Rates of estimation for determinantal point processes

TL;DR

This paper investigates the statistical properties and convergence rates of the maximum likelihood estimator for determinantal point processes, revealing conditions for parametric convergence and highlighting potential high-dimensional challenges.

Contribution

It provides the first detailed analysis of the MLE's local geometry for DPPs, including convergence rates and conditions for parametric efficiency.

Findings

Proves several convergence rates for the MLE in DPPs.

Characterizes when the MLE achieves parametric convergence.

Identifies a potential curse of dimensionality with exponential asymptotic variance.

Abstract

Determinantal point processes (DPPs) have wide-ranging applications in machine learning, where they are used to enforce the notion of diversity in subset selection problems. Many estimators have been proposed, but surprisingly the basic properties of the maximum likelihood estimator (MLE) have received little attention. In this paper, we study the local geometry of the expected log-likelihood function to prove several rates of convergence for the MLE. We also give a complete characterization of the case where the MLE converges at a parametric rate. Even in the latter case, we also exhibit a potential curse of dimensionality where the asymptotic variance of the MLE is exponentially large in the dimension of the problem.