Determinantal Point Processes in the Flat Limit: Extended L-ensembles, Partial-Projection DPPs and Universality Classes

TL;DR

This paper introduces extended L-ensembles for DPPs, enabling analysis of partial-projection DPPs and demonstrating their universality in the flat kernel limit, revealing new insights into their limiting behavior.

Contribution

It presents a novel formalism for DPPs with extended L-ensembles and analyzes their limits, including the universality classes depending on kernel smoothness.

Findings

Extended L-ensembles fix pathologies in traditional DPP formalism.

Partial-projection DPPs arise as limits of L-ensembles with low-rank perturbations.

Limiting processes depend on kernel smoothness and can be universal.

Abstract

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. The contributions of this paper are two-fold. First of all, we introduce the concept of extended L-ensemble, a novel representation of DPPs. These extended L-ensembles are interesting objects because they fix some pathologies in the usual formalism of DPPs, for instance the fact that projection DPPs are not L-ensembles. Every (fixed-size) DPP is an (fixed-size) extended L-ensemble, including projection DPPs. This new formalism enables to introduce and analyze a subclass of DPPs, called partial-projection DPPs. Secondly, with these new definitions in hand, we first show that partial-projection DPPs arise as perturbative limits of L-ensembles, that is, limits in $\varepsilon \rightarrow 0$ of L-ensembles based on matrices of the form $\varepsilon \mathbf{A} + \mathbf{B}$ where $\mathbf{B}$ is low-rank. We generalise this result by showing that partial-projection DPPs also arise as 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 the limiting point process 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.