算法xxx:球面三角形算法:凸壳成员查询的快速Oracle

IF 2.7 1区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING
B. Kalantari, Yikai Zhang
{"title":"算法xxx:球面三角形算法:凸壳成员查询的快速Oracle","authors":"B. Kalantari, Yikai Zhang","doi":"10.1145/3516520","DOIUrl":null,"url":null,"abstract":"<jats:p>\n The\n <jats:italic>Convex Hull Membership</jats:italic>\n (CHM) tests whether\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(p \\in conv(S) \\)</jats:tex-math>\n </jats:inline-formula>\n , where\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(p \\)</jats:tex-math>\n </jats:inline-formula>\n and the\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(n \\)</jats:tex-math>\n </jats:inline-formula>\n points of\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(S \\)</jats:tex-math>\n </jats:inline-formula>\n lie in\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(\\mathbb { R}^m \\)</jats:tex-math>\n </jats:inline-formula>\n . CHM finds applications in Linear Programming, Computational Geometry and Machine Learning. The\n <jats:italic>Triangle Algorithm</jats:italic>\n (TA), previously developed, in\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(O(1/\\varepsilon ^2) \\)</jats:tex-math>\n </jats:inline-formula>\n iterations computes\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(p^{\\prime } \\in conv(S) \\)</jats:tex-math>\n </jats:inline-formula>\n , either an\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(\\varepsilon \\)</jats:tex-math>\n </jats:inline-formula>\n -\n <jats:italic>approximate solution</jats:italic>\n , or a\n <jats:italic>witness</jats:italic>\n certifying\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(p \\not\\in conv(S) \\)</jats:tex-math>\n </jats:inline-formula>\n . We first prove the equivalence of exact and approximate versions of CHM and\n <jats:italic>Spherical</jats:italic>\n -CHM, where\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(p=0 \\)</jats:tex-math>\n </jats:inline-formula>\n and\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(\\Vert v\\Vert =1 \\)</jats:tex-math>\n </jats:inline-formula>\n for each\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(v \\)</jats:tex-math>\n </jats:inline-formula>\n in\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(S \\)</jats:tex-math>\n </jats:inline-formula>\n . If for some\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(M \\ge 1 \\)</jats:tex-math>\n </jats:inline-formula>\n every non-witness with\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(\\Vert p^{\\prime }\\Vert \\gt \\varepsilon \\)</jats:tex-math>\n </jats:inline-formula>\n admits\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(v \\in S \\)</jats:tex-math>\n </jats:inline-formula>\n satisfying\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(\\Vert p^{\\prime } - v\\Vert \\ge \\sqrt {1+\\varepsilon /M} \\)</jats:tex-math>\n </jats:inline-formula>\n , we prove the number of iterations improves to\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(O(M/\\varepsilon) \\)</jats:tex-math>\n </jats:inline-formula>\n and\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(M \\le 1/\\varepsilon \\)</jats:tex-math>\n </jats:inline-formula>\n always holds. Equivalence of CHM and Spherical-CHM implies\n <jats:italic>Minimum Enclosing Ball</jats:italic>\n (MEB) algorithms can be modified to solve CHM. However, we prove\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\((1+ \\varepsilon) \\)</jats:tex-math>\n </jats:inline-formula>\n -approximation in MEB is\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(\\Omega (\\sqrt {\\varepsilon }) \\)</jats:tex-math>\n </jats:inline-formula>\n -approximation in Spherical-CHM. Thus even\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(O(1/\\varepsilon) \\)</jats:tex-math>\n </jats:inline-formula>\n iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the\n <jats:italic>All Vertex Triangle Algorithm</jats:italic>\n (AVTA) for computing all vertices of\n <jats:inline-formula content-type=\"math/tex\">\n <jats:tex-math notation=\"TeX\" version=\"MathJaX\">\\(conv(S) \\)</jats:tex-math>\n </jats:inline-formula>\n . Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank-Wolfe, MEB and LP-Solver.\n </jats:p>","PeriodicalId":50935,"journal":{"name":"ACM Transactions on Mathematical Software","volume":" ","pages":""},"PeriodicalIF":2.7000,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Algorithm xxx: Spherical Triangle Algorithm: A Fast Oracle for Convex Hull Membership Queries\",\"authors\":\"B. Kalantari, Yikai Zhang\",\"doi\":\"10.1145/3516520\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>\\n The\\n <jats:italic>Convex Hull Membership</jats:italic>\\n (CHM) tests whether\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(p \\\\in conv(S) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n , where\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(p \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n and the\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(n \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n points of\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(S \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n lie in\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(\\\\mathbb { R}^m \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n . CHM finds applications in Linear Programming, Computational Geometry and Machine Learning. The\\n <jats:italic>Triangle Algorithm</jats:italic>\\n (TA), previously developed, in\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(O(1/\\\\varepsilon ^2) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n iterations computes\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(p^{\\\\prime } \\\\in conv(S) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n , either an\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(\\\\varepsilon \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n -\\n <jats:italic>approximate solution</jats:italic>\\n , or a\\n <jats:italic>witness</jats:italic>\\n certifying\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(p \\\\not\\\\in conv(S) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n . We first prove the equivalence of exact and approximate versions of CHM and\\n <jats:italic>Spherical</jats:italic>\\n -CHM, where\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(p=0 \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n and\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(\\\\Vert v\\\\Vert =1 \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n for each\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(v \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n in\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(S \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n . If for some\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(M \\\\ge 1 \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n every non-witness with\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(\\\\Vert p^{\\\\prime }\\\\Vert \\\\gt \\\\varepsilon \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n admits\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(v \\\\in S \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n satisfying\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(\\\\Vert p^{\\\\prime } - v\\\\Vert \\\\ge \\\\sqrt {1+\\\\varepsilon /M} \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n , we prove the number of iterations improves to\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(O(M/\\\\varepsilon) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n and\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(M \\\\le 1/\\\\varepsilon \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n always holds. Equivalence of CHM and Spherical-CHM implies\\n <jats:italic>Minimum Enclosing Ball</jats:italic>\\n (MEB) algorithms can be modified to solve CHM. However, we prove\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\((1+ \\\\varepsilon) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n -approximation in MEB is\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(\\\\Omega (\\\\sqrt {\\\\varepsilon }) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n -approximation in Spherical-CHM. Thus even\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(O(1/\\\\varepsilon) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the\\n <jats:italic>All Vertex Triangle Algorithm</jats:italic>\\n (AVTA) for computing all vertices of\\n <jats:inline-formula content-type=\\\"math/tex\\\">\\n <jats:tex-math notation=\\\"TeX\\\" version=\\\"MathJaX\\\">\\\\(conv(S) \\\\)</jats:tex-math>\\n </jats:inline-formula>\\n . Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank-Wolfe, MEB and LP-Solver.\\n </jats:p>\",\"PeriodicalId\":50935,\"journal\":{\"name\":\"ACM Transactions on Mathematical Software\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":2.7000,\"publicationDate\":\"2022-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACM Transactions on Mathematical Software\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1145/3516520\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Mathematical Software","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3516520","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 5

摘要

凸壳隶属度(CHM)测试\(S \)的点\(p \in conv(S) \)、\(p \)和\(n \)是否在\(\mathbb { R}^m \)中。CHM在线性规划、计算几何和机器学习中都有应用。以前开发的三角算法(TA)在\(O(1/\varepsilon ^2) \)迭代中计算\(p^{\prime } \in conv(S) \),要么是\(\varepsilon \) -近似解,要么是证明\(p \not\in conv(S) \)的证人。我们首先证明了精确和近似版本的CHM和球面-CHM的等价性,其中\(S \)中的\(v \)分别为\(p=0 \)和\(\Vert v\Vert =1 \)。如果对于一些\(M \ge 1 \)每一个没有\(\Vert p^{\prime }\Vert \gt \varepsilon \)的证人都承认\(v \in S \)满足\(\Vert p^{\prime } - v\Vert \ge \sqrt {1+\varepsilon /M} \),我们证明迭代次数提高到\(O(M/\varepsilon) \)和\(M \le 1/\varepsilon \)总是成立。基于最小包球法和球面包球法的等价性,可以改进最小包球法求解包球法。然而,我们证明了MEB中的\((1+ \varepsilon) \) -近似是sphericchm中的\(\Omega (\sqrt {\varepsilon }) \) -近似。因此,即使\(O(1/\varepsilon) \)迭代MEB算法也不优于sphere - ta。对于MEB核心集也证明了类似的弱点。Spherical-TA还产生了一个变种的全顶点三角形算法(AVTA),用于计算\(conv(S) \)的所有顶点。对不同问题的大量计算表明,TA和sphere -TA通常比Frank-Wolfe、MEB和LP-Solver等算法具有更高的效率。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Algorithm xxx: Spherical Triangle Algorithm: A Fast Oracle for Convex Hull Membership Queries
The Convex Hull Membership (CHM) tests whether \(p \in conv(S) \) , where \(p \) and the \(n \) points of \(S \) lie in \(\mathbb { R}^m \) . CHM finds applications in Linear Programming, Computational Geometry and Machine Learning. The Triangle Algorithm (TA), previously developed, in \(O(1/\varepsilon ^2) \) iterations computes \(p^{\prime } \in conv(S) \) , either an \(\varepsilon \) - approximate solution , or a witness certifying \(p \not\in conv(S) \) . We first prove the equivalence of exact and approximate versions of CHM and Spherical -CHM, where \(p=0 \) and \(\Vert v\Vert =1 \) for each \(v \) in \(S \) . If for some \(M \ge 1 \) every non-witness with \(\Vert p^{\prime }\Vert \gt \varepsilon \) admits \(v \in S \) satisfying \(\Vert p^{\prime } - v\Vert \ge \sqrt {1+\varepsilon /M} \) , we prove the number of iterations improves to \(O(M/\varepsilon) \) and \(M \le 1/\varepsilon \) always holds. Equivalence of CHM and Spherical-CHM implies Minimum Enclosing Ball (MEB) algorithms can be modified to solve CHM. However, we prove \((1+ \varepsilon) \) -approximation in MEB is \(\Omega (\sqrt {\varepsilon }) \) -approximation in Spherical-CHM. Thus even \(O(1/\varepsilon) \) iteration MEB algorithms are not superior to Spherical-TA. Similar weakness is proved for MEB core sets. Spherical-TA also results a variant of the All Vertex Triangle Algorithm (AVTA) for computing all vertices of \(conv(S) \) . Substantial computations on distinct problems demonstrate that TA and Spherical-TA generally achieve superior efficiency over algorithms such as Frank-Wolfe, MEB and LP-Solver.
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来源期刊
ACM Transactions on Mathematical Software
ACM Transactions on Mathematical Software 工程技术-计算机:软件工程
CiteScore
5.00
自引率
3.70%
发文量
50
审稿时长
>12 weeks
期刊介绍: As a scientific journal, ACM Transactions on Mathematical Software (TOMS) documents the theoretical underpinnings of numeric, symbolic, algebraic, and geometric computing applications. It focuses on analysis and construction of algorithms and programs, and the interaction of programs and architecture. Algorithms documented in TOMS are available as the Collected Algorithms of the ACM at calgo.acm.org.
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