{"id":290,"date":"2023-09-14T02:03:14","date_gmt":"2023-09-13T18:03:14","guid":{"rendered":"http:\/\/106.52.213.145:21080\/?p=290"},"modified":"2023-09-14T02:03:44","modified_gmt":"2023-09-13T18:03:44","slug":"xuexibijibanzhangliangjisemi-tensor-productstp","status":"publish","type":"post","link":"https:\/\/apifj.com\/index.php\/2023\/09\/14\/xuexibijibanzhangliangjisemi-tensor-productstp\/","title":{"rendered":"[\u5b66\u4e60\u7b14\u8bb0]\u534a\u5f20\u91cf\u79ef\uff08Semi-tensor product\uff0cSTP\uff09"},"content":{"rendered":"

\u534a\u5f20\u91cf\u79ef\uff08Semi-tensor product\uff0cSTP\uff09<\/h2>\n

\u4e00\u3001\u534a\u5f20\u91cf\u79ef\u4e8e\u6982\u5ff5<\/h2>\n

\u534a\u5f20\u91cf\u79ef\uff08Semi-tensor product\uff0cSTP\uff09\u662f\u4f20\u7edf\u77e9\u9635\u4e58\u79ef[5]\u7684\u4e00\u79cd\u6cdb\u5316\u5f62\u5f0f\uff0c\u5176\u4e2d\u56e0\u5b50\u77e9\u9635\u53ef\u4ee5\u5177\u6709\u4efb\u610f\u7ef4\u5ea6\u3002\u5728\u672c\u6587\u4e2d\uff0c\u6211\u4eec\u7740\u91cd\u7814\u7a76\u4e86\u5f53\u5de6\u77e9\u9635\u7684\u5217\u6570\u4e0e\u53f3\u77e9\u9635\u7684\u884c\u6570\u6210\u6bd4\u4f8b\u65f6\u7684\u60c5\u51b5\uff0c\u8fd9\u88ab\u79f0\u4e3a\u5de6\u534a\u5f20\u91cf\u79ef\u3002\u8bbea \u2208 \\R^ {1\u00d7\ud835\udc5b\ud835\udc5d}<\/code>\u8868\u793a\u4e00\u4e2a\u884c\u5411\u91cf(\u90a3\u4e48\u6bcf\u4e00\u4e2a\u5411\u91cf\u957f\u5ea6\u5c31\u662fn),\u8bbeb \u2208 \\R^\ud835\udc5d<\/code> \uff1b\u90a3\u4e48 a<\/code> \u53ef\u4ee5\u88ab\u5747\u7b49\u5730\u5206\u5272\u6210 \ud835\udc5d<\/code> \u4e2a\u76f8\u7b49\u5927\u5c0f\u7684\u5757\uff0c\u5373 a\u2081, a\u2082, ..., a_\ud835\udc5d<\/code>\uff0c\u5de6\u534a\u5f20\u91cf\u79ef\u4ee5 \u22c9 \u7b26\u53f7\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n

\na \u22c9 b = \\sum^p_{i=1}a^ib_i \\in\\R^{1 \\times n}\\\\\nb^T \u22c9 a^T = \\sum^p_{i=1} b_i(a^i)^T \\in\\R^{n}\n<\/code><\/pre>\n
\u7531\u56fe\u5f62\u5316\u5982\u4e0b\uff1a<\/p>\n
\"\"<\/p>\n
\u66f4\u4e00\u822c\u7684\u60c5\u51b5\uff1a<\/p>\n
\u5bf9\u4e8e\u4e24\u4e2a\u77e9\u9635 A \u548c B\uff1a<\/p>\n
\nA \u2208 \\R^{\ud835\udc3b\u00d7\ud835\udc5b\ud835\udc43}\uff0c B \u2208 \\R^{\ud835\udc43 \u00d7\ud835\udc44}\n<\/code><\/pre>\n
STP\u7684\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n
\nC = A \u22c9 B\n<\/code><\/pre>\n
\u5176\u4e2d C \u2208 \\R^{1\u00d7\ud835\udc5b}<\/code> \u53ef\u4ee5\u7531\u4ee5\u4e0b\u516c\u5f0f\u8868\u793a\uff1a<\/p>\n
\nC = \u03a3_{\ud835\udc5d,\ud835\udc56}=\\sum_{\ud835\udc5d,\ud835\udc56} a_\ud835\udc56 b_\ud835\udc56 \u2208 \\R^{1\u00d7\ud835\udc5b}\n<\/code><\/pre>\n
C \u5305\u62ec\u4e86 \ud835\udc3b \u00d7 \ud835\udc44 \u4e2a\u5757\uff0c\u6bcf\u4e2a\u5757 C_{\u210e\ud835\udc5e}<\/code> \u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u65b9\u5f0f\u8ba1\u7b97\uff1a<\/p>\n
\nC_{\u210e\ud835\udc5e} = A(\u210e, :) \u22c9 B(:, \ud835\udc5e )\n<\/code><\/pre>\n
\u5176\u4e2d A(\u210e, : ) \u8868\u793a A \u4e2d\u7684\u7b2c \u210e \u884c\uff0cB(:, \ud835\udc5e ) \u8868\u793a B \u4e2d\u7684\u7b2c \ud835\udc5e \u5217\u3002<\/p>\n
\u77e9\u9635 C<\/code> \u5305\u62ec\u4e86 \ud835\udc3b \u00d7 \ud835\udc44<\/code> \u4e2a\u5757\uff0c\u6bcf\u4e2a\u5757 C^{hq}<\/code> \u7684\u5f62\u72b6\u662f (1, \ud835\udc5b)<\/code>\u3002\u56e0\u6b64\uff0c\u6574\u4e2a\u77e9\u9635 C<\/code> \u7684\u5f62\u72b6\u5e94\u8be5\u662f (\ud835\udc3b , \ud835\udc44, \ud835\udc5b)<\/code>\u3002<\/p>\n
\u4e8c\u3001\u8bba\u6587OD-Rec\u4e2d\u7684\u89e3\u91ca<\/h2>\n
\u8bba\u6587\uff1a\u300aOn-Device Next-Item Recommendation with Self-Supervised Knowledge Distillation\u300b\u4e2d\u76844.2\u7ffb\u8bd1<\/p>\n
4.2 \u6a21\u578b\u538b\u7f29<\/p>\n
\u5f88\u663e\u7136\uff0c\u9879\u76ee\u5d4c\u5165\u8868\u5360\u636e\u4e86\u603b\u53ef\u5b66\u4e60\u53c2\u6570\u7684\u7edd\u5927\u90e8\u5206\u3002\u538b\u7f29\u5d4c\u5165\u8868\u662f\u5b9e\u73b0\u8bbe\u5907\u7aef\u63a8\u8350\u7684\u5173\u952e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9996\u5148\u91c7\u7528\u5f20\u91cf\u5217\u8f66\u5206\u89e3\uff08TTD\uff09[13, 25]\u6765\u7f29\u5c0f\u9879\u76ee\u5d4c\u5165\u7684\u5927\u5c0f\u3002\u4e3a\u4e86\u5c06\u5d4c\u5165\u8868\u56e0\u5b50\u5206\u89e3\u4e3a i<\/code> \u4e2a\u8f83\u5c0f\u7684\u5f20\u91cf\uff0c\u6211\u4eec\u8ba9\u9879\u76ee\u603b\u6570\u4e3a|\ud835\udc49 | = \\prod^\ud835\udc51_{\ud835\udc58=1}\ud835\udc3c_\ud835\udc58<\/code> \uff0c\u5d4c\u5165\u7ef4\u5ea6 \ud835\udc41 =  \\prod^\ud835\udc51_ {\ud835\udc58=1} \ud835\udc3d_\ud835\udc58<\/code> \u3002\u5bf9\u4e8e\u4e00\u4e2a\u7279\u5b9a\u7684\u9879\u76ee \ud835\udc63<\/code>\uff0c\u5176\u7d22\u5f15\u4e3a \ud835\udc56<\/code>\uff0c\u6211\u4eec\u6839\u636e [13] \u5c06\u884c\u7d22\u5f15 \ud835\udc56<\/code> \u6620\u5c04\u4e3a \ud835\udc51<\/code> \u7ef4\u5411\u91cf \ud835\udc56 = (\ud835\udc56_1 , ...,\ud835\udc56_\ud835\udc51 )<\/code>\uff0c\u7136\u540e\u4ece TT-\u6838\u5fc3\u4e2d\u83b7\u53d6\u7d22\u5f15\u7684\u7279\u5b9a\u5207\u7247\u6765\u5728\u5207\u7247\u4e0a\u6267\u884c\u77e9\u9635\u4e58\u6cd5 ( \u53c2\u89c1\u516c\u5f0f(1) )\u3002\u901a\u8fc7\u8fd9\u79cd\u5206\u89e3\uff0c\u6a21\u578b\u4e0d\u9700\u8981\u5b58\u50a8\u6d88\u8017\u5927\u91cf\u5185\u5b58\u7684\u5d4c\u5165\u8868\u3002\u76f8\u53cd\uff0c\u6211\u4eec\u53ea\u9700\u5b58\u50a8 TT-\u6838\u5fc3\u4ee5\u5b9e\u73b0\u5bf9\u5d4c\u5165\u8868\u4e2d\u6761\u76ee\u7684\u8fd1\u4f3c\uff0c\u5373\u4e00\u7cfb\u5217\u5c0f\u5f20\u91cf\u7684\u4e58\u6cd5\u3002\u5047\u8bbe\u7ed9\u5b9a\u5c3a\u5bf8\u4e3a \\sum^\ud835\udc51_{\ud835\udc58=1} \ud835\udc45_{\ud835\udc58\u22121} \ud835\udc3c_\ud835\udc58 \ud835\udc3d_\ud835\udc58 \ud835\udc45_\ud835\udc58<\/code>\u7684 {G_\ud835\udc58}<\/code>\uff0c\u538b\u7f29\u7387\u4e3a\uff1a<\/p>\n
\nrate=\\frac{\\prod_{k=1}^{d}I_kJ_k}{I_1J_1R+\\sum_{k=1}^{d-1}I_jJ_k\\frac{R^2}{n^2}+I_dJ_d\\frac{R}{n^2}}\n<\/code><\/pre>\n
\u5176\u4e2d\uff0c\ud835\udc45_0 = \ud835\udc45_\ud835\udc51 = 1<\/code>\u3002\u4e3a\u7b80\u5355\u8d77\u89c1\uff0c\u6211\u4eec\u5728\u672c\u6587\u4e2d\u5c06\u76f8\u540c\u7684\u79e9 \ud835\udc45 \u5206\u914d\u7ed9\u4e2d\u95f4\u7684 TT-\u6838\u5fc3\uff0c\u5373\uff0c\\{\ud835\udc45_\ud835\udc58 \\}^{\ud835\udc51\u22121}_{\ud835\udc58=2} = \ud835\udc45<\/code>\u3002<\/p>\n
\u7136\u800c\uff0c\u4f20\u7edf\u7684TT\u5206\u89e3\u8981\u6c42\u56e0\u5b50\u4e4b\u95f4\u5177\u6709\u4e25\u683c\u7684\u7ef4\u5ea6\u4e00\u81f4\u6027\uff0c\u53ef\u80fd\u5bfc\u81f4\u7ef4\u5ea6\u5197\u4f59\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u5c06\u9879\u76ee\u5d4c\u5165\u77e9\u9635\u5206\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u7684\u5f20\u91cf\uff0c\u5373 X = G\u2081 G\u2082<\/code>\uff0c\u4ee5\u83b7\u53d6\u6bcf\u4e2a\u9879\u76ee\u7684\u5d4c\u5165\uff0c\u90a3\u4e48 G\u2081<\/code> \u7684\u5217\u6570\u5e94\u8be5\u4e0e G\u2082<\/code> \u7684\u884c\u6570\u76f8\u540c\u3002\u8fd9\u79cd\u4e00\u81f4\u6027\u7ea6\u675f\u5bf9\u4e8e\u7075\u6d3b\u9ad8\u6548\u7684\u5f20\u91cf\u5206\u89e3\u6765\u8bf4\u8fc7\u4e8e\u4e25\u683c\uff0c\u963b\u788d\u4e86\u5bf9\u5d4c\u5165\u8868\u8fdb\u4e00\u6b65\u538b\u7f29\u7684\u8fc7\u7a0b\u3002\u56e0\u6b64\uff0c\u53d7\u5230 [54] \u7684\u542f\u53d1\uff0c\u6211\u4eec\u63d0\u51fa\u5c06\u534a\u5f20\u91cf\u79ef\u4e0e\u5f20\u91cf\u5217\u8f66\u5206\u89e3\u76f8\u7ed3\u5408\uff0c\u5176\u4e2d TT-\u6838\u5fc3\u7684\u7ef4\u5ea6\u53ef\u4ee5\u4efb\u610f\u8c03\u6574\u3002\u4f20\u7edf\u7684 TT-\u6838\u5fc3\u4e4b\u95f4\u7684\u5f20\u91cf\u79ef\u88ab\u534a\u5f20\u91cf\u79ef\u6240\u53d6\u4ee3\uff0c\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n
\nX = \\hat{G}_1\u22c9\\hat{G}_2\u22c9\u00b7\u00b7\u00b7\u22c9\\hat{G}_d\n<\/code><\/pre>\n
\u5176\u4e2d \\{G_\ud835\udc58\\}^\ud835\udc51_{\ud835\udc58=1}<\/code> \u662f\u5728\u5e94\u7528\u57fa\u4e8e\u534a\u5f20\u91cf\u79ef\u7684\u5f20\u91cf\u5217\u8f66\u5206\u89e3 (STTD) \u540e\u5f97\u5230\u7684\u6838\u5fc3\u5f20\u91cf\uff0c\u800c G_1 \u2208 \\R^{\ud835\udc3c_1 \ud835\udc3d_1\u00d7\ud835\udc45}<\/code> \uff0c\\{G_k\\}^{\ud835\udc51-1}_{k=2} \u2208 \\R^{\\frac{\ud835\udc45}{\ud835\udc5b} \u00d7 \\frac{\ud835\udc3c_\ud835\udc58 \ud835\udc3d_\ud835\udc58}{n}\u00d7\ud835\udc45}<\/code> \uff0c\u4ee5\u53ca G_\ud835\udc51 \u2208 \\R^{\\frac{\ud835\udc45}{\ud835\udc5b}\u00d7\\frac{\ud835\udc3c_\ud835\udc51 \ud835\udc3d_\ud835\udc51}{n}}<\/code>\u3002<\/p>\n
\"\"<\/p>\n
\u7ed3\u5408\u56fe2\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\uff0c\u6211\u4eec\u5c06\u539f\u6765\u7684user-item\u7684embedding\u77e9\u9635\u7ed9\u4ed6reshape\u4e3a\u4e09\u7ef4\u77e9\u9635\uff0c\u539f\u5148\u6211\u4eec\u662f\u901a\u8fc7\u666e\u901a\u7684TT\u5206\u89e3\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u7b2c\u4e8c\u884c\u7684\u5f62\u5f0f\uff0c\u4ed6\u5f62\u72b6\u8ba1\u7b97\u5982\u4e0b\uff1a<\/p>\n
\n\\R^{12,8}\\stackrel{Reshape}{=}\\R^{4,4,6}=\\R^{4,2} \\times \\R^{2,4,2} \\times \\R^{2,6}\n<\/code><\/pre>\n
\u800c\u6211\u4eec\u7ed3\u5408\u4e86STP\u540e\u6709<\/p>\n
\n\\R^{12,8}\\stackrel{Reshape}{=}\\R^{4,4,6}=\\R^{4,2} \u22c9 \\R^{1,2,2} \u22c9 \\R^{1,3}\n<\/code><\/pre>\n
\u8bba\u6587\u4e2dG_1 \u2208 \\R^{\ud835\udc3c_1 \ud835\udc3d_1\u00d7\ud835\udc45}<\/code> \uff0c\\{G_k\\}^{\ud835\udc51-1}_{k=2} \u2208 \\R^{\\frac{\ud835\udc45}{\ud835\udc5b} \u00d7 \\frac{\ud835\udc3c_\ud835\udc58 \ud835\udc3d_\ud835\udc58}{n}\u00d7\ud835\udc45}<\/code> \uff0c\u4ee5\u53ca G_\ud835\udc51 \u2208 \\R^{\\frac{\ud835\udc45}{\ud835\udc5b}\u00d7\\frac{\ud835\udc3c_\ud835\udc51 \ud835\udc3d_\ud835\udc51}{n}}<\/code>\u3002<\/p>\n
\u4ee5G_1<\/code> \u548c G_2<\/code> \u4e3e\u4f8b,\u8fd9\u91cc\u7ed9\u51fa\u4e86\u4e24\u4e2a\u6838\u5fc3\u5f20\u91cf\u7684\u7ef4\u5ea6\uff1a<\/p>\n
    \n
  1. G_1 \\in \\mathbb{R}^{I_1J_1 \\times R}<\/code><\/li>\n
  2. G_2 \\in \\mathbb{R}^{\\frac{R}{n} \\times \\frac{I_2J_2}{n} \\times R}<\/code><\/li>\n<\/ol>\n
    \u8fd9\u4e24\u4e2a\u5f20\u91cf\u53ef\u4ee5\u901a\u8fc7\u534a\u5f20\u91cf\u79ef\uff08Semi-tensor product, STP\uff09\u8fdb\u884c\u8fd0\u7b97\u3002<\/p>\n
    \u9996\u5148\uff0c\u6211\u4eec\u56de\u987e\u4e00\u4e0b\u534a\u5f20\u91cf\u79ef\u7684\u5b9a\u4e49\uff1a<\/p>\n
    \u5bf9\u4e8e\u4e24\u4e2a\u77e9\u9635 (A \\in \\mathbb{R}^{m \\times n}<\/code>) \u548c (B \\in \\mathbb{R}^{p \\times q}<\/code>)\uff0c\u5b83\u4eec\u7684\u534a\u5f20\u91cf\u79ef (C = A \u22c9 B<\/code>) \u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n
    \nC = \\sum_{i=1}^{p} a_i \\cdot b_i \\in \\mathbb{R}^{m \\times q}\n<\/code><\/pre>\n
    \u5176\u4e2d (a_i<\/code>) \u548c (b_i<\/code>) \u662f\u77e9\u9635 (A<\/code>) \u548c (B<\/code>) \u4e2d\u7684\u5217\u5411\u91cf\u3002<\/p>\n
    \u73b0\u5728\uff0c\u6211\u4eec\u5c06\u5b83\u5e94\u7528\u5230 (G_1<\/code>) \u548c (G_2<\/code>) \u4e2d\uff1a<\/p>\n
      \n
    1. \u5bf9\u4e8e (G_1) \u548c (G_2)\uff0c\u5b83\u4eec\u7684\u534a\u5f20\u91cf\u79ef\u8fd0\u7b97\u5982\u4e0b\uff1a<\/li>\n<\/ol>\n
      \nC = G_1 \u22c9 G_2\n<\/code><\/pre>\n
        \n
      1. \n
        \u5177\u4f53\u5730\uff0c(G_1<\/code>) \u7684\u7ef4\u5ea6\u4e3a (I_1J_1 \\times R<\/code>)\uff0c(G_2<\/code>) \u7684\u7ef4\u5ea6\u4e3a (\\frac{R}{n} \\times \\frac{I_2J_2}{n} \\times R<\/code>)\u3002\u5bf9\u4e8e\u5355\u4e2a\u533a\u5757\u6709<\/p>\n
        \n\nC_{ijk}=G_1(i,:)\u22c9G_2(:,j,k)<\/code><\/pre>\n<\/li>\n<\/ol>\n
           \u5176\u4e2d\uff0c`$$i$$`\u53d6\u503c\u8303\u56f4\u4e3a `$$I_1,J_1$$` ; `$$j$$`\u53d6\u503c\u8303\u56f4\u4e3a `$$\\frac{I_2J_2}{n}$$` \uff1b`$$k$$` \u53d6\u503c\u8303\u56f4\u4e3a`$$R$$` \n\n```katex\n\nG_1(i,:)\\in \\R^{1 \\times R} \\\\ \nG_2(:,j,k)\\in \\R^{\\frac{R}{n} \\times 1\\times1} \\\\\nC_{ijk} \\in \\R^n\n<\/code><\/pre>\n
        for i in I1J1:\n    for k in R:\n      for j in (I2J2)\/n:\n        # \u6bcf\u4e00\u4e2a\u6709n\u7ef4\u5ea6<\/code><\/pre>\n
        \u6240\u4ee5\uff0c\u534a\u5f20\u91cf\u79ef\u8fd0\u7b97\u540e\uff0c\u7ed3\u679c\u5f20\u91cf (`$$C$$`) \u7684\u7ef4\u5ea6\u5c06\u4e3a (`$$I_1J_1 \\times I_2J_2 \\times R$$`)\u3002<\/p>\n
        \u540c\u7406 \uff0c<\/p>\n
        \nX = C \u22c9 G_3\n<\/code><\/pre>\n
        \u5bf9\u4e8e\u5355\u4e2a\u533a\u5757\u6709<\/p>\n
        \nX_{ijk}=C(i,j,:)\u22c9G_3(:,k)\n<\/code><\/pre>\n
        \nC(i,j,:) \\in \\R^{R}\\\\\nG_3(:,k) \\in \\R^{\\frac{R}{n}}\\\\\nX_{ijk}\\in \\R^n\n<\/code><\/pre>\n
        for i in I1J1:\n    for k in I2J2:\n      for j in (I3J3)\/n:\n        # \u6bcf\u4e00\u4e2a\u6709n\u7ef4\u5ea6<\/code><\/pre>\n
        \u6240\u4ee5\u6700\u540e\u7ed3\u679c:<\/p>\n
        \nX \\in \\R^{I_1J_1 \\times I_2J_2 \\times I_3J_3}\n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
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