{"id":1582,"date":"2024-07-27T22:23:14","date_gmt":"2024-07-27T14:23:14","guid":{"rendered":"https:\/\/www.gnn.club\/?p=1582"},"modified":"2024-07-29T20:00:47","modified_gmt":"2024-07-29T12:00:47","slug":"%e7%be%a4%e8%ae%ba","status":"publish","type":"post","link":"http:\/\/gnn.club\/?p=1582","title":{"rendered":"\u7fa4\u8bba"},"content":{"rendered":"<h1><img decoding=\"async\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729195831865.png\" style=\"height:50px;display:inline\">  Deep Learning Math<\/h1>\n<hr \/>\n<h2>\u7fa4\u8bba\uff08group Theory\uff09<\/h2>\n<hr \/>\n<p>\u5728\u5b66\u4e60\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e4b\u524d\uff0c\u5177\u5907\u7fa4\u8bba\u7684\u57fa\u7840\u77e5\u8bc6\u662f\u975e\u5e38\u91cd\u8981\u7684\uff0c\u539f\u56e0\u5728\u4e8e\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u5904\u7406\u7684\u6570\u636e\u548c\u95ee\u9898\u5f80\u5f80\u6d89\u53ca\u5230\u590d\u6742\u7684\u7a7a\u95f4\u53d8\u6362\u548c\u5bf9\u79f0\u6027\uff0c\u4f8b\u5982\uff0c\u5206\u5b50\u7ed3\u6784\u5728\u65cb\u8f6c\u6216\u955c\u50cf\u53cd\u5c04\u540e\u53ef\u80fd\u4fdd\u6301\u4e0d\u53d8\uff0c\u4e09\u7ef4\u7269\u4f53\u7684\u5f62\u72b6\u5728\u4e0d\u540c\u89c6\u89d2\u4e0b\u5177\u6709\u65cb\u8f6c\u5bf9\u79f0\u6027\u3002\u8bc6\u522b\u548c\u5229\u7528\u8fd9\u4e9b\u5bf9\u79f0\u6027\u5bf9\u4e8e\u7406\u89e3\u6570\u636e\u7684\u672c\u8d28\u7279\u5f81\u81f3\u5173\u91cd\u8981\u3002\u800c\u7fa4\u8bba\u6070\u6070\u63d0\u4f9b\u4e86\u4e00\u79cd\u5f3a\u5927\u7684\u6570\u5b66\u5de5\u5177\u6765\u63cf\u8ff0\u548c\u7406\u89e3\u8fd9\u4e9b\u53d8\u6362\u548c\u5bf9\u79f0\u6027\u3002\u901a\u8fc7\u4f7f\u7528\u7fa4\u8bba\u7684\u6982\u5ff5\u8bbe\u8ba1\u7684\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u80fd\u591f\u66f4\u597d\u5730\u7406\u89e3\u548c\u5229\u7528\u6570\u636e\u7684\u5185\u5728\u5bf9\u79f0\u6027\uff0c\u4ece\u800c\u63d0\u9ad8\u6a21\u578b\u5728\u672a\u89c1\u6570\u636e\u4e0a\u7684\u6cdb\u5316\u80fd\u529b\u3002\u5bf9\u4e8e\u53d8\u6362\u4e0d\u53d8\u6027\u548c\u7b49\u53d8\u6027\u7684\u7406\u89e3\u6709\u52a9\u4e8e\u6784\u5efa\u5728\u9762\u5bf9\u6570\u636e\u53d8\u6362\u65f6\u4f9d\u7136\u7a33\u5b9a\u8868\u73b0\u7684\u6a21\u578b\uff0c\u8fd9\u5bf9\u4e8e\u8bb8\u591a\u5b9e\u9645\u5e94\u7528\u81f3\u5173\u91cd\u8981\u3002<\/p>\n<h3><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/bubbles\/50\/000000\/checklist.png\" style=\"height:50px;display:inline\"> Agenda<\/h3>\n<hr \/>\n<ul>\n<li>\n<p>\u7fa4\u7684\u5b9a\u4e49(Definition of Group)<\/p>\n<\/li>\n<li>\n<p>\u5bf9\u79f0\u6027(Symmetry)<\/p>\n<\/li>\n<li>\n<p>\u4e0d\u53d8\u6027(Invariance)<\/p>\n<\/li>\n<li>\n<p>\u7b49\u53d8\u6027(Equivariance)<\/p>\n<\/li>\n<li>\n<p>S0(3) Group<\/p>\n<\/li>\n<li>\n<p>\u7fa4\u7684\u8868\u793a(Representation of Groups)<\/p>\n<\/li>\n<\/ul>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=vXpfyU1CB7sP&format=png&color=000000\" style=\"height:50px;display:inline\"> \u7fa4\u7684\u5b9a\u4e49(Definition of Group)<\/h2>\n<hr \/>\n<p>\u7fa4\u662f\u6570\u5b66\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e00\u79cd\u63cf\u8ff0\u5bf9\u79f0\u6027\u3001\u64cd\u4f5c\u53ca\u5176\u6027\u8d28\u7684\u5f62\u5f0f\u5316\u65b9\u6cd5\u3002\u7fa4\u7684\u4e25\u683c\u6570\u5b66\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n<p>\u4e00\u4e2a\u7fa4\u662f\u4e00\u4e2a\u96c6\u5408 $G$, \u914d\u5408\u4e00\u4e2a\u4e8c\u5143\u8fd0\u7b97, \u6ee1\u8db3\u4ee5\u4e0b\u56db\u4e2a\u6761\u4ef6:<\/p>\n<ol>\n<li>\u5c01\u95ed\u6027(Closure): \u5bf9\u4e8e\u96c6\u5408 $G$ \u4e2d\u4efb\u610f\u4e24\u4e2a\u5143\u7d20 $a$ \u548c $b$, \u8fd0\u7b97\u7684\u7ed3\u679c $a \\cdot b$ \u4e5f\u5728 $G$ \u4e2d\u3002\u5373,\u96c6\u5408\u5bf9\u4e8e\u8fd9\u4e2a\u8fd0\u7b97\u662f\u5c01\u95ed\u7684\u3002<\/li>\n<li>\u7ed3\u5408\u5f8b(Associativity): \u5bf9\u4e8e\u96c6\u5408 $G$ \u4e2d\u4efb\u610f\u4e09\u4e2a\u5143\u7d20 $a \u3001 b$ \u548c $c$, \u8fd0\u7b97\u6ee1\u8db3\u7ed3\u5408\u5f8b $(a \\cdot b) \\cdot c=a \\cdot(b \\cdot c) \u3002$<\/li>\n<li>\u5355\u4f4d\u5143\u7d20(Identity Element): \u5728\u96c6\u5408 $G$ \u4e2d\u5b58\u5728\u4e00\u4e2a\u7279\u6b8a\u7684\u5143\u7d20 $e$, \u5bf9\u4e8e\u96c6\u5408\u4e2d\u7684\u4efb\u610f\u5143\u7d20 $a$, \u5b83\u4e0e $e$ \u7684\u8fd0\u7b97\u7ed3\u679c\u4ecd\u7136\u662f $a: a \\cdot e=e \\cdot a=a$ \u3002\u8fd9\u4e2a\u5143\u7d20 $e$ \u88ab\u79f0\u4e3a\u5355\u4f4d\u5143\u7d20\u6216\u6052\u7b49\u5143\u7d20\u3002<\/li>\n<li>\u9006\u5143\u7d20(Inverse Element): \u5bf9\u4e8e\u96c6\u5408 $G$ \u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20 $a$, \u5b58\u5728\u53e6\u4e00\u4e2a\u5143\u7d20 $b$ \u5728 $G$ \u4e2d, \u4f7f\u5f97 $a$ \u548c $b$ \u7684\u8fd0\u7b97\u7ed3\u679c\u662f\u5355\u4f4d\u5143\u7d20 $e: a \\cdot b=b \\cdot a=e$ \u3002\u8fd9\u4e2a\u5143\u7d20 $b$ \u88ab\u79f0\u4e3a\u5143\u7d20 $a$ \u7684\u9006\u5143\u7d20\u3002<\/li>\n<\/ol>\n<p><strong>\u7b80\u5355\u8bf4\uff0c\u7fa4\u5c31\u662f\u201c\u96c6\u5408+\u8fd0\u7b97\u201d<\/strong><\/p>\n<ul>\n<li>\u8003\u8651\u4e00\u4e2a\u5173\u4e8e\u65f6\u95f4\u7684\u4f8b\u5b50\u6765\u8bf4\u660e\u7fa4\u7684\u6982\u5ff5\u3002<\/li>\n<li>\u60f3\u8c61\u4f60\u6709\u4e00\u4e2a\u975e\u5e38\u7279\u522b\u7684\u624b\u8868\uff0c\u8fd9\u4e2a\u624b\u8868\u6709\u56db\u4e2a\u6309\u94ae\uff0c\u6bcf\u4e2a\u6309\u94ae\u4ee3\u8868\u4e00\u4e2a\u64cd\u4f5c\uff1a\u5feb\u8fdb1\u5c0f\u65f6\uff08A\uff09\uff1b\u540e\u90001\u5c0f\u65f6\uff08B\uff09\uff1b\u65f6\u95f4\u4e0d\u53d8\uff08C\uff09\u3002<\/li>\n<li>\u5728\u8fd9\u4e2a\u8bbe\u7f6e\u4e2d\uff0c\u96c6\u5408\u4e3a{A,B,C}\uff0c\u64cd\u4f5c\u4e3a\u6309\u4e0b\u6309\u94ae\u3002<\/li>\n<li>\u5c01\u95ed\u6027\u7684\u4f53\u73b0\u4e3a\uff1a\u65e0\u8bba\u4f60\u6309\u54ea\u4e24\u4e2a\u6309\u94ae\uff0c\u6216\u8005\u662f\u540c\u4e00\u4e2a\u6309\u94ae\u591a\u6b21\uff0c\u5f97\u5230\u7684\u65f6\u95f4\u53d8\u5316\u90fd\u53ef\u4ee5\u901a\u8fc7\u518d\u6309\u5176\u4ed6\u6309\u94ae\u4e2d\u7684\u67d0\u4e00\u4e2a\u6765\u8fbe\u5230\u3002\u4f8b\u5982\uff0c\u6309\u4e0b\u5feb\u8fdb1\u5c0f\u65f6\uff08A\uff09\u548c\u540e\u90001\u5c0f\u65f6\uff08B\uff09\u7684\u7ec4\u5408\uff0c\u7b49\u540c\u4e8e\u65f6\u95f4\u4e0d\u53d8\uff08C\uff09\u3002\u8fd9\u610f\u5473\u7740\u6240\u6709\u53ef\u80fd\u7684\u65f6\u95f4\u53d8\u5316\u64cd\u4f5c\u90fd\u5305\u542b\u5728\u8fd9\u4e2a\u6309\u94ae\u96c6\u5408\u5185\u3002<\/li>\n<li>\u7ed3\u5408\u5f8b\u4f53\u73b0\u5728\u6309\u94ae\u64cd\u4f5c\u7684\u7ec4\u5408\u987a\u5e8f\u4e0d\u4f1a\u5f71\u54cd\u6700\u7ec8\u7684\u65f6\u95f4\u53d8\u5316\u7ed3\u679c\u3002\u4f8b\u5982\uff0c\u65e0\u8bba\u662f\u5148\u7ec4\u5408A\u548cB\u518d\u5e94\u7528C\uff0c\u8fd8\u662f\u5148\u5c06B\u4e0eC\u7ec4\u5408\u518d\u4e0eA\u7ec4\u5408\uff0c\u6700\u7ec8\u7ed3\u679c\u59cb\u7ec8\u4fdd\u6301\u65f6\u95f4\u4e0d\u53d8\uff08C\uff09\u3002<\/li>\n<li>\u65f6\u95f4\u4e0d\u53d8\u6309\u94ae\uff08G\uff09\u5c31\u662f\u8fd9\u4e2a\u624b\u8868\u64cd\u4f5c\u96c6\u5408\u7684\u5355\u4f4d\u5143\u7d20\u3002\u6309\u4e0b\u8fd9\u4e2a\u6309\u94ae\u4e0d\u4f1a\u6539\u53d8\u5f53\u524d\u7684\u65f6\u95f4\uff0c\u5b83\u786e\u4fdd\u4e86\u65e0\u8bba\u4f60\u4e4b\u524d\u8fdb\u884c\u4e86\u4ec0\u4e48\u64cd\u4f5c\uff0c\u518d\u6309\u4e0b\u65f6\u95f4\u4e0d\u53d8\u6309\u94ae\u90fd\u4e0d\u4f1a\u6539\u53d8\u5f53\u524d\u7684\u65f6\u95f4\u72b6\u6001\u3002<\/li>\n<li>\u5bf9\u4e8e\u8fd9\u4e2a\u624b\u8868\u4e0a\u7684\u6bcf\u4e2a\u6309\u94ae\uff0c\u90fd\u6709\u4e00\u4e2a\u5bf9\u5e94\u7684\u9006\u64cd\u4f5c\u6309\u94ae\u3002\u6bd4\u5982\uff0c\u5feb\u8fdb1\u5c0f\u65f6\uff08A\uff09\u7684\u9006\u64cd\u4f5c\u662f\u540e\u90001\u5c0f\u65f6\uff08B\uff09\u3002\u6309\u4e0b\u4e00\u4e2a\u6309\u94ae\u7136\u540e\u6309\u4e0b\u5b83\u7684\u9006\u64cd\u4f5c\u6309\u94ae\uff0c\u4f1a\u4f7f\u65f6\u95f4\u72b6\u6001\u56de\u5230\u6309\u4e0b\u7b2c\u4e00\u4e2a\u6309\u94ae\u4e4b\u524d\u7684\u72b6\u6001\u3002<\/li>\n<\/ul>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=NWEH8gGsV2I4&format=png&color=000000\" style=\"height:50px;display:inline\"> \u5bf9\u79f0\u6027(Symmetry)<\/h2>\n<hr \/>\n<p>\u7fa4\u8bba\u5728\u63cf\u8ff0\u5bf9\u79f0\u6027\u65b9\u9762\u626e\u6f14\u7740\u6838\u5fc3\u89d2\u8272\uff0c\u63d0\u4f9b\u4e86\u4e00\u5957\u4e25\u683c\u7684\u6570\u5b66\u6846\u67b6\uff0c\u7528\u4ee5\u5206\u6790\u548c\u7406\u89e3\u5bf9\u79f0\u6027\u7684\u672c\u8d28\u3002\u5bf9\u79f0\u6027\u662f\u6307\u67d0\u4e2a\u5bf9\u8c61\u6216\u7cfb\u7edf\u5728\u7ecf\u8fc7\u4e00\u7cfb\u5217\u53d8\u6362\uff08\u5982\u65cb\u8f6c\u3001\u53cd\u5c04\u3001\u5e73\u79fb\u7b49\uff09\u540e\uff0c\u4ecd\u80fd\u4fdd\u6301\u5176\u67d0\u4e9b\u7279\u5f81\u6216\u6027\u8d28\u4e0d\u53d8\u7684\u5c5e\u6027\u3002\u8fd9\u4e9b\u53d8\u6362\u6784\u6210\u4e86\u4e00\u4e2a\u7fa4\uff0c\u79f0\u4e3a\u5bf9\u79f0\u7fa4\uff0c\u5b83\u51c6\u786e\u63cf\u8ff0\u4e86\u7cfb\u7edf\u7684\u5bf9\u79f0\u64cd\u4f5c\u3002<\/p>\n<p>\u5728\u6570\u5b66\u3001\u7269\u7406\u3001\u5316\u5b66\u3001\u751f\u7269\u5b66\u7b49\u4f17\u591a\u79d1\u5b66\u9886\u57df\u5185\uff0c\u5bf9\u79f0\u6027\u4e0d\u4ec5\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u73b0\u8c61\uff0c\u4e5f\u662f\u7406\u89e3\u7cfb\u7edf\u6027\u8d28\u7684\u5173\u952e\u56e0\u7d20\u3002<\/p>\n<p>\u901a\u8fc7\u7fa4\u8bba\uff0c\u6211\u4eec\u80fd\u591f\u5c06\u5bf9\u79f0\u6027\u7684\u6982\u5ff5\u6570\u5b66\u5316\uff0c\u8fdb\u800c\u63ed\u793a\u7cfb\u7edf\u7684\u5185\u5728\u89c4\u5f8b\u548c\u6027\u8d28\u3002<\/p>\n<p>\u4f8b\u5982\uff0c\u6211\u4eec\u8003\u8651\u6570\u5b66\u4e2d\u7684\u7b49\u8fb9\u4e09\u89d2\u5f62\u7684\u5bf9\u79f0\u7fa4D3\uff0c\u6709\u65f6\u4e5f\u79f0\u4e3aS3\uff0c\u5b83\u8868\u793a\u4e09\u89d2\u5f62\u6240\u6709\u53ef\u80fd\u7684\u65cb\u8f6c\u548c\u53cd\u5c04\u64cd\u4f5c\u3002\u8fd9\u4e2a\u5bf9\u79f0\u7fa4\u53ef\u4ee5\u5b8c\u5168\u7531\u4e24\u4e2a\u64cd\u4f5c\u6765\u751f\u6210\uff1a\u65cb\u8f6cR\u548c\u53cd\u5c04F\uff0c\u5982\u56fe<\/p>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729195947714.png\" style=\"height:400px\">\n<\/p>\n<ul>\n<li>\n<p>\u65cb\u8f6c\uff08R\uff09\uff1a\u56fe1-37\u4e2d\u7684R\u4ee3\u8868\u7740\u56f4\u7ed5\u4e09\u89d2\u5f62\u4e2d\u5fc3\u987a\u65f6\u9488\u65cb\u8f6c60\u5ea6\u3002\u65cb\u8f6c\u4e24\u6b21\uff08R2\uff09\u610f\u5473\u7740\u65cb\u8f6c\u4e86120\u5ea6\uff0c\u800c\u65cb\u8f6c\u4e09\u6b21\u76f8\u5f53\u4e8e\u65cb\u8f6c\u4e86180\u5ea6\uff0c\u8fd9\u5b9e\u9645\u4e0a\u4f1a\u4f7f\u4e09\u89d2\u5f62\u56de\u5230\u539f\u6765\u7684\u4f4d\u7f6e\uff08\u5728\u7fa4\u7684\u8bed\u5883\u4e2d\u76f8\u5f53\u4e8e\u505a\u4e86\u4e00\u4e2a\u6052\u7b49\u64cd\u4f5c\uff0c\u6216\u8005\u8bf4\u6ca1\u6709\u6539\u53d8\uff0c\u7528id\u8868\u793a\uff09\u3002<\/p>\n<\/li>\n<li>\n<p>\u53cd\u5c04\uff08F\uff09\uff1a\u4f8b\u5982\uff0cF\u4ee3\u8868\u7740\u6cbf\u4e00\u6761\u901a\u8fc7\u9876\u70b91\u548c\u4e09\u89d2\u5f62\u4e2d\u5fc3\u7684\u7ebf\u8fdb\u884c\u53cd\u5c04\u3002\u8fd9\u79cd\u53cd\u5c04\u4f1a\u6539\u53d8\u9876\u70b92\u548c3\u7684\u4f4d\u7f6e\u3002\u540c\u7406\uff0c\u5728\u9876\u70b92\u548c3\u4e5f\u53ef\u4ee5\u6cbf\u4e2d\u5fc3\u7ebf\u8fdb\u884c\u53cd\u5c04\u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u7b49\u8fb9\u4e09\u89d2\u5f62\u7684\u5bf9\u79f0\u7fa4\u5305\u542b6\u4e2a\u5143\u7d20\uff1a\u6052\u7b49\u64cd\u4f5cid\uff0c\u4e24\u4e2a\u65cb\u8f6c\uff08R\u548cR2\uff09\uff0c\u4ee5\u53ca\u4e09\u4e2a\u53cd\u5c04\uff08F, FR\u548cFR2\uff09\u3002\u8fd9\u4e9b\u5143\u7d20\u7ec4\u6210\u4e86\u7fa4D3\u7684\u6240\u6709\u53ef\u80fd\u64cd\u4f5c\u3002<\/p>\n<p>\u56fe\u7684\u53f3\u8fb9\u662f\u4e00\u4e2a\u4e58\u6cd5\u8868\uff08\u6216\u79f0\u4f5c\u7fa4\u7684\u8fd0\u7b97\u8868\uff09\uff0c\u5c55\u793a\u4e86\u6bcf\u5bf9\u64cd\u4f5c\u7ec4\u5408\u7684\u7ed3\u679c\u3002\u4f8b\u5982\uff0cFR\u5728\u8868\u4e2d\u7684\u610f\u601d\u662f\u5148\u8fdb\u884c\u53cd\u5c04F\u518d\u8fdb\u884c\u65cb\u8f6cR\u3002\u6bcf\u4e00\u884c\u4ee3\u8868\u4e00\u4e2a\u64cd\u4f5c\uff0c\u6bcf\u4e00\u5217\u4e5f\u4ee3\u8868\u4e00\u4e2a\u64cd\u4f5c\uff0c\u4ea4\u70b9\u5219\u4ee3\u8868\u8fd9\u4e24\u4e2a\u64cd\u4f5c\u7684\u7ec4\u5408\u7ed3\u679c\u3002\u4f8b\u5982\uff0c\u5728R\u884c\u548cF\u5217\u7684\u4ea4\u70b9\u662fRF\uff0c\u8fd9\u8868\u660e\u5982\u679c\u6211\u4eec\u5148\u65cb\u8f6c\u518d\u53cd\u5c04\uff0c\u6700\u7ec8\u7684\u6548\u679c\u7b49\u540c\u4e8e\u5355\u4e00\u64cd\u4f5cRF\u3002<\/p>\n<p>\u8fd9\u4e2a\u7fa4\u7684\u7ed3\u6784\u63ed\u793a\u4e86\u7b49\u8fb9\u4e09\u89d2\u5f62\u7684\u6240\u6709\u53ef\u80fd\u5bf9\u79f0\u6027\u3002\u901a\u8fc7\u7814\u7a76\u8fd9\u4e2a\u7fa4\uff0c\u6211\u4eec\u53ef\u4ee5\u4e86\u89e3\u4e09\u89d2\u5f62\u5728\u4e0d\u540c\u64cd\u4f5c\u4e0b\u7684\u884c\u4e3a\uff0c\u4ee5\u53ca\u8fd9\u4e9b\u64cd\u4f5c\u5982\u4f55\u76f8\u4e92\u5173\u8054\u3002\u8be5\u7fa4\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u7b49\u4ef7\u7684\uff0c\u610f\u5473\u7740\u7fa4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u53ef\u4ee5\u901a\u8fc7\u5176\u4ed6\u5143\u7d20\u7684\u7ec4\u5408\u6765\u8868\u8fbe\u3002\u6b64\u5916\uff0c\u8fd9\u4e2a\u7fa4\u662f\u975e\u963f\u8d1d\u5c14\u7fa4\uff08\u975e\u4ea4\u6362\u7fa4\uff09\uff0c\u56e0\u4e3a\u67d0\u4e9b\u64cd\u4f5c\u7684\u7ec4\u5408\u7ed3\u679c\u53d6\u51b3\u4e8e\u5b83\u4eec\u7684\u987a\u5e8f\uff08\u4f8b\u5982\uff0cRF\u4e0d\u540c\u4e8eFR\uff09\u3002<\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=IfbRFKAQvZDQ&format=png&color=5C7CFA\" style=\"height:50px;display:inline\"> \u4e0d\u53d8\u6027\u4e0e\u7b49\u53d8\u6027(Invariance and Equivariance)<\/h2>\n<hr \/>\n<p>\u6b64\u5916\uff0c\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u5bf9\u79f0\u6027\u4e00\u822c\u88ab\u7ec6\u5206\u4e3a\u4e0d\u53d8\u6027\u4e0e\u7b49\u53d8\u6027\u3002<\/p>\n<ul>\n<li>\u4e0d\u53d8\u6027\u610f\u5473\u7740\u6a21\u578b\u8f93\u51fa\u5bf9\u4e8e\u67d0\u79cd\u64cd\u4f5c\u6216\u53d8\u6362\u662f\u4e0d\u53d8\u7684\u3002<\/li>\n<li>\u7b49\u53d8\u6027\u662f\u6307\u5f53\u8f93\u5165\u6570\u636e\u7ecf\u5386\u67d0\u79cd\u53d8\u6362\u65f6\uff0c\u6a21\u578b\u8f93\u51fa\u4e5f\u4f1a\u4ee5\u76f8\u540c\u7684\u65b9\u5f0f\u53d8\u6362\u3002<\/li>\n<\/ul>\n<p>\u5ef6\u7eed\u4e0a\u56fe\u4e2d\u7b49\u53d8\u4e09\u89d2\u5f62\u7684\u4f8b\u5b50\u3002\u5f53\u7ed5\u4e09\u89d2\u5f62\u4e2d\u5fc3\u65cb\u8f6c120\u00b0\uff0c240\u00b0\u6216360\u00b0\u65f6\uff0c\u4e09\u89d2\u5f62\u7684\u5916\u89c2\u548c\u4f4d\u7f6e\u770b\u8d77\u6765\u5b8c\u5168\u4e00\u6837\uff0c\u8fd9\u4f53\u73b0\u4e86\u5176\u5bf9\u79f0\u6027\u3002<\/p>\n<ul>\n<li>\n<p>\u5f53\u4e09\u89d2\u5f62\u7ed5\u4e2d\u5fc3\u65cb\u8f6c120\u00b0\u6216240\u00b0\u65f6\uff0c\u5c3d\u7ba1\u6700\u7ec8\u7684\u5f62\u72b6\u770b\u8d77\u6765\u4e0e\u539f\u59cb\u5f62\u72b6\u76f8\u540c\uff0c\u4f46\u4e09\u89d2\u5f62\u7684\u6bcf\u4e2a\u9876\u70b9\u5b9e\u9645\u4e0a\u5df2\u7ecf\u79fb\u52a8\u5230\u4e86\u65b0\u7684\u4f4d\u7f6e\u3002\u8fd9\u79cd\u9876\u70b9\u4f4d\u7f6e\u7684\u6539\u53d8\u662f\u7b49\u53d8\u6027\u7684\u4f53\u73b0\uff1a\u8f93\u5165\uff08\u4e09\u89d2\u5f62\u7684\u65cb\u8f6c\uff09\u5bfc\u81f4\u4e86\u8f93\u51fa\uff08\u9876\u70b9\u4f4d\u7f6e\uff09\u7684\u5bf9\u5e94\u53d8\u5316\u3002<\/p>\n<\/li>\n<li>\n<p>\u5f53\u65cb\u8f6c360\u00b0\uff0c\u5373\u5b8c\u5168\u65cb\u8f6c\u4e00\u5468\u540e\uff0c\u4e09\u89d2\u5f62\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u6062\u590d\u5230\u4e86\u5176\u539f\u59cb\u4f4d\u7f6e\u3002\u4ece\u6574\u4f53\u4e0a\u770b\uff0c\u4e09\u89d2\u5f62\u770b\u8d77\u6765\u6ca1\u6709\u4efb\u4f55\u53d8\u5316\uff0c\u5373\u4e09\u89d2\u5f62\u7684\u6574\u4f53\u5f62\u72b6\u548c\u4f4d\u7f6e\u4e0e\u539f\u59cb\u72b6\u6001\u76f8\u540c\u3002\u8fd9\u662f\u4e0d\u53d8\u6027\u7684\u4f53\u73b0\uff1a\u5c3d\u7ba1\u8fdb\u884c\u4e86\u64cd\u4f5c\uff08\u65cb\u8f6c360\u00b0\uff09\uff0c\u8f93\u51fa\uff08\u4e09\u89d2\u5f62\u7684\u6574\u4f53\u5f62\u72b6\u548c\u4f4d\u7f6e\uff09\u4fdd\u6301\u4e0d\u53d8\u3002<\/p>\n<\/li>\n<\/ul>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=76996&format=png&color=000000\" style=\"height:50px;display:inline\"> SO(3) Group<\/h2>\n<hr \/>\n<p><strong>SO(3)\u7fa4\u662f\u6240\u6709 $3 \\times 3$ \u7684\u6b63\u4ea4\u77e9\u9635\u7ec4\u6210\u7684\u7fa4<\/strong><\/p>\n<ul>\n<li>\u8fd9\u4e9b\u77e9\u9635\u7684\u884c\u5217\u5f0f\u4e3a 1 \u3002\u6b63\u4ea4\u77e9\u9635\u5177\u6709\u6027\u8d28 $R R^T=I$ \uff0c\u5176\u4e2d $R^T$ \u662f\u77e9\u9635 $R$ \u7684\u8f6c\u7f6e\uff0c $I$ \u662f\u5355\u4f4d\u77e9\u9635\u3002<\/li>\n<li>\u6b63\u4ea4\u77e9\u9635\u4fdd\u8bc1\u4e86\u65cb\u8f6c\u4e0d\u4f1a\u6539\u53d8\u5411\u91cf\u7684\u957f\u5ea6\u3002<\/li>\n<li>\u884c\u5217\u5f0f\u4e3a1\u5219\u4fdd\u8bc1\u4e86\u65cb\u8f6c\u64cd\u4f5c\u662f\u4e0d\u6539\u53d8\u7269\u4f53\u4f53\u79ef\u7684\uff0c\u5373\u6ca1\u6709\u53cd\u5c04\u6216\u7f29\u653e\u53d1\u751f\u3002<\/li>\n<li>\u8fd9\u4e9b\u77e9\u9635\u4ee3\u8868\u4e86\u4e09\u7ef4\u7a7a\u95f4\u4e0b\u7684\u6240\u6709\u65cb\u8f6c\u64cd\u4f5c\uff0c\u5b83\u5305\u62ec\u4e86\u6240\u6709\u53ef\u80fd\u7684\u89d2\u5ea6\u548c\u7ed5\u4efb\u610f\u8f74\u7684\u65cb\u8f6c\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u80fd\u591f\u63cf\u8ff0\u6216\u6a21\u62df\u73b0\u5b9e\u4e16\u754c\u4e2d\u4efb\u4f55\u7269\u4f53\u7684\u65cb\u8f6c\u52a8\u4f5c\u3002<\/li>\n<li>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u4fdd\u6301\u5411\u91cf\u7684\u957f\u5ea6\u548c\u5939\u89d2\uff0c\u4ee5\u53ca\u6574\u4f53\u7684\u7a7a\u95f4\u4f53\u79ef\uff0c\u4f46\u5e76\u4e0d\u4fdd\u8bc1\u65cb\u8f6c\u540e\u7684\u5bf9\u8c61\u4f1a\u4fdd\u6301\u539f\u6709\u7684\u5bf9\u79f0\u6027\u6216\u5916\u89c2\u3002<\/li>\n<\/ul>\n<p><strong>SO(3)\u7fa4\u7684\u6027\u8d28\u5305\u62ec<\/strong>:<\/p>\n<ol>\n<li>\u95ed\u5408\u6027: \u4e24\u4e2a SO(3)\u7fa4\u7684\u5143\u7d20 (\u65cb\u8f6c) \u7684\u7ec4\u5408 (\u5373\u77e9\u9635\u4e58\u6cd5) \u4ecd\u7136\u662fSO(3)\u7fa4\u7684\u5143\u7d20\u3002<\/li>\n<li>\u5355\u4f4d\u5143\u5b58\u5728\u6027: \u5355\u4f4d\u77e9\u9635 $I$ \u662f $\\mathrm{SO}(3)$ \u7fa4\u7684\u5355\u4f4d\u5143\uff0c\u8868\u793a\u65e0\u65cb\u8f6c\u3002<\/li>\n<li>\u9006\u5143\u5b58\u5728\u6027: \u6bcf\u4e2a\u65cb\u8f6c\u77e9\u9635 $R$ \u5728 $\\mathrm{SO}(3)$ \u7fa4\u4e2d\u90fd\u6709\u4e00\u4e2a\u9006\u5143 $R^{-1}=R^T$ \uff0c\u4e5f\u662f\u4e00\u4e2a\u65cb\u8f6c\uff0c\u4f7f\u5f97 $R R^T=I$ \u3002<\/li>\n<li>\u7ed3\u5408\u6027: \u5bf9\u4e8e\u4efb\u610f\u4e09\u4e2a\u5143\u7d20 $R_1, R_2, R_3$, \u6ee1\u8db3 $\\left(R_1 R_2\\right) R_3=R_1\\left(R_2 R_3\\right)$ \u3002<\/li>\n<\/ol>\n<p>\u603b\u7684\u6765\u8bf4\uff0cSO3\u7fa4\u63d0\u4f9b\u4e86\u4e00\u79cd\u7cbe\u786e\u7684\u6570\u5b66\u5de5\u5177\uff0c\u7528\u4e8e\u6a21\u62df\u548c\u7406\u89e3\u771f\u5b9e\u4e16\u754c\u4e2d\u7684\u65cb\u8f6c\u73b0\u8c61\uff0c\u8fd9\u4e9b\u65cb\u8f6c\u73b0\u8c61\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u548c\u9ad8\u79d1\u6280\u884c\u4e1a\u4e2d\u65e0\u5904\u4e0d\u5728\u3002<\/p>\n<p align=\"center\">\n  <img decoding=\"async\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/07\/20240729200030846.png\" style=\"height:400px\">\n<\/p>\n<h2><img decoding=\"async\" src=\"https:\/\/img.icons8.com\/?size=100&id=H4Au8vy8EYkY&format=png&color=000000\" style=\"height:50px;display:inline\">  \u7fa4\u7684\u8868\u793a(Representation of Groups)<\/h2>\n<hr \/>\n<p><strong>\u7fa4\u7684\u8868\u793a\uff08Group Representation\uff09\u63d0\u4f9b\u4e86\u4e00\u79cd\u65b9\u5f0f\u6765\u201c\u5b9e\u73b0\u201d\u7fa4\u7684\u62bd\u8c61\u6982\u5ff5<\/strong>\uff0c\u5c06\u7fa4\u7684\u5143\u7d20\u8868\u73b0\u4e3a\u77e9\u9635\u6216\u7ebf\u6027\u7b97\u5b50\uff0c\u8fd9\u4f7f\u5f97\u7fa4\u7684\u7ed3\u6784\u548c\u7fa4\u4f5c\u7528\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\u7684\u8fd0\u7b97\u548c\u6027\u8d28\u6765\u5206\u6790\u548c\u7406\u89e3\u3002<\/p>\n<p>\u4ee5SO(3)\u7fa4\u4e3a\u4f8b\uff0c\u7fa4\u8868\u793a\u7a7a\u95f4\u662f\u4e09\u7ef4\u7a7a\u95f4\uff0c\u56e0\u4e3aSO(3)\u81ea\u7136\u5730\u4f5c\u7528\u4e8e\u4e09\u7ef4\u7a7a\u95f4\u3002<\/p>\n<ul>\n<li>\n<p>\u5728\u8fd9\u4e2a\u7a7a\u95f4\u4e2d\uff0c\u6bcf\u4e2aSO(3)\u7fa4\u7684\u5143\u7d20\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e2a3\u00d73\u7684\u5b9e\u6b63\u4ea4\u77e9\u9635\u6765\u8868\u793a\uff0c\u8fd9\u4e2a\u77e9\u9635\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u6267\u884c\u4e00\u4e2a\u65cb\u8f6c\u64cd\u4f5c\uff0c\u8868\u793a\u65cb\u8f6c\u7684\u8f74\u548c\u89d2\u5ea6\u3002<\/p>\n<\/li>\n<li>\n<p>\u4e14\u77e9\u9635\u6ee1\u8db3\u4e24\u4e2a\u6761\u4ef6\uff1a\u5176\u8f6c\u7f6e\u7b49\u4e8e\u5176\u9006\u77e9\u9635\uff0c\u4e14\u5176\u884c\u5217\u5f0f\u4e3a1\u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u4e0b\u9762\u5177\u4f53\u5c55\u793a\u5728\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b, \u7ed5\u4e09\u4e2a\u4e3b\u8f74\u4e00\u5373 $x$ \u8f74\u3001 $y$ \u8f74\u548c $z$ \u8f74\u7684 $\\mathrm{SO}(3)$ \u65cb\u8f6c\u77e9\u9635\u7684\u7fa4\u8868\u793a, \u8be5\u8868\u793a\u7ef4\u6570\u4e3a 3 \uff08\u4e09\u7ef4\u7a7a\u95f4\uff09, \u8fd9\u610f\u5473\u7740\u8868\u793a\u7a7a\u95f4\u662f\u7531\u4e09\u4e2a\u57fa\u5411\u91cf\u5f20\u6210\u7684, \u4efb\u4f55\u65cb\u8f6c\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a\u8fd9\u4e09\u4e2a\u57fa\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408, \u5b83\u4eec\u5206\u522b\u662f:<\/p>\n<ol>\n<li>\u7ed5 $z$ \u8f74\u65cb\u8f6c\u7684\u77e9\u9635 $R_3(\\psi)$, \u8868\u793a\u4e3a:<br \/>\n$$<br \/>\nR_3(\\psi)=\\left(\\begin{array}{ccc}<br \/>\n\\cos \\psi &amp; -\\sin \\psi &amp; 0 \\\\<br \/>\n\\sin \\psi &amp; \\cos \\psi &amp; 0 \\\\<br \/>\n0 &amp; 0 &amp; 1<br \/>\n\\end{array}\\right)<br \/>\n$$<\/li>\n<\/ol>\n<p>\u8fd9\u8868\u793a\u4e00\u4e2a\u5bf9\u8c61\u7ed5 $z$ \u8f74\u9006\u65f6\u9488\u65cb\u8f6c\u4e86\u89d2\u5ea6 $\\psi$ \u7684\u65cb\u8f6c\u3002<\/p>\n<ol start=\"2\">\n<li>\u7ed5 $y$ \u8f74\u65cb\u8f6c\u7684\u77e9\u9635 $R_2(\\phi)$, \u8868\u793a\u4e3a:<br \/>\n$$<br \/>\nR_2(\\phi)=\\left(\\begin{array}{ccc}<br \/>\n\\cos \\phi &amp; 0 &amp; \\sin \\phi \\\\<br \/>\n0 &amp; 1 &amp; 0 \\\\<br \/>\n-\\sin \\phi &amp; 0 &amp; \\cos \\phi<br \/>\n\\end{array}\\right)<br \/>\n$$<\/li>\n<\/ol>\n<p>\u8fd9\u8868\u793a\u4e00\u4e2a\u5bf9\u8c61\u7ed5 $y$ \u8f74\u9006\u65f6\u9488\u65cb\u8f6c\u4e86\u89d2\u5ea6 $\\phi$ \u7684\u65cb\u8f6c\u3002<\/p>\n<ol start=\"3\">\n<li>\u7ed5 $x$ \u8f74\u65cb\u8f6c\u7684\u77e9\u9635 $R_1(\\theta)$, \u8868\u793a\u4e3a:<br \/>\n$$<br \/>\nR_1(\\theta)=\\left(\\begin{array}{ccc}<br \/>\n1 &amp; 0 &amp; 0 \\\\<br \/>\n0 &amp; \\cos \\theta &amp; -\\sin \\theta \\\\<br \/>\n0 &amp; \\sin \\theta &amp; \\cos \\theta<br \/>\n\\end{array}\\right)<br \/>\n$$<\/li>\n<\/ol>\n<p>\u8fd9\u8868\u793a\u4e00\u4e2a\u5bf9\u8c61\u7ed5 $x$ \u8f74\u9006\u65f6\u9488\u65cb\u8f6c\u4e86\u89d2\u5ea6 $\\theta$ \u7684\u65cb\u8f6c\u3002<\/p>\n<p>\u8fd9\u4e9b\u77e9\u9635\u662fSO(3)\u7fa4\u7684\u6807\u51c6\u57fa\u7840\u5143\u7d20\uff0c\u4e5f\u662f\u6784\u5efa\u66f4\u590d\u6742\u65cb\u8f6c\u7684\u57fa\u672c\u6784\u4ef6\u3002<\/p>\n<p>\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c<strong>\u4efb\u610f\u7684\u65cb\u8f6c\u90fd\u53ef\u4ee5\u901a\u8fc7\u8fd9\u4e9b\u57fa\u7840\u65cb\u8f6c\u7684\u7ec4\u5408\u6765\u5b9e\u73b0\u3002<\/strong><\/p>\n<ul>\n<li>\u8fd9\u4e9b\u77e9\u9635\u7684\u7279\u70b9\u662f\uff1a\n<ul>\n<li>\u5b83\u4eec\u90fd\u662f\u6b63\u4ea4\u7684\uff08\u5373\u77e9\u9635\u4e58\u4ee5\u5b83\u7684\u8f6c\u7f6e\u7b49\u4e8e\u5355\u4f4d\u77e9\u9635\uff09\u4e14\u884c\u5217\u5f0f\u4e3a1\uff0c\u6b63\u597d\u7b26\u5408SO(3)\u7fa4\u7684\u5b9a\u4e49\u3002<\/li>\n<li>SO(3)\u7fa4\u7684\u8868\u793a\u662f\u540c\u6001\u7684\uff0c\u8fd9\u610f\u5473\u7740\u5982\u679c\u6211\u4eec\u5c06SO(3)\u7fa4\u7684\u4e24\u4e2a\u5143\u7d20\uff08\u4e24\u4e2a\u65cb\u8f6c\u77e9\u9635\uff09\u76f8\u4e58\uff0c\u7ed3\u679c\u4e5f\u5e94\u8be5\u662fSO(3)\u7fa4\u5185\u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u5e76\u4e14\u8fd9\u4e2a\u4e58\u6cd5\u8fd0\u7b97\u5728\u8868\u793a\u7a7a\u95f4\u4e0a\u662f\u8fde\u7eed\u7684\u3002<\/li>\n<li>\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u7684\u7ec4\u5408\u5bf9\u5e94\u4e8e\u8fd9\u4e9b\u65cb\u8f6c\u77e9\u9635\u7684\u77e9\u9635\u4e58\u6cd5\u3002\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0bSO(3)\u65cb\u8f6c\u77e9\u9635\u7684\u7fa4\u8868\u793a\u5c06\u65cb\u8f6c\u8fd9\u4e00\u62bd\u8c61\u6982\u5ff5\u5177\u8c61\u5316\uff0c\u6765\u63cf\u8ff0\u548c\u8ba1\u7b97\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u5bf9\u8c61\u7684\u53d6\u5411\u53d8\u5316\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u53e6\u4e00\u79cd\u63cf\u8ff0\u4e09\u7ef4\u65cb\u8f6c\u7684\u5e38\u89c1\u65b9\u6cd5\u662f<strong>\u6b27\u62c9\u89d2<\/strong>, \u5b83\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f<strong>\u4e00\u7cfb\u5217\u6309\u7279\u5b9a\u987a\u5e8f\u6267\u884c\u7684\u65cb\u8f6c\u7684\u7ec4\u5408<\/strong>\u3002<\/p>\n<ul>\n<li>\n<p>\u901a\u5e38, \u6b27\u62c9\u89d2\u7531\u4e09\u4e2a\u6570\u503c\u7ec4\u6210: $\\alpha, \\beta, \\gamma$, \u5206\u522b\u5bf9\u5e94\u4e8e\u7ed5\u4e0d\u540c\u8f74\u7684\u65cb\u8f6c\u89d2\u5ea6\u3002<\/p>\n<\/li>\n<li>\n<p>\u8fd9\u4e9b\u89d2\u5ea6\u5b9a\u4e49\u4e86\u4ece\u4e00\u4e2a\u5750\u6807\u7cfb\u8f6c\u6362\u5230\u53e6\u4e00\u4e2a\u5750\u6807\u7cfb\u7684\u65cb\u8f6c\u3002<\/p>\n<\/li>\n<li>\n<p>\u5728 $\\mathrm{SO}(3)$ \u7fa4\u7684\u4e0a\u4e0b\u6587\u4e2d, \u6b27\u62c9\u89d2\u63cf\u8ff0\u4e86\u901a\u8fc7\u4e09\u6b21\u65cb\u8f6c\u5c06\u4e00\u4e2a\u521a\u4f53\u5728\u53c2\u8003\u5750\u6807\u7cfb\u7684\u67d0\u4e00\u51e0\u4f55\u72b6\u6001\u65cb\u8f6c\u5230\u5230\u53e6\u4e00\u4e2a\u51e0\u4f55\u72b6\u6001\u8fc7\u7a0b\u3002<\/p>\n<\/li>\n<li>\n<p>\u5177\u4f53\u7684\u65cb\u8f6c\u987a\u5e8f\uff08\u4f8b\u5982 \u201cZ-Y-X\u201d \u6216 \u201cZ-Y-Z\u201d ) \u5c06\u51b3\u5b9a\u8fd9\u4e9b\u89d2\u5ea6\u7684\u51e0\u4f55\u610f\u4e49\u3002<\/p>\n<\/li>\n<li>\n<p>\u6bcf\u4e00\u6b21\u65cb\u8f6c\u90fd\u76f8\u5bf9\u4e8e\u4e00\u4e2a\u7279\u5b9a\u7684\u8f74, \u5e76\u4e14\u6bcf\u6b21\u65cb\u8f6c\u90fd\u662f\u5728\u524d\u4e00\u6b21\u65cb\u8f6c\u4e4b\u540e\u7684\u5750\u6807\u7cfb\u4e2d\u8fdb\u884c\u7684\u3002<\/p>\n<\/li>\n<\/ul>\n<p>\u4ee5 \u201cZ-Y-X\u201d \u6b27\u62c9\u89d2\u4e3a\u4f8b, \u6574\u4e2a\u65cb\u8f6c\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a:<\/p>\n<ol>\n<li>\u9996\u5148\u7ed5\u7740\u521d\u59cb\u5750\u6807\u7cfb\u7684 $z$ \u8f74\u65cb\u8f6c $\\psi$ \u89d2\u5ea6\uff08\u504f\u822a\uff09\u3002<\/li>\n<li>\u63a5\u7740\u5728\u65b0\u5750\u6807\u7cfb\u4e2d\u7ed5 $y$ \u8f74\u65cb\u8f6c $\\theta$ \u89d2\u5ea6 (\u4fef\u4ef0)\u3002<\/li>\n<li>\u6700\u540e\u5728\u518d\u6b21\u66f4\u65b0\u7684\u5750\u6807\u7cfb\u4e2d\u7ed5 $x$ \u8f74\u65cb\u8f6c $\\phi$ \u89d2\u5ea6 (\u6eda\u8f6c)\u3002<\/li>\n<\/ol>\n<p>\u8fd9\u79cd\u65b9\u5f0f\u5b9a\u4e49\u7684\u65cb\u8f6c\u77e9\u9635 $R$ \u662f\u5355\u4e2a\u65cb\u8f6c\u77e9\u9635\u7684\u4e58\u79ef:<\/p>\n<p>$$<br \/>\nR=R_z(\\psi) \\cdot R_y(\\theta) \\cdot R_x(\\phi)<br \/>\n$$<\/p>\n<p>\u5728\u8fd9\u91cc, $R_z(\\psi), R_y(\\theta)$ \u548c $R_x(\\phi)$ \u5206\u522b\u4ee3\u8868\u7ed5 $z, y$ \u548c $x$ \u8f74\u7684\u57fa\u7840\u65cb\u8f6c\u77e9\u9635\u3002\u8fd9\u4e9b\u57fa\u7840\u65cb\u8f6c\u77e9\u9635\u662f\u6b63\u4ea4\u77e9\u9635\uff0c\u5e76\u4e14\u5b83\u4eec\u7684\u884c\u5217\u5f0f\u4e3a1\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u4eec\u6ee1\u8db3SO(3)\u7fa4\u7684\u6761\u4ef6\uff0c\u4e14\u65cb\u8f6c\u64cd\u4f5c\u4e0d\u4f1a\u6539\u53d8\u5bf9\u8c61\u7684\u5927\u5c0f\u548c\u5f62\u72b6\u3002<\/p>\n<p>\u6700\u540e\uff0c\u7b80\u5355\u63d0\u4e00\u4e0b\u7fa4\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\u3002<\/p>\n<p>\u4e0d\u53ef\u7ea6\u8868\u793a\u662f\u6570\u5b66\u4e2d\u7fa4\u8868\u793a\u8bba\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\uff0c<strong>\u5982\u679c\u4e00\u4e2a\u7fa4\u7684\u8868\u793a\u4e0d\u53ef\u4ee5\u88ab\u8fdb\u4e00\u6b65\u5206\u89e3\u4e3a\u66f4\u7b80\u5355\u7684\u5b50\u8868\u793a\uff0c\u90a3\u5c31\u662f\u4e0d\u53ef\u7ea6\u8868\u793a<\/strong>\u3002\u4efb\u4f55\u6709\u9650\u7ef4\u7684\u7fa4\u8868\u793a\u90fd\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e0d\u53ef\u7ea6\u8868\u793a\u7684\u76f4\u548c\u3002\u4e0d\u53ef\u7ea6\u8868\u793a\u53ef\u4ee5\u770b\u4f5c\u662f\u7fa4\u8868\u793a\u7684\u201c\u57fa\u672c\u6784\u4ef6\u201d\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Deep Learning Math \u7fa4\u8bba\uff08group Theory\uff09 \u5728\u5b66\u4e60\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u4e4b\u524d\uff0c\u5177\u5907\u7fa4\u8bba\u7684\u57fa\u7840 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1635,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1582","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-14"],"_links":{"self":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1582","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=1582"}],"version-history":[{"count":6,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1582\/revisions"}],"predecessor-version":[{"id":1713,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/1582\/revisions\/1713"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/media\/1635"}],"wp:attachment":[{"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1582"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1582"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1582"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}