Determinantal Point Processes in the Flat Limit

TL;DR

This paper investigates the behavior of determinantal point processes as the kernel function becomes flat, revealing universal limiting processes that depend on kernel smoothness and do not require length-scale parameters.

Contribution

It characterizes the flat limit of DPPs using extended L-ensembles and partial projection DPPs, highlighting conditions for universality based on kernel smoothness.

Findings

Limiting processes are described by extended L-ensembles and partial projection DPPs.

Universal limits depend only on kernel smoothness, not specific functions.

Flat-limit DPPs are still repulsive and do not need spatial length-scale parameters.

Abstract

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process is even universal, meaning that it does not depend on specifics of the kernel function, but only on its degree of smoothness. Since flat-limit DPPs are still repulsive processes, this implies that practically useful families of DPPs exist that do not require a spatial length-scale parameter.