Testing Determinantal Point Processes

TL;DR

This paper introduces the first algorithm for property testing of determinantal point processes (DPPs), providing both an effective testing method and a matching lower bound on sample complexity, with implications for broader distribution classes.

Contribution

The paper presents the first algorithm for testing whether a distribution is a DPP and establishes a lower bound on the sample complexity, advancing understanding of distribution property testing.

Findings

First algorithm for testing DPPs from samples.

Matching lower bound on sample complexity for DPP testing.

Hardness result for testing log-submodular distributions.

Abstract

Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution $q$ over the subsets of a ground set, we aim to distinguish whether $q$ is a DPP distribution, or $\epsilon$-far from all DPP distributions in $\ell_1$-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions.