{"id":272,"date":"2023-09-13T21:43:42","date_gmt":"2023-09-13T13:43:42","guid":{"rendered":"http:\/\/106.52.213.145:21080\/?p=272"},"modified":"2023-09-17T23:59:41","modified_gmt":"2023-09-17T15:59:41","slug":"gainianlijiejuzhenchengjitaitensor-train-fenjieyudaimashixian","status":"publish","type":"post","link":"https:\/\/apifj.com\/index.php\/2023\/09\/13\/gainianlijiejuzhenchengjitaitensor-train-fenjieyudaimashixian\/","title":{"rendered":"[\u5b66\u4e60\u7b14\u8bb0]\u77e9\u9635\u4e58\u79ef\u6001Tensor Train Decomposition \u4e0e\u4ee3\u7801\u5b9e\u73b0"},"content":{"rendered":"

\u4e00\u3001\u77e9\u9635\u4e58\u79ef\u6001\u4e0eTensor-Train \u5206\u89e3<\/h2>\n

1.1 \u77e9\u9635\u4e58\u79ef\u6001\u4e0eTensor-Train \u6982\u5ff5<\/h3>\n

\u77e9\u9635\u4e58\u79ef\u6001\uff08Matrix Product State\uff0c\u7b80\u79f0MPS\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u8868\u793a\u91cf\u5b50\u591a\u4f53\u7cfb\u7edf\u7684\u65b9\u6cd5\uff0c\u7279\u522b\u5728\u91cf\u5b50\u4fe1\u606f\u3001\u91cf\u5b50\u8ba1\u7b97\u548c\u51dd\u805a\u6001\u7269\u7406\u9886\u57df\u4e2d\u5f97\u5230\u5e7f\u6cdb\u5e94\u7528\u3002\u5b83\u662f\u4e00\u79cd\u5f20\u91cf\u7f51\u7edc\u7684\u8868\u793a\u65b9\u6cd5\uff0c\u7528\u4e8e\u63cf\u8ff0\u91cf\u5b50\u6001\u4ee5\u53ca\u5176\u6f14\u5316\u8fc7\u7a0b\u3002<\/p>\n

Tensor Train\u5206\u89e3\u662f\u4e00\u79cd\u7528\u4e8e\u5206\u89e3\u591a\u7ef4\u5f20\u91cf\u7684\u65b9\u6cd5\uff0c\u7c7b\u4f3c\u4e8e\u77e9\u9635\u5206\u89e3\uff0c\u4f46\u9002\u7528\u4e8e\u66f4\u9ad8\u7ef4\u5ea6\u7684\u6570\u636e\u3002\u5b83\u5c06\u4e00\u4e2a\u591a\u7ef4\u5f20\u91cf\u5206\u89e3\u6210\u4e00\u7cfb\u5217\u7684\u5c0f\u5757\uff08\u6216\u79f0\u4e3a\u6838\uff09\uff0c\u6bcf\u4e2a\u5c0f\u5757\u662f\u4e00\u4e2a\u4f4e\u79e9\u5f20\u91cf\uff0c\u5b83\u4eec\u76f8\u4e92\u8fde\u63a5\u7ec4\u6210\u4e00\u4e2a\u7f51\u7edc\u3002\u8fd9\u79cd\u5206\u89e3\u5141\u8bb8\u9ad8\u6548\u5730\u8868\u793a\u548c\u64cd\u4f5c\u9ad8\u7ef4\u6570\u636e\uff0c\u540c\u65f6\u4fdd\u7559\u4e86\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\u548c\u7279\u5f81\u3002<\/p>\n

\u9635\u4e58\u79ef\u6001\u4e3b\u8981\u5e94\u7528\u4e8e\u91cf\u5b50\u7269\u7406\u9886\u57df\uff0c\u7528\u4e8e\u63cf\u8ff0\u91cf\u5b50\u6001\u7684\u6027\u8d28\u548c\u6f14\u5316\uff0c\u800cTensor-Train \u5206\u89e3\u66f4\u5e7f\u6cdb\u5730\u5e94\u7528\u4e8e\u673a\u5668\u5b66\u4e60\u3001\u6570\u503c\u8ba1\u7b97\u7b49\u9886\u57df\uff0c\u7528\u4e8e\u5904\u7406\u9ad8\u7ef4\u6570\u636e\u7684\u8868\u793a\u548c\u8ba1\u7b97\u3002<\/p>\n

\n\\varphi_{s_0s_1\u00b7\u00b7\u00b7s_{N-1}}=\\sum_{\\{a_*\\}} \\prod_n^{N-1}A^{(n)}_{a_ns_na_{n+1}}I_{a_0a_N}\\\\\n=\\sum_{\\{a_*\\}}A^{(0)}_{a_0s_0a_1} A^{(1)}_{a_1s_1a_2}\u00b7\u00b7\u00b7A^{(N-2)}_{a_{N-2}s_{N-2}a_{N-1}}A^{(N-1)}_{a_{N-1}s_{N-1}a_{N}}\n<\/code><\/pre>\n
\u5176\u4e2d\uff0c<\/p>\n
    \n
  • \\varphi \\in \\R^{s_0 \\times s_1 \u00b7\u00b7\u00b7\\times s_{N-1}}<\/code>  \u662f\u4e00\u4e2a N \u9636\u5f20\u91cf\uff0c\u8868\u793a\u88ab\u5206\u89e3\u7684\u77e9\u9635\u3002\u5b83\u7684\u6bcf\u4e00\u4e2a\u7d22\u5f15 s_i<\/code> \u5bf9\u5e94\u4e8e\u8be5\u5f20\u91cf\u5728\u7b2c i \u4e2a\u7ef4\u5ea6\u4e0a\u7684index\u3002N<\/code> \u8868\u793a\u8be5\u5f20\u91cf\u7684\u9636\u6570\uff0c\u5373\u5b83\u6709\u591a\u5c11\u4e2a\u7ef4\u5ea6\u3002<\/li>\n
  • \\{a_*\\}<\/code> \u662f\u4e00\u7ec4\u7d22\u5f15\uff0c\u8868\u793a\u5728\u5f20\u91cf\u4e58\u6cd5\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u5c0f\u5f20\u91cf\u6838 A^{(n)}<\/code> \u548c I<\/code> \u6240\u8fde\u63a5\u7684\u65b9\u5f0f\u3002<\/li>\n
  • A^{(n)}_{a_ns_na_{n+1}}<\/code> \u662f\u4e00\u4e2a\u4e09\u7ef4\u5f20\u91cf\uff0c\u5176\u4e2d n<\/code> \u662f\u7d22\u5f15\uff0ca_n<\/code>\uff0cs_n<\/code> \u548c a_{n+1}<\/code> \u662f\u5f20\u91cf\u5728\u4e0d\u540c\u7ef4\u5ea6\u4e0a\u7684\u7d22\u5f15\u3002\u8fd9\u4e9b\u7ef4\u5ea6\u6709\u7279\u5b9a\u7684\u5c3a\u5bf8\uff0c\u5176\u4e2d \\chi<\/code> \u662f\u4e00\u4e2a\u53c2\u6570\uff0c\u7528\u4e8e\u63a7\u5236\u5f20\u91cf\u6838\u7684\u79e9\uff08rank\uff09\u3002<\/li>\n
  • A_{a_ns_na_{n+1}}\\in \\R^{\\chi \\times d \\times \\chi}<\/code>  ,\u5176\u4e2d,d<\/code>\u662f\u7269\u7406\u7ef4\u5ea6,\\chi<\/code> \u662f\u865a\u62df\u7ef4\u5ea6\uff08\u5173\u7cfb\u5230\u4e86\u6211\u4eec\u9700\u8981\u538b\u7f29\u7684\u7cfb\u6570\uff09<\/li>\n
  • d<\/code> \u8868\u793a\u6bcf\u4e2a\u7ef4\u5ea6\u6709\u591a\u5c11\u6570\u636e<\/li>\n
  • N<\/code> \u8868\u793a\u5f20\u91cf\u7684\u9636\u6570\uff0c\u5373\u5b83\u6709\u591a\u5c11\u4e2a\u7ef4\u5ea6\u3002<\/li>\n<\/ul>\n
    \u4ece\u7ef4\u5ea6\u65b9\u9762\u770b\u5f85\uff0c\u7ef4\u5ea6\u7684\u53d8\u5316\u5982\u4e0b\uff1a<\/p>\n
    \n\\R^{d^N}=\\R^{\n[(1 \\times d \\times \\chi)\\times (\\chi \\times d \\times \\chi)\u00b7\u00b7\u00b7  (\\chi \\times d \\times 1)]\n}\n<\/code><\/pre>\n
    \u4e5f\u5c31\u662f\u8bf4\uff0c\u4ece\u539f\u6765\u7684\u5355\u4e00\u7684\\R^{d^N}<\/code> \u5206\u89e3\u5230N-2\u4e2a\\R^{\\chi \\times d \\times \\chi}<\/code>  \u548c \\R^{1 \\times d \\times \\chi}<\/code>  \u548c\u4e00\u4e2a\\R^{\\chi \\times d \\times 1}<\/code>\u7684\u77e9\u9635<\/p>\n
    1.2 \u4e3a\u4ec0\u4e48\u5b83\u53ef\u4ee5\u538b\u7f29\uff1f<\/h3>\n
    \u6211\u4eec\u5047\u8bbe\u5979\u7684\u7269\u7406\u7ef4\u5ea6\u4e3ad<\/code> ,\u865a\u62df\u7ef4\u5ea6\u4e3a\\chi<\/code>  \u5219\u539f\u6765\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\\varphi<\/code><\/p>\n
    \n(d \\times d \u00b7\u00b7\u00b7 d)\\sim O(d^N)\n<\/code><\/pre>\n
    \u800c\u5206\u89e3\u540e\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3aA<\/code><\/p>\n
    \n2d\\chi+(N-2)d\\chi^2 \\sim O(Nd\\chi^2)\n<\/code><\/pre>\n
    \u4e5f\u5c31\u662f\u8bf4\uff0c\u901a\u8fc7\u5206\u89e3\u6211\u4eec\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4eceO(d^N)<\/code> \u964d\u5230 O(Nd\\chi^2)<\/code> ,\u5f53\\chi<\/code> \u4e00\u5b9a\u65f6\uff0c\u5b58\u50a8\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u6307\u6570\u7ea7\u522b\u964d\u4f4e\u5230\u7ebf\u6027\u7ea7\u522b\uff0c\u800c\u6709\u4e86A\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\\varphi<\/code> \u6240\u4ee5\u5c31\u8fbe\u5230\u4e86\u538b\u7f29\u7684\u76ee\u7684\u3002<\/p>\n
    1.3 \u56fe\u5f62\u5316\u8868\u793a\u4e58\u79ef\u6001<\/h3>\n
    \"\"<\/p>\n
    \u8fd9\u4e2a\u56fe\u50cf\u4e2d\uff0c\u84dd\u8272\u8868\u793a\u77e9\u9635\uff0c\u5176\u4ed6\u7b26\u53f7\u8868\u793a\u6307\u6807\u3002\u4e5f\u5c31\u662f\u8bf4s_0 \u3001 a_1<\/code> \u4f5c\u7528\u5728A^{(0)}<\/code> \u4e0a\u9762\uff0c\u56fe\u5f62\u65e0\u6cd5\u77e5\u9053\u4f5c\u7528\u987a\u5e8f\uff0c\u5982\u4e0a\u9762\u662f\u4e00\u4e2a\u5faa\u73af\u7684\u4e58\u79ef\u6001\uff0c\u6240\u4ee5\u4e5f\u79f0\u4e3a\u95ed\u8fb9\u754c\u3002\u5171\u540c\u4f5c\u7528\u7684\u6307\u6807\u662f\u53ef\u4ee5\u88ab\u6d88\u9664\u7684(\u7231\u56e0\u65af\u5766\u6c42\u548c\u4e2d\u7684\u5de6\u8fb9\u5171\u6709\u6307\u6807)\uff0c\u6700\u540e\u5f97\u5230\u7684\u662f\u4e0d\u53ef\u88ab\u6d88\u9664\u7684\u6307\u6807\u4f5c\u7528\u4e8e\u6700\u540e\u7ed3\u679c\u4e4b\u4e0a(\u7231\u56e0\u65af\u5766\u6c42\u548c\u4e2d\u7684\u53f3\u8fb9\u90e8\u5206)<\/p>\n
    \u4e8c\u3001\u4e2d\u5fc3\u6b63\u4ea4\u5316\uff08Center Orthogonalization\uff09<\/h2>\n
    \u5728\u5f20\u91cf\u7f51\u7edc\u8868\u793a\u4e2d\uff0c\u7279\u522b\u662f\u50cfTensor Train\uff08TT\uff09\u548cMatrix Product States\uff08MPS\uff09\u8fd9\u6837\u7684\u65b9\u6cd5\u4e2d\uff0c\u5f20\u91cf\u4e4b\u95f4\u7684\u6b63\u4ea4\u6027\u5bf9\u4e8e\u9ad8\u6548\u8868\u793a\u548c\u8ba1\u7b97\u81f3\u5173\u91cd\u8981\u3002\u4e2d\u5fc3\u6b63\u4ea4\u5316\u662f\u4e00\u79cd\u5c06\u5f20\u91cf\u7f51\u7edc\u4e2d\u7684\u5f20\u91cf\u91cd\u65b0\u6392\u5217\u548c\u53d8\u6362\u7684\u8fc7\u7a0b\uff0c\u4ee5\u4fbf\u4f7f\u5f97\u4e00\u90e8\u5206\u4fe1\u606f\u96c6\u4e2d\u5728\u67d0\u4e2a\u6838\u5fc3\uff08\u4e2d\u5fc3\uff09\u5f20\u91cf\u4e2d\uff0c\u4ece\u800c\u63d0\u9ad8\u8868\u793a\u7684\u7d27\u51d1\u6027\u3002<\/p>\n
    \u5728\u4e2d\u5fc3\u6b63\u4ea4\u5316\u8fc7\u7a0b\u4e2d\uff0c\u4e00\u5f00\u59cb\u7684\u5f20\u91cf\u88ab\u5206\u89e3\u6210\u4e00\u5bf9\u6b63\u4ea4\u5f20\u91cf\uff0c\u8fd9\u4e00\u5bf9\u5f20\u91cf\u5206\u522b\u4ee3\u8868\u4e86\u5de6\u8fb9\u548c\u53f3\u8fb9\u7684\u6b63\u4ea4\u90e8\u5206\u3002\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5947\u5f02\u503c\u5206\u89e3\uff08Singular Value Decomposition\uff0cSVD\uff09\u6765\u5b9e\u73b0\u3002<\/p>\n
    \u5bf9\u4e8e\u4e00\u4e9b\u7279\u5b9a\u7684\u91cf\u5b50\u7b97\u6cd5\u6216\u6570\u503c\u6a21\u62df\uff0c\u6b63\u4ea4\u5316\u53ef\u4ee5\u4f7f\u8ba1\u7b97\u8fc7\u7a0b\u66f4\u4e3a\u9ad8\u6548\u3002\u6bd4\u5982\u6c42\u4e8c\u8303\u5f0f\u5219\u53ef\u4ee5\u6c42\u4e2d\u5fc3\u7684\u4e8c\u8303\u5f0f\uff0c\u5927\u5927\u63d0\u9ad8\u6548\u7387\u3002<\/p>\n
    2.1 \u5947\u5f02\u503c\u5206\u89e3\u6cd5\u5206\u89e3\u5411\u91cf<\/h3>\n
    2.1.1 \u5947\u5f02\u503c\u5206\u89e3\u6982\u5ff5<\/h4>\n
    \u5947\u5f02\u503c\u5206\u89e3\uff08Singular Value Decomposition\uff0cSVD\uff09\u662f\u4e00\u79cd\u5e38\u7528\u4e8e\u77e9\u9635\u5206\u89e3\u548c\u964d\u7ef4\u7684\u6570\u5b66\u65b9\u6cd5\u3002\u7ed9\u5b9a\u4e00\u4e2a\u77e9\u9635 M<\/code>\uff0cSVD \u5c06\u5176\u5206\u89e3\u4e3a\u4e09\u4e2a\u77e9\u9635\u7684\u4e58\u79ef\uff1a<\/p>\n
    \nM = U \\Sigma V^\\top\n<\/code><\/pre>\n
    \u5176\u4e2d\uff1a<\/p>\n
      \n
    • U<\/code> \u662f\u4e00\u4e2a\u6b63\u4ea4\u77e9\u9635\uff0c\u5176\u5217\u5411\u91cf\u79f0\u4e3a\u5de6\u5947\u5f02\u5411\u91cf\u3002<\/li>\n
    • \\Sigma<\/code> \u662f\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff0c\u5176\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u79f0\u4e3a\u5947\u5f02\u503c\u3002\u5947\u5f02\u503c\u662f\u6309\u7167\u4ece\u5927\u5230\u5c0f\u6392\u5217\u7684\u3002<\/li>\n
    • V<\/code> \u662f\u53e6\u4e00\u4e2a\u6b63\u4ea4\u77e9\u9635\uff0c\u5176\u884c\u5411\u91cf\u79f0\u4e3a\u53f3\u5947\u5f02\u5411\u91cf\u3002<\/li>\n<\/ul>\n
      \u5728SVD\u5206\u89e3\u4e2d\uff0c\u901a\u5e38\u5c06\u5de6\u5947\u5f02\u5411\u91cf\u77e9\u9635 U<\/code> \u548c\u5305\u542b\u5947\u5f02\u503c\u4ee5\u53ca\u53f3\u5947\u5f02\u5411\u91cf\u7684\u77e9\u9635 V^\\top<\/code> \u5408\u5e76\u5230\u4e00\u8d77\uff0c\u5f97\u5230\u4e00\u4e2a\u7d27\u51d1\u7684\u8868\u793a\u3002<\/p>\n
      \u5177\u4f53\u5730\uff0c\u5c06\u5947\u5f02\u503c\u77e9\u9635 \\Sigma<\/code> \u7684\u975e\u96f6\u5143\u7d20\u6309\u7167\u4ece\u5927\u5230\u5c0f\u6392\u5217\uff0c\u5f97\u5230\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff1a<\/p>\n
      \n\\Sigma = \\begin{bmatrix}\n\\sigma_1 & 0 & 0 & \\dots & 0 \\\\\n0 & \\sigma_2 & 0 & \\dots & 0 \\\\\n0 & 0 & \\sigma_3 & \\dots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\dots & \\sigma_r \\\\\n0 & 0 & 0 & \\dots & 0 \\\\\n\\end{bmatrix}\n<\/code><\/pre>\n
      \u5176\u4e2d r<\/code> \u662f\u77e9\u9635 M<\/code> \u7684\u79e9\uff0c\u4e5f\u662f\u975e\u96f6\u5947\u5f02\u503c\u7684\u6570\u91cf\u3002<\/p>\n
      \u7136\u540e\uff0c\u5c06\u5de6\u5947\u5f02\u5411\u91cf\u77e9\u9635 U<\/code> \u7684\u524d r<\/code> \u5217\u548c\u53f3\u5947\u5f02\u5411\u91cf\u77e9\u9635 V<\/code> \u7684\u524d r<\/code> \u884c\u63d0\u53d6\u51fa\u6765\uff0c\u5f62\u6210\u77e9\u9635 C<\/code> \u548c\u77e9\u9635 D<\/code>\uff1a<\/p>\n
      \nC = U[:, :r]\n\\\\D = \\begin{bmatrix}\n\\sigma_1 & 0 & 0 & \\dots & 0 \\\\\n0 & \\sigma_2 & 0 & \\dots & 0 \\\\\n0 & 0 & \\sigma_3 & \\dots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\dots & \\sigma_r \\\\\n\\end{bmatrix} \\cdot V^\\top[:r, :]\n<\/code><\/pre>\n
      \u8fd9\u6837\uff0cC<\/code> \u5305\u542b\u4e86\u77e9\u9635 M<\/code> \u7684\u4e3b\u8981\u4fe1\u606f\uff0c\u800c D<\/code> \u5305\u542b\u4e86\u5947\u5f02\u503c\u548c\u53f3\u5947\u5f02\u5411\u91cf\u7684\u4fe1\u606f\u3002\u8fd9\u79cd\u5206\u89e3\u5728\u5f88\u591a\u5e94\u7528\u4e2d\u90fd\u975e\u5e38\u6709\u7528\uff0c\u4f8b\u5982\u964d\u7ef4\u3001\u6570\u636e\u538b\u7f29\u3001\u77e9\u9635\u8fd1\u4f3c\u7b49\u3002<\/p>\n
      2.1.2 \u5947\u5f02\u503cTT\u5206\u89e3\u6b65\u9aa4<\/h4>\n
      \u8fd9\u5bf9\u6b63\u4ea4\u5f20\u91cf\u4e2d\u7684\u4e00\u90e8\u5206\u88ab\u5408\u5e76\u5230\u4e00\u4e2a\u4e2d\u5fc3\u5f20\u91cf\u4e2d\uff0c\u540c\u65f6\u4fdd\u6301\u539f\u59cb\u5f20\u91cf\u7684\u4fe1\u606f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e0d\u65ad\u8fed\u4ee3\uff0c\u9010\u6b65\u5c06\u4fe1\u606f\u4ece\u5de6\u53f3\u4e24\u4fa7\u96c6\u4e2d\u5230\u4e2d\u5fc3\uff0c\u4ece\u800c\u8fbe\u5230\u4e2d\u5fc3\u6b63\u4ea4\u5316\u7684\u76ee\u7684\u3002\u5177\u4f53\u8fc7\u7a0b\u5982\u4e0b\uff1a<\/p>\n
      \n
      \u603b\u4f53\u4e0a\u53ef\u4ee5\u5f52\u7eb3\u4e3a\u4ee5\u4e0b\u51e0\u4e2a\u6b65\u9aa4\uff1a<\/p>\n
        \n
      1. \n
        \u521d\u59cb\u5316\uff1a<\/strong> \u5047\u8bbe\u4f60\u6709\u4e00\u4e2a\u5f20\u91cf\u7f51\u7edc\uff0c\u5176\u4e2d\u7684\u5f20\u91cf\u9700\u8981\u8fdb\u884c\u4e2d\u5fc3\u6b63\u4ea4\u5316\u3002\u521d\u59cb\u65f6\uff0c\u8fd9\u4e9b\u5f20\u91cf\u53ef\u80fd\u662f\u4efb\u610f\u6392\u5217\u7684\u3002<\/p>\n<\/li>\n
      2. \n
        \u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\uff1a<\/strong> \u9009\u62e9\u4e00\u4e2a\u5f20\u91cf\uff0c\u5c06\u5176\u5206\u89e3\u4e3a\u4e24\u4e2a\u90e8\u5206\uff1a\u4e00\u4e2a\u6b63\u4ea4\u7684\u5de6\u5f20\u91cf\u548c\u4e00\u4e2a\u6b63\u4ea4\u7684\u53f3\u5f20\u91cf\u3002<\/p>\n<\/li>\n
      3. \n
        \u5408\u5e76\u5230\u4e2d\u5fc3\uff1a<\/strong> \u5c06\u5176\u4e2d\u4e00\u4e2a\u6b63\u4ea4\u5f20\u91cf\u5408\u5e76\u5230\u4e00\u4e2a\u4e2d\u5fc3\u5f20\u91cf\u4e2d\u3002\u8fd9\u4e2a\u4e2d\u5fc3\u5f20\u91cf\u5c06\u4f1a\u79ef\u7d2f\u6765\u81ea\u4e0d\u540c\u5f20\u91cf\u7684\u4fe1\u606f\uff0c\u4ece\u800c\u51cf\u5c11\u603b\u4f53\u7684\u8868\u793a\u590d\u6742\u5ea6\u3002\u901a\u5e38\uff0c\u8fd9\u4e2a\u4e2d\u5fc3\u5f20\u91cf\u4f1a\u5904\u4e8e\u6574\u4e2a\u5f20\u91cf\u7f51\u7edc\u7684\u4e2d\u5fc3\u4f4d\u7f6e\u3002<\/p>\n<\/li>\n
      4. \n
        \u8fed\u4ee3\uff1a<\/strong> \u91cd\u590d\u4e0a\u8ff0\u6b65\u9aa4\uff0c\u9009\u62e9\u4e0b\u4e00\u4e2a\u9700\u8981\u8fdb\u884c\u4e2d\u5fc3\u6b63\u4ea4\u5316\u7684\u5f20\u91cf\uff0c\u5e76\u5c06\u5176\u5206\u89e3\u4e3a\u6b63\u4ea4\u5de6\u5f20\u91cf\u548c\u53f3\u5f20\u91cf\uff0c\u7136\u540e\u5c06\u5176\u4e2d\u4e00\u4e2a\u5408\u5e76\u5230\u4e2d\u5fc3\u5f20\u91cf\u4e2d\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4f1a\u4e0d\u65ad\u8fed\u4ee3\uff0c\u9010\u6e10\u5c06\u4fe1\u606f\u4ece\u8fb9\u7f18\u7684\u5f20\u91cf\u8f6c\u79fb\u5230\u4e2d\u5fc3\u5f20\u91cf\u4e2d\u3002<\/p>\n<\/li>\n
      5. \n
        \u6536\u655b\u6216\u622a\u65ad\uff1a<\/strong> \u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\uff0c\u4f60\u53ef\u4ee5\u9009\u62e9\u5728\u67d0\u4e2a\u6b65\u9aa4\u8fdb\u884c\u622a\u65ad\u64cd\u4f5c\uff0c\u5373\u4fdd\u7559\u90e8\u5206\u6700\u91cd\u8981\u7684\u7279\u5f81\uff0c\u4e22\u5f03\u4e00\u4e9b\u8f83\u5c0f\u7684\u503c\uff0c\u4ee5\u51cf\u5c11\u8ba1\u7b97\u548c\u5b58\u50a8\u6210\u672c\u3002\u622a\u65ad\u540e\uff0c\u7ee7\u7eed\u8fdb\u884c\u8fed\u4ee3\uff0c\u76f4\u5230\u6574\u4e2a\u5f20\u91cf\u7f51\u7edc\u8db3\u591f\u63a5\u8fd1\u4e2d\u5fc3\u6b63\u4ea4\u7684\u72b6\u6001\u3002<\/p>\n<\/li>\n<\/ol>\n<\/blockquote>\n
        \u8fc7\u7a0b\u5927\u6982\u5982\u4e0b\u56fe\u6240\u793a<\/p>\n
        \"\"
        \n\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n
        \nA^{(n)} \\stackrel{SVD}{\\rightarrow} \\hat{A}^{(n)}SV\\\\\n\nSV\\rightarrow A^{n+1}\n<\/code><\/pre>\n
        \u8fd9\u6837\u4e0d\u65ad\u5730\u91cd\u590d\uff0c\u6574\u4f53\u56fe\u5982\u4e0b<\/p>\n
        \"\"<\/p>\n
        2.1.3 \u4ee3\u7801\u5b9e\u73b0<\/h4>\n
        \u5411\u524d\u4f20\u64ad<\/em><\/p>\n
        d<\/code>\u662f\u7269\u7406\u7ef4\u5ea6,\\chi<\/code> \u662f\u865a\u62df\u7ef4\u5ea6\uff0clength<\/code>\u662f\u5206\u89e3\u7684\u957f\u5ea6<\/p>\n
        \u4e2d\u95f4\u7ef4\u5ea6\u662fA_{a_ns_na_{n+1}}\\in \\R^{\\chi \\times d \\times \\chi}<\/code>\uff0c\u5934\u5c3e\u662fA_{a_ns_na_{n+1}}\\in \\R^{1 \\times d \\times \\chi}<\/code>\u548cA_{a_ns_na_{n+1}}\\in \\R^{\\chi \\times d \\times 1}<\/code><\/p>\n
        import torch as tc\n\nprint('\u5efa\u7acb\u5f00\u653e\u8fb9\u754cMPS')\nlength, d, chi = 6, 2, 4\ndtype = tc.float64\n\n#\u968f\u673a\u751f\u6210A\n#\u7b2c\u4e00\u4e2a\u548c\u6700\u540e\u4e00\u4e2a\u4e2a\u7684\u5934\u5c3e\u4e3a1\uff0c\ntensors = [tc.randn(1, d, chi, dtype=dtype)] + \\\n          [tc.randn(chi, d, chi, dtype=dtype)\n           for _ in range(length-2)] + \\\n          [tc.randn(chi, d, 1, dtype=dtype)]<\/code><\/pre>\n
        psi = tensors[0]\nprint('\u7b2c0\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a', tensors[0].shape)\nfor n in range(1, len(tensors)):\n    print('\u7b2c%d\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a' % n, tensors[n].shape)\n    #tensordot\u7528\u6765\u505a\u901a\u7528\u7ef4\u5ea6\u7684\u5f20\u91cf\u76f8\u4e58\uff0c\u7b2c\u4e00\u4e2a\u7b2c\u4e8c\u4e2a\u53c2\u6570\u8868\u793a\u8f93\u5165\u7684\u5f20\u91cf\uff0c\u6700\u540e\u4e00\u4e2a\u8868\u793a\u8981\u8ba1\u7b97\u7684\u7ef4\u5ea6\n    psi = tc.tensordot(psi, tensors[n], [[-1], [0]])\n    print('\u6536\u7f29\u5b8c\u8be5\u5f20\u91cf\u540e\uff0c\u6240\u5f97\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a', psi.shape)\npsi = psi.squeeze()\nprint('\u53bb\u6389\u524d\u540e\u7ef4\u6570\u4e3a1\u7684\u6307\u6807\u5f97\u5230\u6700\u7ec8\u7684\u9ad8\u9636\u5f20\u91cf\uff0c\u5f62\u72b6\u4e3a\uff1a')\nprint(psi.shape)<\/code><\/pre>\n
        \n
        \u7b2c0\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([1, 2, 4])
        \n\u7b2c1\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([4, 2, 4])
        \n\u6536\u7f29\u5b8c\u8be5\u5f20\u91cf\u540e\uff0c\u6240\u5f97\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([1, 2, 2, 4])
        \n\u7b2c2\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([4, 2, 4])
        \n\u6536\u7f29\u5b8c\u8be5\u5f20\u91cf\u540e\uff0c\u6240\u5f97\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([1, 2, 2, 2, 4])
        \n\u7b2c3\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([4, 2, 4])
        \n\u6536\u7f29\u5b8c\u8be5\u5f20\u91cf\u540e\uff0c\u6240\u5f97\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([1, 2, 2, 2, 2, 4])
        \n\u7b2c4\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([4, 2, 4])
        \n\u6536\u7f29\u5b8c\u8be5\u5f20\u91cf\u540e\uff0c\u6240\u5f97\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([1, 2, 2, 2, 2, 2, 4])
        \n\u7b2c5\u4e2a\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([4, 2, 1])
        \n\u6536\u7f29\u5b8c\u8be5\u5f20\u91cf\u540e\uff0c\u6240\u5f97\u5f20\u91cf\u7684\u5f62\u72b6\u4e3a\uff1a torch.Size([1, 2, 2, 2, 2, 2, 2, 1])
        \n\u53bb\u6389\u524d\u540e\u7ef4\u6570\u4e3a1\u7684\u6307\u6807\u5f97\u5230\u6700\u7ec8\u7684\u9ad8\u9636\u5f20\u91cf\uff0c\u5f62\u72b6\u4e3a\uff1a
        \ntorch.Size([2, 2, 2, 2, 2, 2])<\/p>\n<\/blockquote>\n
        \u5206\u89e3\u5411\u91cf<\/em><\/p>\n
        def ttd_tc(x, chi=-1, svd=True):\n    """\n    :param x: \u88ab\u5206\u89e3\u7684\u5f20\u91cf\n    :param chi: \u7ef4\u5ea6\u88c1\u526a. dc=None\u8868\u793a\u4e0d\u88c1\u526a\uff0c\u4f7f\u7528QR\u5206\u89e3;\n                \u5982\u679c\u88c1\u526a\u5219\u4f7f\u7528 SVD \u5206\u89e3 \n    :param svd: use svd or qr\n    :return tensors: tensors in the TT form\n    :return lm: singular values in each decomposition (calculated when dc is not None)\n    """\n    #\u5f62\u72b6\n    dims = x.shape\n    #\u6c42\u6570\u91cf\n    ndim = x.ndimension()\n    #\u8bb0\u5f55\u7b2c\u4e00\u7ef4\u5ea6\u7684\u5927\u5c0f\n    dimL = 1\n\n    tensors = list()\n    lm = list()\n    if chi is None:\n        chi = -1\n\n    for n in range(ndim-1):\n        if (chi < 0) and (not svd):  # \u4e0d\u88c1\u526a\n            #\u5148reshape\u4e3a\u524d\u4e00\u4e2a\u5411\u91cf*\u7ef4\u5ea6\uff0c\u4e5f\u5c31\u662f\u8bf4\u7b2c\u4e00\u7ef4\u5ea6\u662f\u6709\u524d\u9762\u7684n\u4e2a\u5411\u91cf*\u76ee\u524d\u9879\u94fe\u7684\u4e2a\u6570\n            q, x = tc.linalg.qr(x.reshape(dimL*dims[n], -1))\n            dimL1 = x.shape[0]\n        else:\n            q, s, v = tc.linalg.svd(x.reshape(dimL*dims[n], -1))\n\n            #print(q.shape,s.shape,v.shape)\n            #torch.Size([4, 4]) torch.Size([4]) torch.Size([1024, 1024])\n\n            #\u5224\u65ad\u622a\u53d6\u7684\u957f\u5ea6\u4e0d\u5e94\u8be5\u5927\u4e8es\u7684\u957f\u5ea6\n            if chi > 0:\n                dc = min(chi, s.numel())\n            else:\n                #\u5411\u91cf\u4e2a\u6570\n                dc = s.numel()\n\n            #\u622a\u53d6\n            #\u5de6\u5947\u5f02\u503c\u622a\u53d6\u524ddc\u5217\n            q = q[:, :dc]\n            #\u622a\u53d6\u524ddc\u884c\n            s = s[:dc].to(q.dtype)\n            #\u4fdd\u5b58s\n            lm.append(s)\n\n            #\u53f3\u5947\u5f02\u503c\uff08s\u7684\u524ddc\u5217\u7684\u5bf9\u89d2\u5143\u7d20\u00b7v\u7684\u524ddc\u5217\uff09\n            #print(s.shape,tc.diag(s).shape,v[:dc, :].shape)\n            #torch.Size([4]) torch.Size([4, 1024])\n            #diag\u8868\u793a\u628a\u8fd9\u5217\u5143\u7d20\u586b\u5145\u5230\u5bf9\u89d2\u7ebf\u4e0a\n            #[4]-diag->[4,4]\n            #[4,4]\u00b7[4,1024]--mm->[4,1024]\n\n            #x\u662f\u4e0b\u4e00\u4e2a\u5927\u5757\n            x = tc.diag(s).mm(v[:dc, :])\n\n            #print(s)\n            #print(tc.diag(s))\n            #print(x.shape)\n            #torch.Size([dc, 1024]) or torch.Size([s.numel(),1024])\n\n            #\u8bb0\u5f55\u622a\u53d6\u6570\u503c,\u4e3a\u4e0b\u4e00\u4e2adimL*dims[n]\u505a\u51c6\u5907\n            dimL1 = dc\n        #\u8bb0\u5f55\u5f53\u524d\u5757\n        tensors.append(q.reshape(dimL, dims[n], dimL1))\n        #\u8bb0\u5f55\u622a\u53d6\u6570\u503c\n        dimL = dimL1\n    #\u8bb0\u5f55\u6700\u540e\u4e00\u4e2a\n    tensors.append(x.reshape(dimL, dims[-1], 1))\n    # tensors[0] = tensors[0][0, :, :]\n    return tensors, lm\n#\u8fd9\u4e2a\u51fd\u6570\u628a\u6b63\u4ea4\u4e2d\u5fc3\u79fb\u52a8\u5230\u6700\u540e\u4e00\u4e2a<\/code><\/pre>\n
        #\u8bbe\u5b9a\u7ef4\u5ea6\nd = 4\nx = tc.randn((d, d, d, d,d,d), dtype=tc.complex128)\n\ntensors = ttd_tc(x,chi=-1)[0]\nfor n in range(len(tensors)):\n    print('\u7b2c%d\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a' % n,\n          tensors[n].shape)<\/code><\/pre>\n
        \n
        \u7b2c0\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a torch.Size([1, 4, 4])
        \n\u7b2c1\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a torch.Size([4, 4, 16])
        \n\u7b2c2\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a torch.Size([16, 4, 64])
        \n\u7b2c3\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a torch.Size([64, 4, 16])
        \n\u7b2c4\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a torch.Size([16, 4, 4])
        \n\u7b2c5\u4e2a\u5f20\u91cf\u7684\u7ef4\u6570\u4e3a\uff1a torch.Size([4, 4, 1])<\/p>\n<\/blockquote>\n
        def full_tensor(tensors):\n    # \u6ce8\uff1a\u8981\u6c42\u6bcf\u4e2a\u5f20\u91cf\u7b2c0\u4e2a\u6307\u6807\u4e3a\u5de6\u865a\u62df\u6307\u6807\uff0c\u6700\u540e\u4e00\u4e2a\u6307\u6807\u4e3a\u53f3\u865a\u62df\u6307\u6807\n    psi = tensors[0]\n    for n in range(1, len(tensors)):\n        psi = tc.tensordot(psi, tensors[n], [[-1], [0]])\n    if psi.shape[0] > 1:  # \u5468\u671f\u8fb9\u754c\n        psi = psi.permute([0, psi.ndimension()-1] + list(range(1, psi.ndimension()-1)))\n        s = psi.shape\n        psi = tc.einsum('aab->b', psi.reshape(s[0], s[1], -1))\n        psi = psi.reshape(s[2:])\n    else:\n        psi = psi.squeeze()\n    return psi\nx2 = full_tensor(tensors)\nerr = (x - x2).norm() \/ x.norm()\nprint('\u5168\u5c40\u5f20\u91cfTTD\uff08\u6307\u6807\u6b63\u5e8f\uff09\u8bef\u5dee = %g' % err)<\/code><\/pre>\n
        \n
        \u5168\u5c40\u5f20\u91cfTTD\uff08\u6307\u6807\u6b63\u5e8f\uff09\u8bef\u5dee = 4.84283e-15<\/p>\n<\/blockquote>\n
        2.2  \u6b63\u4ea4\u4e09\u89d2\uff08QR\uff09TT\u5206\u89e3<\/h3>\n
        \u8fd9\u91cc\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528QR\u5206\u89e3\u6cd5<\/p>\n
        \u8f6c\u8f7d\u6765\u6e90https:\/\/zhuanlan.zhihu.com\/p\/362248020<\/p>\n
        2.2.1 \u6b63\u4ea4\u4e09\u89d2\uff08QR\uff09\u5206\u89e3\u7684\u5b9a\u4e49<\/strong><\/h4>\n
        \u8bbe \\pmb{A}_{m \\times n}<\/code> \u7684\u79e9\u4e3a n \uff0c\u5219\\pmb{A}<\/code> \u53ef\u4ee5\u552f\u4e00<\/strong>\u5730\u5206\u89e3\u4e3a<\/p>\n
        \n\\pmb{A}_{m \\times n} = \\pmb{Q}_{m \\times n} \\pmb{R}_{n \\times n} \\\\\\\\\n<\/code><\/pre>\n
        \u5176\u4e2d\uff0c\\pmb{Q}_{m \\times n}<\/code> \u662f\u6807\u51c6\u6b63\u4ea4\u5411\u91cf\u7ec4\u77e9\u9635<\/strong>\uff0c \\pmb{R}_{n \\times n}<\/code> \u662f\u6b63\u7ebf\u4e0a\u4e09\u89d2\u9635\u3002<\/strong><\/p>\n
        \uff08\u8865\u5145\u4e00\u4e2a\u5c0f\u77e5\u8bc6\uff1a\\pmb{Q}<\/code>\u5176\u5b9e\u662f\u6e90\u4e8e orthogonal matrices \u6216 orthonormal basis\uff0c\u4e3a\u4e86\u907f\u514d \\pmb{O},\\pmb{0}<\/code>\u4e0d\u597d\u533a\u5206\u7684\u95ee\u9898\uff1b \\pmb{R}<\/code>\u662f\u6307\u4ee3 right triangular matrices\uff09<\/p>\n
        2.2.2 \u6b63\u4ea4\u4e09\u89d2\uff08QR\uff09\u5206\u89e3\u7684\u6c42\u6cd5<\/strong><\/h4>\n
        \u5176\u5b9e\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\\pmb{Q}^T\\pmb{A} = \\pmb{R}<\/code> \u5f97\u51fa \\pmb{R}<\/code>\uff08\u56e0\u4e3a\u6709\u6027\u8d28c\uff1a \\pmb{Q}^T\\pmb{Q}= \\pmb{E}<\/code> \uff0c\u6240\u4ee5\\pmb{Q}^T\\pmb{A} = \\pmb{Q}^T(\\pmb{Q}\\pmb{R}) = \\pmb{E}\\pmb{R} = \\pmb{R}<\/code><\/p>\n
        \u7efc\u4e0a\uff0c\u5f97\u51fa\u6c42\u89e3 \\pmb{A}_{m \\times n}<\/code> \u7684\u6b63\u4ea4\u4e09\u89d2\uff08QR\uff09\u5206\u89e3\\pmb{A}_{m \\times n} = \\pmb{Q}_{m \\times n}\\pmb{R}_{n \\times n}<\/code>\u7684\u6b65\u9aa4\uff1a<\/p>\n
        \u7b2c\u4e00\u6b65\uff1a<\/strong>\u5bf9\u77e9\u9635\\pmb{A}_{m \\times n}<\/code>\u8fdb\u884c\u65bd\u5bc6\u7279\u6807\u51c6\u6b63\u4ea4\u5316\uff0c\u5f97\u51fa\u77e9\u9635\\pmb{Q}_{m \\times n}<\/code> \u3002<\/p>\n
        \u7b2c\u4e8c\u6b65<\/strong>\uff1a\u901a\u8fc7 \\pmb{R}_{n \\times n} = \\pmb{Q}_{m \\times n}^T \\pmb{A}_{m \\times n}<\/code> \u5f97\u51fa\u77e9\u9635\\pmb{R}_{n \\times n} \u3002<\/code><\/p>\n
        2.2.3 \u4f8b\u5b50<\/strong><\/h4>\n
        \u6c42 \\pmb{A}_{4 \\times 3}<\/code> \u7684\u6b63\u4ea4\u4e09\u89d2\uff08QR\uff09\u5206\u89e3\uff0c\u5176\u4e2d<\/p>\n
        \n\\begin{aligned} \\pmb{A}_{4 \\times 3} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1 \\end{bmatrix} \\end{aligned} \\\\\\\\ \n<\/code><\/pre>\n
        \u89e3\uff1a<\/strong><\/p>\n
        \u7b2c\u4e00\u6b65\uff1a<\/strong>\u5bf9\u77e9\u9635\\pmb{A}_{m \\times n}<\/code>\u8fdb\u884c\u65bd\u5bc6\u7279\u6807\u51c6\u6b63\u4ea4\u5316\uff0c\u5f97\u51fa\u77e9\u9635\\pmb{Q}_{m \\times n}<\/code> \u3002<\/p>\n
        \u5bf9 \\pmb{A}_{4 \\times 3}<\/code> \u8fdb\u884c\u6b63\u4ea4\u5316\uff0c\u5f97\uff1a<\/p>\n
        \n\\begin{aligned} \\pmb{v}_1 = \\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{bmatrix}, \\pmb{v}_2 = \\begin{bmatrix} -3 \\\\ 1 \\\\ 1 \\\\ 1 \\end{bmatrix}, \\pmb{v}_3 = \\begin{bmatrix} 0 \\\\ -2\/3 \\\\ 1\/3 \\\\ 1\/3 \\end{bmatrix} \\end{aligned} \\\\\\\\\n<\/code><\/pre>\n
        \u5355\u4f4d\u5316\uff0c\u5f97\uff1a<\/p>\n
        \n\\begin{aligned} \\pmb{Q}_{4 \\times 3} = \\begin{bmatrix} 1\/2 & -3\/\\sqrt{12} & 0\\\\ 1\/2 & 1\/\\sqrt{12} & -2\/\\sqrt{6} \\\\ 1\/2 & 1\/\\sqrt{12} & 1\/\\sqrt{6} \\\\ 1\/2 & 1\/\\sqrt{12} & 1\/\\sqrt{6} \\end{bmatrix} \\end{aligned} \\\\\\\\ \n<\/code><\/pre>\n
        \u7b2c\u4e8c\u6b65<\/strong>\uff1a\u901a\u8fc7\\pmb{R}_{n \\times n} = \\pmb{Q}_{m \\times n}^T \\pmb{A}_{m \\times n}<\/code>\u5f97\u51fa\u77e9\u9635 \\pmb{R}_{n \\times n}<\/code> \u3002<\/p>\n
        \n\\begin{aligned} \\pmb{R}_{3 \\times 3} &= \\pmb{Q}_{4 \\times 3}^T \\pmb{A}_{4 \\times3} \\\\\\\\ &= \\begin{bmatrix} 1\/2 & 1\/2 & 1\/2 & 1\/2\\\\ -3\/\\sqrt{12} & 1\/\\sqrt{12} & 1\/\\sqrt{12} & 1\/\\sqrt{12} \\\\ 0 & -2\/\\sqrt{16} & 1\/\\sqrt{6} & 1\/\\sqrt{6} \\end{bmatrix} \\begin{bmatrix} 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1 \\end{bmatrix} \\\\\\\\ & = \\begin{bmatrix} 2 & 3\/2 & 1 \\\\ 0 & 3\/\\sqrt{12} & 2\/\\sqrt{12} \\\\ 0 & 0 & 2\/\\sqrt{6} \\end{bmatrix} \\end{aligned} \\\\\\\\\n<\/code><\/pre>\n
        **\u89e3\u6bd5\u3002<\/p>\n
        \u6c42 \\pmb{A}_{5 \\times 3}<\/code> \u7684\u6b63\u4ea4\u4e09\u89d2\uff08QR\uff09\u5206\u89e3\uff0c\u5176\u4e2d<\/p>\n
        \n\\begin{aligned} \\pmb{A}_{5 \\times 3} = \\begin{bmatrix} 1 & 2 & 5 \\\\ -1 & 1 & -4 \\\\ -1 & 4 & -3 \\\\ 1 & -4 & 7 \\\\ 1 & 2 & 1 \\end{bmatrix} \\end{aligned} \\\\\\\\\n<\/code><\/pre>\n
        \u89e3\uff1a<\/strong><\/p>\n
        \u7b2c\u4e00\u6b65\uff1a<\/strong>\u5bf9\u77e9\u9635\\pmb{A}_{m \\times n}<\/code> \u8fdb\u884c\u65bd\u5bc6\u7279\u6807\u51c6\u6b63\u4ea4\u5316\uff0c\u5f97\u51fa\u77e9\u9635\\pmb{Q}_{m \\times n}<\/code> \u3002<\/p>\n
        \u5bf9\\pmb{A}_{5 \\times 3}<\/code>\u8fdb\u884c\u6b63\u4ea4\u5316\uff0c\u5f97\uff1a<\/p>\n
        \n\\begin{aligned} \\pmb{v}_1 = \\begin{bmatrix} 1 \\\\ -1 \\\\ -1 \\\\ 1\\\\ 1 \\end{bmatrix} , \\pmb{v}_2 = \\begin{bmatrix} 3 \\\\ 0 \\\\ 3 \\\\ -3\\\\ 3 \\end{bmatrix}, \\pmb{v}_3 = \\begin{bmatrix} 2 \\\\ 0 \\\\ 2 \\\\ 2\\\\ -2 \\end{bmatrix} \\end{aligned} \\\\\\\\ \n<\/code><\/pre>\n
        \u5355\u4f4d\u5316\uff0c\u5f97\uff1a<\/p>\n
        \n\\begin{aligned} \\pmb{Q}_{5 \\times 3} = \\begin{bmatrix} 1\/\\sqrt{5} & 1\/2 & 1\/2 \\\\ -1\/\\sqrt{5} & 0 & 0 \\\\ -1\/\\sqrt{5} & 1\/2 & 1\/2 \\\\ 1\/\\sqrt{5} & -1\/2 & 1\/2 \\\\ 1\/\\sqrt{5} & 1\/2 & -1\/2 \\end{bmatrix} \\end{aligned} \\\\\\\\ \n<\/code><\/pre>\n
        \u7b2c\u4e8c\u6b65<\/strong>\uff1a\u901a\u8fc7\\pmb{R}_{n \\times n} = \\pmb{Q}_{m \\times n}^T \\pmb{A}_{m \\times n}<\/code> \u5f97\u51fa\u77e9\u9635 \\pmb{R}_{n \\times n}<\/code>\u3002<\/p>\n
        \n\\begin{aligned} \\pmb{R}_{3 \\times 3} &= \\pmb{Q}_{5 \\times 3}^T \\pmb{A}_{5 \\times3} \\\\\\\\ &= \\begin{bmatrix} 1\/\\sqrt{5} & -1\/\\sqrt{5} & -1\/\\sqrt{5} & 1\/\\sqrt{5} & 1\/\\sqrt{5} \\\\ 1\/2 & 0 & 1\/2 & -1\/2 & 1\/2 \\\\ 1\/2 & 0 & 1\/2 & 1\/2 & -1\/2 \\end{bmatrix} \\begin{bmatrix} 1 & 2 & 5 \\\\ -1 & 1 & -4 \\\\ -1 & 4 & -3 \\\\ 1 & -4 & 7 \\\\ 1 & 2 & 1 \\end{bmatrix} \\\\\\\\ & = \\begin{bmatrix} \\sqrt{5} & -\\sqrt{5} & 4\\sqrt{5} \\\\ 0 & 6 & -2 \\\\ 0 & 0 & 4 \\end{bmatrix} \\end{aligned} \\\\\\\\ \n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
        \u4e00\u3001\u77e9\u9635\u4e58\u79ef\u6001\u4e0eTensor-Train \u5206\u89e3 1.1 \u77e9\u9635\u4e58\u79ef\u6001\u4e0eTensor-Train \u6982\u5ff5 \u77e9\u9635\u4e58\u79ef\u6001… » \u9605\u8bfb\u5168\u6587<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,7],"tags":[],"class_list":["post-272","post","type-post","status-publish","format-standard","hentry","category-dl","category-xuexibiji"],"_links":{"self":[{"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/posts\/272","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/comments?post=272"}],"version-history":[{"count":13,"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/posts\/272\/revisions"}],"predecessor-version":[{"id":309,"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/posts\/272\/revisions\/309"}],"wp:attachment":[{"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/media?parent=272"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/categories?post=272"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apifj.com\/index.php\/wp-json\/wp\/v2\/tags?post=272"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}