f02\u503c\u548c\u7684\u6bd4\u7387\u6b63\u5feb\u901f\u4e0a\u5347\uff0c\u5728100\u7ef4\u65f6\uff0c\u5947\u5f02\u503c\u5360\u603b\u548c\u768453%<\/p>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    \u6ce8\uff1a<\/strong> \u5747\u65b9\u8bef\u5deeMSE\u6709\u5982\u4e0b\u8ba1\u7b97\u516c\u5f0f\\(\\text{MSE} = \\frac{1}{n}\\left((y_1-y_1′)^2+(y_2-y_2′)^2+\\cdots+(y_n-y_n’)^2\\right)\\).\u6211\u4eec\u5e73\u65f6\u542c\u5230\u7684\\(\\text{RMSE}=\\sqrt{\\text{MSE}}\\)\u3002<\/p>\n\n\n\n
    SVD\u4e0e\u7279\u5f81\u503c\u5206\u89e3\uff08EVD\uff09<\/strong>\u975e\u5e38\u7c7b\u4f3c\uff0c\u5e94\u8be5\u8bf4EVD\u53ea\u662fSVD\u7684\u4e00\u79cd\u7279\u6b8a\u6000\u51b5\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b83\u4eec\u5728\u5b9e\u9645\u7684\u5e94\u7528\u4e2d\u8fd4\u8fc7\u6765\u7406\u89e3\u7279\u5f81\u503c\/\u5947\u5f02\u503c\u7684\u542b\u4e49\uff1a\u7279\u5f81\u503c\/\u5947\u5f02\u503c\u4ee3\u8868\u7740\u6570\u636e\u7684\u4fe1\u606f\u91cf\uff0c\u5b83\u7684\u503c\u8d8a\u5927\uff0c\u4fe1\u606f\u8d8a\u591a\u3002<\/p>\n\n\n\n
    \u5947\u5f02\u503c\u5206\u89e3\u5b9a\u4e49#<\/a><\/h2>\n\n\n\n
    \u6709\u4e00\u4e2a\\(m \\times n\\)\u7684\u5b9e\u6570\u77e9\u9635A\uff0c\u6211\u4eec\u60f3\u8981\u628a\u5b83\u5206\u89e3\u6210\u5982\u4e0b\u7684\u5f62\u5f0f\\(A = U\\Sigma V^T \\tag{2-1}\\)<\/p>\n\n\n\n
    \u5176\u4e2dU\u548cV\u5747\u4e3a\u5355\u4f4d\u6b63\u4ea4\u9635\uff0c\u5373\u6709UU^T=I\u548cVV^T=I\uff0cU\u79f0\u4e3a\u5de6\u5947\u5f02\u77e9\u9635<\/code>\uff0cV\u79f0\u4e3a\u53f3\u5947\u5f02\u77e9\u9635<\/code>\uff0c\\(\\Sigma\\)\u4ec5\u5728\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u6709\u503c\uff0c\u6211\u4eec\u79f0\u5b83\u4e3a\u5947\u5f02\u503c<\/code>\uff0c\u5176\u5b83\u5143\u7d20\u5747\u4e3a0\u3002\u4e0a\u9762\u77e9\u9635\u7684\u7ef4\u5ea6\u5206\u522b\u4e3a\\(U \\in R^{m\\times m},\\ \\Sigma \\in R^{m\\times n},\\ V \\in R^{n\\times n}\\)\u3002<\/p>\n\n\n\n
    \u4e00\u822c\u5730\\(\\Sigma\\)\u6709\u5982\u4e0b\u5f62\u5f0f<\/p>\n\n\n\n
    \\(\\Sigma = \\left[ \\begin{matrix} \\sigma_1 & 0 & 0 & 0 & 0\\\\ 0 & \\sigma_2 & 0 & 0 & 0\\\\ 0 & 0 & \\ddots & 0 & 0\\\\ 0 & 0 & 0 & \\ddots & 0\\\\ \\end{matrix} \\right]_{m\\times n}\\)
    <\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
    \"\"\/
    \u5bf9\u4e8e\u5947\u5f02\u503c\u5206\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e0a\u9762\u7684\u56fe\u5f62\u8c61\u8868\u793a\uff0c\u56fe\u4e2d\u65b9\u5757\u7684\u989c\u8272\u8868\u793a\u503c\u7684\u5927\u5c0f\uff0c\u989c\u8272\u8d8a\u6d45\uff0c\u503c\u8d8a\u5927\u3002\u5bf9\u4e8e\u5947\u5f02\u503c\u77e9\u9635
    \u03a3\uff0c\u53ea\u6709\u5176\u4e3b\u5bf9\u89d2\u7ebf\u6709\u5947\u5f02\u503c\uff0c\u5176\u4f59\u5747\u4e3a0\u3002<\/figcaption><\/figure>\n\n\n\n
    \u5728\u56fe\u50cf\u538b\u7f29\u4e2d\u7684\u5e94\u7528<\/h2>\n\n\n\n
    \u51c6\u5907\u5de5\u5177<\/h3>\n\n\n\n
    \u4e0b\u9762\u7684\u4ee3\u7801\u8fd0\u884c\u73af\u5883\u4e3apython3.6<\/code>+jupyter5.4<\/code><\/p>\n\n\n\n
    SVD\uff08Python\uff09<\/h3>\n\n\n\n
    \u8fd9\u91cc\u6682\u65f6\u7528numpy\u81ea\u5e26\u7684svd\u51fd\u6570\u505a\u56fe\u50cf\u538b\u7f29<\/p>\n\n\n\n
    \u2460\u8bfb\u53d6\u56fe\u7247<\/p>\n\n\n\n
    %matplotlib inline\nimport matplotlib.pyplot as plt\nimport matplotlib.image as mpimg\nimport numpy as np\n\nimg_eg = mpimg.imread(\"..\/img\/beauty.jpg\")\nprint(img_eg.shape)\n<\/code><\/pre>\n\n\n\n
    \u56fe\u7247\u7684\u5927\u5c0f\u662f\\(600\\times 400 \\times 3\\)<\/p>\n\n\n\n
    \u2461\u5947\u5f02\u503c\u5206\u89e3<\/p>\n\n\n\n
    img_temp = img_eg.reshape(600, 400 * 3)\nU,Sigma,VT = np.linalg.svd(img_temp)\n<\/code><\/pre>\n\n\n\n
    \u6211\u4eec\u5148\u5c06\u56fe\u7247\u53d8\u6210\\(600\\times 1200\\)\uff0c\u518d\u505a\u5947\u5f02\u503c\u5206\u89e3\u3002\u4ecesvd<\/code>\u51fd\u6570\u4e2d\u5f97\u5230\u7684\u5947\u5f02\u503csigma<\/code>\u5b83\u662f\u4ece\u5927\u5230\u5c0f\u6392\u5217\u7684\u3002<\/p>\n\n\n\n
    \u2462\u53d6\u524d\u90e8\u5206\u5947\u5f02\u503c\u91cd\u6784\u56fe\u7247<\/p>\n\n\n\n
    # \u53d6\u524d60\u4e2a\u5947\u5f02\u503c\nsval_nums = 60\nimg_restruct1 = (U[:,0:sval_nums]).dot(np.diag(Sigma[0:sval_nums])).dot(VT[0:sval_nums,:])\nimg_restruct1 = img_restruct1.reshape(600,400,3)\n\n# \u53d6\u524d120\u4e2a\u5947\u5f02\u503c\nsval_nums = 120\nimg_restruct2 = (U[:,0:sval_nums]).dot(np.diag(Sigma[0:sval_nums])).dot(VT[0:sval_nums,:])\nimg_restruct2 = img_restruct2.reshape(600,400,3)\n<\/code><\/pre>\n\n\n\n
    \u5c06\u56fe\u7247\u663e\u793a\u51fa\u6765\u770b\u4e00\u4e0b\uff0c\u5bf9\u6bd4\u4e0b\u6548\u679c<\/p>\n\n\n\n
    fig, ax = plt.subplots(1,3,figsize = (24,32))\n\nax[0].imshow(img_eg)\nax[0].set(title = \"src\")\nax[1].imshow(img_restruct1.astype(np.uint8))\nax[1].set(title = \"nums of sigma = 60\")\nax[2].imshow(img_restruct2.astype(np.uint8))\nax[2].set(title = \"nums of sigma = 120\")\n<\/code><\/pre>\n\n\n\n
    <\/a><\/p>\n\n\n\n
    \"\"\/<\/figure>\n\n\n\n
    \u964d\u7ef4\u2014\u2014\u5947\u5f02\u503c\u5206\u89e3<\/h2>\n\n\n\n
          \u5947\u5f02\u503c\u5206\u89e3\uff0c\u5168\u79f0Singular Value Decomposition\uff0c\u7b80\u79f0SVD\u3002\u5b83\u662f\u4e00\u79cd\u77e9\u9635\u56e0\u5f0f\u5206\u89e3\u3002\u901a\u8fc7\u8ba1\u7b97\u5947\u5f02\u503c\u4e2a\u6570\u548c\u5947\u5f02\u5411\u91cf\uff0c\u751f\u6210\u4e00\u4e2a\u53ef\u4ee5\u4ee3\u66ff\u539f\u77e9\u9635\u7684\u8fd1\u4f3c\u77e9\u9635\uff0c\u5c06\u6570\u636e\u96c6\u7684\u5947\u5f02\u503c\u8868\u5f81\u6309\u91cd\u8981\u6027\u6392\u5217\uff0c\u820d\u5f03\u4e0d\u91cd\u8981\u7684\u7279\u5f81\u5411\u91cf\u3002\u53ef\u7528\u6765\u8fbe\u5230\u964d\u7ef4<\/strong>\u7684\u76ee\u7684\uff0c\u4ece\u800c\u627e\u51fa\u6570\u636e\u4e2d\u7684\u4e3b\u6210\u5206\u3002<\/p>\n\n\n\n
          \u5947\u5f02\u503c\u5206\u89e3\u7684\u8ba1\u7b97\u8be5\u65b9\u6cd5\u7c7b\u4f3c\u4e3b\u6210\u5206\u5206\u6790<\/strong>(PCA)\uff0c\u5dee\u522b\u5728\u4e8ePCA\u5229\u7528\u534f\u65b9\u5dee\u77e9\u9635\u8fdb\u884c\u5206\u89e3\uff0c\u800cSVD\u76f4\u63a5\u5728\u539f\u59cb\u77e9\u9635\u4e0a\u8fdb\u884c\u5206\u89e3\u3002\u6240\u4ee5SVD\u4e0d\u8981\u6c42\u88ab\u5206\u89e3\u7684\u77e9\u9635\u662f\u65b9\u9635<\/strong>\uff0c\u53ef\u4ee5\u64cd\u4f5cPCA\u65e0\u6cd5\u64cd\u4f5c\u7684\u6570\u636e\u96c6\uff0c\u8fd9\u4e5f\u662fSVD\u6709\u4ef7\u503c\u7684\u7279\u70b9\u4e4b\u4e00\u3002<\/p>\n\n\n\n
    SVD\u63a8\u8350\u7cfb\u7edf<\/h2>\n\n\n\n
    \u9996\u5148\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u4efb\u4f55\u63a8\u8350\u7cfb\u7edf\u7684\u6574\u4f53\u5927\u6846\u67b6\u90fd\u662f\u4e24\u90e8\u5206\uff1a\u5bf9\u67d0\u4e2a\u7528\u6237user\u800c\u8a00\uff1a\u9996\u5148\u662f\u4ece\u6570\u767e\u4e07\u79cdItem\u4e2d\u7c97\u7565\u7684\u9009\u51fa\u5343\u7ea7\u522b\u7684Item\uff08\u5b9e\u73b0\u8fd9\u79cd\u529f\u80fd\u7684\u65b9\u6cd5\u53eb\u505a\u53ec\u56de\u7b97\u6cd5<\/strong>\uff09\uff0c\u7136\u540e\u4ece\u51e0\u767e\u4e0a\u5343\u7684Item\u4e2d\u7cbe\u7ec6\u7684\u9884\u6d4b\u7b97\u51fa\u7528\u6237\u5bf9\u6bcf\u4e2aItem\u7684\u5177\u4f53\u8bc4\u4ef7\u5206\u6570\uff0c\u7136\u540e\u9009\u51fa\u524d\u5341\u4e2a\u8bc4\u4ef7\u5206\u6570\u6700\u9ad8\u7684Item\u63a8\u8350\u7ed9\u7528\u6237\uff08\u5b9e\u73b0\u8be5\u65b9\u6cd5\u7684\u8fc7\u7a0b\u79f0\u4e3a\u6392\u5e8f\u7b97\u6cd5<\/strong>\uff09<\/p>\n\n\n\n
    \u57fa\u4e8e\u53ec\u56de\u7684\u7b97\u6cd5\uff0c\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u7684\u6709\u4e24\u79cd\uff1a\u57fa\u4e8e\u7528\u6237\u884c\u4e3a\u7684\u4ee5\u53ca\u57fa\u4e8e\u5185\u5bb9\u7684\u3002\u57fa\u4e8e\u7528\u6237\u884c\u4e3a\u7684\u63a8\u8350\u7b97\u6cd5\u6211\u4eec\u4e5f\u79f0\u4e3a\u534f\u540c\u8fc7\u6ee4\u7b97\u6cd5\uff0c\u57fa\u4e8e\u5185\u5bb9\u7684\u63a8\u8350\u7b97\u6cd5\u540e\u671f\u8be6\u7ec6\u8ba8\u8bba\u3002\u800c\u534f\u540c\u8fc7\u6ee4\u7b97\u6cd5\u53c8\u5206\u4e3a\u4e24\u5927\u7c7b\uff0c\u4e00\u7c7b\u662f\u57fa\u4e8e\u90bb\u57df\u7684\u534f\u540c\u8fc7\u6ee4\u7b97\u6cd5\uff0c\u4e00\u7c7b\u662f\u57fa\u4e8eLFM\uff08\u9690\u56e0\u5b50\u6a21\u578b\uff09\uff0c\u800c\u6211\u4eec\u4eca\u5929\u91cd\u70b9\u8ba8\u8bba\u7684SVD\u7b97\u6cd5\u5c31\u662f\u53ec\u56de\u7b97\u6cd5\u91cc\u7684LFM\u7b97\u6cd5\u3002<\/p>\n\n\n\n
    \u6211\u4eec\u5728\u53ec\u56de\u9636\u6bb5\u7684\u76ee\u7684\u662f\u7531\u4e00\u4e2a\u7a00\u758f\u7684\u8bc4\u5206\u77e9\u9635\u63a8\u7b97\u51fa\u5176\u4e2d\u7684\u7a7a\u7f3a\u5206\u6570\u3002\uff08\u8865\u5145\uff1a\u8bc4\u5206\u77e9\u9635\u662f\u6307\u6bcf\u4e2a\u7528\u6237\u5bf9\u5404\u4e2a\u7535\u5f71\u7684\u8bc4\u4ef7\u5206\u6570\uff0cm\u884c\u4ee3\u8868m\u4e2a\u7528\u6237\uff0cn\u884c\u4ee3\u8868n\u4e2a\u7535\u5f71\uff09<\/p>\n\n\n\n
    \u6211\u4eec\u7684\u76ee\u7684\u662f\u5f97\u5230\u7528\u6237\u5bf9\u672a\u8bc4\u4ef7\u7535\u5f71\u7684\u5206\u6570\uff0c\u800c\u8fd9\u79cduser-Item\u7684\u8bc4\u5206\u77e9\u9635\u4e2d\uff0c\u7528\u6237\u5bf9Item\u7684\u5206\u6570\u662f\u6ca1\u6709\u76f4\u63a5\u5173\u7cfb\u7684\uff0c\u6211\u4eec\u9700\u8981\u5bfb\u627e\u4e00\u79cd\u9690\u7a7a\u95f4\uff0c\u4f7f\u5f97\u5c06\u7528\u6237\u548cItem\u8054\u7cfb\u8d77\u6765\u3002\u6bd4\u5982\u6211\u4eec\u65e0\u6cd5\u5f97\u77e5user1\u5bf9\u300a\u5229\u5203\u51fa\u9798\u300b\u8fd9\u90e8\u7535\u5f71\u7684\u5206\u6570\uff0c\u4f46\u662f\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7528\u6237\u7684\u5386\u53f2\u4fe1\u606f\uff08\u5373\u7528\u6237\u5bf9\u522b\u7684\u7535\u5f71\u7684\u5206\u6570\uff09\u5f97\u5230\u7528\u6237\u5bf9\u201c\u60ac\u7591\u201d\u7c7b\u7535\u5f71\u7684\u7231\u597d\u7a0b\u5ea6\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5f97\u5230\u67d0\u4e2a\u7535\u5f71\u7684\u201c\u60ac\u7591\u7c7b\u201d\u7684\u7a0b\u5ea6\u6709\u591a\u5927\uff0c\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u8fd9\u4fe9\u4e2a\u5173\u7cfb\u8054\u7cfb\u5230\u4e00\u8d77\u4e86\uff0c\u6bd4\u5982\u6211\u4eec\u60f3\u7528\u6237A\u5bf9\u300a\u5357\u65b9\u8f66\u7ad9\u7684\u6545\u4e8b\u300b\u8fdb\u884c\u9884\u6d4b\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5f97\u5230\u4ed6\u7684\u5206\u6570\uff0c\u4f46\u662f\u6211\u4eec\u77e5\u9053\u4ed6\u5bf9\u201c\u60ac\u7591\u201d\u7c7b\u7684\u7535\u5f71\u6709\u591a\u559c\u7231\uff0c\u800c\u300a\u5357\u65b9\u8f66\u7ad9\u7684\u6545\u4e8b\u300b\u7684\u201c\u60ac\u7591\u201d\u7a0b\u5ea6\u6709\u591a\u5927\uff0c\u4e24\u4e2a\u503c\u76f8\u4e58\u5373\u53ef\u5f97\u5230A\u5bf9\u300a\u5357\u65b9\u8f66\u7ad9\u7684\u6545\u4e8b\u300b\u7684\u9884\u6d4b\u5206\u6570\uff0c\u8fd9\u91cc\u6211\u4eec\u53ea\u662f\u4f7f\u7528\u5230\u4e86\u4e00\u4e2a\u7ef4\u5ea6\uff1a\u201c\u60ac\u7591\u201d\u7c7b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u77e9\u9635A\u5206\u89e3\u6210m* k\u5217\u7684\u4e00\u5171k\u4e2a\u7ef4\u5ea6\u7684\u77e9\u9635\uff0c\u5c06\u7535\u5f71\u5206\u89e3\u6210k\u00a0*\u00a0<\/strong>n\u5217\u4e00\u5171k\u4e2a\u7ef4\u5ea6\u7684\u77e9\u9635\uff0c\u7136\u540e\u8fd9\u4e24\u4e2a\u77e9\u9635\u76f8\u4e58\u5373\u53ef\u5f97\u5230\u77e9\u9635\u4e2d\u7a7a\u7684A[i][j]\u7684\u503c\u3002<\/p>\n\n\n\n
    \u8bf4\u767d\u4e86\uff0c\u6211\u4eec\u8fd9\u91cc\u4f7f\u7528\u5230\u5f97SVD\u7684\u601d\u60f3\u53ea\u662f\u8bf4\u4efb\u4f55\u4e00\u4e2a\u77e9\u9635\u5747\u53ef\u4ee5\u5206\u89e3\u6210\u4e24\u4e2a\u77e9\u9635\u76f8\u4e58\u7684\u5f62\u5f0f\uff08SVD\u662f\u4e09\u4e2a\uff0c\u4f46\u4e5f\u53ef\u4ee5\u5408\u5e76\u4fe9\u4e2a\u5f97\u5230\u4e24\u4e2a\u77e9\u9635\uff09\uff0c\u8fd9\u4fe9\u4e2a\u5b50\u77e9\u9635\u5206\u522b\u4ee3\u8868\u4e86\u7528\u6237\u7684\u504f\u597d\u7a0b\u5ea6\u548c\u7535\u5f71\u7684\u6210\u5206\u7a0b\u5ea6\uff0c\u5c06\u8fd9\u4e24\u4e2a\u77e9\u9635\u76f8\u4e58\u5373\u53ef\u5f97\u5230\u7528\u6237\u5bf9\u7535\u5f71\u7684\u8bc4\u5206\u77e9\u9635\u3002<\/p>\n\n\n\n
    \u81f3\u6b64\uff0c\u6211\u4eec\u4f7f\u7528SVD\u7684\u601d\u60f3\u5c06\u77e9\u9635A\u5206\u89e3\u6210\u4e24\u4e2a\u77e9\u9635\u76f8\u4e58\u7684\u5f62\u5f0f\uff08B*C\uff09\uff0c\u4f46\u4e8b\u60c5\u6ca1\u6709\u90a3\u4e48\u7b80\u5355\uff0c\u6211\u4eec\u7684\u63a8\u8350\u7cfb\u7edf\u7684\u77e9\u9635A\u662f\u4e00\u4e2a\u7a00\u758f\u77e9\u9635\uff0c\u57fa\u672c\u4e0a\u662f\u4e0d\u80fd\u5206\u89e3\u7684\uff0c\u65e0\u6cd5\u76f4\u63a5SVD\uff0c\u4f46\u5206\u89e3\u7684\u601d\u60f3\u6211\u4eec\u4f9d\u65e7\u8981\u4f7f\u7528\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u673a\u5668\u5b66\u4e60\u7684\u65b9\u6cd5\uff0c\u4f7f\u7528\u6700\u5c0f\u5747\u65b9\u8bef\u5dee\u5b66\u4e60\u5230\u6bcf\u4e2a\u77e9\u9635\u7684\u5404\u4e2a\u5143\u7d20\uff0c\u5177\u4f53\u6765\u8bb2\u5c31\u662f\uff0c\u77e9\u9635B\u7684i\u884c\u4e58\u77e9\u9635C\u7684j\u5217\u5f97\u5230A\u7684ij\u5143\u7d20\uff0c\u6709\u771f\u5b9e\u503c\u7684A[i][j]\u5143\u7d20\u4f5c\u4e3a\u6807\u7b7e\uff0c\u4e0d\u65ad\u5730\u4f7f\u7528RMSE\u51cf\u5c0fA\u4e2d\u6240\u6709\u7684\u6807\u7b7e\uff0c\u7136\u540e\u5f97\u5230\u77e9\u9635B\u548cC\uff0c\u7136\u540e\u4f7f\u7528B\u548cC\u53d6\u9884\u6d4bA\u4e2d\u7684\u7a7a\u767d\u7f3a\u5931\u503c\u3002<\/p>\n\n\n\n
    \u53c2\u8003\uff1a<\/p>\n\n\n\n
    https:\/\/www.cnblogs.com\/endlesscoding\/p\/10033527.html<\/a><\/p>\n\n\n\n
    https:\/\/www.cnblogs.com\/endlesscoding\/p\/10058532.html<\/a><\/p>\n\n\n\n
    https:\/\/www.imooc.com\/article\/267351<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
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