On the minimax spherical designs. Issue 1 (19th April 2022)

• Record Type: Journal Article
• Authors: Fu, Weibo; Wang, Guanyang; Yan, Jun
• Is Part Of: Random structures & algorithms. Volume 62:Issue 1(2023)
• Page Start: 131
• Page End: 154
• Publication Date: 2022-04-19
• Subjects: discrepancy -- minimal energy -- minimax spherical design; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.21087
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: 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.