{"id":46246,"date":"2023-11-01T20:02:00","date_gmt":"2023-11-01T12:02:00","guid":{"rendered":"https:\/\/swarma.org\/?p=46246"},"modified":"2023-11-01T20:02:00","modified_gmt":"2023-11-01T12:02:00","slug":"%e5%bd%93%e6%9c%ba%e5%99%a8%e5%ad%a6%e4%b9%a0%e9%81%87%e8%a7%81%e6%8b%93%e6%89%91%ef%bc%9a%e6%8b%93%e6%89%91%e6%95%b0%e6%8d%ae%e5%88%86%e6%9e%90%e4%b8%8e%e6%8b%93%e6%89%91%e6%b7%b1%e5%ba%a6%e5%ad%a6","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=46246","title":{"rendered":"\u5f53\u673a\u5668\u5b66\u4e60\u9047\u89c1\u62d3\u6251\uff1a\u62d3\u6251\u6570\u636e\u5206\u6790\u4e0e\u62d3\u6251\u6df1\u5ea6\u5b66\u4e60"},"content":{"rendered":"
\n
\n
\n
\n

<\/p>\n<\/section>\n

\n
\n
\n
\n
\n
\n

<\/section>\n<\/section>\n
\n
\n

\u5bfc\u8bed<\/strong><\/p>\n<\/section>\n<\/section>\n

\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n
\n
\u4f5c\u4e3a\u6570\u5b66\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u62d3\u6251\u5b66\u4ee5\u72ec\u7279\u7684\u65b9\u5f0f\u63cf\u8ff0\u7a7a\u95f4\u7684\u6027\u8d28\u548c\u7ed3\u6784\u3002\u8fd1\u5e74\u6765\uff0c\u51e0\u4f55\u548c\u62d3\u6251\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\uff0c\u5c24\u5176\u662f\u62d3\u6251\u6a21\u578b\uff0c\u5728\u6570\u636e\u8868\u793a\u548c\u7279\u5f81\u63d0\u53d6\u65b9\u9762\u6709\u7740\u91cd\u8981\u4f5c\u7528\u3002\u62d3\u6251\u6570\u636e\u5206\u6790\uff08Topological Data Analysis, TDA\uff09\u690d\u6839\u4e8e\u4ee3\u6570\u62d3\u6251\u548c\u8ba1\u7b97\u62d3\u6251\uff0c\u5728\u5904\u7406\u5177\u6709\u7ed3\u6784\u6027\u7684\u6570\u636e\u4e0a\u5f97\u5230\u4e86\u6781\u5927\u7684\u53d1\u5c55\uff0c\u5e76\u9010\u6e10\u6210\u4e3a Math for AI \u7684\u4e00\u4e2a\u91cd\u8981\u65b9\u9762\u3002<\/span><\/strong><\/section>\n

<\/span><\/strong><\/section>\n
\u5728\u96c6\u667a\u4ff1\u4e50\u90e8\u300c\u6570\u5b66\u4e0e\u4eba\u5de5\u667a\u80fd\u8bfb\u4e66\u4f1a<\/a>\u300d\u4e2d\uff0c\u590f\u514b\u6797\u8001\u5e08\u8ba8\u8bba\u4e86\u62d3\u6251\u6570\u636e\u5206\u6790\uff08Topological Data Analysis, TDA\uff09\u7684\u4e3b\u8981\u601d\u60f3\u548c\u6a21\u578b\uff0c\u9996\u5148\u4ecb\u7ecd\u4e86\u57fa\u672c\u7684\u62d3\u6251\u6570\u636e\u8868\u793a\u6a21\u578b\uff0c\u5c24\u5176\u662f\u57fa\u4e8e\u6570\u636e\u7684\u5355\u7eaf\u590d\u5f62\u6784\u9020\uff0c\u4ee5\u53ca\u548c\u4f20\u7edf\u56fe\u6a21\u578b\u7684\u5dee\u5f02\uff0c\u4e4b\u540e\u4ecb\u7ecd\u4e86\u57fa\u4e8e\u5355\u7eaf\u590d\u5f62\u7684\u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\u3002\u62d3\u6251\u6570\u636e\u5206\u6790\u5728\u523b\u753b\u590d\u6742\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u65b9\u9762\u5c55\u793a\u51fa\u6781\u5927\u7684\u4f18\u8d8a\u6027\uff0c\u5c24\u5176\u662f\u5b83\u53ef\u4ee5\u523b\u753b\u4f53\u7cfb\u6700\u672c\u8d28\u7684\u62d3\u6251\u4fe1\u606f\u3002\u62d3\u6251\u6570\u636e\u5206\u6790\u5c06\u8fdb\u4e00\u6b65\u4fc3\u8fdb\u6211\u4eec\u5bf9\u6570\u636e\u7684\u672c\u8d28\u4fe1\u606f\u7684\u6316\u6398\u548c\u523b\u753b\uff0c\u4e3a\u63d0\u9ad8\u673a\u5668\u5b66\u4e60\u6a21\u578b\u7684\u7cbe\u5ea6\u3001\u53ef\u89e3\u91ca\u6027\u3001\u8fc1\u79fb\u6027\u7b49\u6253\u4e0b\u575a\u5b9e\u7684\u6570\u5b66\u57fa\u7840\u3002<\/span><\/strong><\/section>\n<\/section>\n
\n
<\/span><\/strong><\/span><\/strong><\/section>\n<\/section>\n
\n
<\/span><\/strong><\/section>\n
<\/span><\/strong><\/section>\n<\/section>\n
\n
<\/span><\/section>\n

<\/section>\n
\u7814\u7a76\u9886\u57df\uff1a\u590d\u6742\u7cfb\u7edf\uff0cMath for AI\uff0c\u62d3\u6251\u6570\u636e\u5206\u6790\uff0c\u5355\u7eaf\u590d\u5f62\uff0c\u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\uff0c\u56fe\u795e\u7ecf\u7f51\u7edc\uff0c\u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/section>\n<\/section>\n
\n
<\/span><\/strong><\/section>\n<\/section>\n
\n
<\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n
\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n
<\/section>\n<\/section>\n
\n
<\/span><\/strong><\/section>\n
\u590f\u514b\u6797<\/span><\/span><\/span> <\/span>| \u8bb2\u8005<\/span><\/strong><\/span><\/strong><\/section>\n
\u738b\u81f3\u5b8f<\/span> <\/span>| \u6574\u7406<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/section>\n
\u6881\u91d1<\/span> <\/span>| \u7f16\u8f91<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/section>\n
<\/span><\/strong><\/span><\/strong><\/section>\n<\/section>\n

<\/section>\n
\n
\n
\n

\u76ee\u5f55<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
1. \u6570\u636e\u7684\u62d3\u6251\u8868\u793a<\/strong><\/span><\/section>\n
2. \u62d3\u6251\u6570\u636e\u5904\u7406\u7684\u7279\u5f81<\/strong><\/span><\/section>\n
3. \u62d3\u6251\u6df1\u5ea6\u5b66\u4e60<\/strong><\/span><\/section>\n
4. \u57fa\u4e8e\u5355\u7eaf\u590d\u5f62\u7684\u56fe\u795e\u7ecf\u7f51\u7edc<\/strong><\/span><\/section>\n


<\/p>\n


<\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

1. \u6570\u636e\u7684\u62d3\u6251\u8868\u793a<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u672c\u6587\u4ee5\u5206\u5b50\u6570\u636e\u5904\u7406\u4e3a\u5207\u5165\u70b9\uff0c\u63a2\u8ba8\u62d3\u6251\u6570\u636e\u5206\u6790<\/span>\uff08Topological Data Analysis, TDA\uff09<\/span>\u7684\u5e94\u7528\u548c\u7279\u70b9\u3002AI \u5728\u6570\u636e\u5904\u7406\u4e0a\u7684\u4e24\u4e2a\u5173\u952e\u73af\u8282\u2014\u2014\u6570\u636e\u8868\u5f81<\/strong>\u548c\u5efa\u6a21\u5206\u6790\u7279\u6027<\/strong>\uff0c\u4e0e\u62d3\u6251\u6570\u636e\u5206\u6790\u6709\u7740\u7d27\u5bc6\u7684\u8054\u7cfb\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5c55\u5f00\u4ecb\u7ecd\u8fd9\u4e24\u4e2a\u73af\u8282\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe1. \u57fa\u4e8e AI \u7684\u5206\u5b50\u6570\u636e\u5206\u6790<\/span><\/section>\n

<\/section>\n

<\/section>\n
\n
\n
\n

Tauzin, Guillaume, et al. “giotto-tda: A topological data analysis toolkit for machine learning and data exploration.” The Journal of Machine Learning Research 22.1 (2021): 1834-1839.<\/span>
<\/p>\n

\u6587\u7ae0\u603b\u7ed3\u4e86\u62d3\u6251\u6570\u636e\u548c\u673a\u5668\u5b66\u4e60\u7684\u7ed3\u5408\u7684\u76f8\u5173\u7406\u8bba\u3002<\/span><\/section>\n
Chazal, Fr\u00e9d\u00e9ric, and Bertrand Michel. “An introduction to topological data analysis: fundamental and practical aspects for data scientists.” Frontiers in artificial intelligence 4 (2021): 108.<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n

<\/section>\n
1.1 \u6570\u636e\u8868\u5f81<\/span><\/strong><\/span>
<\/section>\n

<\/section>\n
\u5728\u5904\u7406\u56fe\u50cf\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u751f\u6210\u76f8\u5e94\u7684\u6570\u636e\u8868\u793a\u3002\u4f8b\u5982\u901a\u8fc7\u63d0\u53d6\u7279\u5b9a\u7684\u7279\u5f81\u70b9\u6784\u5efa\u7f51\u683c\u6a21\u578b\u8fdb\u884c\u4eba\u8138\u8bc6\u522b\u3002\u9664\u4e86\u7f51\u683c\u6a21\u578b\uff0c\u8fd8\u6709\u5176\u4ed6\u5982\u7279\u5f81\u56fe\u548c\u70ed\u5ea6\u56fe\u7b49\u4e0d\u540c\u7684\u6570\u636e\u8868\u793a\u65b9\u5f0f\u3002\u867d\u7136\u6e90\u81ea\u540c\u4e00\u56fe\u50cf\u6570\u636e\uff0c\u4f46\u4ece\u6570\u5b66\u89d2\u5ea6\u53ef\u4ee5\u5efa\u7acb\u8d77\u4e0d\u540c\u7684\u6a21\u578b\uff1a\u6700\u7b80\u5355\u7684\u77e9\u9635\u6a21\u578b\uff0c\u6216\u8005\u70b9\u9635\u6a21\u578b\u3001\u7f51\u683c\u6a21\u578b\uff0c\u751a\u81f3\u66f4\u590d\u6742\u7684\u51fd\u6570\u6a21\u578b\u3002\u4e00\u65e6\u6570\u5b66\u6a21\u578b\u5efa\u7acb\uff0c\u5c31\u53ef\u4ee5\u57fa\u4e8e\u6a21\u578b\u63d0\u53d6\u7279\u5f81\uff0c\u5e76\u4e0e\u540e\u7eed\u5173\u6ce8\u7684\u4fe1\u606f\u4ea7\u751f\u8054\u7cfb\uff0c\u5982\u901a\u8fc7\u591a\u5c42\u611f\u77e5\u5668<\/span>\uff08MLP\uff09<\/span>\u8fdb\u884c\u9884\u6d4b\u7b49\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe2. \u4eba\u8138\u8bc6\u522b\u6a21\u578b<\/span><\/section>\n

<\/section>\n
\u7c7b\u4f3c\u5730\uff0c\u5728\u5904\u7406\u5206\u5b50\u6570\u636e<\/span>\uff08\u5982\u5c0f\u5206\u5b50\u6570\u636e\u548c\u86cb\u767d\u8d28\u6570\u636e\uff09<\/span>\u65f6\uff0c\u4e5f\u6709\u591a\u79cd\u4e0d\u540c\u7684\u6570\u636e\u8868\u793a\u65b9\u5f0f\u3002\u4e00\u79cd\u5e38\u89c1\u65b9\u6cd5\u662f\u57fa\u4e8e\u5171\u4ef7\u952e\u7684\u56fe\u8868\u5f81<\/strong>\uff0c\u5176\u4e2d\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u539f\u5b50\uff0c\u8fb9\u4ee3\u8868\u5171\u4ef7\u952e\uff0c\u5f62\u6210\u4e00\u79cd\u56fe\u7684\u8868\u793a\u5f62\u5f0f\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe3. \u4e0d\u540c\u7684\u5206\u5b50\u6a21\u578b<\/span><\/section>\n

<\/section>\n
\u9664\u6b64\u4e4b\u5916\uff0c\u8fd8\u6709\u51e0\u4f55\u65b9\u6cd5<\/strong>\uff0c\u4f8b\u5982\u5c06\u539f\u5b50\u770b\u4f5c\u534a\u5f84\u56fa\u5b9a\u7684\u7403\u4f53\uff0c\u89c2\u5bdf\u8005\u53ef\u4ee5\u4ece\u7403\u4f53\u96c6\u5408\u7684\u5916\u90e8\uff0c\u5373\u5206\u5b50\u7684\u8868\u9762\u8fdb\u884c\u7814\u7a76\uff0c\u67e5\u770b\u5176\u8868\u9762\u79ef\u6216\u51f8\u51f9\u533a\u57df\u3002\u8fd9\u4e9b\u51f8\u51f9\u533a\u57df\u4e0e\u539f\u5b50\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u4fe1\u606f\u6709\u5173\uff0c\u8fd9\u79cd\u63cf\u8ff0\u66f4\u504f\u5411\u51e0\u4f55\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u66f4\u8fdb\u4e00\u6b65\uff0c\u8fd8\u53ef\u4ee5\u901a\u8fc7\u5bc6\u5ea6\u6cdb\u51fd\u7406\u8bba\u6765\u8ba1\u7b97\u7535\u5b50\u5bc6\u5ea6\u6216\u7535\u5b50\u51fd\u6570\u5206\u5e03<\/strong>\uff0c\u5c06\u5206\u5b50\u6570\u636e\u8f6c\u5316\u4e3a\u4e00\u79cd\u7a7a\u95f4\u5f62\u6001\u7684\u6570\u636e\u8868\u73b0\u5f62\u5f0f\u3002\u56e0\u6b64\uff0c\u5c3d\u7ba1\u6e90\u81ea\u540c\u6837\u7684\u5206\u5b50\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4ece\u591a\u4e2a\u89d2\u5ea6\u5bf9\u5176\u8fdb\u884c\u8868\u5f81<\/strong>\u3002\u4e00\u65e6\u5b8c\u6210\u8868\u5f81\uff0c\u5c31\u53ef\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u63d0\u53d6\u5404\u79cd\u7279\u6027\uff0c\u5305\u62ec\u5404\u79cd\u6307\u7eb9<\/span>\uff08fingerprint\uff09<\/span>\u548c\u63cf\u8ff0\u7b26<\/span>\uff08descriptor\uff09<\/span>\u7b49\u7b49\u3002\u8fd9\u4e9b\u6027\u8d28\u53ef\u80fd\u548c\u6700\u7ec8\u60f3\u8981\u7406\u89e3\u7684\u529f\u80fd\u4ea7\u751f\u8054\u7cfb\uff0c\u4f8b\u5982\u6c34\u6eb6\u6027\u3001\u8102\u6eb6\u6027\u3001\u6bd2\u6027\u7b49\u3002<\/span><\/section>\n

<\/span><\/section>\n
1.2 \u5206\u5b50\u7ed3\u6784\u7684\u5efa\u6a21<\/span><\/strong><\/span>
<\/section>\n

<\/span><\/section>\n
\u5728\u6784\u5efa\u5173\u4e8e\u5206\u5b50\u529f\u80fd\u6a21\u578b\u7684\u8fc7\u7a0b\u4e2d\uff0c\u5927\u91cf\u4f7f\u7528\u4e86\u7ed3\u6784\u6570\u636e\u3002\u8fd9\u662f\u56e0\u4e3a\u5206\u5b50\u7684\u7ed3\u6784\u548c\u5176\u529f\u80fd\u4e4b\u95f4\u5b58\u5728\u5f3a\u70c8\u7684\u5173\u7cfb\uff0c\u88ab\u79f0\u4e3a\u201c\u7ed3\u6784-\u529f\u80fd\u5173\u7cfb<\/strong>\u201d<\/span>\uff08Structure-Function Relationship\uff09<\/strong><\/span>\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u4f8b\u5982\uff0c\u79bb\u5b50\u901a\u9053\u86cb\u767d\u8d28\u7684\u663e\u8457\u7279\u70b9\u662f\u5b83\u4eec\u4e2d\u5fc3\u6709\u4e00\u4e2a\u6d1e<\/span>\uff08\u56fe4\u5de6\u4e0a\uff09<\/span>\uff0c\u8fd9\u4e2a\u6d1e\u5bf9\u79bb\u5b50\u901a\u9053\u7684\u529f\u80fd\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u5b83\u65b9\u4fbf\u4e86\u7ec6\u80de\u819c\u5916\u90e8\u7684\u79bb\u5b50\u8fdb\u5165\u819c\u5185\uff0c\u6216\u8005\u819c\u5185\u7684\u79bb\u5b50\u79bb\u5f00\u7ec6\u80de\u3002\u53e6\u4e00\u4e2a\u4f8b\u5b50\u662f\u86cb\u767d\u8d28\u7b3c<\/span>\uff08\u56fe4\u53f3\u4e0b\uff09<\/span>\uff0c\u8fd9\u79cd\u86cb\u767d\u8d28\u7684\u8868\u9762\u6709\u4e00\u5b9a\u7ed3\u6784\uff0c\u4f46\u5176\u5185\u90e8\u662f\u7a7a\u7684\uff0c\u5c31\u50cf\u7528\u6765\u88c5\u4e1c\u897f\u7684\u76d2\u5b50\uff0c\u8fd9\u79cd\u7a7a\u5fc3\u7684\u7ed3\u6784\u6709\u5229\u4e8e\u67d0\u4e9b\u7269\u8d28\u7684\u5b58\u50a8\u548c\u8fd0\u8f93\u3002\u6700\u540e\u4e00\u4e2a\u4f8b\u5b50\u662f\u5177\u6709\u4e24\u4e2a\u56fa\u5b9a\u533a\u57df\uff0c\u5e76\u901a\u8fc7\u4e00\u4e2a\u7075\u6d3b\u8fde\u63a5\u533a\u57df\u76f8\u8fde\u7684\u5206\u5b50\u3002\u8fd9\u79cd\u7ed3\u6784\u53ef\u4ee5\u5f62\u6210\u4e00\u79cd\u5f00\u5173\u72b6\u6001\uff0c\u4f7f\u5f97\u5206\u5b50\u80fd\u591f\u5904\u4e8e\u6fc0\u53d1\u6001\u6216\u975e\u6fc0\u53d1\u6001\uff0c\u4ece\u800c\u5f71\u54cd\u5176\u529f\u80fd\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe4. \u86cb\u767d\u8d28\u5206\u5b50\u7ed3\u6784<\/span><\/section>\n

<\/span><\/section>\n
\u65e0\u8bba\u662f\u901a\u8fc7\u5171\u4ef7\u952e\u8fde\u63a5\u8fd8\u662f\u901a\u8fc7\u975e\u5171\u4ef7\u952e\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u90fd\u4f1a\u5f71\u54cd\u6700\u7ec8\u7684\u7a33\u5b9a\u6001\u7ed3\u6784\uff0c\u8fd9\u79cd\u7a33\u5b9a\u6001\u7ed3\u6784\u4e0e\u5206\u5b50\u529f\u80fd\u7d27\u5bc6\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u63cf\u8ff0\u5206\u5b50\u7684\u7ed3\u6784\u5bf9\u4e8e\u7406\u89e3\u5206\u5b50\u529f\u80fd<\/strong>\u5177\u6709<\/strong>\u91cd\u8981\u4f5c\u7528<\/strong>\u3002\u4e3a\u66f4\u597d\u5730\u63cf\u7ed8\u5206\u5b50\u7684\u7ed3\u6784\uff0c\u5927\u91cf\u7684\u63cf\u8ff0\u7b26<\/span>\uff08\u65e0\u8bba\u662f\u7ec4\u5408\u91cf\uff0c\u4ee3\u6570\u91cf\u8fd8\u662f\u51e0\u4f55\u91cf\uff09<\/span>\u88ab\u63d0\u53d6\u4e86\u51fa\u6765\u3002\u5728\u8fd9\u4e9b\u63cf\u8ff0\u7b26\u4e2d\uff0c\u6709\u4e00\u4e9b\u5173\u6ce8\u62d3\u6251\u7279\u6027\uff0c\u6bd4\u5982\u56fe\u4e0a\u5411\u91cf\u3001\u51e0\u4f55\u91cf\u7b49\u7b49\uff0c\u8fd8\u6709\u4e00\u4e9b\u5173\u6ce8\u7ec4\u5408\u6216\u90bb\u8fd1\u4fe1\u606f\u7684\u6307\u7eb9\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe5. \u5316\u5b66\u63cf\u8ff0\u7b26<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u5927\u91cf\u5173\u4e8e\u7ed3\u6784\u7684\u63cf\u8ff0\u91cf\u4e2d\uff0c\u53ef\u80fd\u5b58\u5728\u67d0\u4e9b\u66f4\u672c\u8d28\u3001\u66f4\u5168\u5c40\u7684\u91cf\uff0c\u5b83\u4eec\u80fd\u591f\u66f4\u597d\u5730\u6293\u4f4f\u7ed3\u6784\u7684\u6574\u4f53\u4fe1\u606f\uff0c\u4ece\u800c\u5728\u7406\u89e3\u548c\u63cf\u7ed8\u5206\u5b50\u7684\u529f\u80fd\u65b9\u9762\u8d77\u5230\u66f4\u91cd\u8981\u7684\u4f5c\u7528<\/strong>\u3002\u8fd9\u5c31\u5f15\u51fa\u4e86\u62d3\u6251\u6570\u636e\u5904\u7406\u7684\u6838\u5fc3\uff1a\u901a\u8fc7\u62d3\u6251\u4e0d\u53d8\u91cf\u6765\u63cf\u8ff0\u6570\u636e<\/strong>\u3002<\/span><\/section>\n

<\/section>\n

<\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

2. \u62d3\u6251\u6570\u636e\u5904\u7406\u7684\u7279\u5f81<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n

<\/section>\n

\u62d3\u6251\u6570\u636e\u5206\u6790\u4e0e\u4f20\u7edf\u7684\u5de5\u5177\u76f8\u6bd4\u6709\u4e09\u4e2a\u4e3b\u8981\u7279\u5f81\uff1a<\/span><\/p>\n

1\uff09\u5355\u7eaf\u590d\u5f62<\/span><\/strong>\uff1a\u91c7\u7528\u5355\u7eaf\u590d\u5f62<\/span>\uff08simplicial complex\uff09<\/span>\u7684\u63cf\u8ff0\u65b9\u5f0f\uff0c\u76f8\u6bd4\u56fe\u63cf\u8ff0\u80fd\u6355\u83b7\u6570\u636e\u4e2d\u66f4\u4e30\u5bcc\u7684\u62d3\u6251\u548c\u51e0\u4f55\u4fe1\u606f\u3002<\/span><\/p>\n

2\uff09\u62d3\u6251\u4e0d\u53d8\u91cf<\/span><\/strong>\uff1a\u62d3\u6251\u6570\u636e\u5206\u6790\u4f7f\u7528\u62d3\u6251\u4e0d\u53d8\u91cf\uff0c\u800c\u975e\u4ec5\u4ec5\u4f9d\u8d56\u4e8e\u7edf\u8ba1\u91cf\u6216\u63cf\u8ff0\u6027\u91cf\u3002\u8fd9\u4e9b\u62d3\u6251\u4e0d\u53d8\u91cf\u53ef\u4ee5\u63d0\u4f9b\u5bf9\u6570\u636e\u7684\u6df1\u5ea6\u7406\u89e3\uff0c\u5305\u62ec\u6570\u636e\u7684\u8fde\u63a5\u6027\u3001\u6d1e\u7b49\u590d\u6742\u7ed3\u6784\u3002<\/span><\/p>\n

3\uff09\u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span><\/strong>\uff1a<\/span>\u62d3\u6251\u6570\u636e\u5206\u6790\u5305\u542b\u4e00\u4e2a\u8fc7\u6ee4\u6d41\u8fc7\u7a0b\uff0c\u8fd9\u4e2a\u8fc7\u6ee4\u6d41\u8fc7\u7a0b\u53ef\u4ee5\u4e0e\u7cfb\u7edf\u5185\u7684\u591a\u5c3a\u5ea6\u63cf\u8ff0\u5f88\u597d\u5730\u7ed3\u5408\u3002\u901a\u8fc7\u4ece\u4e0d\u540c\u7684\u5c3a\u5ea6\u53bb\u89c2\u5bdf\u548c\u5206\u6790\u6570\u636e\uff0c\u6211\u4eec\u80fd\u591f\u5f97\u5230\u66f4\u5168\u9762\u7684\u4fe1\u606f\u3002<\/span><\/section>\n

<\/span><\/section>\n
2.1 \u5355\u7eaf\u590d\u5f62<\/span><\/strong><\/span><\/section>\n

<\/section>\n
\u5728\u975e\u6570\u5b66\u9886\u57df\uff0c\u5982\u8ba1\u7b97\u673a\u79d1\u5b66\u3001\u5de5\u7a0b\u548c\u751f\u7269\u5b66\u4e2d\uff0c\u4eba\u4eec\u901a\u5e38\u4f7f\u7528\u56fe\u6765\u8868\u8fbe\u5b9e\u4f53\u4e4b\u95f4\u7684\u8fde\u63a5\u5173\u7cfb\u3002\u7136\u800c\uff0c\u5728\u57fa\u7840\u6570\u5b66\u9886\u57df\u4e2d\uff0c\u66f4\u7ecf\u5e38\u4f7f\u7528\u7684\u662f\u79f0\u4e3a\u5355\u7eaf\u590d\u5f62<\/span>\uff08Simplicial Complex\uff09<\/span>\u7684\u63cf\u8ff0\u65b9\u5f0f\u3002\u4f5c\u4e3a\u9ad8\u7ea7\u7684\u62d3\u6251\u5de5\u5177\uff0c\u5355\u7eaf\u590d\u5f62\u80fd\u66f4\u597d\u5730\u63cf\u7ed8\u590d\u6742\u7cfb\u7edf\u4e2d\u7684\u7ed3\u6784\u4fe1\u606f\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u4e0e\u56fe\u76f8\u6bd4\uff0c\u5355\u7eaf\u590d\u5f62\u6709\u51e0\u4e2a\u91cd\u8981\u7684\u4e0d\u540c\u70b9\uff1a<\/span><\/section>\n

<\/section>\n
1\uff09\u9ad8\u7ef4\u5ea6\u63cf\u7ed8\uff1a<\/span><\/strong>\u9664\u8868\u793a\u8282\u70b9\u548c\u8fb9<\/span>\uff08\u53730\u7ef4\u548c1\u7ef4\u7684\u5bf9\u8c61\uff09<\/span>\uff0c\u5355\u7eaf\u590d\u5f62\u53ef\u4ee5\u8868\u793a\u66f4\u9ad8\u7ef4\u5ea6\u7684\u5bf9\u8c61\u3002\u4f8b\u5982\uff0c\u586b\u5145\u7684\u4e09\u89d2\u5f62\u4ee3\u8868\u4e00\u4e2a2\u7ef4\u7684\u5bf9\u8c61\uff0c\u586b\u5145\u7684\u56db\u9762\u4f53\u4ee3\u88683\u7ef4\u5bf9\u8c61\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/span><\/section>\n
\u56fe6. \u56fe\u4e0e\u5355\u7eaf\u590d\u5f62<\/span><\/section>\n

<\/strong><\/section>\n
2\uff09\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528<\/span><\/strong>\uff1a\u56fe\u4e3b\u8981\u63cf\u8ff0\u4e24\u4e24\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u800c\u901a\u8fc7\u5f15\u5165\u201c\u66f4\u9ad8\u7ef4\u5ea6\u7684\u5355\u5143<\/strong>\u201d\uff0c\u5355\u7eaf\u590d\u5f62\u80fd\u591f\u8868\u8fbe\u51fa\u8d85\u8fc7\u4e24\u4e2a\u5b9e\u4f53\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u4f8b\u5982\uff0c\u586b\u5145\u7684\u4e09\u89d2\u5f62\u8868\u793a\u4e09\u4e2a\u5b9e\u4f53\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u586b\u5145\u7684\u56db\u9762\u4f53\u8868\u793a\u56db\u4e2a\u5b9e\u4f53\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u6ce8\u610f\u4e8c\u8005\u7684\u533a\u522b\uff0c\u7528\u4e00\u4e2a\u5f62\u8c61\u7684\u6bd4\u55bb\u6765\u89e3\u91ca\uff1a\u56fe\u53ef\u4ee5\u8868\u793a\u7236\u4eb2\u4e0e\u5b69\u5b50\u7684\u5173\u7cfb\uff0c\u4ee5\u53ca\u6bcd\u4eb2\u4e0e\u5b69\u5b50\u7684\u5173\u7cfb\uff1b\u800c\u590d\u5f62\u5219\u80fd\u8868\u793a\u7236\u4eb2\u3001\u6bcd\u4eb2\u3001\u5b69\u5b50\u4e09\u8005\u7ec4\u6210\u7684\u5bb6\u5ead\u5355\u4f4d\u7684\u5173\u7cfb\uff0c\u7528\u4e00\u4e2a\u586b\u5145\u7684\u4e8c\u7ef4\u4e09\u89d2\u5f62\u3002<\/span><\/section>\n

<\/strong><\/section>\n
3\uff09\u8ddd\u79bb\u548c\u4f53\u79ef\u7684\u63cf\u8ff0<\/span><\/strong>\uff1a\u56fe\u901a\u5e38\u53ea\u80fd\u63cf\u8ff0\u8def\u5f84\u6216\u8ddd\u79bb\uff0c\u5355\u7eaf\u590d\u5f62\u5219\u53ef\u4ee5\u63cf\u8ff0\u9762\u79ef\uff0c\u4e24\u8fb9\u5939\u89d2<\/span>\uff082\u7ef4\u5355\u5143\uff09<\/span>\u6216\u8005\u4f53\u79ef<\/span>\uff083\u7ef4\u5355\u5143\uff09<\/span>\u3002\u8fd9\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u66f4\u9ad8\u9636\u7684\u4fe1\u606f\uff0c\u4f7f\u6211\u4eec\u80fd\u591f\u6355\u83b7\u5230\u5b9e\u4f53\u95f4\u66f4\u590d\u6742\u7684\u76f8\u4e92\u4f5c\u7528\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u4e3e\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u4ece\u4e00\u7ec4\u70b9\u548c\u56fa\u5b9a\u8ddd\u79bb\u6784\u9020\u4e00\u4e2a\u590d\u5f62<\/span>\uff08Vietoris-Rips\u590d\u5f62\uff09<\/span>\uff1a<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe7. Vietoris-Rips\u590d\u5f62<\/span><\/section>\n

<\/span><\/section>\n
\u6216\u8005\u590d\u6742\u4e9b\uff0c\u5728\u5fae\u5206\u62d3\u6251\u4e2d\uff0c\u6839\u636e\u51fd\u6570\u5bfc\u6570\u53d6\u503c0\u7684\u4fe1\u606f\u53ca\u5176\u6b63\u5b9a\u6027\u5c06\u6d41\u5f62\u5206\u7247<\/span>\uff08Morse\u590d\u5f62\uff09<\/span>\uff1a<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe8. Morse \u590d\u5f62<\/span><\/section>\n

<\/span><\/section>\n
\n
\n
\n

Magillo, Paola, et al. “A discrete approach to compute terrain morphology.” Computer Vision and Computer Graphics. Theory and Applications: International Conference VISIGRAPP 2007, Barcelona, Spain, March 8-11, 2007. Revised Selected Papers. Springer Berlin Heidelberg, 2008.<\/span><\/p>\n

\u63d0\u53d6 Morse \u590d\u5f62\u7684\u79bb\u6563\u65b9\u6cd5<\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


<\/section>\n
2.2 \u62d3\u6251\u4e0d\u53d8\u91cf<\/span><\/strong><\/span>
<\/section>\n

<\/section>\n
\u62d3\u6251\u4e0d\u53d8\u91cf\u662f\u5bf9\u62d3\u6251\u7a7a\u95f4\u6027\u8d28\u7684\u4e00\u79cd\u523b\u753b\uff0c\u5728\u6570\u636e\u5206\u6790\u4e2d\u63d0\u4f9b\u4e86\u5168\u5c40\u548c\u672c\u8d28\u7684\u89c6\u89d2\u3002\u76f8\u8f83\u4e8e\u5982 PCA \u8fd9\u6837\u7684\u7edf\u8ba1\u65b9\u6cd5\uff0c\u5b83\u66f4\u5173\u6ce8\u6574\u4f53\u7684\u6027\u8d28\u3002\u4e3e\u4e24\u4e2a\u7ecf\u5178\u7684\u62d3\u6251\u4e0d\u53d8\u91cf\u4f8b\u5b50\uff1a<\/span><\/section>\n

<\/span><\/section>\n
1\uff09\u6b27\u62c9\u793a\u6027\u6570\uff1a<\/span><\/strong>\u6b27\u62c9\u793a\u6027\u6570\u53d6\u503c\u4e3a\u201c\u9876\u70b9\u6570<\/em> \u51cf\u53bb \u8fb9\u6570<\/em> \u52a0\u4e0a \u9762\u6570<\/em>\u201d<\/span>\uff08V-E+F\uff09<\/span>\u3002\u5bf9\u4e8e\u62d3\u6251\u7b49\u4ef7\u4e8e\u7403\u4f53\u7684\u591a\u9762\u4f53<\/span>\uff08\u4f8b\u5982\u7acb\u65b9\u4f53\u6216\u516b\u9762\u4f53\u7b49\uff09<\/span>\uff0c\u6b27\u62c9\u793a\u6027\u6570\u5747\u4e3a2\u3002\u8fd9\u662f\u56e0\u4e3a\u5728\u62d3\u6251\u610f\u4e49\u4e0a\uff0c\u8fd9\u4e9b\u5f62\u72b6\u90fd\u53ef\u4ee5\u88ab\u8fde\u7eed\u5730\u53d8\u5f62\u4e3a\u7403\u5f62\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe9. \u56db\u9762\u4f53\u4e0e\u516d\u9762\u4f53\u7684\u6b27\u62c9\u793a\u6027\u6570<\/span><\/section>\n

<\/strong><\/section>\n
2\uff09\u8d1d\u8482\u6570\uff1a<\/span><\/strong>\u8d1d\u8482\u6570\u662f\u5728\u62d3\u6251\u6570\u636e\u5206\u6790<\/span>\uff08TDA\uff09<\/span>\u4e2d\u5e38\u7528\u7684\u62d3\u6251\u4e0d\u53d8\u91cf\uff0c\u7528\u6765\u63cf\u8ff0\u62d3\u6251\u7a7a\u95f4\u7684\u590d\u6742\u5ea6<\/strong>\u3002\u96f6\u7ef4\u7684\u8d1d\u8482\u6570\u8868\u793a\u8fde\u901a\u5206\u652f\u7684\u6570\u91cf\uff0c\u4e00\u7ef4\u8d1d\u8482\u6570\u8868\u793a\u72ec\u7acb\u73af\u8def\u7684\u6570\u91cf\uff0c\u4e8c\u7ef4\u8d1d\u8482\u6570\u8868\u793a\u201c\u7a7a\u5fc3\u201d\u7403\u9762\u7684\u6570\u91cf\uff0c\u7b49\u7b49\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/span><\/section>\n
\u56fe10. Betti \u6570\u63cf\u8ff0\u5206\u5b50\u7ed3\u6784<\/span><\/section>\n

<\/span><\/section>\n
2.3 \u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span>\uff08filtration\uff09<\/span>\u662f\u62d3\u6251\u6570\u636e\u5206\u6790\u7684\u6838\u5fc3\u6982\u5ff5\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u53ef\u4ee5\u7406\u89e3\u4e3a\u4e0d\u65ad\u6539\u53d8\u5c3a\u5ea6\u4ee5\u89c2\u5bdf\u590d\u6742\u7cfb\u7edf\u5982\u4f55\u968f\u7740\u5c3a\u5ea6\u7684\u53d8\u5316\u800c\u53d8\u5316<\/strong>\u3002\u8fc7\u6ee4\u6d41\u8fc7\u7a0b\u5728\u4e0d\u540c\u7684\u5c3a\u5ea6\u4e0a\u63cf\u8ff0\u5355\u7eaf\u590d\u5f62\uff0c\u5e76\u4e14\u751f\u6210\u76f8\u5e94\u7684\u6761\u5f62\u7801\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5c3a\u5ea6\u4e0b\u7684\u62d3\u6251\u4fe1\u606f\u3002<\/span><\/section>\n

<\/section>\n

<\/p>\n

\u56fe11. \u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span><\/section>\n

<\/span><\/section>\n
\u4e0a\u56fe\u6700\u5de6\u8fb9\u7684\u5341\u56db\u4e2a\u70b9\u4ee3\u8868\u539f\u59cb\u6570\u636e\uff0c\u6bcf\u70b9\u5468\u56f4\u6709\u4e00\u4e2a\u7403\u4f53\uff0c\u968f\u65f6\u95f4\u63a8\u79fb\uff0c\u8fd9\u4e9b\u7403\u4f53\u7684\u534a\u5f84\u589e\u5927\u3002\u5f53\u4e24\u4e2a\u7403\u4f53\u63a5\u89e6\u65f6\uff0c\u8868\u793a\u4e24\u4e2a\u6570\u636e\u70b9\u4e4b\u95f4\u5b58\u5728\u8fde\u63a5\uff0c\u8fd9\u5c31\u5f62\u6210\u4e86\u4e00\u6761\u8fb9\uff0c\u540c\u65f6\u72ec\u7acb\u5206\u652f\u7684\u6570\u91cf\u51cf\u4e00\u3002\u5728\u8fc7\u6ee4\u6d41\u8fc7\u7a0b\u4e2d\uff0c\u968f\u7740\u7403\u4f53\u534a\u5f84\u7684\u589e\u5927\uff0c\u72ec\u7acb\u5143\u7d20\u9010\u6e10\u51cf\u5c11\uff0c\u540c\u65f6\u51fa\u73b0\u65b0\u7684\u62d3\u6251\u7ed3\u6784<\/span>\uff08\u6bd4\u5982\u73af\u548c\u66f4\u9ad8\u7ef4\u7684\u6d1e\uff09<\/span>\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u4ece\u56fe\u4e2d\u7684\u6761\u5f62\u7801\u53ef\u4ee5\u770b\u51fa\uff0c\u521d\u59cb\u670914\u4e2a\u72ec\u7acb\u8282\u70b9\uff0c\u6240\u4ee5 Betti0<\/sub><\/span> \u4e3a 14 \u4e2a\u3002\u968f\u7740\u65f6\u95f4\u7684\u63a8\u79fb\uff0c\u7403\u4f53\u4e4b\u95f4\u7684\u8fde\u63a5\u589e\u591a\uff0c\u72ec\u7acb\u7684\u8282\u70b9\u6570\u76ee\u51cf\u5c11\u3002\u540c\u65f6\uff0c\u5f53\u51fa\u73b0\u95ed\u5408\u7684\u8def\u5f84\u65f6\uff0c\u5c31\u5f62\u6210\u4e86\u73af\uff0c\u53ef\u4ee5\u5728 Betti1<\/sub><\/span> \u7684\u6761\u5f62\u7801\u4e2d\u770b\u5230\u8fd9\u79cd\u53d8\u5316\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe12. Vietoris-Rips \u590d\u5f62\u4e0e\u5355\u7eaf\u590d\u5f62<\/span><\/section>\n

<\/section>\n
\u901a\u8fc7\u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/strong>\u548c\u5355\u7eaf\u590d\u5f62<\/strong>\uff0c\u6211\u4eec\u53ef\u4ee5\u4ece\u5168\u5c40\u548c\u591a\u5c3a\u5ea6\u7684\u89d2\u5ea6\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u7684\u7ed3\u6784\uff0c\u5e76\u901a\u8fc7 Betti \u6570\u8fd9\u7c7b\u62d3\u6251\u4e0d\u53d8\u91cf<\/strong>\u6765\u91cf\u5316\u8fd9\u4e9b\u6027\u8d28\u3002\u8fd9\u79cd\u65b9\u6cd5\u5728\u673a\u5668\u5b66\u4e60\u3001\u8fc1\u79fb\u5b66\u4e60\u7b49\u9886\u57df\u6709\u7740\u91cd\u8981\u7684\u5e94\u7528\uff0c\u76f8\u6bd4\u4f20\u7edf\u7684\u7edf\u8ba1\u5de5\u5177\uff0c\u5b83\u63d0\u4f9b\u4e86\u5bf9\u6570\u636e\u6df1\u5c42\u672c\u8d28\u7ed3\u6784\u7684\u7406\u89e3\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe13. \u591a\u5c3a\u5ea6\u7684\u5355\u7eaf\u590d\u5f62<\/span><\/section>\n


<\/span><\/h2>\n

<\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

3. \u62d3\u6251\u6df1\u5ea6\u5b66\u4e60<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

3.1 \u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u672c\u6d41\u7a0b<\/span><\/strong><\/span><\/section>\n

<\/section>\n
\u524d\u8fb9\u66f4\u591a\u662f\u4ece\u6570\u5b66\u89d2\u5ea6\u51fa\u53d1\u7684\u8ba8\u8bba\uff0c\u5728\u5904\u7406\u771f\u5b9e\u4e16\u754c\u7684\u95ee\u9898\u65f6\uff0c\u6211\u4eec\u8be5\u5982\u4f55\u5c06\u62d3\u6251\u7406\u8bba\u5e94\u7528\u5230\u5316\u5b66\u5206\u5b50\u7b49\u5177\u4f53\u79d1\u5b66\u95ee\u9898\u4e0a\uff1f<\/strong>\u4ee5\u78b360\u5206\u5b50\u4e3a\u4f8b\uff0cC60<\/sub><\/span> \u662f\u7531 60 \u4e2a\u78b3\u539f\u5b50\u7ec4\u6210\u7684\u5206\u5b50\uff0c\u5176\u5f62\u72b6\u7c7b\u4f3c\u4e8e\u8db3\u7403\uff0c\u5305\u542b\u4e86 12 \u4e2a\u4e94\u5143\u73af\u548c 20 \u4e2a\u516d\u5143\u73af\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6211\u4eec\u7528\u62d3\u6251\u6570\u636e\u5206\u6790\u8fdb\u884c\u5206\u6790\uff0cX \u8f74\u8868\u793a\u76f4\u5f84\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe14. C60<\/sub><\/span> \u5206\u5b50\u6a21\u578b\u968f\u76f4\u5f84\u53d8\u5316\u7684 Betti \u6570<\/span><\/section>\n

<\/span><\/section>\n
 \u2022 \u5728 Betti-0 \u4e2d\uff0c\u6709 60\u4e2a\u6761\u5f62\u7801\uff0c\u5176\u4e2d 30 \u4e2a\u8f83\u77ed\uff0c30 \u4e2a\u8f83\u957f\u3002\u8f83\u77ed\u7684\u4ee3\u8868\u78b3\u78b3\u53cc\u952e\uff0c\u56e0\u4e3a\u53cc\u952e\u8f83\u5f3a\uff0c\u539f\u5b50\u62c9\u5f97\u8fd1\uff1b\u800c\u8f83\u957f\u7684\u5219\u4ee3\u8868\u78b3\u78b3\u5355\u952e\uff0c\u6bd4\u53cc\u952e\u8981\u5f31\uff0c\u56e0\u6b64\u8ddd\u79bb\u7a0d\u957f\u3002\u8fd9\u6837\uff0c<\/span>Betti-0 \u63cf\u8ff0\u7684\u662f\u5171\u4ef7\u952e\u7684\u4fe1\u606f\u3002<\/strong><\/section>\n

<\/span><\/section>\n
 \u2022 \u5728 Betti-1 \u4e2d\uff0c\u6709 32 \u4e2a\u6761\u5f62\u7801\uff0c\u5176\u4e2d 12 \u4e2a\u8f83\u77ed\uff0c20 \u4e2a\u8f83\u957f\u3002\u8f83\u77ed\u7684\u5bf9\u5e94\u4e94\u5143\u73af\uff0c\u800c\u8f83\u957f\u7684\u5219\u5bf9\u5e94\u516d\u5143\u73af\u3002\u6240\u4ee5\uff0c<\/span>Betti-1 \u63cf\u8ff0\u7684\u662f\u73af\u7684\u4fe1\u606f<\/strong>\u3002<\/span><\/section>\n

<\/span><\/section>\n
 \u2022 \u5728 Betti-2 \u4e2d\uff0c\u53ef\u4ee5\u770b\u5230\u4e00\u4e2a\u957f\u6761\u5f62\u7801\uff0c\u8fd9\u4e2a\u5bf9\u5e94 <\/span>C60<\/sub><\/span> \u5206\u5b50\u6574\u4f53\u7684\u7a7a\u5fc3\u7ed3\u6784<\/strong>\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u901a\u8fc7\u8fd9\u4e9b\u7279\u5f81\u4fe1\u606f\uff0c\u6211\u4eec\u5c06\u62d3\u6251\u6570\u636e\u5206\u6790\u4e0e\u673a\u5668\u5b66\u4e60\u76f8\u7ed3\u5408\u3002\u4f8b\u5982\u5728\u6570\u636e\u4e0a\u6784\u5efa\u4e0d\u540c\u7c7b\u578b\u7684\u5355\u7eaf\u590d\u5f62\uff0c\u8fdb\u884c\u8fc7\u6ee4\u6d41\u8fc7\u7a0b\uff0c\u5f97\u5230\u6761\u5f62\u7801\uff0c\u7136\u540e\u63d0\u53d6\u5404\u79cd\u7279\u5f81<\/span>\uff08\u5982\u6700\u957f\u7684\u6761\u5f62\u7801\u3001\u6700\u77ed\u7684\u6761\u5f62\u7801\u3001\u603b\u7684\u6570\u91cf\u7b49\uff09<\/span>\uff0c\u5e76\u5c06\u8fd9\u4e9b\u7279\u5f81\u8f93\u5165\u5230\u673a\u5668\u5b66\u4e60\u6a21\u578b\uff0c\u5982 Random Forest \u6216 Gradient Boosting Tree \u7b49\uff0c\u8fdb\u884c\u529f\u80fd\u9884\u6d4b\u7b49\u4efb\u52a1\u3002\u8fd9\u6837\u5c31\u5b9e\u73b0\u4e86\u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u672c\u6d41\u7a0b\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe15. \u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\u7684\u57fa\u672c\u6d41\u7a0b<\/span><\/section>\n

<\/span><\/section>\n
3.2 \u9886\u57df\u76f8\u5173\u5de5\u4f5c<\/span><\/strong><\/span>
<\/section>\n

<\/span><\/section>\n
\u5728\u62d3\u6251\u6570\u636e\u5206\u6790<\/span>\uff08TDA\uff09<\/span>\u4e0e\u673a\u5668\u5b66\u4e60\u76f8\u7ed3\u5408\u7684\u7814\u7a76\u9886\u57df\u4e2d\uff0c\u9b4f\u56fd\u536b\u6559\u6388\u548c\u4ed6\u7684\u56e2\u961f\u505a\u4e86\u5927\u91cf\u7684\u521b\u65b0\u6027\u5de5\u4f5c\u3002\u4ed6\u4eec\u901a\u8fc7 TDA \u63d0\u53d6\u6570\u636e\u96c6\u7684\u7279\u5f81\uff0c\u5e76\u5c06\u8fd9\u4e9b\u7279\u5f81\u7528\u4e8e\u5404\u79cd\u9884\u6d4b\u4efb\u52a1\u3002\u5728\u8fc7\u53bb\u7684\u51e0\u5e74\u91cc\uff0c\u5728\u56fe\u7f51\u7edc\u5e76\u672a\u5e7f\u6cdb\u5e94\u7528\u3001\u4e14\u53ef\u5904\u7406\u6570\u636e\u91cf\u76f8\u5bf9\u8f83\u5c0f<\/span>\uff08\u901a\u5e38\u5728\u4e09\u5343\u5230\u56db\u5343\u4e4b\u95f4\uff09<\/span>\u7684\u65f6\u4ee3\uff0c\u4ed6\u4eec\u7684\u7814\u7a76\u6210\u679c\u8868\u660e TDA \u80fd\u63d0\u53d6\u51fa\u6bd4\u4f20\u7edf\u7edf\u8ba1\u65b9\u6cd5\u6216\u67d0\u4e9b\u7279\u5b9a\u7ec4\u5408\u66f4\u6709\u6548\u7684\u7279\u5f81\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe16. \u57fa\u4e8e\u62d3\u6251\u5b66\u4e60\u7684\u9884\u6d4b<\/span><\/section>\n

<\/span><\/section>\n
\u4ece\u591a\u4e2abenchmark\u6570\u636e\u96c6\u7684\u7ed3\u679c\u6765\u770b\uff0c\u4ed6\u4eec\u7684\u57fa\u4e8e TDA \u7684\u6a21\u578b\u8868\u73b0\u975e\u5e38\u597d\u3002\u5c24\u5176\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4ed6\u4eec\u5728 D3R \u836f\u7269\u8bbe\u8ba1\u6bd4\u8d5b\u4e2d\uff0c\u901a\u8fc7\u7ed3\u5408TDA\u548c\u673a\u5668\u5b66\u4e60\u7684\u65b9\u6cd5\uff0c\u57282017\u548c2018\u4e24\u5c4a\u6bd4\u8d5b\u4e2d\u90fd\u53d6\u5f97\u4e86\u663e\u8457\u7684\u4f18\u52bf\uff0c\u5e76\u8d85\u8d8a\u4e86\u8bb8\u591a\u4f20\u7edf\u7684\u65b9\u6cd5\u3002\u4ed6\u4eec\u5728TDA\u548c\u673a\u5668\u5b66\u4e60\u7ed3\u5408\u7684\u7814\u7a76\u65b9\u5411\u4e0a\u7684\u65e9\u671f\u5de5\u4f5c\uff0c\u4e3a\u8be5\u9886\u57df\u5960\u5b9a\u4e86\u575a\u5b9e\u7684\u57fa\u7840\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe17. D3R \u836f\u7269\u8bbe\u8ba1\u6bd4\u8d5b<\/span><\/section>\n

<\/span><\/section>\n
\n
\n
\n

Cang, Zixuan, Lin Mu, and Guo-Wei Wei. “Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening.” PLoS computational biology 14.1 (2018): e1005929.<\/p>\n

\u62d3\u6251\u673a\u5668\u5b66\u4e60\u6a21\u578b\u9884\u6d4b\u914d\u4f53\u86cb\u767d\u8d28\u7ed3\u5408\u80fd\u3002<\/p>\n

Nguyen, Duc Duy, et al. “MathDL: mathematical deep learning for D3R Grand Challenge 4.” Journal of computer-aided molecular design 34 (2020): 131-147.<\/p>\n

\u62d3\u6251\u673a\u5668\u5b66\u4e60\u6a21\u578b\u5e94\u7528\u4e8e\u836f\u7269\u8bbe\u8ba1\u3002<\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


<\/section>\n
3.3 Persistent Spectral\uff1a\u8c31\u56fe\u6cd5\u7ed3\u5408\u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span><\/strong><\/span><\/section>\n

<\/h3>\n

<\/section>\n
\u5728\u89c2\u5bdf\u548c\u5206\u6790\u6570\u636e\u65f6\uff0c\u4e3b\u8981\u6709\u4e24\u79cd\u65b9\u5f0f\uff1a\u4e00\u662f\u8003\u8651\u6570\u636e\u7684\u8868\u5f81<\/span>\uff08representation\uff09<\/span>\uff0c\u4e8c\u662f\u5229\u7528\u6570\u636e\u7684\u7279\u6027<\/span>\uff08features\uff09<\/span>\u3002\u524d\u9762\u8ba8\u8bba\u6211\u4eec\u4e3b\u8981\u5173\u6ce8\u62d3\u6251\u7684\u7279\u6027\uff0c\u5305\u62ec\u5404\u79cd\u62d3\u6251\u4e0d\u53d8\u91cf\uff0c\u5b83\u4eec\u63cf\u8ff0\u4e86\u7ed3\u6784\u7684\u590d\u6742\u6027\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5f53\u6211\u4eec\u60f3\u4fdd\u7559\u6570\u636e\u7684\u66f4\u7cbe\u7ec6\u7684\u7279\u5f81\uff0c\u5c31\u9700\u8981\u8003\u8651\u6570\u636e\u7684\u5176\u4ed6\u6570\u5b66\u4e0d\u53d8\u91cf\u3002\u4f8b\u5982\uff0c\u9488\u5bf9\u56fe\u6216\u5355\u7eaf\u590d\u5f62\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u8c31\u56fe\u65b9\u6cd5\u4ee5\u53ca\u5b83\u7684\u6269\u5c55\uff0c\u8fd9\u79cd\u65b9\u6cd5\u57fa\u4e8e\u56fe\u3001\u5355\u7eaf\u590d\u5f62\u6216\u8d85\u56fe\u4e0a\u7684\u79bb\u6563\u62c9\u666e\u62c9\u65af\u7b97\u5b50<\/span>\uff08Hodge Laplacian\uff09<\/span>\uff0c\u5e76\u7528\u5176\u8c31\u7684\u4fe1\u606f\u8fdb\u884c\u6570\u636e\u8868\u793a\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe18. \u8c31\u56fe\u6cd5\u7ed3\u5408\u8fc7\u6ee4\u6d41\u8fc7\u7a0b<\/span><\/section>\n

<\/section>\n
\u4e3a\u5c06\u8fd9\u4e24\u79cd\u601d\u8def\u76f8\u7ed3\u5408\uff0c\u6211\u4eec\u63d0\u51fa\u4e00\u79cd\u65b0\u7684\u6a21\u578b Persistent Spectral<\/strong>\u3002\u8fd9\u4e2a\u6a21\u578b\u7efc\u5408\u5229\u7528\u4e86\u8fc7\u6ee4\u6d41\u8fc7\u7a0b\u548c\u8c31\u56fe\u65b9\u6cd5\uff0c\u5728\u4fdd\u7559\u6570\u636e\u539f\u59cb\u5f62\u6001\u7684\u540c\u65f6\uff0c\u8fd8\u80fd\u63ed\u793a\u5176\u5185\u5728\u7684\u62d3\u6251\u7279\u6027\u3002<\/span><\/section>\n

<\/span><\/section>\n
\n
\n
\n

Edelsbrunner, Herbert, and John Harer. “Persistent Homology-a Survey.” Contemporary mathematics 453.26 (2008): 257-82. <\/p>\n

\u6301\u7eed\u540c\u8c03\uff08Persistent Homology\uff09\u662f\u62d3\u6251\u6570\u636e\u5904\u7406\uff08Topological data analysis, TDA\uff09\u6838\u5fc3\u6a21\u578b\u3002<\/p>\n

Wang, Rui, Duc Duy Nguyen, and Guo\u2010Wei Wei. “Persistent Spectral Graph.” International journal for numerical methods in biomedical engineering 36.9 (2020): e3376.<\/p>\n

\u63d0\u51fa\u4e86\u6301\u7eed\u8c31\u56fe\u6cd5\u3002<\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


<\/section>\n

\u62c9\u666e\u62c9\u65af\u77e9\u9635<\/span><\/strong><\/h4>\n


<\/span><\/p>\n

\u62c9\u666e\u62c9\u65af\u77e9\u9635\u7684\u6982\u5ff5\u6211\u4eec\u53ea\u505a\u5927\u81f4\u4ecb\u7ecd\uff0ck \u7ef4\u62c9\u666e\u62c9\u65af\u77e9\u9635 Lk<\/sub><\/span> \u6709\u5982\u4e0b\u8ba1\u7b97\u516c\u5f0f<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe19. \u62c9\u666e\u62c9\u65af\u77e9\u9635\u8ba1\u7b97\u516c\u5f0f<\/span><\/section>\n

<\/section>\n
\u4e3e\u4e2a\u4f8b\u5b50\uff0c0\u7ef4\u62c9\u666e\u62c9\u65af\u77e9\u9635 L0<\/sub><\/span> \u4ee5\u70b9\u4e3a\u5355\u4f4d\u5bf9\u8c61<\/strong>\uff0c\u5bf9\u89d2\u7ebf\u4e3a\u70b9\u7684\u5ea6\u6570\uff0c\u5f53\u70b9 i \u548c j \u8fde\u63a5\u65f6 L0<\/sub><\/span> \u7684 (i, j) \u4f4d\u7f6e\u53d6 -1 \u5426\u5219\u53d60\u3002\u7c7b\u4f3c\u5730\uff0c\u5728\u590d\u5f62\u4e0a\uff0c\u5c06\u8fb9\u4f5c\u6210\u5355\u4f4d\u5bf9\u8c61<\/strong>\uff0c\u7531\u8fb9\u7684\u5173\u7cfb\u5f97\u5230 1 \u7ef4\u62c9\u666e\u62c9\u65af\u77e9\u9635 L1<\/sub><\/span> \u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe20. \u56fe\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635 L0<\/sub><\/span>, L1<\/sub><\/span>, L2<\/sub><\/span><\/span><\/section>\n

<\/span><\/section>\n
\u5c06\u5f97\u5230\u62c9\u666e\u62c9\u65af\u77e9\u9635\u8fdb\u884c\u7279\u5f81\u503c\u5206\u89e3\uff0c\u5176\u4e2d\u96f6\u7279\u5f81\u503c\u7684\u6570\u76ee\u5bf9\u5e94\u4e86 Betti0<\/sub><\/span>\uff0c\u5176\u53cd\u6620\u56fe\u7684\u8fde\u901a\u5206\u91cf\u7684\u6570\u91cf\u3002\u62c9\u666e\u62c9\u65af\u77e9\u9635\u7684\u975e\u96f6\u7279\u5f81\u503c\u4e5f\u5305\u542b\u6709\u4e30\u5bcc\u7684\u4fe1\u606f\u3002\u6bd4\u5982\u6700\u5c0f\u7684\u975e\u96f6\u7279\u5f81\u503c\uff0c\u4e5f\u88ab\u79f0\u4e3a Fiedler \u503c\uff0c\u5e38\u7528\u6765\u523b\u753b\u56fe\u7684\u8fde\u901a\u6027\uff0c\u5c55\u793a\u5355\u7eaf\u590d\u5f62\u5404\u90e8\u5206\u4e4b\u95f4\u7684\u8fde\u63a5\u5173\u7cfb\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe21. \u96f6\u7279\u5f81\u503c\u6570\u76ee\u4e0e Betti \u6570<\/span><\/section>\n

<\/span><\/section>\n
3.4 Ricci \u66f2\u7387<\/span><\/strong><\/span><\/section>\n

<\/section>\n
\u53e6\u5916\u4e00\u4e2a\u91cd\u8981\u7684\u4e0d\u53d8\u91cf\u662f\u51e0\u4f55\u4e0d\u53d8\u91cf\uff0c\u4f8b\u5982 Ricci \u66f2\u7387\u3002Ricci \u66f2\u7387\u80fd\u591f\u6355\u83b7\u56fe\u6216\u7f51\u7edc\u4e2d\u7684\u793e\u533a\u7ed3\u6784\u6216\u7c07\uff08cluster\uff09\u7ed3\u6784<\/strong>\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5f53\u56fe\u4e2d\u6709\u4e00\u4e2a\u7d27\u5bc6\u8fde\u63a5\u7684\u793e\u533a\u6216\u805a\u7c7b\u65f6\uff0c\u8fd9\u4e2a\u533a\u57df\u7684 Ricci \u66f2\u7387\u901a\u5e38\u662f\u4e00\u4e2a\u8f83\u5927\u7684\u6b63\u503c\u3002\u800c\u8fde\u63a5\u4e24\u4e2a\u4e0d\u540c\u793e\u533a\u6216\u805a\u7c7b\u7684\u6865\u6881\u90e8\u5206\uff0cRicci \u66f2\u7387\u5219\u53ef\u80fd\u4e3a\u8d1f\u503c\u3002\u56e0\u6b64\uff0c\u8bb8\u591a\u7814\u7a76\u8005\u5229\u7528 Ricci \u66f2\u7387\u7684\u8d4b\u503c\u65b9\u6cd5\u6765\u63cf\u8ff0\u7f51\u7edc\u4e2d\u5404\u533a\u57df\u4e4b\u95f4\u7684\u76f8\u4e92\u8fde\u63a5\u6027\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe22. Ricci \u66f2\u7387<\/span><\/section>\n

<\/span><\/section>\n
Ricci \u66f2\u7387\u53ca\u5176\u5b83\u5404\u79cd\u66f2\u7387\u90fd\u662f\u7528\u6765\u63cf\u8ff0\u6574\u4f53\u7ed3\u6784\u3001\u7c07\u7ed3\u6784\u3001\u793e\u533a\u7ed3\u6784\u4ee5\u53ca\u94fe\u63a5\u7ed3\u6784\u95f4\u5173\u7cfb\u7684\u91cd\u8981\u5de5\u5177\uff0c\u80fd\u7528\u4e8e\u63ed\u793a\u7f51\u7edc\u6216\u6570\u636e\u96c6\u5185\u90e8\u4e30\u5bcc\u590d\u6742\u7684\u62d3\u6251\u548c\u51e0\u4f55\u5c5e\u6027\u3002<\/span><\/section>\n
\u5b9e\u9645\u4e0a\uff0c\u4e0a\u8fb9\u63d0\u5230\u7684\u4fe1\u606f\u53ef\u4ee5\u76f8\u4e92\u5173\u8054\u8d77\u6765\u3002\u6bd4\u5982\uff0c\u62d3\u6251\u5b66\u4e2d\u7684 Betti \u6570<\/span>\uff08homology\u4fe1\u606f\uff09<\/span>\u548cHodge Laplacian\u4e2d\u7684\u96f6\u7279\u5f81\u503c\u662f\u4e00\u4e00\u5bf9\u5e94\u7684\u3002\u79bb\u6563\u5f62\u5f0f\u7684 Ricci \u66f2\u7387<\/span>\uff08\u4f8b\u5982Forman Ricci curvature\uff09<\/span>\u4e5f\u53ef\u4ee5\u901a\u8fc7\u4e0eHodge Laplacian\u7684\u67d0\u79cd\u7ec4\u5408<\/span>\uff08\u6bd4\u5982Bochner-Weitzenb\u00f6ck\u516c\u5f0f\uff09<\/span>\u6765\u4ea7\u751f\u8054\u7cfb\u3002<\/span><\/section>\n

<\/section>\n

<\/p>\n

\u56fe23. \u51e0\u4f55\u4e0d\u53d8\u91cf\u7684\u5173\u8054<\/span><\/section>\n

<\/section>\n

\u4f7f\u7528\u8fd9\u4e9b\u5de5\u5177\u4ece\u4e0d\u540c\u89d2\u5ea6\u63cf\u8ff0\u6570\u636e\u7684\u7ed3\u6784\uff1a<\/span><\/p>\n

 \u2022 Ricci<\/strong> <\/strong>\u66f2\u7387\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u6570\u636e\u7684\u51e0\u4f55\u6027\u8d28\uff1b<\/strong><\/p>\n

 \u2022 Betti<\/strong> <\/strong>\u6570\u6216\u8005\u66f4\u4e00\u822c\u7684<\/strong> <\/strong>homology<\/strong> <\/strong>\u4fe1\u606f\u63ed\u793a\u6570\u636e\u7684\u62d3\u6251\u6027\u8d28\uff1b<\/strong><\/p>\n

 \u2022 \u8c31\u65b9\u6cd5\u5219\u80fd\u6355\u83b7\u7f51\u7edc\u6216\u6570\u636e\u96c6\u7684\u5168\u5c40\u7279\u6027\u3002<\/strong><\/section>\n


<\/span><\/h3>\n

3.5 \u5355\u7eaf\u590d\u5f62\u7684\u6784\u9020<\/strong><\/span><\/h3>\n


<\/strong><\/span><\/p>\n

\u4e0a\u8fb9\u4e3b\u8981\u4ecb\u7ecd\u51e0\u79cd\u57fa\u4e8e\u6570\u5b66\u4e0d\u53d8\u91cf\u7684\u6570\u636e\u7684\u7279\u5f81<\/span>\uff08featurization\uff09<\/span>\uff0c\u5305\u62ec Betti\u6570\u3001\u66f2\u7387\u548c\u8c31\u4fe1\u606f\u7b49\u65b9\u5f0f\u3002\u53e6\u4e00\u4e2a\u66f4\u672c\u8d28\u7684\u95ee\u9898\u662f\u6570\u636e\u7684\u8868\u5f81<\/span>\uff08representation\uff09<\/span>\u3002\u6bd4\u5982\u7528\u56fe\u3001\u5355\u7eaf\u590d\u5f62\uff0c\u4ee5\u53ca\u8d85\u56fe\u6765\u8868\u5f81\u6570\u636e\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe24. \u5355\u7eaf\u590d\u5f62\u4e0e\u8d85\u56fe<\/span><\/section>\n

<\/span><\/section>\n
\u8003\u8651\u56fe\u6216\u5355\u7eaf\u590d\u5f62\u7684\u5b50\u7ed3\u6784\uff0c\u6bd4\u5982\u793e\u533a\u3001\u7c07\u6216\u6a21\u5757\u7b49\u3002\u8fd9\u4e9b\u5b50\u7ed3\u6784\u5f80\u5f80\u80fd\u591f\u5c55\u793a\u6570\u636e\u5185\u90e8\u66f4\u52a0\u7cbe\u7ec6\u7684\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u4ece\u800c\u5e2e\u52a9\u6211\u4eec\u66f4\u51c6\u786e\u5730\u7406\u89e3\u548c\u9884\u6d4b\u7cfb\u7edf\u7684\u884c\u4e3a\u3002\u6b64\u5916\uff0c\u4e5f\u53ef\u4ee5\u8003\u8651\u52a8\u6001\u7684\u89c6\u89d2\uff0c\u6bd4\u5982\u65f6\u95f4\u6f14\u5316\u7f51\u7edc\uff0c\u8fd9\u79cd\u89c6\u89d2\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u7406\u89e3\u7cfb\u7edf\u7684\u53d8\u5316\u548c\u53d1\u5c55\u89c4\u5f8b\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u6784\u9020\u5355\u7eaf\u590d\u5f62\u7684\u65b9\u6cd5\u5f88\u591a\uff0c\u9664\u4e86\u5e38\u7528\u7684\u65b9\u6cd5\u5982 Clique complex\uff0cVR complex\uff0cAlpha complex\uff0c\u7b49\u7b49\uff0c\u4e0b\u8fb9\u4f1a\u4ecb\u7ecd\u4e09\u79cd\u65b9\u6cd5\u3002\u4ed6\u4eec\u5728\u62d3\u6251\u5b66\u4e2d\u6709\u7740\u5e7f\u6cdb\u5e94\u7528\u3002\u53e6\u5916\u62d3\u6251\u4fe1\u606f\u8fd8\u53ef\u4ee5\u901a\u8fc7\u5176\u4ed6\u4ee3\u6570\u6a21\u578b\u6765\u8868\u5f81\uff0c\u8fd9\u91cc\u6211\u4eec\u5c06\u4ecb\u7ecd\u4e00\u79cd\u7279\u6b8a\u7684\u4ee3\u6570\u6a21\u578b\uff0cTor-algebra\u3002<\/span><\/section>\n
\n
\n
\n
Bodnar, Cristian. “Topological deep learning: graphs, complexes, sheaves.” PhD diss., University of Cambridge, 2022.<\/span><\/section>\n
\u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\u3002<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n

<\/span><\/section>\n
3.5.1 Neighborhood Complex<\/span><\/strong><\/section>\n

<\/span><\/section>\n
\u6700\u7b80\u5355\u7684\u6784\u9020\u65b9\u5f0f\u662f\u90bb\u57df\u590d\u5f62<\/span>\uff08Neighborhood Complex\uff09<\/span>\uff0c\u57fa\u4e8e\u7ed9\u5b9a\u56fe\u4e2d\u7684\u90bb\u63a5\u5173\u7cfb\u6765\u6784\u5efa\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5047\u8bbe\u6709\u4e00\u4e2a\u70b9\uff0c\u5176\u90bb\u63a5\u70b9\u6709\u4e09\u4e2a\u3002\u6211\u4eec\u5c06\u8fd9\u56db\u4e2a\u70b9\u6784\u6210\u4e00\u4e2a\u56db\u8fb9\u5f62<\/span>\uff08\u79f0\u4e3a2-simplex\uff09<\/span>\u3002\u5982\u679c\u90bb\u63a5\u70b9\u4e2d\u6709\u4e24\u70b9\u4e5f\u76f8\u4e92\u90bb\u63a5\uff0c\u5219\u8fde\u8d77\u8fd9\u4e24\u4e2a\u70b9\u6784\u6210\u4e00\u6761\u8fb9<\/span>\uff081-simplex\uff09<\/span>\u3002\u5982\u679c\u6709\u4e09\u70b9\u76f8\u4e92\u90bb\u63a5\uff0c\u90a3\u4e48\u5c06\u8fd9\u4e09\u70b9\u7ec4\u6210\u4e00\u4e2a\u586b\u5145\u7684\u4e09\u89d2\u5f62<\/span>\uff082-simplex\uff09<\/span>\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\u5c06\u56fe\u8f6c\u5316\u4e3a\u90bb\u57df\u590d\u5f62\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe25. \u90bb\u57df\u590d\u5f62<\/span><\/section>\n

<\/span><\/section>\n
\u8fd9\u79cd\u90bb\u57df\u590d\u5f62\u6240\u63cf\u8ff0\u7684\u62d3\u6251\u4fe1\u606f\u4e0e\u7531\u5176\u4ed6\u65b9\u5f0f<\/span>\uff08\u4f8b\u5982Clique Complex\uff09<\/span>\u6240\u5f97\u5230\u7684\u7ed3\u679c\u4f1a\u6709\u663e\u8457\u7684\u4e0d\u540c\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u53e6\u5916\u4e00\u4e2a\u6709\u8da3\u7684\u5355\u7eaf\u590d\u5f62\u6784\u9020\u65b9\u5f0f\u662f Dowker Complex<\/strong>\u3002<\/span><\/section>\n

<\/span><\/section>\n
3.5.2 Dowker Complex<\/span><\/strong><\/section>\n

<\/section>\n
\u5f53\u7814\u7a76\u4e24\u4e2a\u5b9e\u4f53\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u4f8b\u5982\u4e24\u4e2a\u5206\u5b50\u4e4b\u95f4\u7684\u8fde\u63a5\u65f6\uff0c\u6211\u4eec\u53ef\u80fd\u66f4\u5173\u5fc3\u5206\u5b50\u4e4b\u95f4\u7684\u5168\u5c40\u4ea4\u4e92\u5173\u7cfb\uff0c\u800c\u4e0d\u662f\u5404\u81ea\u5185\u90e8\u7684\u8fde\u63a5\u65b9\u5f0f\u3002\u6b64\u65f6\u4e8c\u90e8\u56fe<\/span>\uff08bipartite graph\uff09<\/span>\u662f\u4e00\u4e2a\u5f88\u597d\u7684\u5de5\u5177\uff0c\u6211\u4eec\u5c06\u5c0f\u5206\u5b50<\/span>\uff08\u4f8b\u5982\u84dd\u70b9\u548c\u7eff\u70b9\uff09<\/span>\u89c6\u4e3a\u56fe\u8282\u70b9\uff0c\u518d\u6839\u636e\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5173\u7cfb\u6dfb\u52a0\u8fb9\u3002<\/span><\/section>\n

<\/section>\n
<\/section>\n
\u56fe26. Dowker \u590d\u5f62<\/span><\/section>\n

<\/section>\n
\u5728\u6b64\u57fa\u7840\u4e0a\u6784\u5efa\u90bb\u57df\u590d\u5f62\u3002\u7531\u4e8e\u84dd\u70b9\u7684\u6240\u6709\u90bb\u63a5\u70b9\u4f4d\u4e8e\u7eff\u70b9\u96c6\u5408\u4e2d\uff0c\u53cd\u4e4b\u4ea6\u7136\uff0c\u56e0\u6b64\u6700\u7ec8\u5f97\u5230\u4e86\u4e24\u4e2a\u5355\u7eaf\u590d\u5f62\uff0c\u5206\u522b\u7531\u84dd\u70b9\u548c\u7eff\u70b9\u7ec4\u6210\uff0c\u501f\u52a9 Dowker \u590d\u5f62\u63a2\u8ba8\u5b9e\u4f53\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5173\u7cfb\u3002<\/span><\/section>\n

<\/span><\/section>\n
3.5.3 Hom \u590d\u5f62<\/span><\/strong><\/section>\n

<\/span><\/strong><\/section>\n
\n
\n
C. H. Dowker, \u201cHomology groups of relations,\u201d Annals of mathematics, pp. 84\u2013 95, 1952. L. Lov\u00e1sz, \u201cKneser\u2019s conjecture, chromatic number, and homotopy,\u201d Journal of Combinatorial Theory, Series A, vol. 25, no. 3, pp. 319\u2013324, 1978.<\/span><\/section>\n<\/section>\n<\/blockquote>\n


<\/p>\n

\u66f4\u590d\u6742\u7684\u573a\u666f\u53ef\u4ee5\u7528\u201cHom Complex\u201d\u6784\u9020\u65b9\u5f0f\uff0c\u8fd9\u79cd\u65b9\u6cd5\u9002\u7528\u4e8e\u7814\u7a76\u4e24\u4e2a\u56fe\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u5176\u6838\u5fc3\u662f\u6784\u5efa\u4e00\u4e2a\u79f0\u4e3aPolyhedral Complex \u7684\u7ed3\u6784\uff0c\u5176\u4e2d\u5143\u7d20\u4e3a\u591a\u91cd\u540c\u6001<\/span>\uff08Multihomomorphisms\uff09<\/span>\u3002<\/span><\/section>\n

<\/section>\n
\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6709\u4e24\u4e2a\u56fe K<\/span>2<\/sub><\/span> \u548c K<\/span>3<\/sub><\/span>\uff0c\u9009\u62e9\u67d0\u79cd\u6620\u5c04\u7b56\u7565\u5c06 K<\/span>2<\/sub><\/span> \u6620\u5c04\u5230  K3 <\/sub><\/span>\u3002\u8fd9\u4e2a\u6620\u5c04\u53ea\u9700\u4fdd\u8bc1\uff1a\u539f\u56fe\u5b58\u5728\u7684\u8fb9\uff0c\u6620\u5c04\u5230\u65b0\u56fe\u4e2d\u4e5f\u5b58\u5728\u5bf9\u5e94\u8fb9\u3002\u6bd4\u5982\u5c06 K<\/span>2<\/sub><\/span> \u7684\u70b9 K<\/span>1<\/sub><\/span> \u6620\u5c04\u5230 K<\/span>2<\/sub><\/span> \u4e2d\u7684\u70b9 a\uff0c\u70b9 x<\/span>2<\/sub><\/span> \u6620\u5c04\u5230\u70b9 b\u3002\u4f46\u5982\u679c\u5c1d\u8bd5\u5c06\u70b9 K<\/span>1 <\/sub><\/span>\u548c x<\/span>2<\/sub><\/span> \u90fd\u6620\u5c04\u5230\u70b9a\uff0c\u90a3\u5c31\u4f1a\u51fa\u73b0\u95ee\u9898\uff0c\u56e0\u4e3a\u5728\u539f\u56fe\u4e2d\u70b9 x1<\/sub><\/span> \u548c x2 <\/sub><\/span>\u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u4f46\u5728\u65b0\u56fe\u4e2d\uff0c\u70b9 a \u65e0\u6cd5\u5f62\u6210\u81ea\u73af\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe27. Hom \u590d\u5f62<\/span><\/section>\n

<\/span><\/section>\n
\u5982\u4e0a\u56fe\u6240\u793a\uff0c\u5c06 x<\/span>2<\/sub><\/span>, x<\/span>2<\/sub><\/span>\u5206\u522b\u6620\u5230\u5355\u70b9\u96c6\u7684\u6620\u5c04\u6784\u6210\u96f6\u5355\u5f62 0-cell\uff0c\u5c06\u5176\u4e2d\u4e00\u70b9\u6620\u5230\u4e8c\u70b9\u96c6\u7684\u6620\u5c04\u6784\u6210\u4e86\u4e00\u5355\u5f62 1-cell\uff0c\u6240\u6709\u8fd9\u4e9b\u6620\u5c04 \u03b7 <\/span>\u6784\u6210\u4e86\u590d\u5f62 Hom(K2<\/sub><\/span>, K3<\/sub><\/span>)\u3002\u5f53\u8003\u8651\u66f4\u590d\u6742\u7684\u8fde\u63a5\u5173\u7cfb\u65f6\uff0c\u6bd4\u5982\u4f7f\u7528\u9ad8\u9636\u6216\u5377\u79ef\u6838\u6837\u5f0f\u7684\u5173\u7cfb\u8fdb\u884c\u6620\u5c04\uff0c\u8fd9\u79cd\u65b9\u6cd5\u80fd\u591f\u5e2e\u52a9\u751f\u6210\u65b0\u7684\u5355\u7eaf\u590d\u5f62\uff0c\u8fdb\u4e00\u6b65\u53cd\u6620\u56fe\u5728\u8fd9\u79cd\u7279\u5b9a\u5185\u6838\u4e0b\u7684\u6df1\u5c42\u8054\u7cfb\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe28. Hom \u590d\u5f62\u793a\u4f8b<\/span><\/section>\n

<\/span><\/section>\n
3.5.4 Tor-algebra<\/span><\/strong><\/section>\n

<\/section>\n
\u6211\u4eec\u8fd8\u80fd\u5c06\u5355\u7eaf\u590d\u5f62\u7684\u7ed3\u6784\u63d0\u5347\u5230\u66f4\u590d\u6742\u7684\u4ee3\u6570\u7ed3\u6784\u8fdb\u884c\u8003\u8651\u3002\u6bd4\u5982\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5355\u7eaf\u590d\u5f62\uff0c\u5b9a\u4e49\u4e00\u7ec4\u591a\u9879\u5f0f\uff0c\u5e76\u5728\u8fd9\u4e9b\u591a\u9879\u5f0f\u4e4b\u95f4\u5efa\u7acb\u7279\u5b9a\u7684\u5173\u7cfb<\/span>\uff08\u4f8b\u5982Stanley-Reisner\u7406\u8bba\uff09<\/span>\uff0c\u8fdb\u800c\u5f97\u5230\u4e00\u4e2a\u7406\u60f3<\/span>\uff08ideal\uff09<\/span>\u7684\u7ed3\u6784\uff0c\u7136\u540e\u7814\u7a76\u8fd9\u4e2a\u7406\u60f3\u7684\u6027\u8d28\uff0c\u4f8b\u5982\u5b83\u7684Tor\u51fd\u5b50\u7b49\u3002\u8fd9\u6837\uff0c\u56fe\u7684\u62d3\u6251\u4fe1\u606f\u5c31\u88ab\u8f6c\u6362\u4e3a\u4e86\u4ee3\u6570\u91cf\uff0c\u5355\u7eaf\u590d\u5f62\u4e0a\u5347\u5230\u4ee3\u6570\u5c42\u9762\uff0c\u5728\u8fd9\u4e2a\u5c42\u9762\u4e0a\u8fdb\u884c\u7814\u7a76\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe29. Tor \u4ee3\u6570<\/span><\/section>\n

<\/section>\n
\n
\n
\n
Xiang, L. I. U., and Kelin Xia. “Persistent Tor-algebra based stacking ensemble learning (PTA-SEL) for protein-protein binding affinity prediction.” ICLR 2022 Workshop on Geometrical and Topological Representation Learning. 2022.<\/span><\/section>\n
Persistent Tor-algebra\uff08PTA\uff09\u4e3a\u751f\u7269\u5b66\u7814\u7a76\u63d0\u4f9b\u4e86\u4e00\u79cd\u5f3a\u5927\u4e14\u6709\u6548\u7684\u65b0\u5de5\u5177<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n


<\/p>\n


<\/section>\n

<\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

4. \u57fa\u4e8e\u5355\u7eaf\u590d\u5f62\u7684\u56fe\u795e\u7ecf\u7f51\u7edc<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u6700\u540e\u4e00\u90e8\u5206\uff0c\u590f\u514b\u6797\u8001\u5e08\u4ecb\u7ecd\u4e86\u57fa\u4e8e\u5355\u7eaf\u590d\u5f62\u7684\u56fe\u795e\u7ecf\u7f51\u7edc\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u79cd\u6269\u5c55\u3002\u5728\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u6838\u5fc3\u7684\u60f3\u6cd5\u662f\u901a\u8fc7\u6d88\u606f\u4f20\u9012<\/span>\uff08message passing\uff09<\/span>\u673a\u5236\uff0c\u5c06\u4e00\u4e2a\u8282\u70b9\u5468\u56f4\u90bb\u5c45\u7684\u4fe1\u606f\u8fdb\u884c\u805a\u96c6\uff0c\u5e76\u4f20\u9012\u5230\u76ee\u6807\u8282\u70b9\uff0c\u7136\u540e\u901a\u8fc7\u8fed\u4ee3\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u5b9e\u73b0\u5bf9\u6574\u4e2a\u56fe\u7ed3\u6784\u7684\u5b66\u4e60\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe30. \u56fe\u795e\u7ecf\u7f51\u7edc<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u62d3\u6251\u6570\u636e\u5904\u7406\u4e2d\uff0c\u6211\u4eec\u4e0d\u518d\u53ea\u662f\u57fa\u4e8e\u56fe\u6765\u64cd\u4f5c\uff0c\u800c\u662f\u5728\u66f4\u9ad8\u7ef4\u5ea6\u7684\u5355\u7eaf\u590d\u5f62\u6216\u8005\u5176\u5b83\u590d\u6742\u7ed3\u6784<\/span>\uff08\u4f8b\u5982Stellar complex\uff09<\/span>\u4e0a\u8fdb\u884c\u64cd\u4f5c\u3002\u4f8b\u5982\uff0c\u9664\u4e86\u5728\u70b9\u7684\u5c42\u9762\u4e0a\u8fdb\u884c\u4fe1\u606f\u4f20\u9012\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5728\u8fb9\u3001\u9762\u6216\u66f4\u9ad8\u7ef4\u5ea6\u7684\u5355\u7eaf\u5f62\u4e0a\u8fdb\u884c\u7c7b\u4f3c\u7684\u64cd\u4f5c\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe31. \u5355\u7eaf\u590d\u5f62\u4e0a\u7684\u4fe1\u606f\u4f20\u9012<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u8fdb\u884c\u8fd9\u79cd\u590d\u6742\u7684\u62d3\u6251\u6570\u636e\u5206\u6790\u65f6\uff0c\u6709\u4e24\u4e2a\u975e\u5e38\u6838\u5fc3\u7684\u6982\u5ff5\uff1a\u8fb9\u754c\u8fd0\u7b97<\/span>\uff08Boundary Operation\uff09<\/span>\u548c\u4f59\u8fb9\u754c\u8fd0\u7b97<\/span>\uff08Coboundary Operation\uff09<\/span>\u3002\u7b80\u5355\u6765\u8bf4\uff0c\u8fb9\u754c\u8fd0\u7b97\u662f\u6307\u4ece\u7ed9\u5b9a\u7684\u5355\u7eaf\u5f62\u4e2d\u627e\u5230\u5176\u6240\u6709\u4f4e\u4e00\u7ef4\u7684\u9762<\/strong>\u3002\u4f8b\u5982\uff0c\u4ece\u4e00\u4e2a\u8fb9<\/span>\uff081-simplex\uff09<\/span>\u51fa\u53d1\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u5b83\u7684\u4e24\u4e2a\u7aef\u70b9<\/span>\uff080-simplex\uff09<\/span>\u3002\u800c\u4f59\u8fb9\u754c\u8fd0\u7b97\u5219\u662f\u53cd\u5411\u64cd\u4f5c\uff0c\u5373\u4ece\u4f4e\u7ef4\u5355\u7eaf\u5f62\u51fa\u53d1\u627e\u9ad8\u4e00\u7ef4\u7684\u5355\u7eaf\u5f62<\/strong>\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe32. \u8fb9\u754c\u8fd0\u7b97\u548c\u90bb\u63a5\u5173\u7cfb<\/span><\/section>\n

<\/span><\/section>\n
\u9664\u6b64\u4e4b\u5916\uff0c\u8fd8\u6709\u4e24\u4e2a\u91cd\u8981\u5173\u7cfb\uff1aLower Adjacency \u548c Upper Adjacency\u3002\u8fd9\u4e24\u4e2a\u5173\u7cfb\u90fd\u662f\u63cf\u8ff0\u56fe\u4e2d\u7684\u90bb\u63a5\u5173\u7cfb\uff0c\u4f46\u65b9\u5f0f\u5404\u5f02\u3002Lower Adjacency\u6307\u7684\u662f\u5f53\u4e24\u6761\u8fb9\u6709\u4e00\u4e2a\u516c\u5171\u9876\u70b9\u65f6\uff0c\u6211\u4eec\u79f0\u8fd9\u4e24\u6761\u8fb9\u662f\u90bb\u63a5\u7684\u3002\u800cUpper Adjacency\u5219\u66f4\u4e3a\u4e25\u683c\uff0c\u53ea\u6709\u5f53\u4e24\u6761\u8fb9\u5171\u4eab\u4e00\u4e2a\u9ad8\u7ef4\u5355\u7eaf\u5f62<\/span>\uff08\u6bd4\u5982\u4e09\u89d2\u5f62\uff09<\/span>\u65f6\uff0c\u6211\u4eec\u624d\u8ba4\u4e3a\u5b83\u4eec\u662f\u90bb\u63a5\u7684\u3002<\/span><\/section>\n
\u901a\u8fc7\u8003\u8651\u4e0d\u540c\u7684\u8fde\u63a5\u65b9\u5f0f\uff0c\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63cf\u7ed8\u51fa\u6570\u636e\u4e2d\u4fe1\u606f\u4f20\u9012\u7684\u4e0d\u540c\u8def\u5f84\uff0c\u5e76\u901a\u8fc7\u5c06\u4e0d\u540c\u7ef4\u5ea6\u7684\u4fe1\u606f\u8026\u5408\u5728\u4e00\u8d77\uff0c\u6784\u5efa\u4e00\u4e2a\u590d\u6742\u7684\u201c\u62d3\u6251\u795e\u7ecf\u7f51\u7edc<\/strong>\u201d\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe33. \u62d3\u6251\u795e\u7ecf\u7f51\u7edc<\/span><\/section>\n

<\/span><\/section>\n
\u8fd9\u79cd\u7ed3\u5408\u4e86\u62d3\u6251\u548c\u6df1\u5ea6\u5b66\u4e60\u7684\u7814\u7a76\u9886\u57df\u8fd8\u76f8\u5bf9\u8f83\u65b0\uff0c\u4f46\u5df2\u7ecf\u88ab\u5e7f\u5927\u5b66\u8005\u6240\u5173\u6ce8\uff0c\u5e76\u6709\u8d8a\u6765\u8d8a\u591a\u7684\u7814\u7a76\u5de5\u4f5c\u5f00\u59cb\u5c1d\u8bd5\u5229\u7528\u62d3\u6251\u6570\u636e\u5206\u6790\u6765\u63d0\u5347\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u7684\u6027\u80fd\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/span><\/section>\n
\n
\n
\n
Hajij, Mustafa, Kyle Istvan, and Ghada Zamzmi. “Cell complex neural networks.” arXiv preprint arXiv:2010.00743 (2020).<\/span><\/section>\n
\u62d3\u6251\u795e\u7ecf\u7f51\u7edc<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n


<\/p>\n

\u601d\u8003\u5ef6\u4f38<\/span><\/strong><\/span><\/h2>\n


<\/span><\/p>\n

\u5728\u62d3\u6251\u6570\u636e\u5206\u6790<\/span>\uff08TDA\uff09<\/span>\u4e2d\uff0c\u6211\u4eec\u7528\u5355\u7eaf\u590d\u5f62\u6765\u8868\u8ff0\u548c\u7406\u89e3\u590d\u6742\u7684\u6570\u636e\u7ed3\u6784\u3002\u7136\u800c\uff0c\u5176\u4ed6\u4e13\u4e1a\u9886\u57df\u7684\u7814\u7a76\u8005\u53ef\u80fd\u5bf9\u8fd9\u6837\u7684\u63cf\u8ff0\u65b9\u5f0f\u611f\u5230\u56f0\u60d1\u3002\u5728\u4ed6\u4eec\u773c\u4e2d\uff0c\u539f\u5b50<\/span>\uff08\u70b9\uff09<\/span>\u548c\u5171\u4ef7\u952e<\/span>\uff08\u8fb9\uff09<\/span>\u5177\u6709\u660e\u786e\u7684\u7269\u7406\u542b\u4e49\uff0c\u800c\u5355\u7eaf\u590d\u5f62\u4e2d\u7684\u4e09\u89d2\u5f62\u770b\u8d77\u6765\u4f3c\u4e4e\u6ca1\u6709\u76f4\u89c2\u7684\u7269\u7406\u610f\u4e49\uff1f\u5b9e\u9645\u4e0a\uff0c\u5728 TDA \u4e2d\uff0c\u4e09\u89d2\u5f62\u6355\u83b7\u4e86\u4e09\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u4e92\u5173\u7cfb\u3002\u5728\u5316\u5b66\u9886\u57df\uff0c\u8fd9\u53ef\u4ee5\u7528\u6765\u8868\u793a\u7531\u4e09\u4e2a\u539f\u5b50\u7ec4\u6210\u7684\u4e8c\u4e2a\u5171\u4ef7\u952e\u4e4b\u95f4\u7684\u89d2\u5ea6\u4fe1\u606f<\/span>\uff08bond angle\uff09<\/span>\u3002\u800c\u4e14\u8fd9\u4e2a\u89d2\u5ea6\u4fe1\u606f\u5728\u5206\u5b50\u52a8\u529b\u5b66\u7684\u6a21\u62df\u4e2d\u6709\u6781\u5176\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7136\u800c\uff0c\u5982\u4f55\u66f4\u597d\u5730\u5b9a\u4e49\u5355\u7eaf\u590d\u5f62\uff0c\u5e76\u7528\u5b83\u6765\u63cf\u8ff0\u4f53\u7cfb\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u4ecd\u7136\u662fTDA\u5efa\u6a21\u4e2d\u7684\u4e00\u4e2a\u4e3b\u8981\u95ee\u9898\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u53e6\u5916\u4e00\u4e2a TDA \u9762\u4e34\u7684\u6311\u6218\u662f\u5982\u4f55\u5c06\u62bd\u8c61\u7684\u6570\u5b66\u4e0d\u53d8\u91cf\u4e0e\u5b9e\u9645\u95ee\u9898\u7d27\u5bc6\u8054\u7cfb\u8d77\u6765<\/strong>\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u7406\u89e3\u8fd9\u4e9b\u62d3\u6251\u7279\u5f81\u6240\u4ee3\u8868\u7684\u5b9e\u9645\u610f\u4e49\u3002\u4f8b\u5982\u6570\u636e\u4e2d\u7684\u73af\u72b6\u7ed3\u6784\u662f\u5426\u53cd\u6620\u51fa\u5b83\u7684\u7269\u7406\u3001\u5316\u5b66\u3001<\/span>\u751f\u7269\uff0c\u6216\u5176\u4ed6\u5b9e\u9645\u4fe1\u606f\u3002\u5c3d\u7ba1 TDA \u4e0e\u4f20\u7edf\u56fe\u7684\u65b9\u6cd5\u5728\u6982\u5ff5\u4e0a\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u5176\u5728\u523b\u753b\u590d\u6742\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u95ee\u9898\u4e2d\u5c55\u793a\u51fa\u4e86\u6781\u5927\u7684\u4f18\u8d8a\u6027\uff0c\u5c24\u5176\u662f\u5b83\u53ef\u4ee5\u523b\u753b\u4f53\u7cfb\u7684\u6700\u672c\u8d28\u7684\u62d3\u6251\u4fe1\u606f\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6784\u9020\u5408\u9002\u7684\u5355\u7eaf\u590d\u5f62\u6765\u63cf\u8ff0\u9ad8\u9636\u4fe1\u606f\uff0c\u5e76\u4e14\u627e\u51fa\u62d3\u6251\u4e0d\u53d8\u91cf\u7684\u5408\u9002\u7684\u5b9e\u9645\u610f\u4e49\uff0c<\/strong>\u8fd9\u6837\u624d\u80fd\u53d1\u6325 TDA \u6a21\u578b\u771f\u6b63\u4f5c\u7528\uff0c<\/strong>\u5e76\u4f7f\u6a21\u578b\u7684\u89e3\u91ca\u6027\u548c\u6027\u80fd\u5f97\u5230\u63d0\u5347\u3002\u8fd9\u5c31\u9700\u8981\u6211\u4eec\u6df1\u5165\u7406\u89e3\u95ee\u9898\u80cc\u666f\uff0c\u5c06\u6570\u5b66\u5de5\u5177\u4e0e\u5b9e\u9645\u95ee\u9898\u7d27\u5bc6\u7ed3\u5408\uff0c\u5e76\u5bfb\u627e\u5230\u4e00\u4e2a\u5408\u9002\u7684\u5e94\u7528\u573a\u666f\u6765\u5c55\u793a\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u3002\u53ea\u6709\u8fd9\u6837\uff0cTDA \u624d\u80fd\u8868\u73b0\u5176\u4ef7\u503c\uff0c\u5e76\u5438\u5f15\u66f4\u591a\u4eba\u5c1d\u8bd5\u4f7f\u7528\u8fd9\u79cd\u65b0\u65b9\u6cd5\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u66f4\u8fdb\u4e00\u6b65\uff0c\u9664\u4e86\u62d3\u6251\u6570\u636e\u5206\u6790\uff0c\u5bf9\u4e8e\u5176\u4ed6\u6570\u5b66\u4e0d\u53d8\u91cf\uff0c\u5305\u62ec\u51e0\u4f55\u4e0d\u53d8\u91cf\u3001\u4ee3\u6570\u4e0d\u53d8\u91cf\u3001\u7ec4\u5408\u4e0d\u53d8\u91cf\u7b49\uff0c\u4e5f\u53ef\u4ee5\u7528\u4e8e\u6570\u636e\u7684\u8868\u5f81\u548c\u7279\u5f81\u63d0\u53d6\uff0c\u8fd9\u4e9b\u6a21\u578b\u5c06\u8fdb\u4e00\u6b65\u4fc3\u8fdb\u6211\u4eec\u5bf9\u6570\u636e\u7684\u672c\u8d28\u4fe1\u606f\u7684\u6316\u6398\u548c\u523b\u753b\u3002\u4e3a\u63d0\u9ad8\u673a\u5668\u5b66\u4e60\u6a21\u578b\u7684\u7cbe\u5ea6\u3001\u53ef\u89e3\u91ca\u6027\u3001\u8fc1\u79fb\u6027\u7b49\u6253\u4e0b\u575a\u5b9e\u7684\u6570\u5b66\u57fa\u7840\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe34. \u5206\u5b50\u6570\u636e\uff0c\u6570\u5b66\u8868\u5f81\uff0c\u6570\u636e\u7279\u6027\u4e0e\u6df1\u5ea6\u5b66\u4e60<\/span><\/section>\n

<\/section>\n

<\/section>\n
\u626b\u63cf\u4e8c\u7ef4\u7801\u89c2\u770b\u8bfb\u4e66\u4f1a\u56de\u653e\u89c6\u9891<\/span><\/section>\n
<\/section>\n
\u8bfb\u4e66\u4f1a\u5730\u5740\uff1ahttps:\/\/pattern.swarma.org\/study_group_issue\/540<\/span><\/section>\n


<\/h3>\n
\n
\n
\n

\u5b66\u8005\u7b80\u4ecb<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

<\/section>\n
\u590f\u514b\u6797<\/span><\/strong>\uff0c\u5357\u6d0b\u7406\u5de5\u5927\u5b66\u526f\u6559\u6388\u30022013\u5e741\u6708\u83b7\u5f97\u4e2d\u56fd\u79d1\u5b66\u9662\u535a\u58eb\u5b66\u4f4d\uff0c\u4e8e2009\u5e7412\u6708\u81f32012\u5e7412\u6708\u5728\u7f8e\u56fd\u5bc6\u6b47\u6839\u5dde\u7acb\u5927\u5b66\u6570\u5b66\u7cfb\u4f5c\u4e3a\u8bbf\u95ee\u5b66\u8005\u3002\u4ece2013\u5e741\u6708\u81f32016\u5e745\u6708\uff0c\u5728\u5bc6\u6b47\u6839\u5dde\u7acb\u5927\u5b66\u62c5\u4efb\u8bbf\u95ee\u52a9\u7406\u6559\u6388\u30022016\u5e746\u6708\uff0c\u52a0\u5165\u5357\u6d0b\u7406\u5de5\u5927\u5b66\uff0c\u5e76\u4e8e2023\u5e743\u6708\u664b\u5347\u4e3a\u526f\u6559\u6388\u3002\u590f\u514b\u6797\u7684\u7814\u7a76\u4e13\u6ce8\u4e8e\u5206\u5b50\u79d1\u5b66\u7684\u6570\u5b66\u4eba\u5de5\u667a\u80fd\uff0c\u5728\u300aSIAM Review\u300b\u3001\u300aScience Advances\u300b\u3001\u300anpj Computational Materials\u300b\u3001\u300aACS nano\u300b\u7b49\u671f\u520a\u4e0a\u53d1\u8868\u4e8670\u591a\u7bc7\u8bba\u6587\u3002<\/span><\/section>\n

<\/section>\n

<\/section>\n
\n
\n
\n
\n

\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u8bfb\u4e66\u4f1a\u542f\u52a8<\/strong>
<\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


<\/section>\n

\u6570\u5341\u5e74\u6765\uff0c\u4eba\u5de5\u667a\u80fd\u7684\u7406\u8bba\u53d1\u5c55\u548c\u6280\u672f\u5b9e\u8df5\u4e00\u76f4\u4e0e\u79d1\u5b66\u63a2\u7d22\u76f8\u4f34\u800c\u751f\uff0c\u5c24\u5176\u5728\u4ee5\u5927\u6a21\u578b\u4e3a\u4ee3\u8868\u7684\u4eba\u5de5\u667a\u80fd\u6280\u672f\u5e94\u7528\u96c6\u4e2d\u7206\u53d1\u7684\u5f53\u4e0b\uff0c\u4eba\u5de5\u667a\u80fd\u6b63\u5728\u52a0\u901f\u7269\u7406\u3001\u5316\u5b66\u3001\u751f\u7269\u7b49\u57fa\u7840\u79d1\u5b66\u7684\u9769\u65b0\uff0c\u800c\u8fd9\u4e9b\u5b66\u79d1\u4e5f\u5728\u53cd\u8fc7\u6765\u542f\u53d1\u4eba\u5de5\u667a\u80fd\u6280\u672f\u521b\u65b0\u3002\u5728\u6b64\u8fc7\u7a0b\u4e2d\uff0c\u6570\u5b66\u4f5c\u4e3a\u517c\u5177\u7406\u8bba\u5c5e\u6027\u4e0e\u5de5\u5177\u5c5e\u6027\u7684\u91cd\u8981\u57fa\u7840\u5b66\u79d1\uff0c\u4e0e\u4eba\u5de5\u667a\u80fd\u5173\u7cfb\u751a\u5bc6\uff0c\u76f8\u8f85\u76f8\u6210\u3002\u4e00\u65b9\u9762\uff0c\u4eba\u5de5\u667a\u80fd\u5728\u89e3\u51b3\u6570\u5b66\u9886\u57df\u7684\u8bf8\u591a\u5de5\u7a0b\u95ee\u9898\u3001\u7406\u8bba\u95ee\u9898\u4e43\u81f3\u5723\u676f\u96be\u9898\u4e0a\u5c61\u521b\u8bb0\u5f55\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b66\u6301\u7eed\u4e3a\u4eba\u5de5\u667a\u80fd\u6784\u7b51\u7406\u8bba\u57fa\u77f3\u5e76\u62d3\u5c55\u5176\u672a\u6765\u7a7a\u95f4\u3002\u8fd9\u4e24\u4e2a\u5173\u952e\u9886\u57df\u7684\u4ea4\u53c9\u878d\u5408\uff0c\u6b63\u5728\u63ed\u5f00\u4e0b\u4e2a\u65f6\u4ee3\u7684\u79d1\u5b66\u4e4b\u5e55\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u4e3a\u4e86\u63a2\u7d22\u6570\u5b66\u4e0e\u4eba\u5de5\u667a\u80fd\u6df1\u5ea6\u878d\u5408\u7684\u53ef\u80fd\u6027\uff0c\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u540c\u6d4e\u5927\u5b66\u7279\u8058\u7814\u7a76\u5458\u9648\u5c0f\u6768\u3001\u6e05\u534e\u5927\u5b66\u4ea4\u53c9\u4fe1\u606f\u5b66\u9662\u52a9\u7406\u6559\u6388\u8881\u6d0b\u3001\u5357\u6d0b\u7406\u5de5\u5927\u5b66\u526f\u6559\u6388\u590f\u514b\u6797\u4e09\u4f4d\u8001\u5e08\uff0c\u5171\u540c\u53d1\u8d77\u201c\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u201d\u8bfb\u4e66\u4f1a\uff0c\u5e0c\u671b\u4ece AI\u3000for\u3000Math\uff0cMath\u3000for\u3000AI \u4e24\u4e2a\u65b9\u9762\u6df1\u5165\u63a2\u8ba8\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u7684\u5bc6\u5207\u8054\u7cfb\u3002\u672c\u8bfb\u4e66\u4f1a\u662f\u201cAI+Science\u201d\u4e3b\u9898\u8bfb\u4e66\u4f1a\u7684\u7b2c\u4e09\u5b63\u3002\u8bfb\u4e66\u4f1a\u81ea9\u670815\u65e5\u5f00\u59cb\uff0c\u6bcf\u5468\u4e94\u665a20:00-22:00\uff0c\u9884\u8ba1\u6301\u7eed\u65f6\u95f48\uff5e10\u5468\u3002\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u62a5\u540d\u53c2\u4e0e\uff01<\/span><\/p>\n


<\/span><\/section>\n
<\/span><\/a><\/span><\/section>\n

<\/span><\/section>\n

\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/p>\n

\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u8bfb\u4e66\u4f1a\u542f\u52a8\uff1aAI for Math\uff0cMath for AI<\/a><\/span>
<\/span><\/p>\n


<\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n

\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/p>\n

1.  <\/strong>\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u524d\u6cbf\u7efc\u8ff0\uff1a\u5982\u4f55\u501f\u52a9 AI \u53d1\u73b0\u6570\u5b66\u89c4\u5f8b\uff1f<\/a><\/strong><\/span><\/section>\n
2. <\/strong>\u6570\u5b66\u63a2\u7d22\u7684\u672a\u6765\uff1a\u4eceAI\u5f15\u5bfc\u4eba\u7c7b\u76f4\u89c9\u5230\u6570\u5b66\u5927\u8bed\u8a00\u6a21\u578b<\/a><\/strong><\/span><\/section>\n
3.<\/strong> \u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\uff1a\u8ba9\u7269\u7406\u4e16\u754c\u62e5\u6709AI<\/a><\/strong><\/span><\/section>\n
4. <\/strong><\/strong><\/span>\u5f20\u6c5f\uff1a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840\u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406 | \u96c6\u667a\u5b66\u56ed\u5168\u65b0\u8bfe\u7a0b<\/a><\/strong><\/strong><\/span><\/section>\n
5. \u6210\u4e3a\u96c6\u667aVIP\uff0c\u89e3\u9501\u5168\u7ad9\u8bfe\u7a0b\uff0f\u8bfb\u4e66\u4f1a<\/a><\/span><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/strong><\/span><\/section>\n
6. <\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/span><\/a><\/section>\n


<\/p>\n


<\/p>\n<\/section>\n

\u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/strong><\/section>\n

<\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\u5bfc\u8bed \u4f5c\u4e3a\u6570\u5b66\u7684\u4e00\u4e2a\u5206\u652f\uff0c\u62d3\u6251\u5b66\u4ee5\u72ec\u7279\u7684\u65b9\u5f0f\u63cf\u8ff0\u7a7a\u95f4\u7684\u6027\u8d28\u548c\u7ed3\u6784\u3002\u8fd1\u5e74\u6765\uff0c\u51e0\u4f55\u548c\u62d3\u6251\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\uff0c\u5c24\u5176\u662f\u62d3\u6251\u6a21\u578b\uff0c\u5728\u6570\u636e\u8868\u793a\u548c\u7279\u5f81\u63d0\u53d6\u65b9\u9762\u6709\u7740\u91cd\u8981\u4f5c\u7528\u3002\u62d3\u6251\u6570\u636e\u5206\u6790\uff08Topological Data Analysis, TDA\uff09\u690d\u6839\u4e8e\u4ee3\u6570\u62d3\u6251\u548c\u8ba1\u7b97\u62d3\u6251\uff0c\u5728\u5904\u7406\u5177\u6709\u7ed3\u6784\u6027\u7684\u6570\u636e\u4e0a\u5f97\u5230\u4e86\u6781…<\/p>\n","protected":false},"author":0,"featured_media":46208,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"special":[],"_links":{"self":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/46246"}],"collection":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"replies":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=46246"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/46246\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/46208"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=46246"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=46246"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=46246"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=46246"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}