A Metropolis random walk algorithm to estimate a lower bound of the star discrepancy

IF 0.8 Q3 STATISTICS & PROBABILITY
Maryam Alsolami, M. Mascagni
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引用次数: 0

Abstract

Abstract In this paper, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s [0,1]^{s} . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.
估计星差下界的Metropolis随机游动算法
摘要在本文中,我们介绍了一种新的算法来估计[0,1]s[0,1]^{s}中任意点集的星差的下界。计算精确的星差是一个NP难题,因此我们一直在寻找有效的近似算法。恒星差异可以被认为是一个称为局部差异的函数的最大值,我们将开发近似算法来最大化这个函数。我们的算法类似于我们之前的一篇论文[M.Alsolami和M。Mascagni,估计星差下界的随机游动算法,蒙特卡罗方法应用。28(2022),44341–348.]。我们通过在每个选定维度上的随机行走中实现Metropolis算法,将统计技术添加到随机行走算法中,以接受或拒绝这种运动。我们称之为Metropolis随机行走算法。与以前所有已知的技术相比,我们的新算法是优越的,尤其是在高维方面。此外,它可以快速确定我们大多数不同大小和维度的数据集中恒星差异的精确值,或者至少确定恒星差异的下限。
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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来源期刊
Monte Carlo Methods and Applications
Monte Carlo Methods and Applications STATISTICS & PROBABILITY-
CiteScore
1.20
自引率
22.20%
发文量
31
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