Determinantal point process models on the sphere

TL;DR

This paper explores determinantal point processes on the sphere, focusing on their properties, construction, and spectral representation, with an emphasis on isotropic models and strategies for developing new models.

Contribution

It characterizes and constructs isotropic DPPs on the sphere, analyzing their spectral properties and proposing new modeling strategies for isotropic covariance functions.

Findings

Characterization of isotropic DPPs on the sphere

Spectral representation for kernel eigenvalues and eigenfunctions

Strategies for developing new isotropic DPP models

Abstract

We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb S^d$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb S^d\times\mathbb S^d$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^d$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.