{"title":"估计流形尺寸的极大极小率","authors":"Jisu Kim, A. Rinaldo, L. Wasserman","doi":"10.20382/jocg.v10i1a3","DOIUrl":null,"url":null,"abstract":"Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}<d_{2}$. With an i.i.d. sample of size $n$, we provide an upper bound on the probability of choosing the wrong dimension of $O\\left( n^{-\\left(d_{2}/d_{1}-1-\\epsilon\\right)n} \\right)$, where $\\epsilon$ is an arbitrarily small positive number. The proof is based on bounding the length of the traveling salesman path through the data points. We also demonstrate a lower bound of $\\Omega \\left( n^{-(2d_{2}-2d_{1}+\\epsilon)n} \\right)$, by applying Le Cam's lemma with a specific set of $d_{1}$-dimensional probability distributions. We then extend these results to get minimax rates for estimating the dimension of well-behaved manifolds. We obtain an upper bound of order $O \\left( n^{-(\\frac{1}{m-1}-\\epsilon)n} \\right)$ and a lower bound of order $\\Omega \\left( n^{-(2+\\epsilon)n} \\right)$, where $m$ is the embedding dimension.","PeriodicalId":54969,"journal":{"name":"International Journal of Computational Geometry & Applications","volume":"21 1","pages":"42-95"},"PeriodicalIF":0.0000,"publicationDate":"2016-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"21","resultStr":"{\"title\":\"Minimax Rates for Estimating the Dimension of a Manifold\",\"authors\":\"Jisu Kim, A. Rinaldo, L. Wasserman\",\"doi\":\"10.20382/jocg.v10i1a3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}<d_{2}$. With an i.i.d. sample of size $n$, we provide an upper bound on the probability of choosing the wrong dimension of $O\\\\left( n^{-\\\\left(d_{2}/d_{1}-1-\\\\epsilon\\\\right)n} \\\\right)$, where $\\\\epsilon$ is an arbitrarily small positive number. The proof is based on bounding the length of the traveling salesman path through the data points. We also demonstrate a lower bound of $\\\\Omega \\\\left( n^{-(2d_{2}-2d_{1}+\\\\epsilon)n} \\\\right)$, by applying Le Cam's lemma with a specific set of $d_{1}$-dimensional probability distributions. We then extend these results to get minimax rates for estimating the dimension of well-behaved manifolds. We obtain an upper bound of order $O \\\\left( n^{-(\\\\frac{1}{m-1}-\\\\epsilon)n} \\\\right)$ and a lower bound of order $\\\\Omega \\\\left( n^{-(2+\\\\epsilon)n} \\\\right)$, where $m$ is the embedding dimension.\",\"PeriodicalId\":54969,\"journal\":{\"name\":\"International Journal of Computational Geometry & Applications\",\"volume\":\"21 1\",\"pages\":\"42-95\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-05-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"21\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Computational Geometry & Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.20382/jocg.v10i1a3\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Computational Geometry & Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.20382/jocg.v10i1a3","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
Minimax Rates for Estimating the Dimension of a Manifold
Many algorithms in machine learning and computational geometry require, as input, the intrinsic dimension of the manifold that supports the probability distribution of the data. This parameter is rarely known and therefore has to be estimated. We characterize the statistical difficulty of this problem by deriving upper and lower bounds on the minimax rate for estimating the dimension. First, we consider the problem of testing the hypothesis that the support of the data-generating probability distribution is a well-behaved manifold of intrinsic dimension $d_1$ versus the alternative that it is of dimension $d_2$, with $d_{1}
期刊介绍：
The International Journal of Computational Geometry & Applications (IJCGA) is a quarterly journal devoted to the field of computational geometry within the framework of design and analysis of algorithms.
Emphasis is placed on the computational aspects of geometric problems that arise in various fields of science and engineering including computer-aided geometry design (CAGD), computer graphics, constructive solid geometry (CSG), operations research, pattern recognition, robotics, solid modelling, VLSI routing/layout, and others. Research contributions ranging from theoretical results in algorithm design — sequential or parallel, probabilistic or randomized algorithms — to applications in the above-mentioned areas are welcome. Research findings or experiences in the implementations of geometric algorithms, such as numerical stability, and papers with a geometric flavour related to algorithms or the application areas of computational geometry are also welcome.
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