Comparison of probabilistic and deterministic point sets

TL;DR

This paper compares probabilistic and deterministic point sets on spheres, showing that certain deterministic spherical t-designs perform as well as probabilistic sets in energy minimization, with asymptotic results for well-separated designs.

Contribution

It establishes asymptotic equalities for the Riesz s-energy of well-separated spherical t-designs and demonstrates their effectiveness compared to probabilistic point sets.

Findings

Deterministic spherical t-designs can match probabilistic sets in energy performance.

Asymptotic formulas for Riesz s-energy of well-separated t-designs are derived.

Existence of well-separated t-designs with N points proportional to t^d is confirmed.

Abstract

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (spherical $t$-designs) are better or as good as probabilistic ones. We find asymptotic equalities for the discrete Riesz $s$-energy of sequences of well separated $t$-designs on the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$, $d\geq2$. The case $d=2$ was studied Hesse and Leopardi. Bondarenko, Radchenko, and Viazovska established, that for $d\geq 2$, there exists a constant $c_{d}$, such that for every $N> c_{d}t^{d}$ there exists a well-separated spherical $t$-design on $\mathbb{S}^{d}$ with $N$ points. For this reason, in our paper we assume, that the sequence of well separated spherical $t$-designs is such that $t$ and $N$ are related by $N\asymp t^{d}$.