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On the minimax spherical designs
Author(s) -
Fu Weibo,
Wang Guanyang,
Yan Jun
Publication year - 2023
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21087
Subject(s) - minimax , point (geometry) , monte carlo method , mathematics , component (thermodynamics) , function (biology) , principal component analysis , mathematical optimization , energy (signal processing) , field (mathematics) , statistical physics , computer science , physics , pure mathematics , geometry , statistics , quantum mechanics , evolutionary biology , biology
Distributing points on a (possibly high‐dimensional) sphere with minimal energy is a long‐standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and combinatorial optimization, and studies its theoretical properties. Our result solves both the exact optimal spherical point configurations in certain cases and the minimal energy asymptotics under general assumptions. Connections between our results and the L1‐principal component analysis and quasi‐Monte Carlo methods are also discussed.