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

2026-03-02

AI Summary

This study investigates the statistical consistency of discrete determinantal point processes (DPPs) as their sample size tends to infinity, focusing on convergence to continuous DPPs under challenging conditions such as unknown kernels, noisy observations, or access only to graph structures. By introducing a notion of “weak consistency,” the work establishes the first non-asymptotic framework for convergence from discrete to continuous DPPs, integrating tools from orthogonal polynomial ensembles, manifold repulsive sampling, and graph limit theory. Key contributions include proving that discrete DPPs yield small coresets with better approximation guarantees than independent sampling, enabling effective repulsive sampling on unknown manifolds, and recovering continuous DPPs solely from observed graph structures—thereby substantially broadening the theoretical and practical applicability of DPPs.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• determinantal point processes
• statistical consistency
• discrete-to-continuous limits
• weak coherency
• limiting behavior

Innovation

• determinantal point processes
• weak coherency
• discrete-to-continuous limit
• repulsive sampling
• non-asymptotic analysis