Asymptotic properties of maximum likelihood estimators for determinantal point processes

TL;DR

This paper establishes strong consistency and Berry-Esseen bounds for maximum likelihood estimators in determinantal point processes, introduces numerical algorithms, and compares MLE with frequency methods for specific matrix structures.

Contribution

It provides new theoretical results on the asymptotic properties of MLEs for DPPs and develops practical algorithms for estimation.

Findings

Proved strong consistency of MLE for DPPs.

Derived Berry-Esseen bounds for the estimator.

Compared MLE with frequency method for 2x2 matrices.

Abstract

We obtain the almost sure strong consistency and the Berry-Esseen type bound for the maximum likelihood estimator Ln of the ensemble L for determinantal point processes (DPPs), strengthening and completing previous work initiated in Brunel, Moitra, Rigollet, and Urschel [BMRU17]. Numerical algorithms of estimating DPPs are developed and simulation studies are performed. Lastly, we give explicit formula and a detailed discussion for the maximum likelihood estimator for blocked determinantal matrix of two by two submatrices and compare it with the frequency method.