Statistical Consistency of Discrete-to-Continuous Limits of Determinantal Point Processes

TL;DR

This paper studies how discrete determinantal point processes (DPPs) converge to continuous DPPs as the sample size grows, providing conditions for this convergence and applications in sampling and graph limits.

Contribution

It introduces a non-asymptotic framework for understanding the convergence of discrete to continuous DPPs, including noisy and inaccessible kernel settings.

Findings

Discrete multivariate orthogonal polynomial ensembles can produce smaller coresets than independent sampling.

A process for repulsive sampling on unknown manifolds from sampled points.

Continuous DPPs can be obtained as limits of random graphs with Bernoulli edges.

Abstract

We investigate the limiting behavior of discrete determinantal point processes (DPPs) towards continuous DPPs when the size of the set to sample from goes to infinity. We propose a non-asymptotic characterization of this limit in terms of the concentration of statistics associated to these processes, which we refer to as "weak coherency". This allows to translate statistical guarantees from the limiting process to the original, discrete one. Our main result describes sufficient conditions for weak coherency to hold. In particular, our study encompasses settings where both the kernel of the continuous process and its underlying space are inaccessible, or when the discrete marginal kernel is a noisy version of its continuous counterpart. We illustrate our theory on several examples. We prove that a discrete multivariate orthogonal polynomial ensemble can be used to produce coresets strictly smaller than independent sampling for the same error. We propose a process achieving repulsive sampling on an unknown manifold from a set of points sampled from an unknown density. Finally, we show that continuous DPPs can be obtained as limits on random graphs with Bernoulli edges, even when only observing the graph structure. We obtain interesting byproduct results along the way.