![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143159986.png\")
Deep Learning<\/h1>\n\n
Deep Learning<\/h1>\ncreate by Arwin Yu<\/p>\n
![](\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/checklist.png\")
Agenda<\/h3>\n![](\"https:\/\/img.icons8.com\/cute-clipart\/64\/000000\/alarm.png\")
\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u672c\u539f\u7406<\/h2>\nGNN\u5de5\u4f5c\u539f\u7406\u57fa\u4e8e\u90bb\u57df\u805a\u5408\uff08\u6216\u6d88\u606f\u4f20\u9012\uff09\u7b56\u7565\uff0c\u5176\u4e2d\u6bcf\u4e2a\u8282\u70b9\u901a\u8fc7\u805a\u5408\u548c\u8f6c\u6362\u5176\u90bb\u5c45\u8282\u70b9\u7684\u4fe1\u606f\u6765\u66f4\u65b0\u81ea\u5df1\u7684\u8868\u793a\u3002<\/p>\n
\u8fd9\u4e2a\u8fc7\u7a0b\u901a\u5e38\u5305\u62ec\u4ee5\u4e0b\u51e0\u4e2a\u6b65\u9aa4\uff1a<\/p>\n
GNN\u5728\u591a\u4e2a\u9886\u57df\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u5206\u6790\u3001\u63a8\u8350\u7cfb\u7edf\u3001\u86cb\u767d\u8d28\u7ed3\u6784\u9884\u6d4b\u3001\u5316\u5b66\u5206\u5b50\u6027\u8d28\u9884\u6d4b\u3001\u77e5\u8bc6\u56fe\u8c31\u63a8\u7406\u4ee5\u53ca\u4ea4\u901a\u7f51\u7edc\u4f18\u5316\u7b49\u3002\u4e0b\u9762\u6211\u4eec\u5c06\u8fdb\u884c\u8be6\u7ec6\u7684\u4ecb\u7ecd\u3002<\/p>\n
\n ![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143313626.png\")
\n<\/p>\n
\n ![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143355287.png\")
\n<\/p>\n
\n ![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143447295.png\")
\n<\/p>\n
\u901a\u8fc7\u6c42\u548c\u64cd\u4f5c\uff0c\u53ef\u4ee5\u628a\u4e24\u4e2a\u8fb9\u5411\u91cf\u7684\u4fe1\u606f\u548c\u5f53\u524d\u8282\u70b9\u5411\u91cf\u7684\u4fe1\u606f\u6c47\u805a\u5728\u4e00\u8d77\u6620\u5c04\u6210\u4e00\u4e2a\u65b0\u7684\u6c47\u805a\u5411\u91cf\u3002<\/p>\n
\u540c\u6837\u7684\u601d\u60f3\u4e5f\u53ef\u4ee5\u8fd0\u7528\u5728\u8fb9\u5411\u91cf\u548c\u5168\u5c40\u5411\u91cf\u4e0a<\/strong>\u3002<\/p>\n \u73b0\u5728\uff0c\u6211\u4eec\u53ef\u4ee5\u5efa\u7acb\u4e00\u4e2a\u7b80\u5355\u7684GNN\u6a21\u578b\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5e76\u901a\u8fc7\u6c47\u805a\u56fe\u4e2d\u4e0d\u540c\u5c5e\u6027\u7684\u4fe1\u606f\u8fdb\u884c\u4e8c\u5206\u7c7b\u9884\u6d4b\u3002\u5176\u4e2d\uff0c\u6c47\u96c6\u64cd\u4f5c\u5c06\u4f5c\u4e3a\u6784\u5efa\u66f4\u590d\u6742\u7684GNN\u6a21\u578b\u7684\u57fa\u77f3\u3002<\/p>\n \n \u8bf7\u6ce8\u610f\uff0c\u5728\u8fd9\u4e2a\u6700\u7b80\u5355\u7684GNN\u8868\u8ff0\u4e2d\uff0c\u5728GNN\u5c42\u5185\u6839\u672c\u6ca1\u6709\u4f7f\u7528\u56fe\u7684\u8fde\u63a5\u6027\uff08\u90bb\u63a5\u77e9\u9635\/\u5217\u8868\uff09<\/strong>\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u662f\u72ec\u7acb\u5904\u7406\u7684\uff0c\u6bcf\u6761\u8fb9\u4e5f\u662f\u5982\u6b64\uff0c\u8fd8\u6709\u5168\u5c40\u73af\u5883\u3002 \u90a3\u4e48\u5e94\u8be5\u5982\u4f55\u878d\u5165\u90bb\u63a5\u77e9\u9635\u7684\u4fe1\u606f\u5462\uff1f\u53ef\u80fd\u5f88\u76f4\u63a5\u5730\u53ef\u4ee5\u60f3\u5230\uff0c\u5c06\u90bb\u63a5\u77e9\u9635\u548c\u7279\u5f81\u5408\u5e76\u5728\u4e00\u8d77\u5e94\u7528\u5728\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u4e0a\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u76f4\u63a5\u5c06\u4e00\u4e2a\u8282\u70b9\u7684\u90bb\u63a5\u77e9\u9635+\u7279\u5f81\u5408\u8d77\u6765\u4f5c\u4e3a\u4e00\u4e2a\u89c2\u6d4b\u3002<\/p>\n \n \u4f46\u662f\uff0c\u8fd9\u79cd\u65b9\u6cd5\u7684\u95ee\u9898\u5728\u4e8e\uff1a<\/p>\n \uff081\uff09\u9700\u8981\u7684\u8f83\u5927\u7684\u53c2\u6570\u590d\u6742\u5ea6\uff1b<\/p>\n \uff082\uff09\u4e0d\u9002\u7528\u4e8e\u4e0d\u540c\u5927\u5c0f\u7684\u56fe\uff1b<\/p>\n \uff083\uff09\u5bf9\u8282\u70b9\u987a\u5e8f\u654f\u611f\u3002<\/p>\n \u89e3\u51b3\u65b9\u6cd5\u662f\u5c06\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u601d\u60f3\u6cdb\u5316\u5230\u56fe\u795e\u7ecf\u7f51\u7edc\u4e0a\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c31\u662f\u5c40\u90e8\u76f8\u5173\u6027\u548c\u5c42\u7ea7\u7ed3\u6784\u3002<\/p>\n \n \u56fe\u7247\u6570\u636e\u662f\u89c4\u5219\u7684\uff0c\u56e0\u6b64\u5728\u5b9a\u4e49\u5c40\u90e8\u76f8\u5173\u6027\u65f6\uff0c\u53ef\u4ee5\u901a\u8fc7\u5b9a\u4e49\u4e00\u4e2a\u5377\u79ef\u6838\u6765\u5b9e\u73b0\uff0c\u5377\u79ef\u6838\u7684\u5c3a\u5bf8\u51b3\u5b9a\u7740\u5c40\u90e8\u7684\u8303\u56f4\u5927\u5c0f\u3002<\/p>\n<\/li>\n \u56fe\u6570\u636e\u7684\u4e0d\u89c4\u5219\u7684\uff0c\u56e0\u6b64\u4e0d\u80fd\u91c7\u7528\u5377\u79ef\u6838\u7684\u65b9\u5f0f\u6765\u5b9e\u73b0\u5c40\u90e8\u76f8\u5173\u6027\u3002\u8fd9\u91cc\u91c7\u7528\u7684\u662f\u805a\u5408\u90bb\u5c45\u8282\u70b9\u7684\u65b9\u5f0f\uff0c\u5373\u6839\u636e\u90bb\u63a5\u77e9\u9635\u7684\u4fe1\u606f\uff0c\u5728\u6bcf\u4e2aGNN\u5c42\u4e2d\u878d\u5408\u8282\u70b9\u90bb\u5c45\u7684\u4fe1\u606f\u3002<\/p>\n<\/li>\n \u5982\u679c\u5b9e\u9645\u4efb\u52a1\u8fd8\u662f\u9700\u8981\u5f88\u591a\u5c42GNN\u7f51\u7edc\uff0c\u90a3\u4e48\u53ef\u4ee5\u5728GNN\u6a21\u578b\u4e2d\u589e\u52a0skip connections\u3002\u8fd9\u4e2a\u60f3\u6cd5\u6765\u6e90\u4e8eCNN\u7b97\u6cd5\u4e2d\u7684ResNet\u6a21\u578b\u3002<\/p>\n<\/li>\n<\/ul>\n \u53e6\u4e00\u65b9\u9762\uff0c\u878d\u5408\u8282\u70b9\u90bb\u5c45\u4fe1\u606f\u8fd9\u4e00\u601d\u60f3\uff0c\u4e0e\u795e\u7ecf\u7f51\u7edc\u7684\u5206\u5c42\u7ed3\u6784\u662f\u975e\u5e38\u76f8\u4f3c\u7684\uff0c\u5982\u56fe<\/p>\n \n \n \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u4e2aMLP\u662f\u5e94\u7528\u5728\u5355\u72ec\u7684\u8282\u70b9\u3001\u8fb9\u6216\u5168\u5c40\u5411\u91cf\u4e0a\u7684\uff0c\u8fd9\u610f\u5473\u7740\u8282\u70b9\u548c\u8fb9\u4e4b\u95f4\u4e0d\u80fd\u901a\u8fc7\u8fd9\u4e2aMLP\u8fdb\u884c\u4fe1\u606f\u7684\u4ea4\u4e92\uff0c\u8fd9\u80af\u5b9a\u662f\u4e0d\u5408\u7406\u7684\u3002<\/p>\n<\/li>\n \u5176\u5b9e\u89e3\u51b3\u529e\u6cd5\u4e5f\u5f88\u7b80\u5355\uff0c\u5c31\u662f\u5728\u5411\u91cf\u8f93\u5165MLP\u4e4b\u524d\uff0c\u5148\u8fdb\u884c\u5728\u8282\u70b9\u548c\u8fb9\u4e4b\u95f4\u8fdb\u884c\u6c47\u805a\u64cd\u4f5c\u3002<\/p>\n<\/li>\n<\/ul>\n \n GNN\u7684\u8bad\u7ec3\u6d41\u7a0b\u5982\u4e0b<\/p>\n \uff081\uff09\u8f93\u5165\u6570\u636e\uff1b<\/p>\n \uff082\uff09\u7528GNN\u8bad\u7ec3\u6570\u636e\uff1b<\/p>\n \uff083\uff09\u5f97\u5230\u8282\u70b9\u5411\u91cf\uff1b<\/p>\n \uff084\uff09\u9001\u5165Predictor\uff08\u672c\u8d28\u662f\u4e00\u4e2aMLP\uff0c\u5c06\u8282\u70b9\u5411\u91cf\u8f6c\u6362\u4e3a\u6700\u7ec8\u9700\u8981\u7684\u9884\u6d4b\u5411\u91cf\uff09\uff1b<\/p>\n \uff085\uff09\u5f97\u5230\u9884\u6d4b\u5411\u91cf\uff1b<\/p>\n \uff086\uff09\u9009\u53d6\u635f\u5931\u51fd\u6570\u548c\u6807\u7b7e\u8ba1\u7b97\u635f\u5931\uff1b<\/p>\n \uff087\uff09\u6839\u636e\u635f\u5931\u66f4\u65b0\u6a21\u578b\u53c2\u6570\u76f4\u5230\u6536\u655b\uff1b<\/p>\n \uff088\uff09\u9009\u53d6\u8bc4\u4f30\u6307\u6807\u6d4b\u8bd5\u6a21\u578b\uff1b<\/p>\n \uff089\uff09\u4f7f\u7528\u6a21\u578b\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u3002<\/p>\n \u5982\u4e0b\u56fe\u6240\u793a<\/p>\n \n Predictor\u5176\u5b9e\u5c31\u7b97\u4e00\u4e2aMLP\uff0c\u7528\u6765\u6539\u53d8\u5411\u91cf\u7ef4\u5ea6\u7684\uff0c\u76ee\u7684\u662f\u53d8\u5316\u6210\u60f3\u8981\u7684\u9884\u6d4b\u5411\u91cf\u7684\u5f62\u72b6\u3002\u8fd9\u91cc\u89e3\u91ca\u4e00\u4e9b\u4e0d\u540c\u7c92\u5ea6\u4efb\u52a1\u4e0b\u7684Predictor\u3002<\/p>\n \u5982\u679c\u662f\u8fb9\u7ea7\u522b\u7684\u9884\u6d4b\u4efb\u52a1<\/strong>\uff0c\u9884\u6d4b\u8282\u70b9\u4e4b\u95f4\u662f\u5426\u6709\u8fb9\u3002<\/p>\n \u8fd9\u65f6\u5019\u53ef\u4ee5\u5c06\u8282\u70b9\u5411\u91cf $\\boldsymbol{u}$ \u548c $\\boldsymbol{v}$ \u8fdb\u884c\u62fc\u63a5, \u62fc\u63a5\u540e\u7684\u5411\u91cf\u5728\u9001\u5165 Predictor \u8fdb\u884c\u7ef4\u5ea6\u53d8\u5316\u5373\u53ef, \u5373 $\\hat{y}_{u v}=\\operatorname{MLP}\\left(\\operatorname{Concat}\\left(h_u^{(L)}, \\mathrm{h}_v^{(L)}\\right)\\right)$ \u3002<\/p>\n \u4e5f\u53ef\u4ee5\u662f\u70b9\u79ef\u64cd\u4f5c\uff08dot product\uff09\u3002\u56e0\u4e3a\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u7ed3\u679c\u610f\u5473\u7740\u4e24\u4e2a\u5411\u91cf\u7684\u76f8\u5173\u7a0b\u5ea6\uff0c\u662f\u4e00\u4e2a\u5e38\u6570\u3002\u5f53\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u7ed3\u679c\u5927\u65f6\uff0c\u610f\u5473\u7740\u8fd9\u4e24\u4e2a\u8282\u70b9\u4e4b\u95f4\u6709\u5f88\u5927\u53ef\u80fd\u5b58\u5728\u8fb9\u3002<\/p>\n \u5982\u679c\u662f\u56fe\u7ea7\u522b\u7684\u4efb\u52a1<\/strong>\uff0c\u53ef\u4ee5\u805a\u5408\u56fe\u4e2d\u6240\u6709\u8282\u70b9\uff08global pooling\uff09\u7684\u8282\u70b9\u5411\u91cf\u6765\u505a\u9884\u6d4b\uff0c\u5176\u4e2d\u805a\u5408\u65b9\u5f0f\u6709\u5f88\u591a\uff1a<\/p>\n (1) global mean pooling: <\/p>\n (2) global max pooling: <\/p>\n (3) global sum pooling: <\/p>\n \u8fd9\u4e9b\u805a\u5408\u65b9\u5f0f\u5176\u5b9e\u6709\u4e00\u5b9a\u7684\u9009\u62e9\u6280\u5de7\uff0c\u4f8b\u5982\uff1a<\/p>\n \u8fd9\u4e9b\u65b9\u6cd5\u90fd\u5728\u5c0f\u56fe\u4e0a\u8868\u73b0\u5f88\u597d\u3002\u4f46\u662f\u5728\u5927\u56fe\u4e0a\u7684global pooling\u65b9\u6cd5\u53ef\u80fd\u4f1a\u9762\u4e34\u4e22\u5931\u4fe1\u606f\u7684\u95ee\u9898\u3002\u4e3e\u4f8b\uff1a<\/p>\n \u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898, \u89e3\u51b3\u65b9\u6cd5\u662f\u5206\u5c42\u805a\u5408\u8282\u70b9\u5411\u91cf (hierarchical global pooling)\u3002<\/p>\n \u5177\u4f53\u6765\u8bf4, \u53ef\u4ee5\u4f7f\u7528 $\\operatorname{ReLU}(\\operatorname{Sum}(\\cdot))$ \u505a\u805a\u5408, \u5148\u5206\u522b\u805a\u5408\u524d\u4e24\u4e2a\u8282\u70b9\u548c\u540e\u4e09\u4e2a\u8282\u70b9\u7684\u5d4c\u5165, \u7136\u540e\u518d\u805a\u5408\u8fd9\u4e24\u4e2a\u5d4c\u5165\u3002\u4e3e\u4f8b\u5982\u4e0b:<\/p>\n G1 :<\/p>\n \u7b2c\u4e00\u8f6e\u805a\u5408: $\\hat{y}_a=\\operatorname{ReLU}(\\operatorname{Sum}(\\lbrace-1,-20\\rbrace))=0, \\hat{y}_b=\\operatorname{ReLU}(\\operatorname{Sum}(\\lbrace0,1,20\\rbrace))=21$<\/p>\n<\/li>\n \u7b2c\u4e8c\u8f6e\u805a\u5408: $\\hat{y}_G=\\operatorname{ReLU}\\left(\\operatorname{Sum}\\left(\\lbrace y_a, y_b\\rbrace\\right)\\right)=21$<\/p>\n<\/li>\n<\/ul>\n<\/li>\n G2:<\/p>\n \u7b2c\u4e00\u8f6e\u805a\u5408: $\\hat{y}_a=\\operatorname{ReLU}(\\operatorname{Sum}(\\lbrace-10,-20\\rbrace))=0, \\hat{y}_b=\\operatorname{ReLU}(\\operatorname{Sum}(\\lbrace0,10,20\\rbrace))=30$<\/p>\n<\/li>\n \u7b2c\u4e8c\u8f6e\u805a\u5408: $\\hat{y}_G=\\operatorname{ReLU}\\left(\\operatorname{Sum}\\left(\\lbrace y_a, y_b\\rbrace\\right)\\right)=30$<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n \u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06G1\u200b\u548cG2\u200b\u4f5c\u51fa\u533a\u5206\u4e86\u3002\u5176\u5b9e\uff0c\u8fd9\u79cd\u5206\u5c42\u805a\u5408\u5f97\u5230\u56fe\u7ea7\u522b\u9884\u6d4b\u7ed3\u679c\u7684\u65b9\u5f0f\uff0c\u67d0\u79cd\u7a0b\u5ea6\u4e0a\u975e\u5e38\u7c7b\u4f3c\u4e8eCNN\u5904\u7406\u56fe\u50cf\u8bc6\u522b\u95ee\u9898\u7684\u5c42\u7ea7\u7ed3\u6784\uff0c\u5982\u56fe<\/p>\n \n \u56fe\u6570\u636e\u7684\u6570\u636e\u589e\u5f3a\uff08Graph Augmentation for GNNs\uff09\u7b80\u79f0\u56fe\u589e\u5f3a\uff0c\u53ef\u4ee5\u4ece\u7279\u5f81\u5c42\u9762\u6216\u7ed3\u6784\u5c42\u9762\u8fdb\u884c\u5c55\u5f00\u3002<\/p>\n \u4e3b\u8981\u89e3\u51b3\u7684\u95ee\u9898\u662f\u73b0\u5b9e\u751f\u6d3b\u4e2d\u5f88\u591a\u56fe\u6570\u636e\u4e0d\u89c4\u8303\uff0c\u4e0d\u80fd\u76f4\u63a5\u9002\u7528GNN\u8fdb\u884c\u5904\u7406\uff0c\u8981\u8fdb\u884c\u4e00\u5b9a\u7684\u6570\u636e\u9884\u5904\u7406\u3002<\/p>\n \u8282\u70b9\u7279\u5f81\u5c42\u9762<\/strong><\/p>\n \u73b0\u5b9e\u751f\u6d3b\u4e2d\uff0c\u56fe\u6570\u636e\u5f80\u5f80\u4f53\u91cf\u5f88\u5e9e\u5927\uff0c\u5f88\u591a\u8282\u70b9\u662f\u6ca1\u6709\u7279\u5f81\u7684\uff0c\u6216\u8005\u53ea\u6709\u4e00\u5c0f\u90e8\u5206\u8282\u70b9\u5b58\u5728\u7279\u5f81\u3002\u6ca1\u6709\u7279\u5f81\u7684\u8bdd\u600e\u4e48\u805a\u5408\u8282\u70b9\u4fe1\u606f\u5462\uff1f<\/p>\n \n \u867d\u7136\u8fd9\u79cd\u65b9\u6cd5\u770b\u8d77\u6765\u597d\u50cf\u6ca1\u4ec0\u4e48\u7528\uff0c\u4f46\u5b9e\u9645\u5e94\u7528\u662f\u6709\u6548\u7684\u3002<\/p>\n \u4e3e\u4f8b\uff1a\u5047\u8bbe\u56fe9-30\u4e2d\u5404\u4e2a\u8282\u70b9\u91c7\u6837\u4e24\u5c42\u7684\u90bb\u5c45\u8282\u70b9\uff1aA\u8282\u70b9\u7684\u4e00\u9636\u90bb\u5c45\u662fBCD\uff0c\u4e8c\u9636\u90bb\u5c45\u662fAC\u3001AEF\u3001A\u3002<\/p>\n \u6240\u4ee5\u5982\u679c\u4f7f\u7528\u52a0\u6cd5\u4f5c\u4e3a\u805a\u5408\u4fe1\u606f\u7684\u65b9\u6cd5\u7684\u8bdd\uff0c\u800c\u4e14\u8fd8\u6709\u805a\u5408\u5176\u672c\u8eab\u7684\u8bdd\uff0c\u6700\u540e\u7684\u7ed3\u679c\u5c31\u662f\uff081+1+1\uff09+\uff081+1+1+1+1\uff09+\uff081+1\uff09+ 1 = 11\u3002\u8fd9\u4e2a\u6570\u503c\u6709\u70b9\u50cf\u662f\u8282\u70b9\u5ea6\u6570\uff0c\u662f\u6839\u636e\u56fe\u7ed3\u6784\u4fe1\u606f\u805a\u5408\u5f97\u5230\u7684\u3002<\/p>\n \u76f8\u6bd4\u4e8e\u8d4b\u503c\u5e38\u91cf\uff0c\u72ec\u70ed\u7f16\u7801\u65b9\u6cd5\u7684\u8868\u8fbe\u6027\u66f4\u5f3a\uff0c\u5728\u6700\u5f00\u59cb\u65f6\u5c31\u80fd\u4f53\u73b0\u4e0d\u540c\u7684\u8282\u70b9\u3002\u4f46\u662f\uff0c\u4e00\u4e2a\u96be\u4ee5\u5904\u7406\u7684\u5730\u65b9\u662f\u8f93\u5165one-hot\u7684\u7ef4\u5ea6\u662f\u53d6\u51b3\u4e8e\u8f93\u5165\u56fe\u7684\u8282\u70b9\u6570\u91cf\uff0c\u5982\u679c\u6709\u65b0\u8282\u70b9\u52a0\u5165\uff0c\u53ef\u80fd\u4f1a\u6539\u53d8\u4e4b\u524d\u6240\u6709\u8282\u70b9\u7684\u5411\u91cf\u957f\u5ea6\u3002<\/p>\n \u9664\u4e86\u4e0a\u9762\u4e24\u79cd\u7b80\u5355\u7684\u65b9\u6cd5\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u56fe\u7684\u5404\u79cd\u5c5e\u6027<\/strong>\u6765\u8fdb\u884c\u8282\u70b9\u5411\u91cf\u7684\u521d\u59cb\u5316\u3002<\/p>\n \u8fd9\u4e9b\u5c5e\u6027\u53ef\u4ee5\u8ba9\u8282\u70b9\u66f4\u597d\u7684\u8868\u793a\u56fe\u7684\u7ed3\u6784\u4fe1\u606f\u3002<\/p>\n \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u7279\u5b9a\u7684\u7ed3\u6784\u96be\u4ee5\u88abGNN\u5b66\u4e60\uff0c\u5982\u4e0b\u56fe\u6240\u793a\uff0cv1\u7684\u5ea6\u6570\u90fd\u662f2\uff0c\u4f46\u662f\u6240\u5728\u7684\u56fe\u5b8c\u5168\u4e0d\u540c\uff0c\u8fd9\u5c31\u5bfc\u81f4\u8f93\u5165\u57fa\u672c\u7279\u5f81\u96be\u4ee5\u5b66\u4e60\u5230\u8fd9\u4e2a\u7279\u6027\u3002<\/p>\n \n \u8fd9\u79cd\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6cd5\u4e5f\u5f88\u7b80\u5355\uff1a\u5177\u4f53\u95ee\u9898\u5177\u4f53\u5bf9\u5f85\u3002<\/p>\n \u53ef\u4ee5\u901a\u8fc7\u73af\u8ba1\u6570\u6765\u533a\u5206\u4e0a\u9762\u7684\u4e24\u4e2av1\uff0c\u56fe\u5de6\u4fa7\u7684\u73af\u8ba1\u6570\u4e3a3\uff0c\u56fe\u53f3\u4fa7\u7684\u73af\u8ba1\u6570\u4e3a4\uff0c\u5c063\u548c4\u8fd9\u4e24\u4e2a\u5e38\u91cf\u4f5c\u4e3a\u7279\u5f81\u5206\u522b\u6dfb\u52a0\u5230\u8282\u70b9\u7684\u7279\u5f81\u5411\u91cf\u4e2d\u5373\u53ef\u3002<\/p>\n \u8fd9\u4e2a\u65b9\u6cd5\u53ef\u4ee5\u6269\u5c55\u5230\u5176\u5b83\u5177\u4f53\u95ee\u9898\u4e2d\uff0c\u4f8b\u5982\uff0c\u5728\u86cb\u767d\u8d28\u5206\u5b50\u6a21\u578b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u539f\u5b50\uff0c\u53ef\u4ee5\u628a\u539f\u5b50\u7684\u7535\u8377\u6570\u91cf\u3001\u8d28\u91cf\u3001\u4f53\u79ef\u7b49\u90fd\u4f5c\u4e3a\u7279\u5f81\u5411\u91cf\u7684\u5143\u7d20\u3002<\/p>\n \u56fe\u7ed3\u6784\u5c42\u9762<\/strong><\/p>\n \u56fe\u7ed3\u6784\u5c42\u9762\u7684\u6570\u636e\u589e\u5f3a\u4e3b\u8981\u662f\u7528\u6765\u89e3\u51b3\u56fe\u7ed3\u6784\u8fc7\u4e8e\u7a00\u758f\u6216\u8fc7\u4e8e\u7a20\u5bc6\u7684\u95ee\u9898\u3002<\/p>\n \u56fe\u7ed3\u6784\u8fc7\u5ea6\u7a00\u758f\u53ef\u80fd\u5bfc\u81f4\u4fe1\u606f\u4f20\u9012\u6548\u7387\u4f4e\uff0c\u56e0\u4e3a\u8fb9\u592a\u5c11\u4e86\uff0c\u5f88\u6709\u53ef\u80fd\u51fa\u73b0\u56fe\u4e2d\u4e24\u4e2a\u8282\u70b9\u76f8\u8ddd\u592a\u8fdc\uff0c\u4fe1\u606f\u4ea4\u4e92\u56f0\u96be\u3002<\/p>\n \u53cd\u4e4b\uff0c\u56fe\u8fc7\u5ea6\u7a20\u5bc6\u5219\u4f1a\u5bfc\u81f4\u4fe1\u606f\u4f20\u9012\u7684\u8ba1\u7b97\u4ee3\u4ef7\u592a\u9ad8\uff0c\u6bcf\u6b21\u8ba1\u7b97\u90fd\u9700\u8981\u5bf9\u597d\u51e0\u4e2a\u8282\u70b9\u505a\u8fd0\u7b97\uff0c\u56e0\u4e3a\u56fe\u7a20\u5bc6\u610f\u5473\u7740\u90bb\u63a5\u8282\u70b9\u591a\u3002<\/p>\n \u9488\u5bf9\u56fe\u7a00\u758f\u95ee\u9898\uff0c\u53ef\u4ee5\u4e3a\u4e24\u4e2a\u8df3\u90bb\u5c45\u548c\u76ee\u6807\u8282\u70b9\u4e4b\u95f4\u589e\u52a0\u4e00\u4e2a\u865a\u62df\u8fb9\uff1a<\/p>\n \n \u5f53\u7136\uff0c\u4e5f\u53ef\u4ee5\u589e\u52a0\u865a\u62df\u70b9\uff1a<\/p>\n \n \u4e0d\u7ba1\u662f\u589e\u52a0\u8fb9\uff0c\u8fd8\u662f\u589e\u52a0\u8282\u70b9\uff0c\u90fd\u53ef\u4ee5\u63d0\u5347\u5728\u7a00\u758f\u56fe\u4e2d\u7684\u4fe1\u606f\u4f20\u9012\u80fd\u529b\u3002<\/p>\n \u5bf9\u4e8e\u8fc7\u4e8e\u7a20\u5bc6\u7684\u56fe\u6570\u636e<\/strong>\uff0c\u5982\u679c\u805a\u5408\u6240\u6709\u7684\u90bb\u5c45\u8282\u70b9\u7684\u8bdd\uff0c\u6700\u540e\u8ba1\u7b97\u7684\u6210\u672c\u53ef\u80fd\u8f83\u9ad8\u3002<\/p>\n \u4e3a\u4e86\u89e3\u51b3\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u4f7f\u7528\u91c7\u6837\u65b9\u6cd5\uff0c\u5373\u6bcf\u6b21\u53ea\u968f\u673a\u91c7\u6837\u4e2aN\u90bb\u5c45\uff0c\u7136\u540e\u5f00\u59cb\u805a\u5408<\/strong>\u3002<\/p>\n \u8fd9\u4e2a\u60f3\u6cd5\u5728\u56fe\u6a21\u578bGraph Sample and Aggregate Network\uff08GraphSAGE\uff09\u5c31\u6709\u5e94\u7528\u3002<\/p>\n \u4f18\u70b9\u662f\u53ef\u4ee5\u6bcf\u6b21\u62bd\u6837\u4e0d\u540c\u7684\u90bb\u5c45\uff0c\u4ee5\u589e\u52a0\u6a21\u578b\u9c81\u68d2\u6027\uff0c\u5e76\u4e14\u53ef\u4ee5\u63a7\u5236\u53c2\u6570\uff0c\u6765\u63a7\u5236\u8ba1\u7b97\u91cf\uff1b<\/p>\n \u7f3a\u70b9\u4e5f\u5f88\u660e\u663e\uff0c\u53ef\u80fd\u4f1a\u635f\u5931\u91cd\u8981\u4fe1\u606f\uff0c\u56e0\u4e3a\u6709\u7684\u90bb\u5c45\u76f4\u63a5\u4e0d\u4f7f\u7528\u3002<\/p>\n \n Given:<\/p>\n Adjacency matrix $A$ : Feature matrix $H$ : We will perform the matrix multiplication $A H$ :<\/p>\n $$\\begin{aligned} \u76f8\u4e58\u7684\u7ed3\u679c\u662f\u4e00\u4e2a\u65b0\u7684\u77e9\u9635 $A X$ \uff0c\u5b83\u8868\u793a\u56fe\u4e2d\u6bcf\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u5411\u91cf\u5728\u4e0e\u5176\u90bb\u5c45\u7684\u7279\u5f81\u95ee\u91cf\u7ed3\u5408\u540e\u7684\u7ed3\u679c\u3002<\/strong><\/p>\n \u4e0b\u9762\uff0c\u7ecf\u8fc7\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u5c42\u7684\u6620\u5c04\uff08\u6682\u65f6\u5ffd\u7565\u504f\u7f6e\u548c\u6fc0\u6d3b\u51fd\u6570\uff09\uff1a<\/p>\n \n \u7136\u800c\uff0c\u8fd9\u79cd\u4f20\u64ad\u89c4\u5219\u5b58\u5728\u4e00\u4e9b\u95ee\u9898\u3002<\/p>\n \u4f8b\u5982\uff0c\u6211\u4eec\u671f\u671b\u8282\u70b9\u7684\u7279\u5f81\u4f1a\u5bf9\u8282\u70b9\u672c\u8eab\u7684\u8f93\u51fa\u4ea7\u751f\u4e00\u4e9b\u5f71\u54cd<\/strong>\u3002\u7136\u800c....<\/p>\n \n \u6211\u4eec\u7684\u90bb\u63a5\u77e9\u9635\u4e2d\u6ca1\u6709\u4efb\u4f55\u81ea\u5faa\u73af<\/p>\n \u8fd9\u610f\u5473\u7740\u8282\u70b9\u4e0d\u4f1a\u5c06\u5176\u7279\u5f81\u4f20\u64ad\u7ed9\u81ea\u5df1\u3002<\/p>\n \u8fd9\u4e2a\u95ee\u9898\u5f88\u5bb9\u6613\u89e3\u51b3\u3002<\/p>\n \u73b0\u5728 $\\tilde{A}$ \u662f\u5305\u542b\u81ea\u73af\u7684\u90bb\u63a5\u77e9\u9635\uff01 \n \u53e6\u4e00\u4e2a\u95ee\u9898\u662f\u7279\u5f81\u77e9\u9635A\u6ca1\u6709\u6807\u51c6\u5316<\/strong>\uff0c\u8fd9\u6539\u53d8\u4e86\u4e58\u6cd5\u8fc7\u7a0b\u4e2d\u7279\u5f81\u7684\u5c3a\u5ea6\u3002<\/p>\n \u8fd9\u5c06\u4f7f\u4f18\u5316\u53d8\u5f97\u56f0\u96be\uff0c\u5e76\u4e14\u5c42\u6570\u8d8a\u591a\uff0c\u503c\u5c31\u8d8a\u5927\u3002<\/p>\n \n \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8282\u70b9\u5ea6\u6765\u6807\u51c6\u5316 $A$ \u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898<\/strong>\u3002<\/p>\n \n \u6211\u4eec\u79f0\u4e0a\u8ff0\u77e9\u9635\u4e3a\u5bf9\u89d2\u5ea6\u77e9\u9635$\\tilde{D}$\uff08\u5b83\u662f$\\tilde{A}$\u7684\u5bf9\u89d2\u5ea6\u77e9\u9635\uff0c\u5373$\\tilde{A}=A+I$\uff09\u3002<\/p>\n \u4e3a\u4e86\u6807\u51c6\u5316 $A$\uff0c\u6211\u4eec\u73b0\u5728\u5fc5\u987b\u5bf9\u76f8\u90bb\u8282\u70b9\u7279\u5f81\u8fdb\u884c\u5e73\u5747\u3002<\/p>\n \u8fd9\u53ef\u4ee5\u8868\u793a\u4e3a $\\tilde{D}^{-1}A$<\/p>\n \u4f46\u662f\uff0c\u73b0\u5728\u7684\u6807\u51c6\u5316\u53ea\u5173\u6ce8\u6bcf\u4e2a\u8282\u70b9\u7684\u51fa\u5ea6\uff08\u5373\u8fde\u63a5\u5230\u5176\u4ed6\u8282\u70b9\u7684\u6570\u91cf\uff09\uff0c\u79f0\u4e3a\u884c\u6807\u51c6\u5316\u3002<\/p>\n \u884c\u6807\u51c6\u5316\u6ca1\u6709\u8003\u8651\u8282\u70b9\u7684\u5165\u5ea6\uff08\u5373\u88ab\u5176\u4ed6\u8282\u70b9\u8fde\u63a5\u7684\u6570\u91cf\uff09\u3002\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5728\u67d0\u4e9b\u56fe\u7ed3\u6784\u4e2d\u4fe1\u606f\u4f20\u64ad\u7684\u4e0d\u5e73\u8861\uff0c\u5c24\u5176\u662f\u5728\u6709\u5411\u56fe\u4e2d\u3002<\/p>\n \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u518d\u6b21\u6267\u884c\u5217\u6807\u51c6\u5316\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff1a<\/p>\n $\\Large D^{-\\frac{1}{2}}AD^{-\\frac{1}{2}}$<\/p>\n \uff08\u6709\u5173\u6807\u51c6\u5316\u7684\u66f4\u8be6\u7ec6\u89e3\u91ca\u6b64\u5904<\/a>\uff09<\/p>\n \u73b0\u5728\u6211\u4eec\u5df2\u7ecf\u638c\u63e1\u4e86\u7406\u89e3 [Kipf & Welling \u5728\u5176 2017 \u5e74 ICLR \u8bba\u6587GCN\u4e2d] (https:\/\/arxiv.org\/pdf\/1609.02907.pdf<\/a>) \u5f15\u5165\u7684\u4f20\u64ad\u89c4\u5219\u6240\u9700\u7684\u4e00\u5207\uff1a<\/p>\n $f(H^{(l)},A)= \\sigma(\\tilde{D}^{\\frac{1}{2}}\\tilde{A}\\tilde{D}^{\\frac{1}{2}}H^{(l)}W^{(l)})$<\/p>\n \n \u7ed9\u5b9a\u4e00\u4e2a\u56fe $G$ \uff0c\u8ba9\u6211\u4eec\u56fa\u5b9a\u56fe $G$ \u7684 $n$ \u4e2a\u8282\u70b9\u7684\u4efb\u610f\u987a\u5e8f\u3002\u6211\u4eec\u7528 $A$ \u8868\u793a $G$ \u7684 0-1 \u90bb\u63a5\u77e9\u9635\uff0c\u53ef\u4ee5\u6784\u5efa $G$ \u7684\u5bf9\u89d2\u5ea6\u77e9\u9635 $D$ : \u5176\u4e2d\uff0c\u8282\u70b9 $v$ \u7684\u5ea6\u6570\u662f\u4e0e $v$ \u76f8\u8fde\u7684\u8fb9\u7684\u6570\u91cf\u3002 $A_{v u}$ \u8868\u793a\u77e9\u9635 $A$ \u4e2d\u5bf9\u5e94\u4e8e\u884c $v$ \u548c\u5217 $u$ \u7684\u5143\u7d20\u3002\u6211\u4eec\u5c06\u5728\u672c\u8282\u4e2d\u4f7f\u7528\u8fd9\u79cd\u7b26\u53f7\u3002 \n \u65e2\u7136\u6211\u4eec\u5df2\u7ecf\u4e86\u89e3\u4e86\u56fe\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u662f\u4ec0\u4e48\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u5982\u4e0b\u5f62\u5f0f\u7684\u591a\u9879\u5f0f: \u8fd9\u79cd\u5f62\u5f0f\u7684\u6bcf\u4e2a\u591a\u9879\u5f0f\u53ef\u4ee5\u7528\u5176\u7cfb\u6570\u5411\u91cf $w=\\left[w_0, \\ldots, w_d\\right]$ \u6765\u8868\u793a\u3002\u6ce8\u610f\uff0c\u5bf9\u4e8e\u6bcf\u4e2a$w \uff0c p_w(L)$\u662f\u4e00\u4e2a $n \\times n$ \u77e9\u9635\uff0c\u5c31\u50cf $L$ \u4e00\u6837\u3002<\/p>\n \u8fd9\u4e9b\u591a\u9879\u5f0f\u53ef\u4ee5\u88ab\u8ba4\u4e3a\u662f\u5377\u79ef\u795e\u7ecf\u7f51\u7edc (CNN) \u4e2d\u201c\u5377\u79ef\u6838\u201d\u7684\u7b49\u4ef7\u7269\uff0c\u800c\u7cfb\u6570 $w$ \u5219\u662f\u201c\u5377\u79ef\u6838\u201d\u7684\u6743\u91cd\u3002<\/p>\n \u4e3a\u4e86\u4fbf\u4e8e\u8bf4\u660e\uff0c\u6211\u4eec\u5c06\u91cd\u70b9\u5173\u6ce8\u8282\u70b9\u5177\u6709\u4e00\u7ef4\u7279\u5f81\u7684\u60c5\u51b5\uff1a\u5bf9\u4e8e $v \\in V$ \u7684\u6bcf\u4e2a $x_v$ \u53ea\u662f\u4e00\u4e2a\u5b9e\u6570\u3002\u5f53\u6bcf\u4e2a $x_v$ \u662f\u9ad8\u7ef4\u5411\u91cf\u65f6\uff0c\u540c\u6837\u7684\u601d\u60f3\u4e5f\u9002\u7528\u3002<\/p>\n \u4f7f\u7528\u4e4b\u524d\u9009\u5b9a\u7684\u8282\u70b9\u987a\u5e8f\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6240\u6709\u8282\u70b9\u7279\u5f81$x_v$\u5806\u53e0\u8d77\u6765\u5f97\u5230\u4e00\u4e2a\u5411\u91cf $x \\in \\mathbb{R}^n$ \u3002<\/p>\n \n \u4e00\u65e6\u6211\u4eec\u6784\u5efa\u4e86\u7279\u5f81\u5411\u91cf $x$ \uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u5b83\u4e0e\u591a\u9879\u5f0f\u6ee4\u6ce2\u5668 $p_w$ \u7684\u5377\u79ef\u4e3a: $$ \u4e3a\u4e86\u7406\u89e3\u7cfb\u6570 $w$ \u5982\u4f55\u5f71\u54cd\u5377\u79ef\uff0c\u8ba9\u6211\u4eec\u5148\u8003\u8651\u201c\u6700\u7b80\u5355\u7684\u201c\u591a\u9879\u5f0f: \u5f53 $w_0=1$ \u4e14\u6240\u6709\u5176\u4ed6\u7cfb\u6570\u4e3a 0 \u65f6\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c $x^{\\prime}$ \u5c31\u662f $x$ : \u73b0\u5728\uff0c\u5982\u679c\u6211\u4eec\u589e\u52a0\u591a\u9879\u5f0f\u7684\u9636\u6570\uff0c\u5e76\u8003\u8651 $w_1=1$ \u4e14\u6240\u6709\u5176\u4ed6\u7cfb\u6570\u4e3a 0 \u7684\u60c5\u51b5\u3002\u90a3\u4e48\uff0c $x^{\\prime}$ \u5c31\u662f $L x$ , \u56e0\u6b64: \u8fd9\u91cc\uff0c $\\mathcal{N}(v)$ \u8868\u793a\u8282\u70b9 $v$ \u7684\u90bb\u5c45\u96c6\u5408\u3002<\/p>\n \u5bf9\u4e8e\u4e0a\u8ff0\u63a8\u5bfc\u7684\u6700\u540e\u4e00\u6b65\uff0c\u4e0b\u9762\u8be6\u7ec6\u89e3\u91ca\u4e00\u4e0b\uff1a<\/p>\n \u5ea6\u77e9\u9635 $D$ \u662f\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635\uff0c\u5176\u5143\u7d20 $D_{v u}$ \u53ea\u6709\u5728 $v=u$ \u65f6\u975e\u96f6\uff0c\u5373 $D_{v v}=D_v$ (\u8282\u70b9 $v$ \u7684\u5ea6\u6570\uff09\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u5c06\u4e0a\u5f0f\u8fdb\u4e00\u6b65\u5206\u89e3\uff1a<\/p>\n \u5f53 $v=u$ \u65f6\uff0c $D_{v u}=D_v$ \uff1b \u90bb\u63a5\u77e9\u9635 $A$ \u7684\u5143\u7d20 $A_{v u}$ \u53ea\u6709\u5f53\u8282\u70b9 $v$ \u548c\u8282\u70b9 $u$ \u662f\u90bb\u5c45\u65f6\u4e3a 1 \uff0c\u5426\u5219\u4e3a 0 \u3002\u56e0\u6b64\uff0c\u53ea\u6709\u5f53 $u$ \u662f $v$\u7684\u90bb\u5c45\u65f6\uff0c $A_{v u}$ \u624d\u4e3a 1 \u3002<\/p>\n \u800c\uff0c\u6c42\u548c\u53ea\u9700\u8981\u8003\u8651 $u$ \u662f $v$ \u7684\u90bb\u5c45\u7684\u60c5\u51b5\uff0c\u5373 $u \\in \\mathcal{N}(v)$ : \u6700\u540e\uff0c\u5c06\u8fd9\u4e24\u90e8\u5206\u5408\u5e76\u5728\u4e00\u8d77\uff0c\u6211\u4eec\u5f97\u5230: \u8fd9\u4e00\u6b65\u9aa4\u6e05\u695a\u5730\u5c55\u793a\u4e86\u56fe\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u5982\u4f55\u5e73\u6ed1\u8282\u70b9\u7684\u7279\u5f81\uff0c\u901a\u8fc7\u51cf\u53bb\u90bb\u5c45\u7684\u7279\u5f81\u5b9e\u73b0\u4e86\u7279\u5f81\u7684\u5747\u8861\u548c\u4f20\u64ad\u3002<\/p>\n \u8fd9\u4e2a\u51cf\u53f7\u53cd\u6620\u4e86\u6bcf\u4e2a\u8282\u70b9 $v$ \u7684\u7279\u5f81 $x_v$ \u4e0e\u5176\u90bb\u5c45\u8282\u70b9\u7279\u5f81 $x_u$\u7684\u5dee\u5f02\uff0c\u8d77\u5230\u4e00\u79cd\u5e73\u6ed1\u4f5c\u7528\uff0c\u51cf\u5c0f\u76f8\u90bb\u8282\u70b9\u4e4b\u95f4\u7279\u5f81\u503c\u7684\u5dee\u5f02\u3002\u8fd9\u5728\u56fe\u4fe1\u53f7\u5904\u7406\u4e2d\u662f\u5f88\u6709\u7528\u7684\uff0c\u56e0\u4e3a\u5b83\u5e2e\u52a9\u6355\u6349\u5c40\u90e8\u7ed3\u6784\u4fe1\u606f\u5e76\u9632\u6b62\u7279\u5f81\u503c\u7684\u5267\u70c8\u6ce2\u52a8\u3002<\/p>\n \u53e6\u4e00\u65b9\u9762\uff0c\u5728\u56fe\u5377\u79ef\u795e\u7ecf\u7f51\u7edc (GCNs) \u4e2d\uff0c\u901a\u5e38\u7684\u64cd\u4f5c\u662f\u5bf9\u6bcf\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u8fdb\u884c\u52a0\u6743\u6c42\u548c\uff0c\u8fd9\u53ef\u4ee5\u770b\u4f5c\u662f\u5bf9\u4fe1\u606f\u7684\u878d\u5408\u3002\u4f8b\u5982\uff0c\u5728\u6bcf\u4e00\u5c42\u5377\u79ef\u4e2d\uff0c\u8282\u70b9 $v$ \u7684\u65b0\u7684\u7279\u5f81\u8868\u793a$h_v^{\\prime}$ \u662f\u901a\u8fc7\u5b83\u81ea\u8eab\u7684\u7279\u5f81\u548c\u90bb\u5c45\u8282\u70b9\u7684\u7279\u5f81\u52a0\u6743\u6c42\u548c\u5f97\u5230\u7684<\/p>\n \u7ed3\u5408\u4e24\u8005\u7684\u7406\u89e3\u6765\u8bf4\uff0c\u56fe\u62c9\u666e\u62c9\u65af\u77e9\u9635\u4e2d\u7684\u51cf\u53f7\u548cGCN\u4e2d\u7684\u52a0\u53f7\u5728\u6982\u5ff5\u4e0a\u662f\u4e0d\u540c\u7684\uff0c\u524d\u8005\u662f\u57fa\u4e8e\u56fe\u4fe1\u53f7\u5904\u7406\u4e2d\u7684\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u5b9a\u4e49\uff0c\u800c\u540e\u8005\u662f\u57fa\u4e8e\u795e\u7ecf\u7f51\u7edc\u4e2d\u4fe1\u606f\u878d\u5408\u7684\u9700\u6c42\u3002<\/p>\n \u8fd8\u5b58\u5728\u4e00\u4e2a\u91cd\u8981\u95ee\u9898\uff1a\u591a\u9879\u5f0f\u7684\u9636\u6570 $d$ \u5982\u4f55\u5f71\u54cd\u5377\u79ef\u7684\u884c\u4e3a\uff1f<\/strong><\/p>\n \u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635 $A$ \u6216\u62c9\u666e\u62c9\u65af\u77e9\u9635 $L$ \uff0c\u5e42\u6b21\u77e9\u9635 $L^i$ \u7684\u542b\u4e49\u4e0e\u56fe\u4e2d\u8282\u70b9\u4e4b\u95f4\u7684\u8fde\u901a\u6027\u548c\u8def\u5f84\u957f\u5ea6\u6709\u5173\u3002\u5177\u4f53\u6765\u8bf4:<\/p>\n \u867d\u7136\u62c9\u666e\u62c9\u65af\u77e9\u9635 $L$ \u7684\u5e42\u6b21\u4e0d\u80fd\u76f4\u63a5\u88ab\u89e3\u91ca\u4e3a\u8def\u5f84\u7684\u6570\u91cf\uff0c\u4f46\u662f\u5b83\u53cd\u6620\u4e86\u8282\u70b9\u4e4b\u95f4\u7684\u591a\u8df3 (hop) \u8fde\u63a5\u5173\u7cfb\u7684\u5f71\u54cd\u3002\u62c9\u666e\u62c9\u65af\u77e9\u9635\u7684\u5e42\u6b21 $L^i$ \u8868\u793a\u7684\u662f\u4ece\u8282\u70b9 $v$ \u5230\u8282\u70b9 $u$ \u7684 $i$ \u8df3\u5185\u7684\u7d2f\u79ef\u5f71\u54cd\u6216\u4f20\u64ad\u5f3a\u5ea6\u3002<\/p>\n \u5728\u56fe\u5377\u79ef\u4e2d\uff0c\u6211\u4eec\u5bf9\u56fe\u62c9\u666e\u62c9\u65af\u77e9\u9635 $L$ \u8fdb\u884c\u591a\u9879\u5f0f\u5377\u79ef\uff1a \u5176\u4e2d\uff0c \u8fd9\u610f\u5473\u7740\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u8282\u70b9 $v$ \uff0c\u5176\u7279\u5f81 $x_v^{\\prime}$ \u662f\u901a\u8fc7\u5176\u81ea\u8eab\u548c\u6700\u591a $d$ \u8df3\u90bb\u5c45\u7684\u7279\u5f81\u7684\u52a0\u6743\u6c42\u548c\u5f97\u5230\u7684\u3002\u516c\u5f0f\u4e2d\u7684 $i$ \u7684\u53d6\u503c\u8303\u56f4\u662f 0 \u5230 $d$ \uff0c\u8fd9\u8868\u660e\u8282\u70b9\u4e4b\u95f4\u7684\u4f20\u64ad\u548c\u5f71\u54cd\u662f\u5c40\u90e8\u5316\u7684\uff0c\u5e76\u4e14\u53d7\u9650\u4e8e\u591a\u9879\u5f0f\u7684\u9636\u6570 $d$ \u3002<\/p>\n \u8fd9\u4e2a\u5c40\u90e8\u5316\u7684\u9650\u5236\uff08\u5373\u5f71\u54cd\u8303\u56f4\uff09\u53ef\u4ee5\u901a\u8fc7\u8282\u70b9\u7684\u5ea6\u6570\u6765\u7406\u89e3\u3002\u8282\u70b9\u7684\u5ea6\u6570\u8d8a\u5927\uff0c\u5b83\u7684\u76f4\u63a5\u90bb\u5c45\u548c\u8fdb\u4e00\u6b65\u90bb\u5c45\u7684\u6570\u91cf\u5c31\u8d8a\u591a\u3002\u8fd9\u4e5f\u662f\u5408\u7406\u7684\uff0c\u56e0\u4e3a\u4e00\u4e2a\u8282\u70b9\u5ea6\u6570\u8d8a\u5927\uff0c\u4e00\u822c\u610f\u5473\u8d8a\u91cd\u8981\uff0c\u6b64\u65f6\u8003\u8651\u66f4\u591a\u7684\u8df3\u6570\u662f\u5408\u7406\u7684\u3002<\/p>\n \u5b9e\u9645\u4e0a\uff0c\u7ecf\u5178\u7684GCN\u5b9a\u4e49\u662f\u4e0a\u8ff0\u57fa\u4e8e\u62c9\u666e\u62c9\u65af\u77e9\u9635\u7684\u591a\u9879\u5f0f\u56fe\u5377\u79ef\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u63a8\u5bfc\u5982\u4e0b\uff1a<\/p>\n \u9996\u5148\uff0c\u6211\u4eec\u8003\u8651\u56fe\u62c9\u666e\u62c9\u65af\u77e9\u9635 $L$ \u7684\u5b9a\u4e49: \u5176\u4e2d\uff0c $D$ \u662f\u8282\u70b9\u5ea6\u77e9\u9635\uff0c $A$ \u662f\u90bb\u63a5\u77e9\u9635\u3002 \u5176\u4e2d\uff0c $\\tilde{A}=A+I$ \u662f\u52a0\u4e0a\u81ea\u73af\u540e\u7684\u90bb\u63a5\u77e9\u9635\uff0c $\\tilde{D}$ \u662f $\\tilde{A}$ \u7684\u5ea6\u77e9\u9635\u3002\u6211\u4eec\u7528\u5bf9\u79f0\u5f52\u4e00\u5316\u7684\u90bb\u63a5\u77e9\u9635\u6765\u66ff\u4ee3\u56fe\u62c9\u666e\u62c9\u65af\u77e9\u9635\u8fdb\u884c\u5377\u79ef: \u73b0\u5728\uff0c\u8003\u8651\u591a\u9879\u5f0f\u5377\u79ef\u7684\u5f62\u5f0f: \u5bf9\u4e8e\u7ecf\u5178 GCN\uff0c\u6211\u4eec\u53ea\u8003\u8651\u5230 $L$ \u7684\u4e00\u6b21\u5e42\uff0c\u5373: \u4f7f\u7528\u5bf9\u79f0\u5f52\u4e00\u5316\u7684\u90bb\u63a5\u77e9\u9635\u66ff\u4ee3 $L$ : \u8fdb\u4e00\u6b65\u7b80\u5316:<\/strong><\/p>\n \u5bf9\u4e8e\u7ecf\u5178 $\\mathrm{GCN}$ \uff0c\u5047\u8bbe $w_0=1$ \u548c $w_1=1$ \uff0c\u6211\u4eec\u6709: \u56e0\u4e3a $I x$ \u4ee3\u8868\u7684\u662f\u81ea\u73af\uff08\u6bcf\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u4fdd\u6301\u4e0d\u53d8\uff09\uff0c\u8fd9\u5728\u52a0\u81ea\u73af\u7684\u90bb\u63a5\u77e9\u9635 $\\tilde{A}$ \u4e2d\u5df2\u7ecf\u5305\u542b\u4e86\u81ea\u73af\uff0c\u56e0\u6b64: \u6dfb\u52a0\u7ebf\u6027\u53d8\u6362\u548c\u6fc0\u6d3b\u51fd\u6570: \u81f3\u6b64\uff0c\u6211\u4eec\u5f97\u5230\u7ecf\u5178GCN\u7684\u5b9a\u4e49: \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5c3d\u7ba1\u5355\u5c42GCN\u53ea\u8003\u8651\u4e00\u8df3\u90bb\u5c45\uff0c\u901a\u8fc7\u591a\u5c42\u7f51\u7edc\u7684\u53e0\u52a0\uff0c\u53ef\u4ee5\u5b9e\u73b0\u9010\u5c42\u6269\u5c55\u8282\u70b9\u7684\u611f\u53d7\u91ce\uff0c\u8fdb\u800c\u6355\u83b7\u66f4\u8fdc\u7684\u90bb\u5c45\u4fe1\u606f\u3002<\/strong><\/p>\n \u8fd9\u79cd\u9010\u5c42\u5806\u53e0\u7684\u65b9\u5f0f\u6709\u6548\u5730\u5229\u7528\u4e86\u5c40\u90e8\u5316\u5377\u79ef\uff0c\u540c\u65f6\u4e5f\u8fbe\u5230\u4e86\u6269\u5c55\u8282\u70b9\u611f\u53d7\u91ce\u7684\u6548\u679c\u3002<\/p>\n \u53c2\u6570:<\/p>\n GraphSAGE\u4e0eGCN\u7684\u533a\u522b\u6709\u4e24\u4e2a\uff1a\u4e00\u4e2a\u662f\u7ed3\u5408\u8282\u70b9\u81ea\u8eab\u4fe1\u606f\u7684\u65b9\u5f0f\u4e0d\u540c\uff0c\u7b2c\u4e8c\u4e2a\u662f\u6cdb\u5316\u805a\u5408\u90bb\u5c45\u70b9\u65f6\u6240\u91c7\u7528\u7684\u805a\u5408\u51fd\u6570\u4e0d\u540c\u3002<\/p>\n GraphSAGE\u8282\u70b9\u4fe1\u606f\u7684\u66f4\u65b0\u8fc7\u7a0b\u4e3b\u8981\u5206\u4e09\u6b65\uff1a<\/p>\n \uff081\uff09\u805a\u5408\u90bb\u5c45\u8282\u70b9\u7684\u4fe1\u606f\uff0c\u8fd9\u4e2a\u805a\u5408\u51fd\u6570\u6709\u4e09\u79cd\uff0c\u5c06\u5728\u4e0b\u5348\u5c55\u5f00\u89e3\u91ca\uff1b<\/p>\n \uff082\uff09\u5c06\u805a\u5408\u540e\u7684\u4fe1\u606f\u4e0e\u81ea\u8eab\u7684\u8282\u70b9\u4fe1\u606f\u8fdb\u884c\u62fc\u63a5\uff0c\u8fdb\u884c\u7279\u5f81\u7684\u878d\u5408\uff1b<\/p>\n \uff083\uff09\u9001\u5165\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u4e2d\u8fdb\u884c\u6620\u5c04\uff0c\u5f97\u5230\u66f4\u65b0\u540e\u7684\u8282\u70b9\u4fe1\u606f\u3002<\/p>\n \u4e3e\u4f8b\uff1a\u56fe\u6570\u636e\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u73b0\u5728\u4f7f\u7528GraphSAGE\u5bf9\u8282\u70b91\u8fdb\u884c\u66f4\u65b0\u3002<\/p>\n \n \u8282\u70b91\u7279\u5f81\u5411\u91cf\u7684\u66f4\u65b0\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n \uff081\uff09\u805a\u5408\u90bb\u5c45\u8282\u70b9: $h_{\\mathrm{N}(1)}^1 \\leftarrow \\operatorname{AGGREGATE}\\left(h_3^0, h_4^0, h_5^0, h_6^0\\right)$;<\/p>\n \uff082\uff09\u62fc\u63a5\u81ea\u8eab\u4fe1\u606f\uff1a $h_1^1 \\leftarrow \\operatorname{CONCAT}\\left(h_1^0, h_{\\mathrm{N}(1)}^0\\right)$;<\/p>\n \uff083\uff09\u7ecf\u8fc7\u795e\u7ecf\u7f51\u7edc\u6620\u5c04\uff1a $h_1^1 \\leftarrow s\\left(\\boldsymbol{W}^1 \\cdot \\operatorname{CONCAT}\\left(h_1^0, h_{\\mathrm{N}(1)}^0\\right)\\right)$ \u3002<\/p>\n \u5047\u8bbe\u6b65\u9aa4 1 \u4e2d\u805a\u5408\u51fd\u6570 AGGREGATE \u662f Mean \u51fd\u6570, \u5219\u4ee3\u6570\u5f97: \u53e6\u5916\uff0c\u5728\u8fd9\u4e2a\u8ba1\u7b97\u6d41\u7a0b\u4e2d\u6709\u4e24\u4e2a\u5730\u65b9\u9700\u8981\u989d\u5916\u6ce8\u610f\u3002<\/p>\n \u7b2c\u4e00\uff0cGraphSAGE\u5728\u805a\u5408\u67d0\u8282\u70b9\u90bb\u5c45\u4fe1\u606f\u7684\u65f6\u5019\uff0c\u5e76\u4e0d\u662f\u805a\u5408\u5168\u90e8\u7684\u90bb\u5c45\uff0c\u800c\u662f\u805a\u5408\u4e2a K \u90bb\u5c45\uff0cK \u662f\u4e00\u4e2a\u8d85\u53c2\u6570\u3002<\/p>\n \u7b2c\u4e8c\uff0c\u4e2a\u9700\u8981\u6ce8\u610f\u7684\u662fGraphSAGE\u5b9a\u4e49\u4e86\u4e09\u79cd\u4e0d\u540c\u7684\u805a\u5408\u51fd\u6570\uff1a<\/p>\n \u603b\u4f53\u800c\u8a00\uff0cGraphSAGE\u901a\u8fc7\u90bb\u5c45\u91c7\u6837\u548c\u7075\u6d3b\u7684\u805a\u5408\u51fd\u6570\u8bbe\u8ba1\uff0c\u63d0\u4f9b\u4e86\u4e00\u79cd\u66f4\u9ad8\u6548\u548c\u53ef\u6269\u5c55\u7684\u56fe\u8868\u793a\u5b66\u4e60\u65b9\u6cd5\uff0c\u5c24\u5176\u5728\u5904\u7406\u5927\u89c4\u6a21\u56fe\u6570\u636e\u65f6\u8868\u73b0\u51fa\u663e\u8457\u4f18\u52bf\uff08\u4e3b\u8981\u5f97\u76ca\u4e8e\u968f\u673a\u91c7\u6837\uff09\u3002\u800cGCN\u66f4\u9002\u5408\u5c0f\u89c4\u6a21\u56fe\u7684\u573a\u666f\uff0c\u5176\u57fa\u4e8e\u5168\u56fe\u7684\u5377\u79ef\u64cd\u4f5c\u53ef\u4ee5\u5728\u8f83\u5c0f\u7684\u56fe\u4e0a\u83b7\u5f97\u826f\u597d\u7684\u6027\u80fd\u3002<\/p>\n \u53c2\u6570:<\/p>\n \u8ba9\u6211\u4eec\u518d\u770b\u770b\u6211\u4eec\u539f\u6765\u7684 GCN \u5c42\u4f20\u64ad\u89c4\u5219\uff1a<\/p>\n $\\Large h_{v_{i}}^{(l+1)} = \\sigma \\left(\\sum_{j} \\frac{1}{c_{ij}} h_{v_{j}}^{(l)} W^{(l)}\\right)$<\/p>\n \u5982\u524d\u6240\u8ff0\uff0c$\\large c_{ij}$ \u662f\u57fa\u4e8e\u56fe\u7ed3\u6784\u7684\u5f52\u4e00\u5316\u5e38\u6570\u3002\u5373 $\\large c_{ij}=\\tilde{D}^{\\frac{1}{2}} \\tilde{A} \\tilde{D}^{\\frac{1}{2}}$<\/p>\n \u5728 GAT<\/a> \u4e2d\uff0c\u8fd9\u79cd \u9759\u6001<\/em> \u5f52\u4e00\u5316\u7684\u5377\u79ef\u8fd0\u7b97\u88ab\u6ce8\u610f\u529b\u673a\u5236\u53d6\u4ee3\u3002<\/p>\n \u6211\u4eec\u8be5\u600e\u4e48\u505a\u5462\ud83e\udd14<\/p>\n \u7c7b\u4f3c\u4e8eTransformer\u4e2d\u7684\u6ce8\u610f\u529b\u673a\u5236\uff0cGAT\u7684\u8ba1\u7b97\u4e5f\u5206\u4e3a\u4e24\u6b65\uff1a\u8ba1\u7b97\u6ce8\u610f\u529b\u7cfb\u6570\uff08attention coefficient\uff09\u548c\u52a0\u6743\u6c42\u548c\uff08aggregate\uff09\u3002<\/p>\n \u663e\u7136\uff0c\u5b66\u4e60\u8282\u70b9 $i, j$ \u4e4b\u95f4\u7684\u76f8\u5173\u6027, \u5c31\u662f\u901a\u8fc7\u53ef\u5b66\u4e60\u7684\u53c2\u6570 $W$ \u548c $\\alpha(\\cdot)$ \u6620\u5c04\u5b8c\u6210\u7684<\/strong>, \u672c\u8d28\u4e0a\u662f\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u5b9e\u73b0\u7684\u3002<\/p>\n \u6709\u4e86\u76f8\u5173\u7cfb\u6570, \u79bb\u6ce8\u610f\u529b\u7cfb\u6570\u5c31\u5dee\u5f52\u4e00\u5316\u4e86\uff01\u5176\u5b9e\u5c31\u662f\u4f7f\u7528 softmax<\/strong> \u505a\u4e00\u6b65\u6620\u5c04, \u516c\u5f0f\u5982\u4e0b:<\/p>\n $$ \u81f3\u4e8e\u52a0\u6743\u6c42\u548c\u7684\u5b9e\u73b0\u4e5f\u5f88\u7b80\u5355\uff0c\u6839\u636e\u8ba1\u7b97\u597d\u7684\u6ce8\u610f\u529b\u7cfb\u6570\uff0c\u628a\u7279\u5f81\u52a0\u6743\u6c42\u548c\u805a\u5408\uff08aggregate\uff09\u4e00\u4e0b\u3002\u5373\uff1a \u5176\u4e2d $K$ \u662f\u6ce8\u610f\u529b\u673a\u5236\u7684\u5934\u6570, \u6bcf\u4e2a\u5934\u90fd\u4f1a\u7ef4\u62a4\u66f4\u65b0\u81ea\u5df1\u7684\u53c2\u6570, \u8ba1\u7b97\u5f97\u5230\u81ea\u5df1\u7684\u7ed3\u679c, $|_{k-1}^K$\u200b\u8868\u793a\u5c06\u6240\u6709\u5934\u7684\u8ba1\u7b97\u7ed3\u679c\u8fdb\u884c\u62fc\u63a5\uff08concat\uff09\u5f97\u5230\u6700\u540e\u66f4\u65b0\u597d\u7684\u65b0\u8282\u70b9\u5411\u91cf\u3002<\/p>\n \u591a\u5934\u6ce8\u610f\u529b\u673a\u5236\u4e5f\u53ef\u4ee5\u7406\u89e3\u6210\u7528\u4e86\u96c6\u6210\u5b66\u4e60\u7684\u65b9\u6cd5\uff0c\u5c31\u50cf\u5377\u79ef\u4e2d\uff0c\u4e5f\u8981\u9760\u5927\u91cf\u7684\u5377\u79ef\u6838\u624d\u80fd\u5927\u663e\u795e\u5a01\u4e00\u6837\uff01<\/p>\n \u6700\u540e\u901a\u8fc7\u4e00\u4e2a\u793a\u4f8b\u6765\u590d\u4e60\u4e00\u4e0bGAT\u7684\u8ba1\u7b97\u8fc7\u7a0b\uff0c\u56fe\u6570\u636e\u5982\u56fe\uff1a<\/p>\n \n \u5047\u8bbe\u53c2\u6570 $\\boldsymbol{W}$ \u548c $\\boldsymbol{\\alpha}$ \u7684\u503c\u4e3a $\\boldsymbol{W}=[1,1], \\boldsymbol{\\alpha}=[1,1,1,1]$ \u3002\u6ce8\u610f, \u8fd9\u4e9b\u53c2\u6570\u90fd\u662f\u53ef\u5b66\u4e60\u7684, \u968f\u7740\u7f51\u7edc\u7684\u8bad\u7ec3\u800c\u66f4\u65b0\u3002 $e_{14}$ \u7531\u4e8e\u5355\u5411\u6027, \u5373\u8282\u70b9 1 \u6307\u5411 2 , \u56e0\u6b64\u5728\u8ba1\u7b97\u65f6, \u76f8\u5173\u6027\u4e3a\u96f6\u3002\u7136\u540e\u901a\u8fc7\u516c\u5f0f \u7136\u540e\u901a\u8fc7\u52a0\u6743\u6c42\u548c\u5bf9\u67d0\u8282\u70b9\u7684\u90bb\u5c45\u505a\u91cd\u8981\u7a0b\u5ea6\u7684\u91cd\u5206\u914d, \u5373: \u4ee5\u8282\u70b9 1 \u4e3a\u4f8b: \u53c2\u6570:<\/p>\n Planetoid\uff08Cora\u3001CiteSeer\u3001PubMed\uff09\u6570\u636e\u96c6\u662f\u56fe\u795e\u7ecf\u7f51\u7edc\u7814\u7a76\u4e2d\u5e38\u7528\u7684\u57fa\u51c6\u6570\u636e\u96c6\u4e4b\u4e00\uff0c\u7528\u4e8e\u5b66\u672f\u8bba\u6587\u5206\u7c7b\u4efb\u52a1\u3002\u6bcf\u4e2a\u6570\u636e\u96c6\u5305\u542b\u4e00\u7ec4\u8bba\u6587\u548c\u5b83\u4eec\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\uff0c\u76ee\u6807\u662f\u6839\u636e\u5f15\u7528\u7f51\u7edc\u548c\u8282\u70b9\u7279\u5f81\uff08\u5982\u8bba\u6587\u7684\u8bcd\u5411\u91cf\uff09\u5bf9\u6bcf\u7bc7\u8bba\u6587\u8fdb\u884c\u5206\u7c7b\u3002<\/p>\n Cora: \u5305\u542b2708\u7bc7\u673a\u5668\u5b66\u4e60\u9886\u57df\u7684\u8bba\u6587\uff0c\u5206\u4e3a7\u4e2a\u7c7b\u522b\u3002<\/p>\n \n Deep Learning create by Arwin Yu Tutorial 08 – Graph Ne […]<\/p>\n","protected":false},"author":1,"featured_media":2177,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18,24],"tags":[19],"class_list":["post-2170","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-18","category-24","tag-19"],"_links":{"self":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2170","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2170"}],"version-history":[{"count":13,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2170\/revisions"}],"predecessor-version":[{"id":2211,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2170\/revisions\/2211"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/media\/2177"}],"wp:attachment":[{"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2170"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2170"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143528668.png\")
\n<\/p>\n
\n\u53ea\u5728\u6c47\u96c6\u4fe1\u606f\u8fdb\u884c\u9884\u6d4b\u65f6\u4f7f\u7528\u8fde\u63a5\u6027\u3002<\/strong><\/p>\n\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u5c42\u7ed3\u6784\u4e0e\u8fde\u63a5\u6027<\/h3>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143617137.png\")
\n<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143704220.png\")
\n<\/p>\n\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143814760.png\")
\n<\/p>\n\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926143852935.png\")
\n<\/p>\n\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926144209890.png\")
\n<\/p>\n\u56fe\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u7684\u8bad\u7ec3<\/h3>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926144247859.png\")
\n<\/p>\n\n
\n
\n
\n
\n
\n
\n
\n
\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926144834355.png\")
\n<\/p>\n\u56fe\u6570\u636e\u7684\u6570\u636e\u589e\u5f3a<\/h3>\n
\n
\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926144911469.png\")
\n<\/p>\n\n
\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926144955672.png\")
\n<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145030264.png\")
\n<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145105984.png\")
\n<\/p>\n \u56fe\u5377\u79ef\u7f51\u7edc\uff08GCN\uff09<\/h2>\n
\nimport warnings\n\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\nimport networkx as nx\nimport numpy as np\nfrom typing import Any, Dict, Tuple, Union, List\n\nwarnings.simplefilter(action="ignore", category=FutureWarning)\n\nplt.style.use("default")\n\nG = nx.Graph()\n# our example graph consists of 4 nodes with 2-dimensional features\nG.add_nodes_from(\n [\n (0, {"x": [-1.3, 2.2]}),\n (1, {"x": [0.6, -1.1]}),\n (2, {"x": [2.0, 1.4]}),\n (3, {"x": [5.1, 1.2]}),\n ]\n)\n# create the edges\nedge_list = [(0, 2), (1, 2), (1, 3), (2, 3)]\nG.add_edges_from(edge_list)\n\npos = nx.spring_layout(G)\n\ndef nudge(pos: Dict[int, np.ndarray], x_shift: float, y_shift: float):\n """Nudge position to position attributes"""\n return {n: (x + x_shift, y + y_shift) for n, (x, y) in pos.items()}\n\ndef draw_graph_with_attributes(\n G: nx.Graph,\n props: Dict[int, Any] = None,\n pos: Dict[int, np.ndarray] = None,\n figsize: Tuple[int, int] = (6, 6),\n x_nudge: float = 0.0,\n y_nudge: float = 0.07,\n ax: mpl.axes.Axes = None,\n font_color: str = "green",\n edge_color: str = "black",\n node_color: Union[str, List] = "lightblue",\n):\n """Draw a graph with node labels and attributes"""\n if ax is None:\n fig, ax = plt.subplots(1, 1, figsize=figsize)\n\n if pos is None:\n pos = nx.spring_layout(G)\n\n nx.draw_networkx(G, pos=pos, with_labels=True, ax=ax, edge_color=edge_color, node_color=node_color)\n pos_nudged = nudge(pos, x_nudge, y_nudge)\n if props is None:\n props = nx.get_node_attributes(G, "x")\n props = {\n node_id: np.array2string(np.array(x), precision=2, separator=",")\n for node_id, x in props.items()\n }\n nx.draw_networkx_labels(\n G, pos=pos_nudged, labels=props, ax=ax, font_color=font_color\n )\n ax.set_ylim(tuple(i * 1.1 for i in ax.get_ylim()))\n ax.spines["top"].set_visible(False)\n ax.spines["right"].set_visible(False)\n ax.spines["bottom"].set_visible(False)\n ax.spines["left"].set_visible(False)\n\ndraw_graph_with_attributes(G, pos=pos)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145137771.png\")
\n<\/p>\n\u90bb\u63a5\u77e9\u9635<\/h3>\n
A = nx.adjacency_matrix(G).toarray()\nA<\/code><\/pre>\narray([[0, 0, 1, 0],\n [0, 0, 1, 1],\n [1, 1, 0, 1],\n [0, 1, 1, 0]])<\/code><\/pre>\n\u8282\u70b9\u7279\u5f81\u77e9\u9635<\/h3>\n
props = nx.get_node_attributes(G, "x")\nX = np.array([props[i] for i in range(len(G))])\nX<\/code><\/pre>\narray([[-1.3, 2.2],\n [ 0.6, -1.1],\n [ 2. , 1.4],\n [ 5.1, 1.2]])<\/code><\/pre>\n \u8003\u8651\u90bb\u63a5\u77e9\u9635A\uff0c\u4e0e\u8282\u70b9\u7279\u5f81X\u76f8\u4e58\u662f\u5426\u6709\u610f\u4e49\uff1f<\/h4>\n
\n
\n$$
\n\\left[\\begin{array}{llll}
\n0 & 0 & 1 & 0 \\\\
\n0 & 0 & 1 & 1 \\\\
\n1 & 1 & 0 & 1 \\\\
\n0 & 1 & 1 & 0
\n\\end{array}\\right]
\n$$<\/p>\n
\n$$
\n\\left[\\begin{array}{cc}
\n-1.3 & 2.2 \\\\
\n0.6 & -1.1 \\\\
\n2.0 & 1.4 \\\\
\n5.1 & 1.2
\n\\end{array}\\right]
\n$$<\/p>\n
\n&AH=\\begin{bmatrix}0*(-1.3)+0*0.6+1*2.0+0*5.1&0*2.2+0*(-1.1)+1*1.4+0*1.2\\\\0*(-1.3)+0*0.6+1*2.0+1*5.1&0*2.2+0*(-1.1)+1*1.4+1*1.2\\\\1*(-1.3)+1*0.6+0*2.0+1*5.1&1*2.2+1*(-1.1)+0*1.4+1*1.2\\\\0*(-1.3)+1*0.6+1*2.0+0*5.1&0*2.2+1*(-1.1)+1*1.4+0*1.2\\end{bmatrix} \\\\
\n&=\\begin{bmatrix}2.0&1.4\\\\7.1&2.6\\\\4.4&2.3\\\\2.6&0.3\\end{bmatrix}
\n\\end{aligned}$$<\/p>\nW = np.random.rand(2, 2)\nW<\/code><\/pre>\narray([[0.63156147, 0.31418133],\n [0.79517293, 0.08585469]])<\/code><\/pre>\ndef gcn_layer(H: np.ndarray, A: np.ndarray, W: np.ndarray):\n return A @ H @ W<\/code><\/pre>\nfrom typing import Callable, Dict, Tuple\n\ndef features_and_adjacency(G: nx.Graph) -> Tuple[np.ndarray, np.ndarray]:\n """Get features and adjacency matrix from graph."""\n props = nx.get_node_attributes(G, "x")\n X = np.array([props[i] for i in range(len(G))])\n A = nx.adjacency_matrix(G).toarray()\n return X, A\n\ndef feature_array_to_dict(X: np.ndarray):\n """Create feature dictionary"""\n\n return {node_id: {"x": x} for node_id, x in enumerate(X)}\n\ndef propagate_and_draw(\n G: nx.Graph,\n X: np.ndarray,\n A: np.ndarray,\n W: np.ndarray,\n gcn_layer_func: Callable,\n pos: Dict[int, np.ndarray] = None,\n figsize: Tuple[int, int] = (16, 6),\n) -> Tuple[nx.Graph, np.ndarray]:\n """Apply propagation rule and draw before and after"""\n fig, ax = plt.subplots(1, 2, figsize=figsize)\n if pos is None:\n pos = nx.spring_layout(G)\n draw_graph_with_attributes(G, pos=pos, ax=ax[0])\n G_1 = G.copy()\n H_1 = gcn_layer_func(X, A, W)\n nx.set_node_attributes(G_1, feature_array_to_dict(H_1))\n draw_graph_with_attributes(G_1, pos=pos, ax=ax[1], font_color="red")\n return G_1, H_1\n\nX, A = features_and_adjacency(G)\nG_1, H_1 = propagate_and_draw(G, X, A, W, gcn_layer, pos=pos)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145254822.png\")
\n<\/p>\nnx.set_node_attributes(G, {2: {"x": [10, 10]}})\nG_1, H_1 = propagate_and_draw(G, X, A, W, gcn_layer, pos=pos)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145325420.png\")
\n<\/p>\nI = np.eye(4, 4)\nI<\/code><\/pre>\narray([[1., 0., 0., 0.],\n [0., 1., 0., 0.],\n [0., 0., 1., 0.],\n [0., 0., 0., 1.]])<\/code><\/pre>\nA_tilde = A + I\nA_tilde<\/code><\/pre>\narray([[1., 0., 1., 0.],\n [0., 1., 1., 1.],\n [1., 1., 1., 1.],\n [0., 1., 1., 1.]])<\/code><\/pre>\n
\n\u8ba9\u6211\u4eec\u518d\u6b21\u5c1d\u8bd5\u6211\u4eec\u7684\u793a\u4f8b\u3002<\/p>\nG_1, H_1 = propagate_and_draw(G, X, A_tilde, W, gcn_layer, pos=pos)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145357294.png\")
\n<\/p>\nG_2, X_2 = propagate_and_draw(G_1, H_1, A_tilde, W, gcn_layer, pos=pos)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145419224.png\")
\n<\/p>\ndegree_dict = {node_id: degree for node_id, degree in nx.degree(G)}\ndegree_dict<\/code><\/pre>\n{0: 1, 1: 2, 2: 3, 3: 2}<\/code><\/pre>\ndraw_graph_with_attributes(\n G, pos=pos, props=degree_dict, y_nudge=0.06, font_color="green"\n)<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145446482.png\")
\n<\/p>\ndegree_dict = nx.degree(G)\n# + 1 to add self-loops into degree\nD_tilde = np.array([degree_dict[i] + 1 for i in range(len(G))])\nD_tilde = D_tilde * I # create diagonal degree by point-wise multiplication\nD_tilde<\/code><\/pre>\narray([[2., 0., 0., 0.],\n [0., 3., 0., 0.],\n [0., 0., 4., 0.],\n [0., 0., 0., 3.]])<\/code><\/pre>\nfrom numpy.linalg import matrix_power\n\nD_power_minus_1 = matrix_power(D_tilde, -1)\nD_power_minus_1<\/code><\/pre>\narray([[0.5 , 0. , 0. , 0. ],\n [0. , 0.33333333, 0. , 0. ],\n [0. , 0. , 0.25 , 0. ],\n [0. , 0. , 0. , 0.33333333]])<\/code><\/pre>\nD_power_minus_1 @ A_tilde<\/code><\/pre>\narray([[0.5 , 0. , 0.5 , 0. ],\n [0. , 0.33333333, 0.33333333, 0.33333333],\n [0.25 , 0.25 , 0.25 , 0.25 ],\n [0. , 0.33333333, 0.33333333, 0.33333333]])\n\n\u5f52\u4e00\u5316\u540e\uff0c\u73b0\u5728\u6bcf\u884c\u7684\u603b\u548c\u4e3a\u201c1\u201d<\/code><\/pre>\nimport scipy\n\nD_power_minus_1_2 = scipy.linalg.fractional_matrix_power(D_tilde, -1 \/ 2)\nD_power_minus_1_2<\/code><\/pre>\narray([[0.70710678, 0. , 0. , 0. ],\n [0. , 0.57735027, 0. , 0. ],\n [0. , 0. , 0.5 , 0. ],\n [0. , 0. , 0. , 0.57735027]])<\/code><\/pre>\nD_power_minus_1_2 @ A_tilde @ D_power_minus_1_2<\/code><\/pre>\narray([[0.5 , 0. , 0.35355339, 0. ],\n [0. , 0.33333333, 0.28867513, 0.33333333],\n [0.35355339, 0.28867513, 0.25 , 0.28867513],\n [0. , 0.33333333, 0.28867513, 0.33333333]])<\/code><\/pre>\n GCN\u4e2d\u7684\u5377\u79ef\u4f53\u73b0\u5728\u54ea\u91cc\uff1f\u4e0eMPNN\u7684\u4e0d\u540c\u662f\u4ec0\u4e48\uff1f<\/h4>\n
\n\u57fa\u4e8e\u8c31\u56fe\u7406\u8bba\u770b\u5f85GCN<\/h2>\n
\u5377\u79ef\u7684\u672c\u8d28\u662f\u5c40\u90e8<\/h3>\n
![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145547477.png\")
\n<\/p>\n\u56fe\u62c9\u666e\u62c9\u65af\u7b97\u5b50<\/h3>\n
\n$$
\nD_v=\\sum_u A_{v u}
\n$$<\/p>\n
\n\u7136\u540e\uff0c\u56fe\u62c9\u666e\u62c9\u65af\u7b97\u5b50 $L$ \u662f\u5b9a\u4e49\u4e3a $L=D-A$ \u7684 $n \\times n$ \u77e9\u9635\u3002<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926145641233.png\")
\n<\/p>\n\u62c9\u666e\u62c9\u65af\u591a\u9879\u5f0f<\/h3>\n
\n$$
\np_w(L)=w_0 I_n+w_1 L+w_2 L^2+\\cdots+w_d L^d=\\sum_{i=0}^d w_i L^i \\text {. }
\n$$<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926150011191.png\")
\n<\/p>\n
\n$$
\nx^{\\prime}=p_w(L) x
\n$$<\/p>\n
\np_w(L)=w_0 I_n+w_1 L+w_2 L^2+\\cdots+w_d L^d=\\sum_{i=0}^d w_i L^i \\text {. }
\n$$<\/p>\n
\n$$
\nx^{\\prime}=p_w(L) x=\\sum_{i=0}^d w_i L^i x=w_0 I_n x=x .
\n$$<\/p>\n
\n$$
\n\\begin{aligned}
\n& x_v^{\\prime}=(L x)_v=L_v x \\\\
\n& =\\sum_{u \\in G} L_{v u} x_u \\\\
\n& =\\sum_{u \\in G}\\left(D_{v u}-A_{v u}\\right) x_u \\\\
\n& =D_v x_v-\\sum_{u \\in \\mathcal{N}(v)} x_u
\n\\end{aligned}
\n$$<\/p>\n
\n\u5f53 $v \\neq u$ \u65f6\uff0c $D_{v u}=0$ \u3002
\n\u56e0\u6b64\uff0c\u4e0a\u5f0f\u53ef\u4ee5\u5199\u6210\u4e24\u90e8\u5206\u4e4b\u548c:
\n$$
\n(L x)_v=D_v x_v-\\sum_{u \\neq v} A_{v u} x_u
\n$$<\/p>\n
\n$$
\n\\sum_{u \\neq v} A_{v u} x_u=\\sum_{u \\in \\mathcal{N}(v)} x_u
\n$$<\/p>\n
\n$$
\n(L x)_v=D_v x_v-\\sum_{u \\in \\mathcal{N}(v)} x_u
\n$$<\/p>\n\n
\n$$
\nx^{\\prime}=p_w(L) x
\n$$<\/p>\n
\n$$
\np_w(L)=w_0 I_n+w_1 L+w_2 L^2+\\cdots+w_d L^d=\\sum_{i=0}^d w_i L^i
\n$$<\/p>\n\u6700\u540e\u4e00\u4e2a\u95ee\u9898\uff1a\u4e0a\u8ff0\u57fa\u4e8e\u62c9\u666e\u62c9\u65af\u77e9\u9635\u7684\u591a\u9879\u5f0f\u56fe\u5377\u79ef\uff0c\u4e0e\u76f4\u63a5\u7ed9\u51fa\u7684GCN\u5b9a\u4e49\u4e3a\u5565\u76f8\u5dee\u8fd9\u4e48\u591a\uff1f<\/h3>\n
\n$$
\nL=D-A
\n$$<\/p>\n
\n\u5bf9\u4e8e\u7ecf\u5178GCN\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u7684\u662f\u5bf9\u79f0\u5f52\u4e00\u5316\u7684\u56fe\u62c9\u666e\u62c9\u65af\u77e9\u9635:
\n$$
\n\\tilde{L}=I-\\tilde{D}^{-1 \/ 2} \\tilde{A} \\tilde{D}^{-1 \/ 2}
\n$$<\/p>\n
\n$$
\n\\tilde{A}_{\\text {sym }}=\\tilde{D}^{-1 \/ 2} \\tilde{A} \\tilde{D}^{-1 \/ 2}$$<\/p>\n
\n$$
\nx^{\\prime}=p_w(L) x=\\sum_{i=0}^d w_i L^i x
\n$$<\/p>\n
\n$$
\nx^{\\prime}=w_0 I x+w_1 L x
\n$$<\/p>\n
\n$$
\nx^{\\prime}=w_0 I x+w_1 \\tilde{A}_{s y m} x
\n$$<\/p>\n
\n$$
\nx^{\\prime}=I x+\\tilde{A}_{s y m} x
\n$$<\/p>\n
\n$$
\nx^{\\prime}=\\tilde{A}_{\\text {sym }} x
\n$$<\/p>\n
\n\u73b0\u5728\uff0c\u6211\u4eec\u6dfb\u52a0\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362 $W$ \u548c\u6fc0\u6d3b\u51fd\u6570 $\\sigma$ \uff0c\u5bf9\u4e8e\u56fe\u5377\u79ef\u7f51\u7edc\u6765\u8bf4\uff0c\u7ebf\u6027\u53d8\u6362 $W$ \u4ee3\u8868\u8282\u70b9\u7279\u5f81\u7684\u7ebf\u6027\u6620\u5c04:
\n$$
\nH^{(l+1)}=\\sigma\\left(\\tilde{A}_{s y m} H^{(l)} W^{(l)}\\right)
\n$$<\/p>\n
\n$$
\nf\\left(H^{(l)}, A\\right)=\\sigma\\left(\\tilde{D}^{-1 \/ 2} \\tilde{A} \\tilde{D}^{-1 \/ 2} H^{(l)} W^{(l)}\\right)
\n$$<\/p>\n![](\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/fire-element.png\")
Pytorch\uff1aconv = GCNConv()<\/h3>\n\n
GraphSAGE<\/h2>\n
\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926151235272.png\")
\n<\/p>\n
\n$$
\nh_{\\mathrm{N}(1)}^1 \\leftarrow \\operatorname{AGGREGATE}\\left(h_3^0, h_4^0, h_5^0, h_6^0\\right)=\\operatorname{Mean}([0.3,0.4],[0.2,0.2],[0.7,0.8],[0.5,0.6])
\n$$<\/p>\n\n
\n
\n
Pytorch\uff1aconv = SAGEConv()<\/h3>\n\n
torch_geometric.nn.aggr<\/code>\u7684\u4efb\u4f55\u805a\u5408\u65b9\u6cd5\uff0c\u4f8b\u5982 "mean", "max", \u6216 "Istm"\u3002(\u9ed8\u8ba4\u503c: "mean")<\/li>\n GAT<\/h3>\n
\n\n
\n\\alpha_{i j}=\\frac{\\exp \\left(\\operatorname{LeakyReLU}\\left(e_{i j}\\right)\\right)}{\\sum_{k \\in N_i} \\exp \\left(\\operatorname{LeakyReLU}\\left(e_{i k}\\right)\\right)}
\n$$<\/p>\n
\n$$
\nh_i^{\\prime}=\\sigma\\left(\\sum_{j \\in N_j} \\alpha_{i j} W h_j\\right)
\n$$
\n$h_i^{\\prime}$ \u5c31\u662f GAT \u8f93\u51fa\u7684\u5bf9\u4e8e\u6bcf\u4e2a\u8282\u70b9 $i$ \u7684\u65b0\u7279\u5f81, \u8fd9\u4e2a\u65b0\u7279\u5f81\u7684\u5411\u91cf\u8868\u793a\u878d\u5408\u4e86\u90bb\u57df\u4fe1\u606f, $\\sigma(\\cdot)$ \u662f\u6fc0\u6d3b\u51fd\u6570\u3002\u6700\u540e, \u4e0e Transformer\u4e00\u6837, GAT \u4e5f\u53ef\u4ee5\u7528\u591a\u5934\u6ce8\u610f\u529b\u673a\u5236\u6765\u8fdb\u5316\u589e\u5f3a:
\n$$
\nh_i^{\\prime}(K)=|_{k=1}^K \\sigma\\left(\\sum_{j \\in N_i} \\alpha_{i j}{ }^k W^k h_j\\right)
\n$$<\/p>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926151918823.png\")
\n<\/p>\n
\n\u9996\u5148, \u8ba1\u7b97\u6ce8\u610f\u529b\u7cfb\u6570 $e_{i j}=\\boldsymbol{\\alpha}\\left(\\boldsymbol{W} h_i, \\boldsymbol{W} h_j\\right)$, \u4ee5\u8282\u70b9 1 \u4e3a\u4f8b, \u4e0e\u5176\u5b83\u8282\u70b9\u7684\u76f8\u5173\u6027\u7cfb\u6570\u4e3a:
\n$$
\n\\begin{aligned}
\n& e_{12}=\\alpha \\cdot[0.1,0.2,0.2,0.2]=0.7 \\\\
\n& e_{13}=\\alpha \\cdot[0.1,0.2,0.25,0.2]=0.75 \\\\
\n& e_{14}=0 \\\\
\n& e_{15}=\\alpha \\cdot[0.1,0.2,0.3,0.8]=1.4 \\\\
\n& e_{16}=\\alpha \\cdot[0.1,0.2,0.5,0.6]=1.4
\n\\end{aligned}
\n$$<\/p>\n
\n$$
\n\\alpha_{12}=\\frac{\\exp \\left(\\operatorname{LeakyReLU}\\left(e_{12}\\right)\\right)}{\\exp \\left(\\operatorname{LeakyReLU}\\left(e_{12}\\right)\\right)+\\exp \\left(\\operatorname{LeakyReLU}\\left(e_{13}\\right)\\right)+\\ldots+\\exp \\left(\\operatorname{LeakyReLU}\\left(e_{16}\\right)\\right)}
\n$$<\/p>\n
\n$$
\n\\overrightarrow{h_i^{\\prime}}=\\sigma\\left(\\sum_{j \\in N_i} \\alpha_{i j} W \\vec{h}_j\\right)
\n$$<\/p>\n
\n$$
\n\\vec{h}_1^{\\prime T}=\\sigma\\left(\\alpha_{12} \\cdot W \\cdot \\vec{h}_2+\\alpha_{13} \\cdot W \\cdot \\vec{h}_3 \\cdots\\right)=\\sigma\\left(\\alpha_{12} \\cdot W \\cdot[0.2,0.2]+\\cdots\\right)
\n$$<\/p>\n Pytorch\uff1aconv = GATConv()<\/h3>\n\n
Demo\u9879\u76ee\uff1aCora<\/h3>\n
\nimport torch\nimport torch.nn.functional as F\nfrom torch_geometric.nn import SAGEConv, GATConv, GCNConv\nfrom torch_geometric.datasets import Planetoid\nfrom torch_geometric.transforms import NormalizeFeatures\nimport matplotlib.pyplot as plt\n\n# \u52a0\u8f7d\u5e76\u5f52\u4e00\u5316 Cora \u6570\u636e\u96c6\ndataset = Planetoid(root='data\/Planetoid', name='Cora', transform=NormalizeFeatures())\ndata = dataset[0]\n\n# \u5b9a\u4e49\u6a21\u578b\u67b6\u6784\nclass GraphModel(torch.nn.Module):\n def __init__(self, conv_layer, in_channels, hidden_channels, out_channels):\n super(GraphModel, self).__init__()\n self.conv1 = conv_layer(in_channels, hidden_channels)\n self.conv2 = conv_layer(hidden_channels, out_channels)\n\n def forward(self, x, edge_index):\n x = self.conv1(x, edge_index)\n x = F.relu(x)\n x = self.conv2(x, edge_index)\n return F.log_softmax(x, dim=1)\n\n# \u8bad\u7ec3\u548c\u6d4b\u8bd5\u51fd\u6570\ndef train(model, optimizer):\n model.train()\n optimizer.zero_grad()\n out = model(data.x, data.edge_index)\n loss = F.nll_loss(out[data.train_mask], data.y[data.train_mask])\n loss.backward()\n optimizer.step()\n return loss.item()\n\ndef test(model):\n model.eval()\n logits, accs = model(data.x, data.edge_index), []\n for mask in [data.train_mask, data.val_mask, data.test_mask]:\n pred = logits[mask].max(1)[1]\n acc = pred.eq(data.y[mask]).sum().item() \/ mask.sum().item()\n accs.append(acc)\n return accs\n\n# \u5b9a\u4e49\u8bbe\u5907\u548c\u4f18\u5316\u5668\ndevice = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n\n# \u521d\u59cb\u5316\u6a21\u578b\u548c\u4f18\u5316\u5668\ndef initialize_model_and_optimizer(conv_layer):\n model = GraphModel(conv_layer, dataset.num_features, 16, dataset.num_classes).to(device)\n data.to(device)\n optimizer = torch.optim.Adam(model.parameters(), lr=0.01, weight_decay=5e-4)\n return model, optimizer\n\n# \u8bad\u7ec3\u548c\u8bc4\u4f30\u6a21\u578b\ndef train_and_evaluate(conv_layer, epochs=50):\n model, optimizer = initialize_model_and_optimizer(conv_layer)\n train_losses, train_accs, val_accs, test_accs = [], [], [], []\n\n for epoch in range(epochs):\n loss = train(model, optimizer)\n train_acc, val_acc, test_acc = test(model)\n train_losses.append(loss)\n train_accs.append(train_acc)\n val_accs.append(val_acc)\n test_accs.append(test_acc)\n print(f'Epoch: {epoch:03d}, Loss: {loss:.4f}, Train Acc: {train_acc:.4f}, Val Acc: {val_acc:.4f}, Test Acc: {test_acc:.4f}')\n\n return train_losses, train_accs, val_accs, test_accs\n\n# \u8fd0\u884c\u5e76\u6536\u96c6\u7ed3\u679c\nresults = {}\nfor name, conv_layer in [('SAGEConv', SAGEConv), ('GATConv', GATConv), ('GCNConv', GCNConv)]:\n print(f'\\nRunning {name}...\\n')\n results[name] = train_and_evaluate(conv_layer)\n\n# \u53ef\u89c6\u5316\u7ed3\u679c\ndef plot_results(results):\n fig, axes = plt.subplots(3, 1, figsize=(6, 12))\n\n for i, (name, (train_losses, train_accs, val_accs, test_accs)) in enumerate(results.items()):\n axes[i].plot(train_losses, label='Train Loss')\n axes[i].plot(train_accs, label='Train Accuracy')\n axes[i].plot(val_accs, label='Validation Accuracy')\n axes[i].plot(test_accs, label='Test Accuracy')\n axes[i].set_title(f'{name} Results')\n axes[i].set_xlabel('Epochs')\n axes[i].set_ylabel('Metrics')\n axes[i].legend()\n\n plt.tight_layout()\n plt.show()\n\nplot_results(results)\n<\/code><\/pre>\nRunning SAGEConv...\n\nEpoch: 000, Loss: 1.9586, Train Acc: 0.1429, Val Acc: 0.0720, Test Acc: 0.0910\nEpoch: 001, Loss: 1.9421, Train Acc: 0.1429, Val Acc: 0.0720, Test Acc: 0.0910\nEpoch: 002, Loss: 1.9220, Train Acc: 0.1429, Val Acc: 0.0720, Test Acc: 0.0910\nEpoch: 003, Loss: 1.8996, Train Acc: 0.2000, Val Acc: 0.0760, Test Acc: 0.0910\nEpoch: 004, Loss: 1.8744, Train Acc: 0.4929, Val Acc: 0.1680, Test Acc: 0.1850\nEpoch: 005, Loss: 1.8461, Train Acc: 0.6786, Val Acc: 0.3020, Test Acc: 0.2880\nEpoch: 006, Loss: 1.8150, Train Acc: 0.8214, Val Acc: 0.3360, Test Acc: 0.3290\nEpoch: 007, Loss: 1.7815, Train Acc: 0.8000, Val Acc: 0.3280, Test Acc: 0.3420\nEpoch: 008, Loss: 1.7457, Train Acc: 0.8214, Val Acc: 0.3580, Test Acc: 0.3570\nEpoch: 009, Loss: 1.7073, Train Acc: 0.8357, Val Acc: 0.3740, Test Acc: 0.3770\nEpoch: 010, Loss: 1.6664, Train Acc: 0.8571, Val Acc: 0.4000, Test Acc: 0.3910\nEpoch: 011, Loss: 1.6231, Train Acc: 0.8571, Val Acc: 0.4240, Test Acc: 0.4070\nEpoch: 012, Loss: 1.5773, Train Acc: 0.8714, Val Acc: 0.4400, Test Acc: 0.4190\nEpoch: 013, Loss: 1.5292, Train Acc: 0.8786, Val Acc: 0.4560, Test Acc: 0.4350\nEpoch: 014, Loss: 1.4788, Train Acc: 0.8857, Val Acc: 0.4700, Test Acc: 0.4510\nEpoch: 015, Loss: 1.4264, Train Acc: 0.9214, Val Acc: 0.4780, Test Acc: 0.4740\nEpoch: 016, Loss: 1.3722, Train Acc: 0.9357, Val Acc: 0.4900, Test Acc: 0.4980\nEpoch: 017, Loss: 1.3165, Train Acc: 0.9429, Val Acc: 0.5060, Test Acc: 0.5100\nEpoch: 018, Loss: 1.2596, Train Acc: 0.9429, Val Acc: 0.5160, Test Acc: 0.5210\nEpoch: 019, Loss: 1.2017, Train Acc: 0.9643, Val Acc: 0.5340, Test Acc: 0.5270\nEpoch: 020, Loss: 1.1433, Train Acc: 0.9786, Val Acc: 0.5460, Test Acc: 0.5390\nEpoch: 021, Loss: 1.0846, Train Acc: 0.9786, Val Acc: 0.5580, Test Acc: 0.5540\nEpoch: 022, Loss: 1.0260, Train Acc: 0.9857, Val Acc: 0.5680, Test Acc: 0.5700\nEpoch: 023, Loss: 0.9680, Train Acc: 0.9857, Val Acc: 0.5960, Test Acc: 0.5820\nEpoch: 024, Loss: 0.9108, Train Acc: 0.9929, Val Acc: 0.6080, Test Acc: 0.5980\nEpoch: 025, Loss: 0.8549, Train Acc: 0.9929, Val Acc: 0.6260, Test Acc: 0.6110\nEpoch: 026, Loss: 0.8005, Train Acc: 0.9929, Val Acc: 0.6500, Test Acc: 0.6240\nEpoch: 027, Loss: 0.7479, Train Acc: 1.0000, Val Acc: 0.6520, Test Acc: 0.6350\nEpoch: 028, Loss: 0.6975, Train Acc: 1.0000, Val Acc: 0.6680, Test Acc: 0.6460\nEpoch: 029, Loss: 0.6495, Train Acc: 1.0000, Val Acc: 0.6740, Test Acc: 0.6620\nEpoch: 030, Loss: 0.6040, Train Acc: 1.0000, Val Acc: 0.6880, Test Acc: 0.6800\nEpoch: 031, Loss: 0.5612, Train Acc: 1.0000, Val Acc: 0.7120, Test Acc: 0.6880\nEpoch: 032, Loss: 0.5212, Train Acc: 1.0000, Val Acc: 0.7320, Test Acc: 0.7020\nEpoch: 033, Loss: 0.4840, Train Acc: 1.0000, Val Acc: 0.7320, Test Acc: 0.7210\nEpoch: 034, Loss: 0.4496, Train Acc: 1.0000, Val Acc: 0.7420, Test Acc: 0.7330\nEpoch: 035, Loss: 0.4180, Train Acc: 1.0000, Val Acc: 0.7500, Test Acc: 0.7470\nEpoch: 036, Loss: 0.3892, Train Acc: 1.0000, Val Acc: 0.7600, Test Acc: 0.7530\nEpoch: 037, Loss: 0.3629, Train Acc: 1.0000, Val Acc: 0.7620, Test Acc: 0.7560\nEpoch: 038, Loss: 0.3392, Train Acc: 1.0000, Val Acc: 0.7660, Test Acc: 0.7600\nEpoch: 039, Loss: 0.3177, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7630\nEpoch: 040, Loss: 0.2984, Train Acc: 1.0000, Val Acc: 0.7680, Test Acc: 0.7700\nEpoch: 041, Loss: 0.2810, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7690\nEpoch: 042, Loss: 0.2654, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7700\nEpoch: 043, Loss: 0.2514, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7690\nEpoch: 044, Loss: 0.2389, Train Acc: 1.0000, Val Acc: 0.7720, Test Acc: 0.7660\nEpoch: 045, Loss: 0.2277, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7670\nEpoch: 046, Loss: 0.2177, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7690\nEpoch: 047, Loss: 0.2088, Train Acc: 1.0000, Val Acc: 0.7680, Test Acc: 0.7710\nEpoch: 048, Loss: 0.2008, Train Acc: 1.0000, Val Acc: 0.7700, Test Acc: 0.7730\nEpoch: 049, Loss: 0.1938, Train Acc: 1.0000, Val Acc: 0.7740, Test Acc: 0.7760\n\nRunning GATConv...\n\nEpoch: 000, Loss: 1.9456, Train Acc: 0.3571, Val Acc: 0.2020, Test Acc: 0.1930\nEpoch: 001, Loss: 1.9399, Train Acc: 0.6357, Val Acc: 0.3100, Test Acc: 0.3490\nEpoch: 002, Loss: 1.9333, Train Acc: 0.7357, Val Acc: 0.4460, Test Acc: 0.4510\nEpoch: 003, Loss: 1.9235, Train Acc: 0.7929, Val Acc: 0.5180, Test Acc: 0.5250\nEpoch: 004, Loss: 1.9123, Train Acc: 0.8286, Val Acc: 0.6640, Test Acc: 0.6770\nEpoch: 005, Loss: 1.9006, Train Acc: 0.8071, Val Acc: 0.6900, Test Acc: 0.7000\nEpoch: 006, Loss: 1.8888, Train Acc: 0.8429, Val Acc: 0.6960, Test Acc: 0.6880\nEpoch: 007, Loss: 1.8761, Train Acc: 0.8643, Val Acc: 0.6860, Test Acc: 0.6790\nEpoch: 008, Loss: 1.8624, Train Acc: 0.8857, Val Acc: 0.6980, Test Acc: 0.6820\nEpoch: 009, Loss: 1.8479, Train Acc: 0.9000, Val Acc: 0.6980, Test Acc: 0.6920\nEpoch: 010, Loss: 1.8324, Train Acc: 0.9143, Val Acc: 0.6980, Test Acc: 0.6960\nEpoch: 011, Loss: 1.8162, Train Acc: 0.9214, Val Acc: 0.6960, Test Acc: 0.7000\nEpoch: 012, Loss: 1.7993, Train Acc: 0.9286, Val Acc: 0.7020, Test Acc: 0.7140\nEpoch: 013, Loss: 1.7816, Train Acc: 0.9286, Val Acc: 0.7180, Test Acc: 0.7260\nEpoch: 014, Loss: 1.7634, Train Acc: 0.9429, Val Acc: 0.7260, Test Acc: 0.7330\nEpoch: 015, Loss: 1.7446, Train Acc: 0.9429, Val Acc: 0.7360, Test Acc: 0.7340\nEpoch: 016, Loss: 1.7251, Train Acc: 0.9429, Val Acc: 0.7360, Test Acc: 0.7350\nEpoch: 017, Loss: 1.7051, Train Acc: 0.9429, Val Acc: 0.7340, Test Acc: 0.7330\nEpoch: 018, Loss: 1.6844, Train Acc: 0.9429, Val Acc: 0.7320, Test Acc: 0.7290\nEpoch: 019, Loss: 1.6632, Train Acc: 0.9357, Val Acc: 0.7300, Test Acc: 0.7350\nEpoch: 020, Loss: 1.6413, Train Acc: 0.9357, Val Acc: 0.7360, Test Acc: 0.7400\nEpoch: 021, Loss: 1.6188, Train Acc: 0.9429, Val Acc: 0.7340, Test Acc: 0.7470\nEpoch: 022, Loss: 1.5958, Train Acc: 0.9571, Val Acc: 0.7360, Test Acc: 0.7500\nEpoch: 023, Loss: 1.5722, Train Acc: 0.9571, Val Acc: 0.7460, Test Acc: 0.7570\nEpoch: 024, Loss: 1.5480, Train Acc: 0.9643, Val Acc: 0.7540, Test Acc: 0.7590\nEpoch: 025, Loss: 1.5232, Train Acc: 0.9643, Val Acc: 0.7500, Test Acc: 0.7600\nEpoch: 026, Loss: 1.4980, Train Acc: 0.9643, Val Acc: 0.7520, Test Acc: 0.7590\nEpoch: 027, Loss: 1.4722, Train Acc: 0.9643, Val Acc: 0.7500, Test Acc: 0.7620\nEpoch: 028, Loss: 1.4458, Train Acc: 0.9643, Val Acc: 0.7500, Test Acc: 0.7630\nEpoch: 029, Loss: 1.4190, Train Acc: 0.9643, Val Acc: 0.7600, Test Acc: 0.7680\nEpoch: 030, Loss: 1.3917, Train Acc: 0.9643, Val Acc: 0.7620, Test Acc: 0.7700\nEpoch: 031, Loss: 1.3639, Train Acc: 0.9643, Val Acc: 0.7640, Test Acc: 0.7680\nEpoch: 032, Loss: 1.3356, Train Acc: 0.9643, Val Acc: 0.7660, Test Acc: 0.7650\nEpoch: 033, Loss: 1.3070, Train Acc: 0.9643, Val Acc: 0.7660, Test Acc: 0.7640\nEpoch: 034, Loss: 1.2779, Train Acc: 0.9643, Val Acc: 0.7680, Test Acc: 0.7640\nEpoch: 035, Loss: 1.2486, Train Acc: 0.9643, Val Acc: 0.7660, Test Acc: 0.7640\nEpoch: 036, Loss: 1.2189, Train Acc: 0.9714, Val Acc: 0.7660, Test Acc: 0.7660\nEpoch: 037, Loss: 1.1890, Train Acc: 0.9714, Val Acc: 0.7620, Test Acc: 0.7620\nEpoch: 038, Loss: 1.1588, Train Acc: 0.9714, Val Acc: 0.7660, Test Acc: 0.7610\nEpoch: 039, Loss: 1.1285, Train Acc: 0.9714, Val Acc: 0.7640, Test Acc: 0.7640\nEpoch: 040, Loss: 1.0981, Train Acc: 0.9714, Val Acc: 0.7640, Test Acc: 0.7640\nEpoch: 041, Loss: 1.0675, Train Acc: 0.9714, Val Acc: 0.7660, Test Acc: 0.7640\nEpoch: 042, Loss: 1.0369, Train Acc: 0.9714, Val Acc: 0.7640, Test Acc: 0.7690\nEpoch: 043, Loss: 1.0064, Train Acc: 0.9714, Val Acc: 0.7640, Test Acc: 0.7680\nEpoch: 044, Loss: 0.9759, Train Acc: 0.9714, Val Acc: 0.7660, Test Acc: 0.7690\nEpoch: 045, Loss: 0.9455, Train Acc: 0.9714, Val Acc: 0.7640, Test Acc: 0.7680\nEpoch: 046, Loss: 0.9153, Train Acc: 0.9714, Val Acc: 0.7640, Test Acc: 0.7670\nEpoch: 047, Loss: 0.8854, Train Acc: 0.9714, Val Acc: 0.7620, Test Acc: 0.7670\nEpoch: 048, Loss: 0.8559, Train Acc: 0.9714, Val Acc: 0.7620, Test Acc: 0.7670\nEpoch: 049, Loss: 0.8269, Train Acc: 0.9786, Val Acc: 0.7660, Test Acc: 0.7660\n\nRunning GCNConv...\n\nEpoch: 000, Loss: 1.9455, Train Acc: 0.2643, Val Acc: 0.1700, Test Acc: 0.1870\nEpoch: 001, Loss: 1.9385, Train Acc: 0.4714, Val Acc: 0.2500, Test Acc: 0.2590\nEpoch: 002, Loss: 1.9311, Train Acc: 0.7357, Val Acc: 0.4840, Test Acc: 0.4840\nEpoch: 003, Loss: 1.9216, Train Acc: 0.7786, Val Acc: 0.5240, Test Acc: 0.5270\nEpoch: 004, Loss: 1.9112, Train Acc: 0.8071, Val Acc: 0.5380, Test Acc: 0.5430\nEpoch: 005, Loss: 1.9001, Train Acc: 0.7929, Val Acc: 0.5380, Test Acc: 0.5560\nEpoch: 006, Loss: 1.8885, Train Acc: 0.8143, Val Acc: 0.5620, Test Acc: 0.5840\nEpoch: 007, Loss: 1.8761, Train Acc: 0.8286, Val Acc: 0.5880, Test Acc: 0.6000\nEpoch: 008, Loss: 1.8627, Train Acc: 0.8357, Val Acc: 0.5920, Test Acc: 0.6050\nEpoch: 009, Loss: 1.8487, Train Acc: 0.8429, Val Acc: 0.5820, Test Acc: 0.6010\nEpoch: 010, Loss: 1.8342, Train Acc: 0.8357, Val Acc: 0.5800, Test Acc: 0.5940\nEpoch: 011, Loss: 1.8190, Train Acc: 0.8500, Val Acc: 0.5920, Test Acc: 0.6020\nEpoch: 012, Loss: 1.8031, Train Acc: 0.8571, Val Acc: 0.5920, Test Acc: 0.6060\nEpoch: 013, Loss: 1.7865, Train Acc: 0.8571, Val Acc: 0.5960, Test Acc: 0.6110\nEpoch: 014, Loss: 1.7693, Train Acc: 0.8571, Val Acc: 0.5980, Test Acc: 0.6120\nEpoch: 015, Loss: 1.7515, Train Acc: 0.8571, Val Acc: 0.5960, Test Acc: 0.6110\nEpoch: 016, Loss: 1.7331, Train Acc: 0.8571, Val Acc: 0.5980, Test Acc: 0.6140\nEpoch: 017, Loss: 1.7142, Train Acc: 0.8571, Val Acc: 0.6000, Test Acc: 0.6190\nEpoch: 018, Loss: 1.6947, Train Acc: 0.8714, Val Acc: 0.6060, Test Acc: 0.6240\nEpoch: 019, Loss: 1.6746, Train Acc: 0.8786, Val Acc: 0.6120, Test Acc: 0.6340\nEpoch: 020, Loss: 1.6537, Train Acc: 0.8786, Val Acc: 0.6160, Test Acc: 0.6420\nEpoch: 021, Loss: 1.6324, Train Acc: 0.8786, Val Acc: 0.6240, Test Acc: 0.6550\nEpoch: 022, Loss: 1.6106, Train Acc: 0.8857, Val Acc: 0.6340, Test Acc: 0.6610\nEpoch: 023, Loss: 1.5882, Train Acc: 0.8929, Val Acc: 0.6480, Test Acc: 0.6730\nEpoch: 024, Loss: 1.5652, Train Acc: 0.9000, Val Acc: 0.6580, Test Acc: 0.6850\nEpoch: 025, Loss: 1.5417, Train Acc: 0.9143, Val Acc: 0.6720, Test Acc: 0.6940\nEpoch: 026, Loss: 1.5177, Train Acc: 0.9286, Val Acc: 0.6820, Test Acc: 0.7040\nEpoch: 027, Loss: 1.4934, Train Acc: 0.9286, Val Acc: 0.6860, Test Acc: 0.7100\nEpoch: 028, Loss: 1.4687, Train Acc: 0.9214, Val Acc: 0.6960, Test Acc: 0.7130\nEpoch: 029, Loss: 1.4437, Train Acc: 0.9214, Val Acc: 0.6980, Test Acc: 0.7250\nEpoch: 030, Loss: 1.4183, Train Acc: 0.9286, Val Acc: 0.7060, Test Acc: 0.7320\nEpoch: 031, Loss: 1.3928, Train Acc: 0.9286, Val Acc: 0.7160, Test Acc: 0.7430\nEpoch: 032, Loss: 1.3670, Train Acc: 0.9286, Val Acc: 0.7180, Test Acc: 0.7470\nEpoch: 033, Loss: 1.3411, Train Acc: 0.9286, Val Acc: 0.7240, Test Acc: 0.7490\nEpoch: 034, Loss: 1.3151, Train Acc: 0.9357, Val Acc: 0.7240, Test Acc: 0.7500\nEpoch: 035, Loss: 1.2891, Train Acc: 0.9357, Val Acc: 0.7360, Test Acc: 0.7520\nEpoch: 036, Loss: 1.2631, Train Acc: 0.9357, Val Acc: 0.7380, Test Acc: 0.7590\nEpoch: 037, Loss: 1.2371, Train Acc: 0.9357, Val Acc: 0.7400, Test Acc: 0.7620\nEpoch: 038, Loss: 1.2112, Train Acc: 0.9357, Val Acc: 0.7420, Test Acc: 0.7640\nEpoch: 039, Loss: 1.1855, Train Acc: 0.9429, Val Acc: 0.7460, Test Acc: 0.7640\nEpoch: 040, Loss: 1.1600, Train Acc: 0.9429, Val Acc: 0.7480, Test Acc: 0.7680\nEpoch: 041, Loss: 1.1346, Train Acc: 0.9429, Val Acc: 0.7500, Test Acc: 0.7770\nEpoch: 042, Loss: 1.1096, Train Acc: 0.9429, Val Acc: 0.7500, Test Acc: 0.7800\nEpoch: 043, Loss: 1.0849, Train Acc: 0.9429, Val Acc: 0.7520, Test Acc: 0.7800\nEpoch: 044, Loss: 1.0606, Train Acc: 0.9500, Val Acc: 0.7540, Test Acc: 0.7800\nEpoch: 045, Loss: 1.0366, Train Acc: 0.9500, Val Acc: 0.7540, Test Acc: 0.7800\nEpoch: 046, Loss: 1.0131, Train Acc: 0.9500, Val Acc: 0.7580, Test Acc: 0.7870\nEpoch: 047, Loss: 0.9900, Train Acc: 0.9500, Val Acc: 0.7580, Test Acc: 0.7910\nEpoch: 048, Loss: 0.9673, Train Acc: 0.9500, Val Acc: 0.7560, Test Acc: 0.7910\nEpoch: 049, Loss: 0.9452, Train Acc: 0.9500, Val Acc: 0.7620, Test Acc: 0.7920<\/code><\/pre>\n![](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/09\/20240926152235914.png\")
\n<\/p>\n![](\"https:\/\/img.icons8.com\/dusk\/64\/000000\/prize.png\")
Credits<\/h2>\n
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