Finite frames, frame potentials and determinantal point processes on the sphere

TL;DR

This paper investigates the behavior of determinantal point processes on the sphere, showing they converge faster to finite unit norm tight frames than Poisson processes, with implications for spherical ensemble design.

Contribution

It provides a comparative analysis of DPPs and Poisson processes on the sphere, highlighting the rapid convergence of DPPs towards FUNTFs.

Findings

DPPs on the sphere converge faster to FUNTFs than Poisson processes.

Different types of DPPs (spherical, harmonic, jittered) are analyzed.

The results improve understanding of point process behavior on spheres.

Abstract

Herein, we address the expectations of frame potentials of three types of determinantal point processes(DPPs) on the d-dimensional unit sphere: (i) spherical ensembles on the 2-dimensional unit sphere; (ii) harmonic ensembles on the d-dimensional unit sphere and (iii) jittered sampling point processes on the d-dimensional unit sphere. The random point configurations generated by such DPPs converge more rapidly towards finite unit norm tight frames(FUNTFs) than the Poisson point processes on the unit sphere.