{"id":23439,"date":"2026-03-01T17:30:22","date_gmt":"2026-03-01T09:30:22","guid":{"rendered":"https:\/\/www.orczhou.com\/?p=23439"},"modified":"2026-03-01T19:16:55","modified_gmt":"2026-03-01T11:16:55","slug":"diffusion-model-foundation-variational-autoencoder","status":"publish","type":"post","link":"https:\/\/www.orczhou.com\/index.php\/2026\/03\/diffusion-model-foundation-variational-autoencoder\/","title":{"rendered":"\u6269\u6563\uff08diffusion\uff09\u6a21\u578b\u57fa\u7840\uff1aVAE \u6a21\u578b"},"content":{"rendered":"\n\n\n\n

\u5728\u6700\u521d\u5c1d\u8bd5\u4e86\u89e3 diffusion \u6a21\u578b\u7684\u65f6\u5019\uff0c\u539f\u672c\u662f\u6253\u7b97\u8df3\u8fc7 VAE \uff08Variational Autoencoder \u53d8\u5206\u81ea\u7f16\u7801 \uff09\u6a21\u578b\u7684\uff0c\u540e\u9762\u53d1\u73b0\u6709\u70b9\u8df3\u4e0d\u8fc7\u53bb\u3002\u518d\u82b1\u4e86\u4e00\u4e9b\u65f6\u95f4\u53bb\u4e86\u89e3 VAE\uff0c\u624d\u610f\u8bc6\u5230\u5176\u5b9e\u4e0d\u5e94\u8be5\u8df3\u8fc7\u53bb\uff0c\u76f8\u53cd\u7684\uff0c\u4e86\u89e3 VAE \u5b9e\u73b0\u7684\u4e00\u4e9b\u67b6\u6784\u3001\u539f\u7406\u3001\u76f4\u89c9\u3001\u521d\u8877\u3001\u6570\u5b66\u539f\u7406\u5219\u53ef\u4ee5\u5927\u5927\u5e2e\u52a9\u7406\u89e3\u540e\u7eed\u7684\u751f\u6210\u6a21\u578b\u3002<\/p>\n\n\n\n

1. \u4e3a\u4ec0\u4e48\u73b0\u5728\u4f60\u4f9d\u65e7\u9700\u8981\u4e86\u89e3 VAE<\/h3>\n\n\n\n

\u201c\u6f5c\u7a7a\u95f4\u201d\uff08latent space\uff09\u7684\u5904\u7406\u4f9d\u65e7\u662f\u73b0\u4ee3\uff08SOTA\uff09\u751f\u6210\u6a21\u578b\u6700\u4e3a\u6838\u5fc3\u7684\u7ec4\u4ef6\u3002\u800c\u4eceautoencoder\u6a21\u578b\uff0c\u6269\u5c55\u5230 VAE \u6a21\u578b\uff0c\u5219\u662f\u751f\u6210\u6a21\u578b\u8d70\u51fa\u7684\u5173\u952e\u6b65\u9aa4\u4e4b\u4e00\u3002\u5c06\u6f5c\u7a7a\u95f4\u9650\u5236\u5728\u4e00\u4e2a\u6b63\u6001\u5206\u5e03\u7684\u7a7a\u95f4\u5185\uff0c\u7136\u540e\uff0c\u5728\u8fd9\u4e2a\u7a7a\u95f4\u8fdb\u884c\u91c7\u6837\u540e\u8fdb\u884c decoder \u7684\u751f\u6210\u601d\u60f3\uff0c\u5219\u662f\u73b0\u4ee3\u751f\u6210\u6a21\u578b\u5f88\u591a\u601d\u60f3\u6765\u65f6\u7684\u8def\u3002<\/p>\n\n\n\n

\u5982\u679c\u8df3\u8fc7\u8fd9\u4e00\u6bb5\uff0c\u5f88\u591a\u7684\u6982\u5ff5\u5219\u4f1a\u663e\u5f97\u975e\u5e38\u7a81\u7136\u3002<\/p>\n\n\n\n

2. VAE \u5173\u952e\u76f4\u89c9\u4e0e\u4e3b\u8981\u601d\u60f3<\/h3>\n\n\n\n

2.1 \u5173\u952e\u76f4\u89c9<\/h4>\n\n\n\n

\u5728\u8ba1\u7b97\u673a\u4e0e\u6570\u5b66\u79d1\u5b66\u4e2d\u6709\u51e0\u4e2a\u6982\u5ff5\u662f\u53cd\u590d\u51fa\u73b0\u7684\uff0c\u5176\u4e2d\u4e4b\u4e00\u5c31\u662f\u201c\u9ad8\u65af\u5206\u5e03\u201d\uff08\u6216\u8005\u53eb\u201c\u6b63\u6001\u5206\u5e03\u201d\uff09\u3002\u5728\u56fe\u50cf\u751f\u6210\u6a21\u578b\u4e2d\u975e\u5e38\u5173\u952e\u7684\uff0c\u5219\u662f\u5c06\u6f5c\u7a7a\u95f4\u9650\u5236\u5728\u4e86\u4e00\u4e2a\u6b63\u6001\u5206\u5e03\u4e4b\u4e2d\uff0c\u4e3a\u4ec0\u4e48\u4f1a\u8fd9\u6837\uff1f\u4f3c\u4e4e\u5e76\u6ca1\u6709\u4eba\u53bb\u8bf4\u660e\u8fd9\u4e00\u70b9\uff0c\u8fd9\u91cc\u505a\u4e00\u4e2a\u7b80\u5355\u7684\u9610\u8ff0\u548c\u7406\u89e3\u3002\u9ad8\u65af\u5206\u5e03\u53ef\u4ee5\u7406\u89e3\u4e3a\u5927\u91cf\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u6781\u9650\u5f62\u5f0f\uff0c\u73b0\u5b9e\u4e16\u754c\u7684\u5206\u5e03\u6211\u4eec\u901a\u5e38\u4f1a\u5047\u8bbe\u5176\u4e3a\u5927\u91cf\u5fae\u5c0f\u56e0\u5b50\u7684\u5171\u540c\u4f5c\u7528\u4e0b\u7684\u5b8f\u89c2\u8868\u73b0\uff0c\u6545\u901a\u5e38\u5047\u8bbe\u5176\u6ee1\u8db3\u9ad8\u65af\u5206\u5e03\u3002<\/p>\n\n\n\n

\u90a3\u8fd9\u4e0e VAE \u6709\u4ec0\u4e48\u5173\u7cfb\u5462\uff1f\u5728\u7ecf\u5178\u7684 Guassian VAE \u4e2d\uff0c\u4e00\u4e2a\u5173\u952e\u5047\u8bbe\u662f\uff1a\u201c\u6f5c\u7a7a\u95f4\u201d \\(z \\) \u7b26\u5408\u9ad8\u65af\u5206\u5e03\u7684\uff08\u901a\u8fc7\u8bad\u7ec3\u6765\u5c06\u5176\u62c9\u5411\u9ad8\u65af\u5206\u5e03\uff09\u3002\u7136\u540e\u518d\u5bf9\u6f5c\u7a7a\u95f4\u91c7\u6837\u540e\uff0c\u5c31\u53ef\u4ee5\u751f\u6210\u8f83\u4e3a\u201c\u903c\u771f\u201d\u7684\u56fe\u50cf\u4e86\u3002\u90a3\u4e48\u5047\u8bbe\u201c\u6f5c\u7a7a\u95f4\u201d \\(z \\) \u7b26\u5408\u9ad8\u65af\u5206\u5e03\u6709\u4ec0\u4e48\u6df1\u610f\u5417\uff1f\u8fd8\u662f\u53ea\u662f\u4e3a\u4e86\u65b9\u4fbf\u91c7\u6837\uff1f<\/p>\n\n\n\n

\u4ece\u76f4\u89c9\u6784\u5efa\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u8ba4\u4e3a\uff1a\u6f5c\u7a7a\u95f4\u7684\u6bcf\u4e2a\u201c\u7ef4\u5ea6\u201d\u90fd\u6709\u67d0\u8fd9\u975e\u5e38\u5f3a\u7684\u610f\u4e49\uff0c\u4f8b\u5982\u8272\u5f69\u3001\u98ce\u683c\u3001\u7c7b\u578b\u7b49\uff0c\u90a3\u4e48\u5219\u6709\u7406\u7531\u76f8\u4fe1\uff0c\u4e00\u4e2a\u5bf9\u8c61\u5728\u6bcf\u4e00\u4e2a\u201c\u7ef4\u5ea6\u201d\u4e0a\u7684\u5206\u5e03\u4e5f\u662f\u7b26\u5408\u6b63\u6001\u5206\u5e03\u7684\u3002\u5373\u201c\u6f5c\u7a7a\u95f4\u201d\u662f\u4e00\u4e2a\u7b26\u5408\u6b63\u6001\u5206\u5e03\u7684\u7a7a\u95f4\uff0c\u90a3\u4e48\u4ece\u8fd9\u4e2a\u6837\u672c\u7a7a\u95f4\u8fdb\u884c\u91c7\u6837\u65f6\uff0c\u4e5f\u5c31\u66f4\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u6709\u4ee3\u8868\u610f\u4e49\u7684\u70b9\u3002<\/p>\n\n\n\n

2.2 VAE \u7684\u4e3b\u8981\u601d\u60f3<\/h4>\n\n\n\n
\"\"<\/figure>\n\n\n\n

\u5982\u4f55\u4f7f\u7528\u201c\u795e\u7ecf\u7f51\u7edc\u201d\u751f\u6210\u201c\u903c\u771f\u201d\u7684\u6570\u636e\uff08\u4f8b\u5982\u56fe\u7247\u3001\u89c6\u9891\u7b49\uff09\u5462\uff1f\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\uff0c\u4e00\u4e2a\u6bd4\u8f83\u81ea\u7136\u7684\u63a2\u7d22\u662f\u4ece\u201cautoencoder\u201d\u67b6\u6784\u53bb\u505a\u4e00\u4e9b\u5c1d\u8bd5\uff1a\u201cautoencoder\u201d \u7531\u4e00\u4e2a\u786e\u5b9a\u7684encoder \u5c06\u8f93\u5165\u538b\u7f29\u5230\u4e00\u4e2a\u4f4e\u7ef4\u7684\u201c\u6f5c\u7f16\u7801\u201d\uff08latent code\uff09\u7a7a\u95f4\u4e2d\uff0c\u7136\u540e\u518d\u7531decoder\u6839\u636e\u201clatent code\u201d\u91cd\u5efa\u6570\u636e\u3002<\/p>\n\n\n\n

\u5982\u53f3\u56fe\u6240\u793a\uff0c\u5bf9\u4e8e\u8fd9\u6837\u7684\u795e\u7ecf\u7f51\u7edc\u8bbe\u8ba1\u5176\u4f18\u52bf\u662f\u975e\u5e38\u660e\u663e\u7684\uff0c\u8fd9\u53ef\u4ee5\u662f\u975e\u5e38\u597d\u7684\u201c\u65e0\u76d1\u7763\u5b66\u4e60\u201d\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u5bf9\u6bd4\u6700\u7ec8\u8f93\u51fa\u548c\u8f93\u5165\u7684\u6570\u636e\u5373\u53ef\u4ee5\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\u3002<\/p>\n\n\n\n

\u4f46\uff0c\u5728\u6570\u636e\u751f\u6210\uff08\u4f8b\u5982\u65b0\u7684\u56fe\u50cf\u751f\u6210\uff09\u4e0a\uff0c\u8fd9\u6837\u7684\u8bbe\u8ba1\u5728\u5b9e\u8df5\u4e2d\u4e5f\u6709\u7740\u975e\u5e38\u660e\u663e\u7684\u9650\u5236\uff1a\u5373\u5bf9\u4e8e\u5982\u679c\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u201clatent code\u201d\uff0c\u901a\u5e38\u53ea\u80fd\u751f\u6210\u4e00\u4e9b\u65e0\u610f\u4e49\u7684\u6570\u636e\u3002<\/p>\n\n\n\n

2013\u5e74\u7684 \u201cVariational Autoencoder (VAE)\u201d (Kingma and Welling, 2013) \u67b6\u6784\u5219\u5c1d\u8bd5\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff1a\u5c06\u201c\u6f5c\u7a7a\u95f4\u201d\uff08latent space\uff09\u8bbe\u8ba1\u4e3a\u67d0\u4e2a\u7b26\u5408\u67d0\u4e2a\u6982\u7387\u5206\u5e03\u7ed3\u6784\u3002\u7ec8\u4e8e\uff0c\u53ef\u4ee5\u751f\u6210\u51fa\u8272\u7684\u3001\u903c\u771f\u7684\u6570\u636e\uff08\u4f8b\u5982\u201c\u56fe\u50cf\u201d\uff09\u3002<\/p>\n\n\n\n

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

\u4e0a\u8ff0\u7684\u63cf\u8ff0\u53ef\u80fd\u8fd8\u662f\u6bd4\u8f83\u62bd\u8c61\u7684\uff0c\u4e0b\u9762\u5c06\u5b9e\u73b0\u4e00\u4e2a\u57fa\u7840\u7684\u201cVAE\u201d\u6a21\u578b\uff0c\u4ece\u800c\u89c2\u6d4b\u8be5\u6a21\u578b\u7684\u5404\u4e2a\u6a21\u5757\u3002<\/p>\n\n\n\n

3. \u4e00\u4e2a\u7b80\u5355\u7684 VAE \u6a21\u578b\u7684\u89c2\u6d4b<\/h3>\n\n\n\n

3.1 \u6a21\u578b\u67b6\u6784\u6982\u8ff0<\/h4>\n\n\n\n
\"\"<\/figure>\n\n\n\n

\u8fd9\u91cc\u5b9e\u73b0\u4e86\uff08\u4e3b\u8981\u9760Gemini\uff09\u4e00\u4e2a\u7531\u591a\u4e2a\u5168\u8fde\u63a5\u5c42\u6784\u6210 VAE \u6a21\u578b\u3002\u8f93\u5165\u662f MNIST \u6570\u636e\u96c6\uff0c\u8f93\u5165\u5c42\u7ef4\u5ea6\u662f \\(28 \\times 28 = 784 \\)\uff0c\u4e2d\u95f4\u7684\u6f5c\u7a7a\u95f4\uff08Latent Space\uff09\u7ef4\u5ea6\u4e3a 20 \uff0c\u5305\u62ec 20 \u4e2a\u5747\u503c\u548c20\u4e2a\u65b9\u5dee\uff0c\u7136\u540e\u662f\u4e00\u4e2a\u57fa\u672c\u5bf9\u79f0\u7684decoder\uff0c\u5305\u62ec\u4e00\u4e2a400\u4e2a\u795e\u7ecf\u5143\u7684\u5168\u8fde\u63a5\u5c42\u548c\u4e00\u4e2a784\u4e2a\u795e\u7ecf\u5143\u7684\u8f93\u51fa\u5c42\uff0c\u624b\u7ed8\u63cf\u8ff0\u5982\u53f3\u56fe\u3002<\/p>\n\n\n\n

\u76f8\u5173\u7684\u4ee3\u7801\u5b9e\u73b0\u53c2\u8003\uff1aVAE-my-practice.ipynb<\/a>\u3002<\/p>\n\n\n\n

\u8fd9\u4e2a\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u4e4b\u7b80\u5355\u3001\u751f\u6210\u6548\u679c\u4e4b\u60ca\u4eba\uff0c\u6211\u81ea\u5df1\u662f\u6709\u70b9\u88ab\u9707\u60ca\u4e86\u7684\uff0c\u76f8\u6bd4\u4e8eAutoencoder \uff0c\u8fd9\u91cc\u5c06\u201c\u6f5c\u7a7a\u95f4\u201d\u9650\u5236\u5728\u4e86\u67d0\u4e2a\u6982\u7387\u5206\u5e03\u4e4b\u4e2d\uff0c\u66f4\u4e3a\u5177\u4f53\u7684\u662f\u5c06\u6bcf\u4e00\u4e2a\u6837\u672c\u90fd\u201c\u5c1d\u8bd5\u201d\u6620\u5c04\u5230\u6b63\u6001\u5206\u5e03\u4e2d\uff08\u6ce8\uff1a\\(p(z|x) \\)\u662f\u6b63\u6001\u5206\u5e03\uff09\u3002\u8fd9\u91cc\u7684\u201c\u5c1d\u8bd5\u201d\u7684\u505a\u6cd5\u662f\uff0c\u5148\u8ba9\u795e\u7ecf\u7f51\u7edc\u8f93\u51fa\u201c\u5747\u503c\u201d\u548c\u201c\u65b9\u5dee\u201d\uff0c\u7136\u540e\u518d\u505a\u4e00\u6b21\u91cd\u53c2\u6570\u5316\u7684\u201c\u91c7\u6837\u201d\uff08Sampling\uff09\u5f97\u5230 \\(z \\)\u3002<\/p>\n\n\n\n

\u8bf4\u660e\uff1a\\(p(z|x) \\)\u662f\u6b63\u6001\u5206\u5e03\uff1b\u4ece\u6a21\u578b\u67b6\u6784\u4e0a\u770b\uff0c\\(p(z) \\) \u4f3c\u4e4e\u5e76\u4e0d\u662f\u6b63\u6001\u5206\u5e03\uff0c\u4f46\u6700\u7ec8\u7531\u8bad\u7ec3\u7684\u76ee\u6807\u51fd\u6570\u5c06\\(p(z) \\) \u201c\u62c9\u5411\u201d\u4e00\u4e2a\u6b63\u6001\u5206\u5e03\u3002<\/p>\n\n\n\n

3.2 \u6a21\u578b\u6548\u679c\u89c2\u6d4b\uff1a\u67d0\u4e2a\u70b9\u9644\u8fd1\u7684\u6570\u636e<\/h4>\n\n\n\n
\"\"<\/figure>\n\n\n\n

\u8fd9\u91cc\u4f7f\u7528\u4e0a\u8ff0\u6a21\u578b\uff0c\u5c06 MNIST \u6570\u636e\u96c6\u4e2d\u7684\u4e00\u5f20\u539f\u59cb\u56fe\u7247\u8ba1\u7b97\u51fa\u5bf9\u5e94\u7684\u6b63\u6001\u5206\u5e03\uff0c\u5373\u5747\u503c\u548c\u65b9\u5dee\uff0c\u7136\u540e\u4f7f\u7528\u8be5\u5747\u503c\u548c\u65b9\u5dee\u8fdb\u884c\u4e86 11 \u4e00\u6b21\u91c7\u6837\uff0c\u518d\u751f\u6210 11 \u5f20\u56fe\u7247\uff0c\u53ef\u4ee5\u770b\u5230\u8fd9\u4e9b\u751f\u6210\u7684\u56fe\u7247\u548c\u539f\u59cb\u56fe\u7247\u6709\u975e\u5e38\u9ad8\u7684\u76f8\u4f3c\u5ea6\uff0c\u540c\u65f6\u53c8\u4e0e\u539f\u59cb\u56fe\u7247\u4e0d\u540c\u3002<\/p>\n\n\n\n

\u5373\u8fbe\u5230\u4e86\u751f\u6210\u201c\u903c\u771f\u201d\u56fe\u7247\u7684\u6548\u679c\u3002\u8fd9\u79cd\u5b9e\u73b0\u8303\u5f0f\uff0c\u5c55\u73b0\u51fa\u4e86\u975e\u5e38\u5f3a\u7684\u56fe\u7247\u751f\u6210\u6f5c\u529b\u3002\u4e5f\u662f\u540e\u7eed diffusion \u6a21\u578b\u7684\u91cd\u8981\u57fa\u7840\u4e4b\u4e00\u3002<\/p>\n\n\n\n

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

\u518d\u4ece \\(z \\sim \\mathcal{N}(0,I) \\) \u968f\u673a\u91c7\u6837\u4e00\u4e9b\u503c\uff0c\u518d\u770b\u770b\u8fd9\u4e9b\u968f\u673a\u751f\u6210\u7684\u6548\u679c\uff0c\u5982\u53f3\u56fe\u3002<\/p>\n\n\n\n

\u53ef\u4ee5\u770b\u5230\uff0c\u57fa\u672c\u4e0a\u80fd\u591f\u751f\u6210\u4e00\u4e9b\u6709\u610f\u4e49\u7684\u56fe\u3002\u8fd9\u91cc\u53ea\u662f\u4f7f\u7528\u4e86\u4e00\u4e9b\u5168\u8fde\u63a5\u5c42\uff0c\u5373\u53ef\u4ee5\u4ea7\u751f\u975e\u5e38\u597d\u7684\u6548\u679c\uff0c\u53ef\u4ee5\u9884\u671f\uff0c\u5982\u679c\u4f7f\u7528\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff0c\u662f\u53ef\u4ee5\u6709\u66f4\u597d\u7684\u6548\u679c\u7684\u3002<\/p>\n\n\n\n

3.3 \u6a21\u578b\u6548\u679c\u89c2\u6d4b\uff1a\u7a7a\u95f4\u4e2d\u4ea4\u63a5\u5904\u7684\u60c5\u51b5<\/h4>\n\n\n\n

\u4ece\u6784\u5efa\u76f4\u89c9\u7684\u89d2\u5ea6\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u7406\u89e3\uff0cencoder \u5c42\u5c06\u539f\u59cb\u7684\u201c\u5411\u91cf\u7a7a\u95f4\u201d\u6620\u5c04\u5230\u4e86\u4e00\u4e2a\u201c\u6f5c\u7a7a\u95f4\u201d\uff08latent space\uff09\uff0c\u8be5\u6f5c\u7a7a\u95f4\u4e2d\u7684\u6570\u636e\u7b26\u5408\u6807\u51c6\u6b63\u6001\u5206\u5e03\u3002\u6240\u4ee5\uff0c\u8fd9\u5c31\u4f7f\u5f97\u6211\u4eec\u5728\u201c\u6f5c\u7a7a\u95f4\u201d\u53d8\u91cf \\(z \\) \u4e2d\u505a\u4e00\u4e2a\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u91c7\u6837\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 decoder \u751f\u6210\u4e00\u4e2a\u6709\u610f\u4e49\u7684\u56fe\u7247\u3002\u5e76\u4e14\uff0c\u4ece\u6784\u5efa\u76f4\u89c9\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3adecoder \u5c06\u6b63\u6001\u5206\u5e03\u4e2d\u7684\u70b9\u6620\u5c04\u5230\u201c10\u4e2a\u201d\u6570\u5b57\u7684\u201c\u7a7a\u95f4\u201d\uff08784\u7ef4\uff09\u4e2d\uff0c\u751a\u81f3\u53ef\u4ee5\u8ba4\u4e3a\uff08\u76f4\u89c9\u89d2\u5ea6\uff09\u76ee\u6807\u7a7a\u95f4\u4e2d\u6709\u4e00\u7247\u201c\u8fde\u7eed\u201d\u7684\u7a7a\u95f4\u5373\u4e3a\u201c\u50cf\u7a7a\u95f4\u201d\uff0c\u8fd9\u4e9b\u7a7a\u95f4\u67d0\u79cd\u7a0b\u5ea6\u53ef\u4ee5\u201c\u805a\u7c7b\u201d\u4e3a\u201c10\u7c7b\u201d\uff0c\u8fd9\u4e9b\u805a\u7c7b\u7684\u8fb9\u7f18\u5219\u662f\u4e00\u4e9b\u4ecb\u4e8e\u4e0d\u540c\u6570\u5b57\u4e4b\u95f4\u7684\u96be\u4ee5\u8fa8\u8ba4\u7684\u6570\u5b57\u3002<\/p>\n\n\n\n

\u4f8b\u5982\uff0c\u6211\u4eec\u8003\u8651\u8fd9\u6837\u89c2\u6d4b\u8fd9\u4e9b\u201c\u805a\u7c7b\u201d\u4e2d\u6570\u5b57\u201c1\u201d\u548c\u6570\u5b57\u201c0\u201d\u4e4b\u95f4\u7684\u56fe\u50cf\u3002\u6211\u4eec\u5148\u4f7f\u7528encoder\u8ba1\u7b97\uff08\u627e\u4e00\u5f20\u56fe\u7247\u4f5c\u4e3a\u8f93\u5165\uff09\u51fa\u6570\u5b57\u201c1\u201d\u3001\u201c0\u201d\u5bf9\u5e94\u7684\u5747\u503c\uff0c\u5e76\u4ee5\u6b64\u4f5c\u4e3a\u5bf9\u5e94\u6570\u5b57\u5728\u201c\u6f5c\u7a7a\u95f4\u201d\u4e2d\u7684\u4e2d\u5fc3\uff08\u66f4\u4e3a\u4e25\u683c\u7684\u53ef\u4ee5\u591a\u8ba1\u7b97\u4e00\u4e9b\u76f8\u540c\u7684\u6570\u5b57\uff0c\u518d\u8ba1\u7b97\u5747\u503c\uff09\uff0c\u7136\u540e\u5728\u201c1\u201d\u3001\u201c1\u201d\u5bf9\u5e94\u7684\u8d28\u5fc3\u4e4b\u95f4\u8fdb\u884c\u63d2\u503c\uff0c\u7136\u540e\u89c2\u6d4b\u8fd9\u4e9b\u201c\u63d2\u503c\u201d\u7ecf\u8fc7encoder\u540e\u7684\u56fe\u7247\u3002<\/p>\n\n\n\n

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

\u5728\u653e\u592733% ~ 66% \u8fd9\u4e2a\u5173\u952e\u9636\u6bb5\uff1a<\/p>\n\n\n\n

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

3.4 \u6a21\u578b\u5c0f\u7ed3<\/h4>\n\n\n\n

\u5982\u4e0a\u6240\u8ff0\uff0cencoder \u6700\u7ec8\u628a\u201c\u539f\u59cb\u7684\u6570\u636e\u201d\u6620\u5c04\u5230\u4e00\u4e2a\u201c\u6f5c\u7a7a\u95f4\u201d\uff08latent space\uff09\uff0c\u201c\u6f5c\u7a7a\u95f4\u201d\u4e2d\u7684\u6570\u636e\uff08\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d\u7c7b\u4f3c\u7684\u6982\u5ff5\u79f0\u4e3a\u201c\u50cf\u7a7a\u95f4\u201d\uff09\u7b26\u5408\u6b63\u6001\u5206\u5e03\u3002\u4ece\u4ece\u6784\u5efa\u76f4\u89c9\u89d2\u5ea6\u6211\u4eec\u53ef\u4ee5\u6709\u5982\u4e0b\u7684\u4e00\u4e9b\u7406\u89e3\u548c\u7591\u95ee\u3002<\/p>\n\n\n\n

VAE \u6a21\u578b\u5c1d\u8bd5\u628a\u201c\u6f5c\u7a7a\u95f4\u201d\u9650\u5236\u5728\u4e00\u4e2a\u975e\u5e38\u5c0f\u533a\u57df\uff0c\u5373\u4ee5\u201c\u539f\u70b9\u201d\u4e3a\u4e2d\u5fc3\u7684\u4e00\u4e2a\u5c0f\u201c\u7403\u201d\u4e2d\uff08\u60f3\u60f3 3 \\(\\sigma \\) \u5185\u5c31\u53ef\u80fd\u8986\u76d6\u4e86 99.73% \u7684\u70b9\uff09\uff0c\u66f4\u4e3a\u51c6\u786e\u7684\u662f\u4e00\u7247\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u4e91\u5f53\u4e2d\u3002\u95ee\u9898\uff1a\u8fd9\u79cd\u628a\u201c\u50cf\u7a7a\u95f4\u201d\u9650\u5236\u5728\u5c0f\u533a\u57df\u5185\uff0c\u662f\u5426\u662f\u5b9e\u73b0\u201c\u903c\u771f\u751f\u6210\u201d\u7684\u5173\u952e\u56e0\u7d20\uff1f\u5c1d\u8bd5\u56de\u7b54\uff1a\u50cf\u7a7a\u95f4\u7684\u5927\u5c0f\u4e5f\u8bb8\u5e76\u4e0d\u662f\u5173\u952e\u7684\uff0c\u6bd5\u7adf\uff0c\u4ece\u6570\u5b66\u89d2\u5ea6\u6765\u770b\uff0c\u5355\u4f4d\u201c\u7403\u201d\u53ef\u80fd\u4e0e\u6574\u4e2a\u7a7a\u95f4\u662f\u4e00\u6837\u5927\u7684\uff0c\u53ea\u662f\u770b\u8d77\u6765\u5c0f\u4e86\u800c\u5df2\uff1b\u90a3\u4e48\uff0c\u5173\u952e\u53ef\u80fd\u5728\u4e8e\u628a\u201c\u50cf\u7a7a\u95f4\u201d\u73b0\u5728\u5728\u4e86\u67d0\u79cd\u7ed3\u6784\u4e0a\uff0c\u8fd9\u91cc\u7684\u7ed3\u6784\u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7a7a\u95f4\u3002\u95ee\u9898\uff1a\u4e3a\u4ec0\u4e48\u628a\u201c\u50cf\u7a7a\u95f4\u201d\u9650\u5236\u5728\u67d0\u79cd\u7ed3\u6784\u4e0a\uff0c\u5c31\u80fd\u591f\u8fbe\u5230\u8fd9\u6837\u7684\u6548\u679c\u5462\uff1f<\/p>\n\n\n\n

\u5728\u4e0a\u8ff0\u793a\u4f8b\u7684\u201c\u6982\u7387\u4e91\u201d\u4e2d\uff0c\u6240\u6709\u7684\u70b9\u5e76\u4e0d\u662f\u201c\u5bf9\u7b49\u7684\u201d\uff0c\u6982\u7387\u5bc6\u5ea6\u76f8\u7b49\u7684\u70b9\uff0c\u5176\u610f\u4e49\u4e5f\u6709\u7740\u975e\u5e38\u5927\u7684\u533a\u522b\u3002\u4ece\u76f4\u89c9\u7684\u89d2\u5ea6\uff0c\u4f8b\u5982\u6700\u67d0\u4e2a\u6570\u5b571\u7684\u70b9 \\(z_1 \\in z \\)\uff0c\u5728\u70b9 \\(z_1 \\) \u9644\u8fd1\u7684\u70b9\u5219\u6620\u5c04\u5230\u7684\u201c\u50cf\u201d\u4e5f\u662f\u66f4\u4e3a\u63a5\u8fd1\u7684\u4e0e\u6570\u5b57\u201c1\u201d\u7684\u70b9\uff08\u8fd9\u91cc\u7684 \\(z_1 = q_{\\theta}(x) \\quad \\text{where} \\, x \\in \\text{Samples} \\)\uff09\u3002<\/p>\n\n\n\n

4. VAE \u5b9e\u73b0\u7684\u4e00\u4e9b\u7ec6\u8282<\/h3>\n\n\n\n

4.1 VAE\u7684\u8bad\u7ec3\u76ee\u6807<\/h4>\n\n\n\n

VAE \u6a21\u578b\u5c06\u6f5c\u7a7a\u95f4\u9650\u5236\u5728\u4e86\u4e00\u4e2a\u201c\u6b63\u6001\u5206\u5e03\u201d\u4e4b\u4e2d\uff0c\u901a\u8fc7\u5728\u6b63\u6001\u5206\u5e03\u4e2d\u91c7\u6837\u7684\u65b9\u5f0f\u7528\u6765\u751f\u6210\u65b0\u7684\u56fe\u7247\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5e76\u4e0d\u662f\u5f88\u597d\u7406\u89e3\uff0c\u8fd9\u91cc\u4ecb\u7ecd\u4e00\u4e0b VAE \u7684\u6570\u5b66\u5efa\u6a21\u601d\u8def\u3002<\/p>\n\n\n\n

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

\u6a21\u578b\u7684\u8bad\u7ec3\u76ee\u6807\uff08ELBO\uff09\uff1a<\/p>\n\n\n

$$
\n\\min_{\\theta,\\phi} \\mathbb{E}_{q_{\\theta}(z|x)}[\\frac{1}{2\\sigma^2}|| x-\\mu_{\\phi}(z) ||^2] + \\mathcal{D}_{KL}(q_{\\theta}(z|x)||p(z) ) \\tag{1}
\n$$<\/p>\n\n\n\n

\u5173\u4e8e\u8fd9\u4e2a\u5f0f\u5b50\u7684\u5b8c\u6574\u7684\u6570\u5b66\u63a8\u5bfc\uff0c\u5728\u7f51\u4e0a\u6bd4\u8f83\u597d\u627e\u5230\uff0c\u6709\u4e00\u4e9b\u590d\u6742\u3001\u4e5f\u4e0d\u662f\u5f88\u597d\u7406\u89e3\u3002\u4f46\u662f\uff0c\u8fd9\u4e2a\u5f0f\u5b50\u7684\u610f\u4e49\u5374\u6bd4\u8f83\u660e\u786e\uff1a<\/p>\n\n\n\n

    \n
  • \u524d\u4e00\u90e8\u5206\uff0c\u5bf9\u4e8e\u67d0\u4e2a\u8f93\u5165 \\(x \\)\uff0c\u9700\u8981\u795e\u7ecf\u7f51\u7edc\u7684\u53c2\u6570 \\(\\theta , \\phi \\) \u751f\u6210\u7684\u56fe\u50cf\\(x’ \\)\u80fd\u591f\u4e0e\u539f\u59cb\u56fe\u50cf\u63a5\u8fd1<\/li>\n\n\n\n
  • \u540e\u534a\u90e8\u5206\uff0c\u5219\u8ba9\u795e\u7ecf\u7f51\u7edc\u53c2\u6570\\(\\theta \\) \u662f\u7684\u5206\u5e03 \\(q_{\\theta}(z|x) \\) \u9760\u8fd1 \\(p(z) \\)\u5373\u6807\u51c6\u6b63\u6001\u5206\u5e03<\/li>\n<\/ul>\n\n\n\n

    4.2 VAE\u7684\u6570\u5b66\u6a21\u578b<\/h4>\n\n\n\n

    \u7406\u89e3 VAE \u6bd4\u8f83\u56f0\u96be\u7684\u5927\u6982\u662f\u8fd9\u80cc\u540e\u7684\u6570\u5b66\u6a21\u578b\uff0c\u800c\u7406\u89e3\u4e86\u8fd9\u4e2a\u201c\u6570\u5b66\u6a21\u578b\u201d\u624d\u53ef\u80fd\u8fdb\u4e00\u6b65\u7406\u89e3\u4e0a\u8ff0\u7684\u516c\u5f0f\uff0c\u4ee5\u53ca\u66f4\u591a\u7684VAE\u4ee5\u53ca\u540e\u7eed\u751f\u6210\u6a21\u578b\u7684\u601d\u60f3\u3002\u6240\u4ee5\u8fd9\u91cc\u603b\u7ed3\u4e00\u4e9b\u6211\u5bf9\u8fd9\u4e2a\u201c\u6570\u5b66\u6a21\u578b\u201d\u7684\u7406\u89e3\uff1a<\/p>\n\n\n\n

    Encoder \u90e8\u5206 \\(q_{\\theta} \\) <\/mark>\u662f\u5c06\u4e00\u4e2a\u56fe\u7247 \\(x \\) \u6620\u5c04\u5230\u4e00\u4e2a\u6982\u7387\u5206\u5e03\uff08\u800c\u4e0d\u662f\u67d0\u4e2a\u5177\u4f53\u7684\u503c\uff09\uff0c\u901a\u5e38\u662f \\(\\mathcal{N}(\\mu,\\sigma^2) \\)\u3002<\/mark>\u90a3\u4e48\uff0c\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u5982\u4f55\u5c06\u4e00\u4e2a\u5177\u4f53\u7684\u6837\u672c\/\u6570\u636e \\(x \\) \u6620\u5c04\u5230\u4e00\u4e2a\u6982\u7387\u5206\u5e03\u5462\uff1f\u662f\u8fd9\u6837\u7684\uff1a\u795e\u7ecf\u7f51\u7edc\u5219\u9700\u8981\u8f93\u51fa\/\u62df\u5408\/\u56de\u5f52\u51fa\u8be5\u5206\u5e03\u7684\u5173\u952e\u53c2\u6570\u5373\u53ef\uff0c\u4f8b\u5982\u5bf9\u4e8e\u6b63\u6001\u5206\u5e03\u53ea\u9700\u8981\u7ed9\u5747\u503c\u548c\u65b9\u5dee\u5373\u53ef\u3002\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u5728\u4e0a\u8ff0\u7684\u201c\u793a\u4f8bVAE\u201d\u4e2d Encoder \u90e8\u5206\u8f93\u51fa\u5373\u4e3a\u5206\u5e03 \\(z \\sim p(z) \\) \u7684\u5747\u503c\u4e0e\u65b9\u5dee\u3002\u90a3\u4e48\uff0c\u8981\u4ece\u6f5c\u7a7a\u95f4 \\(z \\) \u4e2d\u53d6\u4e00\u4e2a\u503c\uff08\u5373\u91c7\u6837\u4e00\u4e2a\u503c\uff09\u7684\u65f6\u5019\uff0c\u5219\u9700\u8981\u201c\u968f\u673a\u201d\u7684\u505a\u4e00\u6b21\u751f\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u968f\u673a\u201d\u5219\u4f7f\u7528\u4e86\u201c\u91cd\u53c2\u6570\u5316\u201d\u6280\u5de7\u8fdb\u884c\u5904\u7406\u4ece\u800c\u89e3\u51b3\u8bad\u7ec3\u65f6\u5411\u540e\u4f20\u64ad\u7684\u968f\u673a\u6027\u95ee\u9898\u3002\u66f4\u4e3a\u4e00\u822c\u7684\u5373\uff1a$$ q_{\\theta}(z|x) := \\mathcal{N}(z;\\mu_{\\theta}(x),\\text{diag}(\\sigma_{\\theta}^2(x))) \\tag{2} $$<\/p>\n\n\n\n

    Decoder \u90e8\u5206 \\(p_{\\phi} \\) \u5219\u662f\u5c06\u6f5c\u7a7a\u95f4\u7684\u91c7\u6837 \\(z \\) \u6620\u5c04\u5230\u4e00\u4e2a\u6982\u7387<\/mark>\u5206\u5e03\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u503c\uff0c\u4f46\u8fd9\u91cc\u53ef\u80fd\u611f\u89c9\u662f\u4e00\u4e2a\u5177\u4f53\u6307\u7684\u539f\u56e0\u662f\uff0c\u901a\u5e38\u8f93\u51fa\u5c31\u662f\u8fd9\u4e2a\u5206\u5e03\u7684\u5747\u503c\uff08\u6216\u671f\u671b\uff09\uff0c\u800c\u4e0d\u518d\u53e6\u5916\u8fdb\u884c\u91c7\u6837\u64cd\u4f5c\u3002<\/mark>\u5728\u5e38\u89c1\u7684 Gaussian VAE \u4e2d\uff0c\u66f4\u4e3a\u4e00\u822c\u7684\u6709\uff1a<\/p>\n\n\n\n

    $$ p_{\\phi}(x|z) := \\mathcal{N}(x;\\mu_{\\phi}(z),\\sigma^2 I) \\tag{3} $$<\/p>\n\n\n\n

    \u6700\u540e<\/mark>\uff0c\u6709\u4e86\u8fd9\u4e24\u4e2a\u5b9a\u4e49\u516c\u5f0f(2)\u3001(3)\uff0c\u5c31\u53ef\u4ee5\u8ba1\u7b97\u4e0a\u8ff0\u7684\u635f\u5931\u51fd\u6570\u516c\u5f0f(1)\u4e86\u3002<\/p>\n\n\n\n

    5. VAE \u4e2d\u7684\u6570\u5b66\u63a8\u5bfc<\/h3>\n\n\n\n

    \u8fd9\u5927\u6982\u662f\u6700\u96be\u7406\u89e3\u7684\u90e8\u5206\uff0c\u751a\u81f3\u6ce8\u610f\u5230\u5f88\u591a\u9886\u57df\u5185\u7684\u4e13\u5bb6\uff08\u66f4\u52a0\u6ce8\u91cd\u5b9e\u8df5\u80fd\u529b\uff09\uff0c\u5bf9\u4e8e\u8fd9\u90e8\u5206\u4e5f\u5f88\u5934\u75bc\uff0c\u8fd9\u91cc\u505a\u4e00\u4e9b\u5c1d\u8bd5\u3002<\/p>\n\n\n\n

    5.1 VAE \u57fa\u672c\u6570\u5b66\u6982\u7387\u6a21\u578b\u4e0e\u8bad\u7ec3\u76ee\u6807<\/h4>\n\n\n\n

    \u6211\u4eec\u6709\u4e00\u4e2a\u4e0a\u8ff0\u7684 VAE \u6a21\u578b\uff0c\u5176\u4e2d\uff1a<\/p>\n\n\n\n

      \n
    • \\(\\theta \\) \u4e3aencoder\u90e8\u5206\u53c2\u6570\uff0c\\(\\phi \\) \u4e3adecoder \u7684\u53c2\u6570<\/li>\n\n\n\n
    • \\(p_{\\theta}(z|x) \\) \u8868\u793a\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u8f93\u5165 \\(x \\)\uff0cencoder \u5c06\u8f93\u5165\u6620\u5c04\u4e3a\u5206\u5e03\\(p_{\\theta}(z|x) \\) \uff0c\u901a\u5e38\u6309\u7167\u4e0a\u8ff0\u7684\u516c\u5f0f(2)\u5b9a\u4e49<\/li>\n\n\n\n
    • \\(p_{\\phi}(x|z) \\) \u8868\u793a\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u6f5c\u7a7a\u95f4\u91c7\u6837\u503c \\(z \\)\uff0cdecoder \u5c06\u5176\u6620\u5c04\u4e3a\u5206\u5e03\\(p_{\\phi}(x|z) \\) \uff0c\u901a\u5e38\u6309\u7167\u4e0a\u8ff0\u7684\u516c\u5f0f(3)\u5b9a\u4e49<\/li>\n<\/ul>\n\n\n\n

      \u5728\u6570\u5b66\u5f62\u5f0f\u4e0a\uff0c\u5c31\u53ef\u4ee5\u6709\u76ee\u6807\u7aef\uff08\u795e\u7ecf\u7f51\u7edc\u7684\u8f93\u51fa\uff09\\(x \\) \u7684\u5206\u5e03\uff1a<\/p>\n\n\n

      $$
      \np_{\\phi}(x) = \\int p_{\\phi}(x|z)p(z)dz
      \n$$<\/p>\n\n\n\n

      \u8fd9\u662f\u4e00\u4e2a\u5f62\u5f0f\u4e0a\u7684\u8868\u8fbe\uff0c\u5e76\u4e0d\u80fd\u76f4\u63a5\u8ba1\u7b97\uff0c\u8fd9\u662f\u56e0\u4e3a \\(p_{\\phi}(x|z) \\) \u662f\u4e00\u4e2a\u7531\u795e\u7ecf\u7f51\u7edc\u5b9a\u4e49\u7684\u6620\u5c04\u6216\u5206\u5e03\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a \\(z \\)\uff0c\u5e76\u6ca1\u6709\u4e00\u4e2a\u7b80\u5355\u7684\u8868\u8fbe\u5f0f\u7ed9\u51fa\\(p_{\\phi}(x|z) \\)\u7684\u5f62\u5f0f\uff0c\u6216\u8005\u66f4\u4e3a\u76f4\u63a5\u7684\uff0c\u5728\u516c\u5f0f(3)\u4e2d\u7684 \\(\\mu_{\\phi}(z) \\) \u7531\u795e\u7ecf\u7f51\u7edc\u5b9a\u4e49\uff0c\u5e76\u6ca1\u6709\u6570\u5b66\u5f62\u5f0f\u7684\u8868\u8fbe\u3002<\/p>\n\n\n\n

      \u8fd9\u4e2a\u5f0f\u5b50\u786e\u5b9e\u4e0d\u53ef\u8ba1\u7b97\uff0c\u5728\u5b9e\u9645\u8bad\u7ec3\u4e2d\uff0c\u5219\u505a\u4e86\u4e00\u4e9b\u6570\u5b66\u201c\u63a8\u5bfc\u201d\uff0c\u8bad\u7ec3\u53e6\u4e00\u4e2a\u4e0e\u6b64\u76f8\u5173\u7684\u5f0f\u5b50\uff08ELBO\uff09\uff0c\u8fd9\u4e2a\u6211\u4eec\u540e\u7eed\u518d\u770b\u3002<\/p>\n\n\n\n

      \u6240\u4ee5\uff0c\u5bf9\u4e8e VAE \u6a21\u578b\u7684\u8bad\u7ec3\uff0c\u5373\u9700\u8981\u6700\u5927\u5316\uff1a<\/p>\n\n\n

      $$
      \n\\max_{\\phi}\\sum_\\limits{i} \\log p_{\\phi}(x_i) \\quad \\forall x_i \\in \\text{Sample Data} \\tag{4}
      \n$$<\/p>\n\n\n\n

      \u5373\u5bf9\u4e8e\u6700\u7ec8\u7684\u6982\u7387\u5206\u5e03\uff08PDF\uff09\\(p_{\\phi}(x) \\)\uff0c\u4efb\u4f55\u7684\u771f\u5b9e\u7684\u56fe\u50cf\uff08\u5bf9\u5e94\u7684 \\(\\forall x \\in {\\text{data}} \\)\uff09\u6240\u5bf9\u5e94\u7684\u5bc6\u5ea6\u503c\u53d6\u503c\u90fd\u9700\u8981\u5f88\u5927\uff0c\u624d\u80fd\u6700\u7ec8\u4f7f\u5f97\u4e0a\u8ff0\u5f0f\u5b50\u53d6\u503c\u6700\u5927\u3002<\/p>\n\n\n\n

      \u8fd9\u4f3c\u4e4e\u8fd8\u6bd4\u8f83\u62bd\u8c61\uff0c\u8fd9\u91cc\u6211\u4eec\u8fdb\u4e00\u6b65\u89e3\u91ca\u4e00\u4e0b\u4e0a\u8ff0\u7684\u5f0f\u5b50(4)\u3002\u8003\u8651\u771f\u5b9e\u5206\u5e03\u4e3a \\(p_{\\text{data}}(x) \\)\uff0c\u901a\u5e38\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 KL \u6563\u5ea6\u8861\u91cf\u4e24\u4e2a\u5206\u5e03\u7684\u8ddd\u79bb\uff0c\u90a3\u4e48\uff0c\u5c31\u9700\u8981\u6700\u5c0f\u5316\u8fd9\u4e2a\u6563\u5ea6\u503c\uff1a<\/p>\n\n\n

      $$
      \n\\begin{aligned}
      \n\\mathcal{D}_{\\text{KL}} &= \\int p_{\\text{data}}(x) \\log \\frac{p_{\\text{data}}(x)}{ p_{\\phi}(x)}dx \\\\[0.8em]
      \n&= \\int p_{\\text{data}}(x) \\log p_{\\text{data}}(x)dx – \\int p_{\\text{data}}(x) \\log p_{\\phi}(x)dx
      \n\\end{aligned}
      \n$$<\/p>\n\n\n\n

      \u4e0a\u8ff0\u5f0f\u5b50\u4e2d\u7684\u524d\u9879\u662f\u4e00\u4e2a\u5e38\u6570\uff08\u8ba4\u4e3a\u771f\u5b9e\u4e16\u754c\u7684\u5206\u5e03\u662f\u786e\u5b9a\u7684\uff0c\u662f\u62df\u5408\u7684\u76ee\u6807\uff0c\u5176\u5b9e\u5c31\u662f\u8fd9\u4e2a\u5206\u5e03\u7684\u201c\u71b5\u201d\uff09\uff0c\u90a3\u4e48\u8981\u8fd9\u4e2a\u6563\u5ea6\u503c\u6700\u5c0f\uff0c\u5c31\u662f\u8ba9\u4e0a\u8ff0\u5f0f\u5b50\u4e2d\u540e\u4e00\u9879\u6700\u5927\uff0c\u5373 \\(\\int p_{\\text{data}}(x)p_{\\phi}(x)dx \\) \u6700\u5927\uff1b\u5728\u79bb\u6563\u573a\u666f\uff0c\u79ef\u5206\u5219\u53d8\u6210\u6c42\u548c\uff0c\u5e76\u4e14 \\(p_{\\text{data}}(x) \\)\u901a\u5e38\u4e0d\u53ef\u77e5\uff0c\u5219\u901a\u5e38\u4f7f\u7528\\(\\frac{1}{N} \\)\u66ff\u4ee3\u6216\u8005\u76f4\u63a5\u5ffd\u7565\uff0c\u5373\u6709\u4e0a\u9762\u7684\u5f0f\u5b50(4)\u3002<\/p>\n\n\n\n

      5.2 \u4f7f\u7528 ELBO \u5bf9\u4e0a\u8ff0\u76ee\u6807\u8fdb\u884c\u8fd1\u4f3c<\/h4>\n\n\n\n

      \u5173\u4e8e\u201cEvidence Lower Bound\u201d \uff08\u7b80\u79f0 ELBO \uff09\u867d\u7136\u7f51\u4e0a\u6709\u5f88\u591a\u7684\u8bf4\u660e\uff0c\u4f46\u603b\u662f\u4e0d\u591f\u7ec6\u81f4\uff08\u6216\u8005\u8981\u6c42\u5bf9\u6982\u7387\u8bba\u6709\u975e\u5e38\u6df1\u7684\u7406\u89e3\uff09\uff0c\u8fd9\u91cc\u505a\u4e00\u4e2a\u6700\u4e3a\u8be6\u7ec6\u7684\u63a8\u5bfc\u548c\u8bf4\u660e\u5982\u4e0b\uff08\u5e94\u8be5\u662f\u4e92\u8054\u7f51\u4e0a\u80fd\u591f\u627e\u5230\u7684\u6700\u4e3a\u8be6\u7ec6\u7684\u8bf4\u660e\u4e86\uff09\u3002<\/p>\n\n\n\n

      ELBO\u7684\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n

      $$
      \n\\mathcal{L}_{\\text{ELBO}}
      \n=
      \n\\underbrace{
      \n\\mathbb{E}_{z \\sim q_\\theta(z|x)}
      \n\\left[ \\log p_\\phi(x|z) \\right]
      \n}_{\\text{Reconstruction Term}}
      \n–
      \n\\underbrace{
      \nD_{\\mathrm{KL}}\\!\\left( q_\\theta(z|x) \\,\\|\\, p(z) \\right)
      \n}_{\\text{Latent Regularization}}
      \n$$<\/p>\n\n\n\n

      \u53ef\u4ee5\u901a\u8fc7\u4e25\u683c\u7684\u8bc1\u660e\uff0c\u5982\u4e0b\u4e0d\u7b49\u5f0f\u6210\u7acb\uff1a<\/p>\n\n\n

      $$
      \n\\begin{equation}
      \n\\log p_\\phi(x) \\ge \\mathcal{L}_{\\text{ELBO}}(\\theta, \\phi; x)
      \n\\end{equation}
      \n$$<\/p>\n\n\n\n

      \u8bc1\u660e\u5982\u4e0b\uff1a<\/p>\n\n\n

      $$
      \n\\begin{aligned}
      \n\\log p_\\phi(x) &\\stackrel{(1)}{=} \\log \\int p_\\phi(x, z) \\, dz \\\\[0.8em]
      \n&\\stackrel{(2)}{=} \\log \\int q_\\theta(z|x) \\frac{p_\\phi(x, z)}{q_\\theta(z|x)} \\, dz \\\\[0.8em]
      \n&\\stackrel{(3)}{=} \\log \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\frac{p_\\phi(x, z)}{q_\\theta(z|x)} \\right] \\\\[0.8em]
      \n&\\stackrel{(4)}{\\geq} \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\log \\frac{p_\\phi(x, z)}{q_\\theta(z|x)} \\right] \\\\[0.8em]
      \n&\\stackrel{(5)}{=} \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\log \\frac{p_\\phi(x, z) p(z)}{q_\\theta(z|x)p(z)} \\right] \\\\[0.8em]
      \n&\\stackrel{(6)}{=} \\mathbb{E}_{z \\sim q_\\theta(z|x) p(z)} \\left[ \\log \\frac{p_{\\phi}(x|z)p(z)}{q_\\theta(z|x)} \\right] \\\\[0.8em]
      \n&\\stackrel{(7)}{=} \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\log {p_{\\phi}(x|z)} – \\log { \\frac{q_\\theta(z|x)}{p(z)} } \\right] \\\\[0.8em]
      \n&\\stackrel{(8)}{=} \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\log {p_{\\phi}(x|z)} \\right] – \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\log { \\frac{q_\\theta(z|x)}{p(z)} } \\right] \\\\[0.8em]
      \n&\\stackrel{(9)}{=} \\mathbb{E}_{z \\sim q_\\theta(z|x)} \\left[ \\log {p_{\\phi}(x|z)}] – \\mathcal{D}_{\\mathrm{KL}}(q_\\theta(z|x)||p(z)) \\right] \\\\[0.8em]
      \n\\end{aligned}
      \n$$<\/p>\n\n\n\n

      \u5404\u6b65\u9aa4\u7684\u8bf4\u660e\uff1a<\/p>\n\n\n\n

        \n
      • \u6b65\u9aa4 (1) \u4e3a\u8fb9\u7f18\u6982\u7387\u516c\u5f0f<\/li>\n\n\n\n
      • \u6b65\u9aa4(2) \u5206\u5b50\u3001\u5206\u6bcd\u90fd\u4e58\u4ee5\u4e86 \\(q_\\theta(z|x) \\)<\/li>\n\n\n\n
      • \u6b65\u9aa4(3) \u4f7f\u7528\u671f\u671b\u7684\u5f62\u5f0f\u6539\u5199\u4e0a\u8ff0\u516c\u5f0f\uff08\u53ef\u4ee5\u60f3\u4e00\u60f3\u8fde\u7eed\u5f62\u5f0f\u7684\u6982\u7387\u671f\u671b\u8ba1\u7b97\uff09\uff0c\u8fd9\u4e00\u6b65\u662f\u4e3a\u4e86\u8f6c\u5316\u4e3a\u4e0b\u4e00\u6b65\u53ef\u4ee5\u4f7f\u7528 \u201cJensen\u201d \u4e0d\u7b49\u5f0f<\/li>\n\n\n\n
      • \u6b65\u9aa4(4) \u8fd9\u91cc\u4f7f\u7528 \u201cJensen\u201d \u4e0d\u7b49\u5f0f\uff0c\u5bf9\u4e8e\u51f9\u51fd\u6570 \\(\\varphi \\)\u6709 \\(\\varphi(\\mathbb{E}[X]) \\geq \\mathbb{E}[\\varphi(X)] \\) \uff0c\u51fd\u6570 \\(f(x) = \\log (x) \\) \u5373\u4e3a\u51f9\u51fd\u6570<\/li>\n\n\n\n
      • \u6b65\u9aa4(5) \u5206\u5b50\u3001\u5206\u6bcd\u90fd\u4e58\u4ee5 \\(p(z) \\)<\/li>\n\n\n\n
      • \u6b65\u9aa4(6) \u4f7f\u7528\u57fa\u7840\u7684\u6761\u4ef6\u6982\u7387\u516c\u5f0f \\(p_{\\phi}(x|z)p(z) = p_{\\phi}(x,z) \\)<\/li>\n\n\n\n
      • \u6b65\u9aa4(7) \u4f7f\u7528\u5bf9\u6570\u51fd\u6570\u57fa\u672c\u6027\u8d28 \\(\\log \\frac{ab}{c} = \\log {a} – \\log \\frac{c}{b} \\)<\/li>\n\n\n\n
      • \u6b65\u9aa4(8) \u4f7f\u7528\u671f\u671b\u516c\u5f0f\u7684\u7ebf\u6027\u7279\u6027 \\(\\mathbb{E}[X+Y] = \\mathbb{E}[X] + \\mathbb{E}[Y] \\)<\/li>\n\n\n\n
      • \u6b65\u9aa4(9) \u4f7f\u7528\u4e86KL\u6563\u5ea6\u7684\u5b9a\u4e49\uff0c\u7b80\u5355\u7684\u8bf4\uff0c\u5373\u6982\u7387\u5bc6\u5ea6\u5bf9\u6570\u7684\u5dee\u7684\u671f\u671b\u5373\u4e3a\u6563\u5ea6<\/li>\n<\/ul>\n\n\n\n

        \u6b65\u9aa4(9) \u4e5f\u6700\u7ec8\u4e0e\u524d\u8ff0\u7684 \\(\\mathcal{L}_{\\mathrm{ELBO}} \\) \u5b9a\u4e49\u5b8c\u5168\u76f8\u540c\uff0c\u6545\u5f97\u8bc1\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

        \u5728\u6700\u521d\u5c1d\u8bd5\u4e86\u89e3 diffusion \u6a21\u578b\u7684\u65f6\u5019\uff0c\u539f\u672c\u662f\u6253\u7b97\u8df3\u8fc7 VAE \uff08Variational Autoencoder \u53d8\u5206\u81ea\u7f16\u7801 \uff09\u6a21\u578b\u7684\uff0c\u540e\u9762\u53d1\u73b0\u6709\u70b9\u8df3\u4e0d\u8fc7\u53bb\u3002\u518d\u82b1\u4e86\u4e00\u4e9b\u65f6\u95f4\u53bb\u4e86\u89e3 VAE\uff0c\u624d\u610f\u8bc6\u5230\u5176\u5b9e\u4e0d\u5e94\u8be5\u8df3\u8fc7\u53bb\uff0c\u76f8\u53cd\u7684\uff0c\u4e86\u89e3 VAE \u5b9e\u73b0\u7684\u4e00\u4e9b\u67b6\u6784\u3001\u539f\u7406\u3001\u76f4\u89c9\u3001\u521d\u8877\u3001\u6570\u5b66\u539f\u7406\u5219\u53ef\u4ee5\u5927\u5927\u5e2e\u52a9\u7406\u89e3\u540e\u7eed\u7684\u751f\u6210\u6a21\u578b\u3002 1. \u4e3a\u4ec0\u4e48\u73b0\u5728\u4f60\u4f9d\u65e7\u9700\u8981\u4e86\u89e3 VAE \u201c\u6f5c\u7a7a\u95f4\u201d\uff08latent space\uff09\u7684\u5904\u7406\u4f9d\u65e7\u662f\u73b0\u4ee3\uff08SOTA\uff09\u751f\u6210\u6a21\u578b\u6700\u4e3a\u6838\u5fc3\u7684\u7ec4\u4ef6\u3002\u800c\u4eceautoencoder\u6a21\u578b\uff0c\u6269\u5c55\u5230 VAE \u6a21\u578b\uff0c\u5219\u662f\u751f\u6210\u6a21\u578b\u8d70\u51fa\u7684\u5173\u952e\u6b65\u9aa4\u4e4b\u4e00\u3002\u5c06\u6f5c\u7a7a\u95f4\u9650\u5236\u5728\u4e00\u4e2a\u6b63\u6001\u5206\u5e03\u7684\u7a7a\u95f4\u5185\uff0c\u7136\u540e\uff0c\u5728\u8fd9\u4e2a\u7a7a\u95f4\u8fdb\u884c\u91c7\u6837\u540e\u8fdb\u884c decoder \u7684\u751f\u6210\u601d\u60f3\uff0c\u5219\u662f\u73b0\u4ee3\u751f\u6210\u6a21\u578b\u5f88\u591a\u601d\u60f3\u6765\u65f6\u7684\u8def\u3002 \u5982\u679c\u8df3\u8fc7\u8fd9\u4e00\u6bb5\uff0c\u5f88\u591a\u7684\u6982\u5ff5\u5219\u4f1a\u663e\u5f97\u975e\u5e38\u7a81\u7136\u3002 2. VAE \u5173\u952e\u76f4\u89c9\u4e0e\u4e3b\u8981\u601d\u60f3 2.1 \u5173\u952e\u76f4\u89c9 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\u90a3\u8fd9\u4e0e VAE \u6709\u4ec0\u4e48\u5173\u7cfb\u5462\uff1f\u5728\u7ecf\u5178\u7684 Guassian VAE \u4e2d\uff0c\u4e00\u4e2a\u5173\u952e\u5047\u8bbe\u662f\uff1a\u201c\u6f5c\u7a7a\u95f4\u201d \\(z \\) \u7b26\u5408\u9ad8\u65af\u5206\u5e03\u7684\uff08\u901a\u8fc7\u8bad\u7ec3\u6765\u5c06\u5176\u62c9\u5411\u9ad8\u65af\u5206\u5e03\uff09\u3002\u7136\u540e\u518d\u5bf9\u6f5c\u7a7a\u95f4\u91c7\u6837\u540e\uff0c\u5c31\u53ef\u4ee5\u751f\u6210\u8f83\u4e3a\u201c\u903c\u771f\u201d\u7684\u56fe\u50cf\u4e86\u3002\u90a3\u4e48\u5047\u8bbe\u201c\u6f5c\u7a7a\u95f4\u201d \\(z \\) \u7b26\u5408\u9ad8\u65af\u5206\u5e03\u6709\u4ec0\u4e48\u6df1\u610f\u5417\uff1f\u8fd8\u662f\u53ea\u662f\u4e3a\u4e86\u65b9\u4fbf\u91c7\u6837\uff1f \u4ece\u76f4\u89c9\u6784\u5efa\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u8ba4\u4e3a\uff1a\u6f5c\u7a7a\u95f4\u7684\u6bcf\u4e2a\u201c\u7ef4\u5ea6\u201d\u90fd\u6709\u67d0\u8fd9\u975e\u5e38\u5f3a\u7684\u610f\u4e49\uff0c\u4f8b\u5982\u8272\u5f69\u3001\u98ce\u683c\u3001\u7c7b\u578b\u7b49\uff0c\u90a3\u4e48\u5219\u6709\u7406\u7531\u76f8\u4fe1\uff0c\u4e00\u4e2a\u5bf9\u8c61\u5728\u6bcf\u4e00\u4e2a\u201c\u7ef4\u5ea6\u201d\u4e0a\u7684\u5206\u5e03\u4e5f\u662f\u7b26\u5408\u6b63\u6001\u5206\u5e03\u7684\u3002\u5373\u201c\u6f5c\u7a7a\u95f4\u201d\u662f\u4e00\u4e2a\u7b26\u5408\u6b63\u6001\u5206\u5e03\u7684\u7a7a\u95f4\uff0c\u90a3\u4e48\u4ece\u8fd9\u4e2a\u6837\u672c\u7a7a\u95f4\u8fdb\u884c\u91c7\u6837\u65f6\uff0c\u4e5f\u5c31\u66f4\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u6709\u4ee3\u8868\u610f\u4e49\u7684\u70b9\u3002 2.2 VAE \u7684\u4e3b\u8981\u601d\u60f3 \u5982\u4f55\u4f7f\u7528\u201c\u795e\u7ecf\u7f51\u7edc\u201d\u751f\u6210\u201c\u903c\u771f\u201d\u7684\u6570\u636e\uff08\u4f8b\u5982\u56fe\u7247\u3001\u89c6\u9891\u7b49\uff09\u5462\uff1f\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\uff0c\u4e00\u4e2a\u6bd4\u8f83\u81ea\u7136\u7684\u63a2\u7d22\u662f\u4ece\u201cautoencoder\u201d\u67b6\u6784\u53bb\u505a\u4e00\u4e9b\u5c1d\u8bd5\uff1a\u201cautoencoder\u201d \u7531\u4e00\u4e2a\u786e\u5b9a\u7684encoder \u5c06\u8f93\u5165\u538b\u7f29\u5230\u4e00\u4e2a\u4f4e\u7ef4\u7684\u201c\u6f5c\u7f16\u7801\u201d\uff08latent code\uff09\u7a7a\u95f4\u4e2d\uff0c\u7136\u540e\u518d\u7531decoder\u6839\u636e\u201clatent code\u201d\u91cd\u5efa\u6570\u636e\u3002 \u5982\u53f3\u56fe\u6240\u793a\uff0c\u5bf9\u4e8e\u8fd9\u6837\u7684\u795e\u7ecf\u7f51\u7edc\u8bbe\u8ba1\u5176\u4f18\u52bf\u662f\u975e\u5e38\u660e\u663e\u7684\uff0c\u8fd9\u53ef\u4ee5\u662f\u975e\u5e38\u597d\u7684\u201c\u65e0\u76d1\u7763\u5b66\u4e60\u201d\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u5bf9\u6bd4\u6700\u7ec8\u8f93\u51fa\u548c\u8f93\u5165\u7684\u6570\u636e\u5373\u53ef\u4ee5\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\u3002 \u4f46\uff0c\u5728\u6570\u636e\u751f\u6210\uff08\u4f8b\u5982\u65b0\u7684\u56fe\u50cf\u751f\u6210\uff09\u4e0a\uff0c\u8fd9\u6837\u7684\u8bbe\u8ba1\u5728\u5b9e\u8df5\u4e2d\u4e5f\u6709\u7740\u975e\u5e38\u660e\u663e\u7684\u9650\u5236\uff1a\u5373\u5bf9\u4e8e\u5982\u679c\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u201clatent code\u201d\uff0c\u901a\u5e38\u53ea\u80fd\u751f\u6210\u4e00\u4e9b\u65e0\u610f\u4e49\u7684\u6570\u636e\u3002 2013\u5e74\u7684 \u201cVariational Autoencoder (VAE)\u201d 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