On the Minimax Spherical Designs

TL;DR

This paper introduces a new energy function for distributing points on high-dimensional spheres, providing theoretical insights, exact solutions in some cases, and asymptotic behavior, with connections to statistical and optimization methods.

Contribution

It proposes a novel energy function for spherical point distributions, solves exact configurations in specific cases, and analyzes asymptotic minimal energy behavior.

Findings

Exact optimal configurations identified in certain cases

Asymptotic minimal energy characterized under general assumptions

Connections established with PCA and Quasi-Monte Carlo methods

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.