Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus

TL;DR

This paper investigates determinantal point processes on spheres and tori, analyzing their energy, discrepancy, and optimal transport properties, revealing they outperform i.i.d. points in Wasserstein distance and extending known energy results.

Contribution

It extends Riesz energy results to negative regimes and analyzes discrepancy and optimal transport for determinantal processes on manifolds, providing new bounds and insights.

Findings

Expected $L^2$ discrepancy computed for harmonic ensemble

Determinantal processes achieve optimal $N^{-1/2}$ rate in Wasserstein distance

Determinantal point processes outperform i.i.d. points in transport metrics

Abstract

Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so-called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$, and the so-called spherical ensemble on $\mathbb{S}^2$, which originates in random matrix theory. We extend results of Beltr\'an, Marzo and Ortega-Cerd\`a on the Riesz $s$-energy of the harmonic ensemble to the nonsingular regime $s<0$, and as a corollary find the expected value of the spherical cap $L^2$ discrepancy via the Stolarsky invariance principle. We find the expected value of the $L^2$ discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on $\mathbb{T}^d$. We also show that the spherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and $\mathbb{T}^2$ with $N$ points attain the optimal rate $N^{-1/2}$ in expectation in the Wasserstein metric $W_2$, in contrast to i.i.d. random points, which are known to lose a factor of $(\log N)^{1/2}$.