Likelihood Geometry of Determinantal Point Processes

TL;DR

This paper investigates the likelihood geometry of determinantal point processes (DPPs) using algebraic statistics, counting critical points of the likelihood function and providing counterexamples to a previous conjecture.

Contribution

It introduces a novel algebraic approach to analyze DPP likelihoods and disproves a prior conjecture about their critical points.

Findings

Counted critical points of the likelihood function for small DPP models.

Disproved a conjecture by Brunel, Moitra, Rigollet, and Urschel.

Provided algebraic insights into the likelihood geometry of DPPs.

Abstract

We study determinantal point processes (DPP) through the lens of algebraic statistics. We count the critical points of the log-likelihood function, and we compute them for small models, thereby disproving a conjecture of Brunel, Moitra, Rigollet and Urschel.