{"id":1867,"date":"2022-06-25T23:00:07","date_gmt":"2022-06-25T15:00:07","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1867"},"modified":"2025-02-26T11:34:46","modified_gmt":"2025-02-26T03:34:46","slug":"cayley_klein_metric","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/06\/25\/cayley_klein_metric\/","title":{"rendered":"Cayley-Klein\u5ea6\u91cf"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u8fd9\u5468\u5728\u5de5\u4f5c\u4e0a\u88ab\u4e00\u4e2a\u6a21\u578b\u5339\u914d\u7684\u95ee\u9898\u5361\u4e86\u4e24\u4e09\u5929, \u6709\u70b9\u96be\u53d7, \u867d\u7136\u540e\u9762\u53d1\u73b0\u66f4\u591a\u7684\u662f\u6765\u81ea\u4e8eHoudini Api\u7684\u95ee\u9898, \u5e0c\u671b\u4e0b\u5468\u80fd\u591f\u5c3d\u5feb\u89e3\u51b3\u53ed~ \u672c\u6587\u4e3b\u8981\u7814\u7a76\u4e0a\u7bc7\u6587\u7ae0\u4e2d\u63d0\u5230\u7684\u4f7f\u7528Klein\u6a21\u578b(\u4ea6\u79f0\u5c04\u5f71\u6a21\u578b) \u80fd\u591f\u83b7\u5f97\u66f4\u9ad8\u7684\u8ba1\u7b97\u6027\u80fd\u7684\u6280\u5de7.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Cayley%E2%80%93Klein_metric\">Cayley\u2013Klein metric<\/a><br \/>\n2. Bi Y, Fan B, Wu F. Beyond mahalanobis metric: cayley-klein metric learning[C]\/\/Proceedings of the IEEE conference on computer vision and pattern recognition. 2015: 2339-2347.<br \/>\n3. <a href=\"https:\/\/blog.csdn.net\/weixin_44580210\/article\/details\/90319594\">\u591a\u89c6\u56fe\u51e0\u4f55\u603b\u7ed3\u2014\u2014\u7b49\u8ddd\u53d8\u6362\u3001\u76f8\u4f3c\u53d8\u6362\u3001\u4eff\u5c04\u53d8\u6362\u548c\u5c04\u5f71\u53d8\u6362<\/a><\/p>\n<p>\u5176\u4e2d, \u53c2\u8003\u6750\u65991\u662f\u4e00\u7bc7\u5173\u4e8eCayley-Klein\u5ea6\u91cf\u7684Wiki, \u8fd9\u7bc7Wiki\u662f\u4ece\u7edd\u5bf9\u51e0\u4f55\u7684\u89d2\u5ea6\u51fa\u53d1\u7684, \u7531\u4e8e\u81ea\u5df1\u6ca1\u5b66\u8fc7\u7edd\u5bf9\u51e0\u4f55, \u56e0\u6b64\u8fd9\u7bc7Wiki\u770b\u5f97\u4e00\u8138\u61f5\u903c; \u800c\u53c2\u8003\u6750\u65992\u662f\u4e00\u7bc7\u5173\u4e8eCayley-Klein\u5ea6\u91cf\u7684\u5e94\u7528\u7684\u8bba\u6587, \u65e0\u8bba\u662f\u4e13\u4e1a\u6027\u8fd8\u662f\u6613\u8bfb\u6027\u90fd\u662f\u9ad8\u4e8e\u524d\u8005\u7684, \u5bf9\u4e8e\u5e94\u7528\u4e13\u4e1a\u51fa\u8eab\u7684\u81ea\u5df1\u4e5f\u6bd4\u8f83\u53cb\u597d, \u56e0\u6b64\u672c\u6587\u5185\u5bb9\u4e5f\u4e3b\u8981\u53c2\u8003\u81ea\u53c2\u8003\u6750\u65992.<\/p>\n<p><strong>1. \u5b9a\u4e49<\/strong><\/p>\n<p>\u7ed9\u5b9a\u4e00\u4e2a\u53ef\u9006\u7684\u5bf9\u79f0\u77e9\u9635$\\Psi \\in R^{(n + 1) \\times (n + 1)}$, $x, y \\in R^n$\u7684\u53cc\u7ebf\u6027\u8868\u793a$\\psi(x, y)$\u4e3a:$$\\psi(x, y) = (x^T, 1) \\Psi \\binom{y}{1}, \\forall x, y \\in R^n.$$\u4e3a\u4e86\u65b9\u4fbf\u4e66\u5199, \u6211\u4eec\u63a5\u4e0b\u6765\u5c06$\\psi(x, y)$\u8bb0\u4e3a$\\psi_{xy}$.<br \/>\n$\\\\$ \u82e5\u77e9\u9635$\\Psi$\u662f\u6b63\u5b9a\u77e9\u9635, \u5219$\\psi_{xy} > 0$, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$\\rho_E(x, y): R^n \\times $$ R^n \\to R^+$\u4e3a:$$\\rho_E(x, y) = \\frac{k}{2i} log(\\frac{\\psi_{xy} + \\sqrt{\\psi_{xy}^2 &#8211; \\psi_{xx}\\psi_{yy}}}{\\psi_{xy} &#8211; \\sqrt{\\psi_{xy}^2 &#8211; \\psi_{xx}\\psi_{yy}}}), k > 0.$$\u82e5\u77e9\u9635$\\Psi$\u662f\u4e0d\u5b9a\u77e9\u9635, \u4ee4$B^n = \\{ x \\in R^n | \\psi_{xx} < 0 \\}$, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$\\rho_H(x, y): $$ B^n \\times B^n \\to R^+$\u4e3a:$$\\rho_H(x, y) = \\frac{k}{2} log(\\frac{\\psi_{xy} + \\sqrt{\\psi_{xy}^2 - \\psi_{xx}\\psi_{yy}}}{\\psi_{xy} - \\sqrt{\\psi_{xy}^2 - \\psi_{xx}\\psi_{yy}}}), k > 0.$$\u53ef\u4ee5\u8bc1\u660e, $\\rho_E(x, y)$\u4e0e$\\rho_H(x, y)$\u5206\u522b\u4e3a$R^n$\u4e0e$B^n$\u7684\u5ea6\u91cf, \u56e0\u4e3a\u5b83\u4eec\u5747\u6ee1\u8db3\u5982\u4e0b\u4e09\u6761\u5ea6\u91cf\u516c\u7406:<br \/>\n$\\cdot$ $\\rho(x, y) \\ge 0$(\u975e\u8d1f\u6027).<br \/>\n$\\cdot$ $\\rho(x, y) = 0 \\Leftrightarrow x = y$(\u975e\u9000\u5316\u6027).<br \/>\n$\\cdot$ $\\rho(x, y) = \\rho(y, x)$(\u5bf9\u79f0\u6027).<br \/>\n$\\cdot$ $\\rho(x, z) \\le \\rho(x, y) + \\rho(y, z)$(\u4e09\u89d2\u4e0d\u7b49\u5f0f\u6027).<br \/>\n$\\\\$ $(R^n, \\rho_E)$\u88ab\u79f0\u4e3a\u692d\u5706\u51e0\u4f55\u7a7a\u95f4, $(R^n, \\rho_H)$\u88ab\u79f0\u4e3a\u53cc\u66f2\u51e0\u4f55\u7a7a\u95f4. $\\rho_E$\u4e0e$\\rho_H$\u4e00\u8d77\u88ab\u79f0\u4e3aCaley-Klein\u5ea6\u91cf. \u4e3a\u65b9\u4fbf\u8d77\u89c1, \u53ef\u7edf\u4e00\u5199\u4e3a\u5982\u4e0b\u5f62\u5f0f:$$\\rho(x, y) = \\frac{k}{2} \\left | log(\\frac{\\psi_{xy} + \\sqrt{\\psi_{xy}^2 &#8211; \\psi_{xx}\\psi_{yy}}}{\\psi_{xy} &#8211; \\sqrt{\\psi_{xy}^2 &#8211; \\psi_{xx}\\psi_{yy}}}) \\right | , k > 0,$$\u5176\u4e2d$1 \/ k$($-1 \/ k$) \u4e0e\u692d\u5706(\u53cc\u66f2) \u7a7a\u95f4\u7684\u66f2\u7387\u6709\u5173.<br \/>\n$\\\\$ \u6839\u636e\u4e0a\u8ff0\u5b9a\u4e49, Caley-Klein\u5ea6\u91cf\u53ea\u4f9d\u8d56\u4e8e\u5bf9\u79f0\u77e9\u9635$\\Psi$. \u6362\u53e5\u8bdd\u8bf4, \u7ed9\u5b9a\u4e00\u4e2a\u5bf9\u79f0\u77e9\u9635, \u4fbf\u53ef\u4ee5\u8bf1\u5bfc\u51fa\u4e00\u4e2a\u7279\u5b9a\u7684Caley-Klein\u5ea6\u91cf. \u56e0\u6b64, \u5bf9\u79f0\u77e9\u9635$\\Psi$\u88ab\u79f0\u4e3aCaley-Klein\u5ea6\u91cf\u77e9\u9635.<\/p>\n<p><strong>2. \u4e0d\u53d8\u6027<\/strong><\/p>\n<p>\u51e0\u4f55\u5b66\u7684\u4e3b\u8981\u5185\u5bb9\u662f\u7814\u7a76\u5728\u5404\u79cd\u53d8\u6362\u7fa4\u4f5c\u7528\u4e0b\u7684\u51e0\u4f55\u4e0d\u53d8\u91cf. \u5176\u4e2d, \u7b49\u8ddd\u53d8\u6362\u7684\u4e0d\u53d8\u91cf\u4e3a\u957f\u5ea6, \u89d2\u5ea6\u4e0e\u9762\u79ef; \u76f8\u4f3c\u53d8\u6362\u7684\u4e0d\u53d8\u91cf\u4e3a\u957f\u5ea6\u7684\u6bd4\u7387, \u89d2\u5ea6\u7684\u6bd4\u7387\u4e0e\u9762\u79ef\u7684\u6bd4\u7387; \u4eff\u5c04\u53d8\u6362\u7684\u4e0d\u53d8\u91cf\u4e3a\u5e73\u884c\u7ebf\u6bb5\u7684\u957f\u5ea6\u6bd4, \u5e73\u884c\u7ebf\u4e0e\u9762\u79ef\u6bd4, \u4eff\u5c04\u53d8\u6362\u662f\u4fdd\u6301\u65e0\u7a77\u8fdc\u70b9\u7ebf\u4e0d\u53d8\u5f62\u7684\u6700\u4e00\u822c\u7684\u7ebf\u6027\u53d8\u6362, \u8fd9\u53e5\u8bdd\u7684\u610f\u601d\u5c31\u662f\u8bf4, \u4f8b\u5982\u5c04\u5f71\u53d8\u6362\u662f\u4f1a\u5c06\u65e0\u7a77\u8fdc\u70b9\u53d8\u6210\u6709\u9650\u70b9, \u6b63\u56e0\u4e3a\u5982\u6b64, \u5c04\u5f71\u53d8\u6362\u53ef\u4ee5\u5b8c\u6210\u6d88\u9664\u900f\u89c6\u5931\u771f\u7684\u64cd\u4f5c, \u4f7f\u5f97\u539f\u672c\u5e73\u884c\u7684\u76f4\u7ebf\u4e0d\u518d\u5e73\u884c, \u800c\u4eff\u5c04\u53d8\u6362\u4e4b\u540e\u5e73\u884c\u76f4\u7ebf\u4ecd\u7136\u5e73\u884c, \u56e0\u4e3a\u5176\u4e0d\u6539\u53d8\u65e0\u7a77\u8fdc\u70b9\u7684\u6027\u8d28; \u5c04\u5f71\u53d8\u6362\u662f\u975e\u9f50\u6b21\u5750\u6807\u7684\u4e00\u822c\u7684\u975e\u5947\u5f02\u7ebf\u6027\u53d8\u6362\u4e0e\u5e73\u79fb\u7684\u590d\u5408, \u5176\u4e0d\u53d8\u91cf\u4e3a\u5171\u70b9, \u5171\u7ebf, \u63a5\u89e6\u7684\u9636\u8fd8\u6709\u957f\u5ea6\u7684\u6bd4\u7387\u7684\u6bd4\u7387. \u6b64\u5904\u9644\u4e0a\u4e00\u5f20\u591a\u89c6\u56fe\u51e0\u4f55\u4e2d\u5173\u4e8e\u51e0\u79cd\u53d8\u6362\u7684\u603b\u7ed3\u8868.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/invariant_properties.png\" alt=\"\" width=\"642\" height=\"406\" class=\"aligncenter size-full wp-image-1893\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/invariant_properties.png 642w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/invariant_properties-300x190.png 300w\" sizes=\"(max-width: 642px) 100vw, 642px\" \/><\/p>\n<p>\u63a5\u4e0b\u6765\u4ecb\u7ecd\u51e0\u4e2a\u5173\u4e8eCaley-Klein\u5ea6\u91cf\u7684\u4e0d\u53d8\u91cf\u7684\u547d\u9898.<br \/>\n<strong>\u547d\u98981<\/strong> \u7ed9\u5b9a\u4e24\u70b9$x, y \\in R^n(B^n)$, \u4ee4$z_+$\u4e0e$z_-$\u4e3a$x, y$\u6240\u5728\u7684\u76f4\u7ebf\u4e0e\u4e8c\u6b21\u66f2\u9762$\\Omega = \\{ z | \\psi(z, z) = 0 \\}$\u7684\u4e24\u4e2a\u4ea4\u70b9, \u5219$$\\rho(x, y) = \\frac{k}{2} \\left | log \\ r(xy, z_+ z_-) \\right |,$$\u5176\u4e2d$r(xy, z_+ z_-)$\u4e3a\u56db\u70b9$\\{ x, y, z_+, z_- \\}$\u7684\u4ea4\u53c9\u6bd4:$$r(xy, z_+ z_-) = \\frac{(x &#8211; z_+)(y &#8211; z_-)}{(x &#8211; z_-)(y &#8211; z_+)}.$$\u7ed9\u5b9aCaley-Klein\u5ea6\u91cf\u77e9\u9635$\\Psi$, \u8003\u8651\u5982\u4e0b\u77e9\u9635\u7fa4:$$G(\\Psi) = \\{ G \\in R^{(n + 1) \\times (n + 1)} | G^{-T} \\Psi G^{-1} = \\Psi \\}.$$\u5bf9\u4e8e\u4efb\u610f\u77e9\u9635$G = (g_{ij}) \\in G(\\Psi)$, \u53ef\u4ee5\u5b9a\u4e49\u5982\u4e0b\u4e00\u4e2a\u5206\u5f0f\u7ebf\u6027\u53d8\u6362$x&#8217; = g(x)$:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/linear_fractional_transformation.png\" alt=\"\" width=\"268\" height=\"127\" class=\"aligncenter size-full wp-image-1901\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/linear_fractional_transformation.png 535w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/linear_fractional_transformation-300x142.png 300w\" sizes=\"(max-width: 268px) 100vw, 268px\" \/><\/p>\n<p>\u53ef\u4ee5\u8bc1\u660e\u4e8c\u6b21\u66f2\u9762$\\Omega = \\{ x | \\psi(x, x) = 0 \\}$\u5728\u4e0a\u8ff0\u5206\u5f0f\u7ebf\u6027\u53d8\u6362\u4e0b\u662f\u4e0d\u53d8\u7684, \u5373$$\\psi(x, x) = 0 \\Leftrightarrow \\psi(g(x), g(x)) = 0, \\forall G \\in G(\\Psi).$$\u7531\u4e8e\u5206\u5f0f\u7ebf\u6027\u53d8\u6362\u80fd\u591f\u4fdd\u6301\u4ea4\u53c9\u6bd4\u4e0d\u53d8, \u7ed3\u5408\u547d\u98981, \u6211\u4eec\u53ef\u4ee5\u77e5\u9053$$\\forall x, y \\in R^n(B^n), \\rho(g(x), g(y)) = \\rho(x, y), \\forall G \\in G(\\Psi).$$\u56e0\u6b64, Caley-Klein\u5ea6\u91cf\u662f\u53d8\u6362\u7fa4$G(\\Psi)$\u7684\u4e0d\u53d8\u91cf.<\/p>\n<p><strong>\u547d\u98982<\/strong> \u5bf9\u4e8e\u4efb\u610f$G \\in G(\\Psi)$, \u5b58\u5728\u4e00\u4e2a$(n + 1)$\u7ef4\u53cd\u5bf9\u79f0\u77e9\u9635$W$\u6ee1\u8db3$$G = (\\Psi + W)^{-1}(\\Psi &#8211; W).$$\u56e0\u6b64, \u53d8\u6362\u7fa4$G(\\Psi)$\u4e3b\u8981\u662f\u7531$n(n + 1)\/2$\u4e2a\u53c2\u6570\u51b3\u5b9a\u7684.<\/p>\n<p>\u901a\u8fc7\u4f7f\u7528\u5206\u5f0f\u7ebf\u6027\u53d8\u6362, \u4e0e\u4e0a\u8ff0\u5173\u4e8eCaley-Klein\u5ea6\u91cf\u7684\u4e0d\u53d8\u91cf\u7684\u547d\u9898, \u6211\u4eec\u53ef\u4ee5\u7b80\u5316Caley-Klein\u5ea6\u91cf\u7684\u8ba1\u7b97. \u4e14\u5bf9\u4e8e\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u6d89\u53ca\u5230\u7684\u4e09\u7ef4\u77e9\u9635\u4e0e\u4e09\u7ef4\u5411\u91cf\u7684\u4e58\u6cd5\u8fd0\u7b97, \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u56fe\u5f62\u786c\u4ef6\u5c06\u8ba1\u7b97\u6027\u80fd\u8fdb\u884c\u8fdb\u4e00\u6b65\u5730\u63d0\u5347.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u5728\u5de5\u4f5c\u4e0a\u88ab\u4e00\u4e2a\u6a21\u578b\u5339\u914d\u7684\u95ee\u9898\u5361\u4e86\u4e24\u4e09\u5929, \u6709\u70b9\u96be\u53d7, \u867d\u7136\u540e\u9762\u53d1\u73b0\u66f4\u591a\u7684\u662f\u6765\u81ea\u4e8eHoudini Api\u7684\u95ee &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/06\/25\/cayley_klein_metric\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">Cayley-Klein\u5ea6\u91cf<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1867"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1867"}],"version-history":[{"count":44,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1867\/revisions"}],"predecessor-version":[{"id":3646,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1867\/revisions\/3646"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1867"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}