Nonparametric estimation of continuous DPPs with kernel methods

TL;DR

This paper develops a nonparametric maximum likelihood estimation method for continuous determinantal point processes using kernel methods, providing a finite-dimensional solution and interpretability enhancements.

Contribution

It introduces a novel nonparametric MLE approach for continuous DPPs via RKHS representer theorem, with a practical fixed point algorithm and interpretability improvements.

Findings

Finite-dimensional MLE problem derived

Proposed fixed point algorithm effective

Enhanced interpretability of DPP kernels

Abstract

Determinantal Point Process (DPPs) are statistical models for repulsive point patterns. Both sampling and inference are tractable for DPPs, a rare feature among models with negative dependence that explains their popularity in machine learning and spatial statistics. Parametric and nonparametric inference methods have been proposed in the finite case, i.e. when the point patterns live in a finite ground set. In the continuous case, only parametric methods have been investigated, while nonparametric maximum likelihood for DPPs -- an optimization problem over trace-class operators -- has remained an open question. In this paper, we show that a restricted version of this maximum likelihood (MLE) problem falls within the scope of a recent representer theorem for nonnegative functions in an RKHS. This leads to a finite-dimensional problem, with strong statistical ties to the original MLE. Moreover, we propose, analyze, and demonstrate a fixed point algorithm to solve this finite-dimensional problem. Finally, we also provide a controlled estimate of the correlation kernel of the DPP, thus providing more interpretability.