估计流形尺寸的极大极小率

Q4 Mathematics

International Journal of Computational Geometry & Applications Pub Date : 2016-05-03 DOI:10.20382/jocg.v10i1a3

Jisu Kim, A. Rinaldo, L. Wasserman

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引用次数: 21

摘要

机器学习和计算几何中的许多算法都需要流形的内在维数作为输入，以支持数据的概率分布。这个参数很少为人所知，因此必须加以估计。我们通过推导估计维数的极大极小率的上界和下界来表征这个问题的统计难度。首先，我们考虑检验假设的问题，即数据生成概率分布的支持是一个固有维数$d_1$的良好流形，而不是它是维数$d_2$的替代品，具有$d_{1}本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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}

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来源期刊

International Journal of Computational Geometry & Applications 数学-计算机：理论方法

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

>12 weeks

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.