{"id":35249,"date":"2024-12-21T14:59:57","date_gmt":"2024-12-21T06:59:57","guid":{"rendered":"https:\/\/17aitech.com\/?p=35249"},"modified":"2024-12-21T14:59:57","modified_gmt":"2024-12-21T06:59:57","slug":"%e5%a6%82%e4%bd%95%e8%ae%a9%e7%ad%89%e5%8f%98%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%e5%8f%af%e8%a7%a3%e9%87%8a%e6%80%a7%e6%9b%b4%e5%bc%ba%ef%bc%9f%e8%af%95%e8%af%95%e5%b0%86%e5%ae%83%e5%88%86%e8%a7%a3","status":"publish","type":"post","link":"https:\/\/17aitech.com\/?p=35249","title":{"rendered":"\u5982\u4f55\u8ba9\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u53ef\u89e3\u91ca\u6027\u66f4\u5f3a\uff1f\u8bd5\u8bd5\u5c06\u5b83\u5206\u89e3\u6210\u300c\u7b80\u5355\u8868\u793a\u300d"},"content":{"rendered":"<p>\u6587\u7ae0\u6765\u6e90\u4e8e\u4e92\u8054\u7f51:<a href=\"https:\/\/www.jiqizhixin.com\/articles\/2024-08-23\" target=\"_blank\">\u5982\u4f55\u8ba9\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u53ef\u89e3\u91ca\u6027\u66f4\u5f3a\uff1f\u8bd5\u8bd5\u5c06\u5b83\u5206\u89e3\u6210\u300c\u7b80\u5355\u8868\u793a\u300d<\/a><\/p>\n<p>\u795e\u7ecf\u7f51\u7edc\u662f\u4e00\u79cd\u7075\u6d3b\u4e14\u5f3a\u5927\u7684\u51fd\u6570\u8fd1\u4f3c\u65b9\u6cd5\u3002\u800c\u8bb8\u591a\u5e94\u7528\u90fd\u9700\u8981\u5b66\u4e60\u4e00\u4e2a\u76f8\u5bf9\u4e8e\u67d0\u79cd\u5bf9\u79f0\u6027\u4e0d\u53d8\u6216\u7b49\u53d8\u7684\u51fd\u6570\u3002\u56fe\u50cf\u8bc6\u522b\u4fbf\u662f\u4e00\u4e2a\u5178\u578b\u793a\u4f8b \u2014\u2014 \u5f53\u56fe\u50cf\u53d1\u751f\u5e73\u79fb\u65f6\uff0c\u60c5\u51b5\u4e0d\u4f1a\u53d1\u751f\u53d8\u5316\u3002\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\uff08equivariant neural network\uff09\u53ef\u4e3a\u5b66\u4e60\u8fd9\u4e9b\u4e0d\u53d8\u6216\u7b49\u53d8\u51fd\u6570\u63d0\u4f9b\u4e00\u4e2a\u7075\u6d3b\u7684\u6846\u67b6\u3002<\/p>\n<p>\u800c\u8981\u7814\u7a76\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\uff0c\u53ef\u4f7f\u7528\u8868\u793a\u8bba\uff08representation theory\uff09\u8fd9\u79cd\u6570\u5b66\u5de5\u5177\u3002\uff08\u8bf7\u6ce8\u610f\uff0c\u300c\u8868\u793a\u300d\u8fd9\u4e00\u6570\u5b66\u6982\u5ff5\u4e0d\u540c\u4e8e\u673a\u5668\u5b66\u4e60\u9886\u57df\u4e2d\u7684\u300c\u8868\u5f81\u300d\u7684\u5178\u578b\u542b\u4e49\u3002\u672c\u8bba\u6587\u4ec5\u4f7f\u7528\u8be5\u672f\u8bed\u7684\u6570\u5b66\u610f\u4e49\u3002\uff09<\/p>\n<p>\u8fd1\u65e5\uff0cJoel Gibson\u3001Daniel Tubbenhauer \u548c Geordie Williamson \u4e09\u4f4d\u7814\u7a76\u8005\u5bf9\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u8fdb\u884c\u4e86\u63a2\u7d22\uff0c\u5e76\u7814\u7a76\u4e86\u5206\u6bb5\u7ebf\u6027\u8868\u793a\u8bba\u5728\u5176\u4e2d\u7684\u4f5c\u7528\u3002<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-6d9005e2720fa133c63100e7afbe223a.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-6d9005e2720fa133c63100e7afbe223a.png\"><\/a><\/p>\n<ul>\n<li>\n<p>\u8bba\u6587\u6807\u9898\uff1aEquivariant neural networks and piecewise linear representation theory<\/p>\n<\/li>\n<li>\n<p>\u8bba\u6587\u5730\u5740\uff1ahttps:\/\/arxiv.org\/pdf\/2408.00949<\/p>\n<\/li>\n<\/ul>\n<p>\u5728\u8868\u793a\u8bba\u4e2d\uff0c\u7b80\u5355\u8868\u793a\uff08simple representation\uff09\u662f\u6307\u8be5\u7406\u8bba\u7684\u4e0d\u53ef\u7ea6\u7b80\u7684\u539f\u5b50\u3002\u5728\u89e3\u51b3\u95ee\u9898\u65f6\uff0c\u8868\u793a\u8bba\u7684\u4e00\u4e2a\u4e3b\u8981\u7b56\u7565\u662f\u5c06\u8be5\u95ee\u9898\u5206\u89e3\u6210\u7b80\u5355\u8868\u793a\uff0c\u7136\u540e\u5206\u522b\u57fa\u4e8e\u8fd9\u4e9b\u57fa\u672c\u7247\u6bb5\u7814\u7a76\u8be5\u95ee\u9898\u3002\u4f46\u5bf9\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u800c\u8a00\uff0c\u8fd9\u4e00\u7b56\u7565\u5e76\u4e0d\u594f\u6548\uff1a\u5b83\u4eec\u7684\u975e\u7ebf\u6027\u6027\u8d28\u5141\u8bb8\u7b80\u5355\u8868\u793a\u4e4b\u95f4\u53d1\u751f\u4e92\u52a8\uff0c\u800c\u7ebf\u6027\u4e16\u754c\u65e0\u6cd5\u505a\u5230\u8fd9\u4e00\u70b9\u3002<\/p>\n<p>\u4f46\u662f\uff0c\u8be5\u56e2\u961f\u53c8\u8bba\u8bc1\u8868\u660e\uff1a\u5c06\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u7684\u5c42\u5206\u89e3\u6210\u7b80\u5355\u8868\u793a\u4f9d\u7136\u80fd\u5e26\u6765\u597d\u5904\u3002\u7136\u540e\u5f88\u81ea\u7136\u5730\uff0c\u4ed6\u4eec\u53c8\u8fdb\u4e00\u6b65\u7814\u7a76\u4e86\u7b80\u5355\u8868\u793a\u4e4b\u95f4\u7684\u5206\u6bb5\u7ebf\u6027\u6620\u5c04\u548c\u5206\u6bb5\u7ebf\u6027\u8868\u793a\u8bba\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8fd9\u79cd\u5206\u89e3\u6210\u7b80\u5355\u8868\u793a\u7684\u8fc7\u7a0b\u80fd\u4e3a\u795e\u7ecf\u7f51\u7edc\u7684\u5c42\u6784\u5efa\u4e00\u4e2a\u65b0\u7684\u57fa\u7840\uff0c\u8fd9\u662f\u5bf9\u5085\u7acb\u53f6\u53d8\u6362\u7684\u6cdb\u5316\u3002<\/p>\n<p>\u8be5\u56e2\u961f\u8868\u793a\uff1a\u300c\u6211\u4eec\u5e0c\u671b\u8fd9\u79cd\u65b0\u57fa\u7840\u80fd\u4e3a\u7406\u89e3\u548c\u89e3\u8bfb\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u63d0\u4f9b\u4e00\u4e2a\u6709\u7528\u7684\u5de5\u5177\u3002\u300d<\/p>\n<p><strong>\u8be5\u8bba\u6587\u8bc1\u660e\u4e86\u4ec0\u4e48\uff1f<\/strong><\/p>\n<p>\u5728\u4ecb\u7ecd\u8be5\u8bba\u6587\u7684\u4e3b\u8981\u7ed3\u679c\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u5374\u975e\u5e73\u51e1\u7684\u793a\u4f8b\u3002<\/p>\n<p>\u4ee5\u4e00\u4e2a\u5c0f\u578b\u7684\u7b80\u5355\u795e\u7ecf\u7f51\u7edc\u4e3a\u4f8b\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-1a4fa17bad2b362a7e76a2112957f919.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-1a4fa17bad2b362a7e76a2112957f919.png\"><\/a><\/p>\n<p>\u5176\u4e2d\u6bcf\u4e2a\u8282\u70b9\u90fd\u662f \u211d \u7684\u4e00\u4e2a\u526f\u672c\uff0c\u6bcf\u4e2a\u7bad\u5934\u90fd\u6807\u8bb0\u4e86\u4e00\u4e2a\u6743\u91cd w\uff0c\u5e76\u4e14\u5c42\u4e4b\u95f4\u7684\u6bcf\u4e2a\u7ebf\u6027\u6620\u5c04\u7684\u7ed3\u679c\u90fd\u7531\u4e00\u4e2a\u975e\u7ebf\u6027\u6fc0\u6d3b\u51fd\u6570 \ud835\udc53 \u7ec4\u6210\uff0c\u7136\u540e\u518d\u8fdb\u5165\u4e0b\u4e00\u5c42\u3002<\/p>\n<p>\u4e3a\u4e86\u6784\u5efa\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\uff0c\u53ef\u5c06 \u211d \u548c w \u66ff\u6362\u6210\u5177\u6709\u66f4\u591a\u5bf9\u79f0\u6027\u7684\u66f4\u590d\u6742\u5bf9\u8c61\u3002\u6bd4\u5982\u53ef\u4ee5\u8fd9\u6837\u66ff\u6362\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-3562233df761da3177f3a8e3c4d7a489.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-3562233df761da3177f3a8e3c4d7a489.png\"><\/a><\/p>\n<p>\u5176\u53ef\u88ab\u63cf\u8ff0\u4e3a\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-8c84a0940bd9e2f82a0018c629ebc825.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-8c84a0940bd9e2f82a0018c629ebc825.png\"><\/a><\/p>\n<p>\u4e0d\u8fc7\uff0c\u8981\u60f3\u5728\u8ba1\u7b97\u673a\u4e0a\u771f\u6b63\u5b9e\u73b0\u8fd9\u4e2a\u7ed3\u6784\uff0c\u5374\u6839\u672c\u4e0d\u53ef\u80fd\uff0c\u4f46\u8fd9\u91cc\u5148\u5ffd\u7565\u8fd9\u4e00\u70b9\u3002<\/p>\n<p>\u73b0\u5728\u6682\u65f6\u5047\u8bbe\u51fd\u6570\u662f\u5468\u671f\u6027\u7684\uff0c\u5468\u671f\u4e3a 2\u03c0\u3002\u5f53\u7528\u5085\u91cc\u53f6\u7ea7\u6570\u5c55\u5f00\u795e\u7ecf\u7f51\u7edc\u65f6\uff0c\u6211\u4eec\u5f88\u81ea\u7136\u5c31\u4f1a\u95ee\u53d1\u751f\u4e86\u4ec0\u4e48\u3002\u5728\u5085\u91cc\u53f6\u7406\u8bba\u4e2d\uff0c\u5377\u79ef\u7b97\u5b50\u4f1a\u5728\u5085\u91cc\u53f6\u57fa\u4e2d\u53d8\u6210\u5bf9\u89d2\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u7406\u89e3\u4fe1\u53f7\u6d41\u8fc7\u4e0a\u8ff0\u795e\u7ecf\u7f51\u7edc\u7684\u65b9\u5f0f\uff0c\u8fd8\u9700\u8981\u7406\u89e3\u6fc0\u6d3b\u51fd\u6570\u5728\u57fa\u9891\u4e0a\u7684\u5de5\u4f5c\u65b9\u5f0f\u3002<\/p>\n<p>\u4e00\u4e2a\u57fa\u672c\u5374\u5173\u952e\u7684\u89c2\u5bdf\u662f\uff1a\ud835\udc53(sin (x)) \u7684\u5085\u91cc\u53f6\u7ea7\u6570\u4ec5\u6d89\u53ca\u8f83\u9ad8\u5171\u632f\u9891\u7387\u7684\u9879\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-47d12735931d69461c95f8dee52b10f2.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-47d12735931d69461c95f8dee52b10f2.png\"><\/a><\/p>\n<p>\uff08\u8fd9\u91cc\u5c55\u793a\u4e86\u5f53 \ud835\udc53 \u662f ReLU \u65f6\uff0c\ud835\udc53(sin (x)) \u7684\u524d\u51e0\u4e2a\u5085\u91cc\u53f6\u7ea7\u6570\u9879\u3002\uff09\u8fd9\u4e0e\u6211\u4eec\u62e8\u52a8\u5409\u4ed6\u7434\u5f26\u65f6\u53d1\u751f\u7684\u60c5\u51b5\u975e\u5e38\u76f8\u4f3c\uff1a\u4e00\u4e2a\u97f3\u7b26\u5177\u6709\u4e0e\u6240\u5f39\u594f\u97f3\u7b26\u76f8\u5bf9\u5e94\u7684\u57fa\u9891\uff0c\u4ee5\u53ca\u66f4\u9ad8\u7684\u9891\u7387\uff08\u6cdb\u97f3\uff0c\u7c7b\u4f3c\u4e8e\u4e0a\u9762\u5e95\u90e8\u7684\u4e09\u5f20\u56fe\u7247\uff09\uff0c\u5b83\u4eec\u7ed3\u5408\u5728\u4e00\u8d77\u5f62\u6210\u4e86\u5409\u4ed6\u72ec\u7279\u7684\u97f3\u8272\u3002<\/p>\n<p>\u8be5\u56e2\u961f\u7684\u7814\u7a76\u8868\u660e\uff1a\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u5728\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u4fe1\u606f\u4f1a\u4ece\u66f4\u4f4e\u5171\u632f\u9891\u7387\u6d41\u5411\u66f4\u9ad8\u5171\u632f\u9891\u7387\uff0c\u4f46\u53cd\u4e4b\u5219\u4e0d\u7136\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-d7bcce99e9e5daea60e6f50b64c19115.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-d7bcce99e9e5daea60e6f50b64c19115.png\"><\/a><\/p>\n<p>\u8fd9\u5bf9\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u6709\u4e24\u4e2a\u5177\u4f53\u5f71\u54cd\uff1a<\/p>\n<ol>\n<li>\n<p>\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u7684\u5927\u90e8\u5206\u590d\u6742\u6027\u90fd\u51fa\u73b0\u5728\u9ad8\u9891\u533a\uff0c<\/p>\n<\/li>\n<li>\n<p>\u5982\u679c\u60f3\u5b66\u4e60\u4e00\u4e2a\u4f4e\u9891\u51fd\u6570\uff0c\u90a3\u4e48\u53ef\u4ee5\u5ffd\u7565\u795e\u7ecf\u7f51\u7edc\u4e2d\u4e0e\u9ad8\u9891\u76f8\u5bf9\u5e94\u7684\u5927\u90e8\u5206\u3002<\/p>\n<\/li>\n<\/ol>\n<p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5982\u679c\u4f7f\u7528\u5178\u578b\u7684\u6d41\u5f0f\u793a\u610f\u56fe\uff08\u79f0\u4e3a\u4ea4\u4e92\u56fe \/interaction graph\uff09\u8868\u793a\uff0c\u4e00\u4e2a\u57fa\u4e8e\uff088 \u9636\u5faa\u73af\u7fa4\uff09\u6784\u5efa\u7684\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u662f\u8fd9\u6837\u7684\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-9e8ca3b24430fbf5e5ccba369def91cf.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-9e8ca3b24430fbf5e5ccba369def91cf.png\"><\/a><\/p>\n<p>\u5176\u4e2d\u7684\u8282\u70b9\u662f C_8 \u7684\u7b80\u5355\u8868\u793a\uff0c\u8282\u70b9\u4e2d\u7684\u503c\u8868\u793a\u751f\u6210\u5668\u7684\u52a8\u4f5c\u3002\u5728\u6b64\u56fe\u4e2d\uff0c\u300c\u4f4e\u9891\u300d\u7b80\u5355\u8868\u793a\u4f4d\u4e8e\u9876\u90e8\uff0c\u4fe1\u606f\u4ece\u4f4e\u9891\u6d41\u5411\u9ad8\u9891\u3002\u8fd9\u610f\u5473\u7740\u5728\u5927\u578b\u7f51\u7edc\u4e2d\uff0c\u9ad8\u9891\u5c06\u5360\u636e\u4e3b\u5bfc\u5730\u4f4d\u3002<\/p>\n<p><strong>\u4e3b\u8981\u8d21\u732e<\/strong><\/p>\n<p>\u8be5\u56e2\u961f\u505a\u51fa\u4e86\u4e00\u4e9b\u91cd\u8981\u7684\u7406\u8bba\u8d21\u732e\uff0c\u4e3b\u8981\u5305\u62ec\uff1a<\/p>\n<ol>\n<li>\n<p>\u4ed6\u4eec\u6307\u51fa\u5c06\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u5206\u89e3\u6210\u7b80\u5355\u8868\u793a\u662f\u6709\u610f\u4e49\u4e14\u6709\u7528\u7684\u3002<\/p>\n<\/li>\n<li>\n<p>\u4ed6\u4eec\u8bba\u8bc1\u8868\u660e\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u5fc5\u987b\u901a\u8fc7\u7f6e\u6362\u8868\u793a\u6784\u5efa\u3002<\/p>\n<\/li>\n<li>\n<p>\u4ed6\u4eec\u8bc1\u660e\u5206\u6bb5\u7ebf\u6027\uff08\u4f46\u5e76\u975e\u7ebf\u6027\uff09\u7684\u7b49\u53d8\u6620\u5c04\u7684\u5b58\u5728\u53d7\u63a7\u4e8e\u7c7b\u4f3c\u4e8e\u4f3d\u7f57\u74e6\u7406\u8bba\u7684\u6b63\u89c4\u5b50\u7fa4\u3002<\/p>\n<\/li>\n<li>\n<p>\u4ed6\u4eec\u8ba1\u7b97\u4e86\u4e00\u4e9b\u793a\u4f8b\uff0c\u5c55\u793a\u4e86\u7406\u8bba\u7684\u4e30\u5bcc\u6027\uff0c\u5373\u4f7f\u5728\u5faa\u73af\u7fa4\u7b49\u300c\u7b80\u5355\u300d\u793a\u4f8b\u4e2d\u4e5f\u662f\u5982\u6b64\u3002<\/p>\n<\/li>\n<\/ol>\n<p><strong>\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u548c\u5206\u6bb5\u7ebf\u6027\u8868\u793a<\/strong><\/p>\n<p>\u8be5\u56e2\u961f\u5728\u8bba\u6587\u4e2d\u9996\u5148\u7b80\u8981\u4ecb\u7ecd\u4e86\u8868\u793a\u8bba\u548c\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u7840\u77e5\u8bc6\uff0c\u8fd9\u91cc\u53d7\u9650\u4e8e\u7bc7\u5e45\uff0c\u6211\u4eec\u7565\u8fc7\u4e0d\u8868\uff0c\u8be6\u89c1\u539f\u8bba\u6587\u3002\u6211\u4eec\u4ec5\u91cd\u70b9\u4ecb\u7ecd\u6709\u5173\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u548c\u5206\u6bb5\u7ebf\u6027\u8868\u793a\u7684\u7814\u7a76\u6210\u679c\u3002<\/p>\n<p><strong>\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\uff1a\u4e00\u4e2a\u793a\u4f8b<\/strong><\/p>\n<p>\u8fd9\u7bc7\u8bba\u6587\u7684\u51fa\u53d1\u70b9\u662f\uff1a\u5b66\u4e60\u5173\u4e8e\u67d0\u79cd\u5bf9\u79f0\u6027\u7684\u7b49\u53d8\u6620\u5c04\u662f\u6709\u7528\u7684\u3002\u4e3e\u4e9b\u4f8b\u5b50\uff1a<\/p>\n<ol>\n<li>\n<p>\u56fe\u50cf\u8bc6\u522b\u7ed3\u679c\u901a\u5e38\u4e0d\u4f1a\u968f\u5e73\u79fb\u53d8\u5316\uff0c\u6bd4\u5982\u8bc6\u522b\u56fe\u50cf\u4e2d\u7684\u300c\u51b0\u6dc7\u6dcb\u300d\u65f6\u4e0e\u51b0\u6dc7\u6dcb\u6240\u5728\u7684\u4f4d\u7f6e\u65e0\u5173\uff1b<\/p>\n<\/li>\n<li>\n<p>\u6587\u672c\u8f6c\u8bed\u97f3\u65f6\uff0c\u300c\u51b0\u6dc7\u6dcb\u300d\u8fd9\u4e2a\u8bcd\u4e0d\u7ba1\u5728\u6587\u672c\u4e2d\u7684\u4ec0\u4e48\u4f4d\u7f6e\uff0c\u90fd\u5e94\u8be5\u751f\u6210\u4e00\u6837\u7684\u97f3\u9891\uff1b<\/p>\n<\/li>\n<li>\n<p>\u5de5\u7a0b\u5b66\u548c\u5e94\u7528\u6570\u5b66\u9886\u57df\u7684\u8bb8\u591a\u95ee\u9898\u90fd\u9700\u8981\u5206\u6790\u70b9\u4e91\u3002\u8fd9\u91cc\uff0c\u4eba\u4eec\u611f\u5174\u8da3\u7684\u901a\u5e38\u662f\u5bf9\u70b9\u4e91\u96c6\u5408\u7684\u8d28\u91cf\u8bc4\u4f30\uff0c\u800c\u4e0e\u987a\u5e8f\u65e0\u5173\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8fd9\u6837\u7684\u95ee\u9898\u4e0d\u4f1a\u968f\u70b9\u7684\u6392\u5217\u987a\u5e8f\u53d8\u5316\u800c\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u8fd9\u91cc\u7684\u5b66\u4e60\u95ee\u9898\u5728\u5bf9\u79f0\u7fa4\u4e0b\u662f\u4e0d\u53d8\u7684\u3002<\/p>\n<\/li>\n<\/ol>\n<p>\u4e3a\u4e86\u89e3\u91ca\u6784\u5efa\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u7684\u65b9\u5f0f\uff0c\u8be5\u56e2\u961f\u4f7f\u7528\u4e86\u4e00\u4e2a\u57fa\u4e8e\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u7b80\u5355\u793a\u4f8b\uff0c\u5176\u8981\u5904\u7406\u4e00\u5f20\u5e26\u5468\u671f\u6027\u7684\u56fe\u50cf\u3002<\/p>\n<p>\u8fd9\u91cc\uff0c\u8fd9\u5f20\u5468\u671f\u6027\u56fe\u50cf\u53ef\u8868\u793a\u6210\u4e00\u4e2a n \u00d7 n \u7684\u7f51\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u70b9\u90fd\u662f\u4e00\u4e2a\u5b9e\u6570\u3002\u5982\u679c\u8bbe\u5b9a n=10\uff0c\u518d\u5c06\u8fd9\u4e9b\u5b9e\u6570\u8868\u793a\u6210\u7070\u5ea6\u503c\uff0c\u5219\u53ef\u5f97\u5230\u5982\u4e0b\u6240\u793a\u7684\u56fe\u50cf\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-7699f6c40e955df5db2963845c64a32d.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-7699f6c40e955df5db2963845c64a32d.png\"><\/a><\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u5728\u8fd9\u5f20\u56fe\u4e0a\u4e0b\u5de6\u53f3\u8fdb\u884c\u91cd\u590d\uff0c\u4f7f\u4e4b\u5177\u6709\u5468\u671f\u6027\uff0c\u4e5f\u5c31\u76f8\u5f53\u4e8e\u8fd9\u5f20\u56fe\u5728\u4e00\u4e2a\u73af\u9762\u4e0a\u3002\u4ee4 C_n = \u2124\/n\u2124 \u4e3a n \u9636\u5faa\u73af\u7fa4\uff0cC^2_n = C_n \u00d7 C_n\u3002\u7528\u6570\u5b66\u672f\u8bed\u6765\u8bf4\uff0c\u4e00\u5f20\u5468\u671f\u6027\u56fe\u50cf\u662f\u4ece\u7fa4 C^2_n \u5230 \u211d \u7684\u6620\u5c04\u7684 \u211d \u5411\u91cf\u7a7a\u95f4\u7684\u4e00\u4e2a\u5143\u7d20\uff1a<a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-c3b4f14b42492df6bb833d7ca079b344.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-c3b4f14b42492df6bb833d7ca079b344.png\"><\/a>\u3002\u5728\u8fd9\u4e2a\u5468\u671f\u6027\u56fe\u50cf\u7684\u6a21\u578b\u4e2d\uff0cV \u662f\u4e00\u4e2a\u300cC^2_n \u8868\u793a\u300d\u3002\u4e8b\u5b9e\u4e0a\uff0c\u7ed9\u5b9a (a, b) \u2208 C^2_n \u548c \ud835\udc53 \u2208 V\uff0c\u53ef\u901a\u8fc7\u79fb\u52a8\u5750\u6807\u5f97\u5230\u4e00\u5f20\u65b0\u7684\u5468\u671f\u6027\u56fe\u50cf\uff1a<\/p>\n<ul>\n<li>\n<p>((a, b)\u30fbf)(x, y) = f (x + a, y + b)<\/p>\n<\/li>\n<\/ul>\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e73\u79fb\u5468\u671f\u6027\u56fe\u50cf\u4f1a\u5f97\u5230\u65b0\u7684\u5468\u671f\u6027\u56fe\u50cf\uff0c\u4f8b\u5982\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-bf304d8ebcd98aad68504ab5f0d1e810.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-bf304d8ebcd98aad68504ab5f0d1e810.png\"><\/a><\/p>\n<p>\u5f97\u5230\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u4e2a\u5173\u952e\u89c2\u5bdf\u662f\uff1a\u4ece V \u5230 V \u7684\u6240\u6709\u7ebf\u6027\u6620\u5c04\u7684 \u211d \u5411\u91cf\u7a7a\u95f4\u7684\u7ef4\u5ea6\u4e3a n^4\uff0c\u800c\u6240\u6709 C^2_n \u8868\u793a\u7ebf\u6027\u6620\u5c04\u7684 \u211d \u5411\u91cf\u7a7a\u95f4\u7684\u7ef4\u5ea6\u4e3a n^2\u3002<\/p>\n<p>\u4e0b\u9762\u6765\u770b\u4e00\u4e2a C^2_n \u7b49\u53d8\u6620\u5c04\u3002\u5bf9\u4e8e<a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-6ab3705d879e68518daf0897e7704250.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-6ab3705d879e68518daf0897e7704250.png\"><\/a>\uff0c\u53ef\u901a\u8fc7\u4e00\u4e2a\u5377\u79ef\u578b\u516c\u5f0f\u5f97\u5230 C^2_n \u7b49\u53d8\u6620\u5c04 V \u2192 V\uff1a<a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-920453abf7bf8493f5ac20331c8ac05e.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-920453abf7bf8493f5ac20331c8ac05e.png\"><\/a><\/p>\n<p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5982\u679c\u4ee4 c = 1\/4 ((1, 0) + (0, 1) + (\u22121, 0) + (0, \u22121))\u3002\u5219 c\u30fb\ud835\udc53 \u662f\u5468\u671f\u6027\u56fe\u50cf\u4e14\u5176\u50cf\u7d20 (a, b) \u5904\u7684\u503c\u662f\u5176\u76f8\u90bb\u50cf\u7d20 (a+1, b)\u3001(a, b+1)\u3001(a\u22121, b) \u548c (a, b\u22121) \u7684\u503c\u7684\u5e73\u5747\u503c\u3002\u7528\u56fe\u50cf\u8868\u793a\u5373\u4e3a\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-b16a6d237cafcc510ac452975ff5c388.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-b16a6d237cafcc510ac452975ff5c388.png\"><\/a><\/p>\n<p>\u66f4\u4e00\u822c\u5730\uff0c\u4e0d\u540c c \u7684\u5377\u79ef\u53ef\u5bf9\u5e94\u56fe\u50cf\u5904\u7406\u4e2d\u5e7f\u6cdb\u4f7f\u7528\u7684\u5404\u79cd\u6620\u5c04\u3002<\/p>\n<p>\u73b0\u5728\uff0c\u5c31\u53ef\u4ee5\u5b9a\u4e49\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684 C^2_n \u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u4e86\u3002\u5176\u7ed3\u6784\u5982\u4e0b\uff1a<\/p>\n<p><a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-78f0755fac3b3e48137ddef9b34482ff.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-78f0755fac3b3e48137ddef9b34482ff.png\"><\/a><\/p>\n<p>\u5176\u4e2d\u6bcf\u4e2a\u7bad\u5934\u90fd\u662f\u4e00\u4e2a\u5377\u79ef\u3002\u6b64\u5916\uff0cW \u901a\u5e38\u662f \u211d \u6216 V\u3002\u4e0a\u56fe\u662f\u4e00\u5f20\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\uff08\u7ecf\u8fc7\u7b80\u5316\u7684\uff09\u56fe\u50cf\uff0c\u800c\u8be5\u7f51\u7edc\u5728\u673a\u5668\u5b66\u4e60\u9886\u57df\u5177\u6709\u91cd\u8981\u5730\u4f4d\u3002\u5bf9\u4e8e\u8be5\u7f51\u7edc\u7684\u6784\u5efa\u65b9\u5f0f\uff0c\u503c\u5f97\u6ce8\u610f\u7684\u4e3b\u8981\u6982\u5ff5\u662f\uff1a<\/p>\n<ol>\n<li>\n<p>\u6b64\u795e\u7ecf\u7f51\u7edc\u7684\u7ed3\u6784\u4f1a\u8feb\u4f7f\u5f97\u5230\u7684\u6620\u5c04 V \u2192 W \u4e3a\u7b49\u53d8\u6620\u5c04\u3002<\/p>\n<\/li>\n<li>\n<p>\u6240\u6709\u6743\u91cd\u7684\u7a7a\u95f4\u6bd4\u4f20\u7edf\u7684\uff08\u5168\u8fde\u63a5\uff09\u795e\u7ecf\u7f51\u7edc\u5c0f\u5f97\u591a\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u7b49\u53d8\u795e\u7ecf\u7f51\u7edc\u6240\u80fd\u5904\u7406\u7684\u6837\u672c\u6bd4\u300c\u539f\u59cb\u300d\u795e\u7ecf\u7f51\u7edc\u6240\u80fd\u5904\u7406\u7684\u5927\u5f97\u591a\u3002\uff08\u8fd9\u4e00\u73b0\u8c61\u4e5f\u88ab\u673a\u5668\u5b66\u4e60\u7814\u7a76\u8005\u79f0\u4e3a\u6743\u91cd\u5171\u4eab\u3002\uff09<\/p>\n<\/li>\n<\/ol>\n<p>\u8be5\u56e2\u961f\u8fd8\u6307\u51fa\u4e0a\u56fe\u9690\u5f0f\u5730\u5305\u542b\u4e86\u6fc0\u6d3b\u56fe\uff0c\u800c\u4ed6\u4eec\u6700\u559c\u6b22\u7684\u9009\u62e9\u662f 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\u96c6\u5408 X \u548c Y\uff0c\u6709<a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-a46a9d0bf3d9a84ab3e7a7aca5674f2f.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-a46a9d0bf3d9a84ab3e7a7aca5674f2f.png\"><\/a>\uff0c\u5176\u4e2d Fun_G (X \u00d7 Y, \u211d) \u8868\u793a G \u4e0d\u53d8\u51fd\u6570 X\u00d7Y \u2192\u211d\u3002<\/p>\n<p>\u6839\u636e\u8be5\u5f15\u7406\uff0c<a href=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-ec8ab912aa00596317d8fe5a15ae5f8e.png\" data-fancybox=\"images\" data-fancybox=\"gallery\"><img decoding=\"async\" src=\"https:\/\/17aitech.com\/wp-content\/uploads\/2024\/08\/frc-ec8ab912aa00596317d8fe5a15ae5f8e.png\"><\/a>\uff0c\u5e76\u4e14 G = S_n 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