Kernel Matrix Estimation of a Determinantal Point Process from a Finite Set of Samples: Properties and Algorithms

Abstract

Determinantal point processes (DPPs) on finite sets have recently gained popularity because of their ability to promote diversity among selected elements in a given subset. The probability distribution of a DPP is defined by the determinant of a positive semi-definite, real-valued matrix. When estimating the DPP parameter matrix, it is often more convenient to express the maximum likelihood criterion using the framework of L-ensembles. However, the resulting optimization problem is non-convex and N P-hard to solve. In this paper, we establish conditions under which the maximum likelihood criterion has a well-defined optimum for a given finite set of samples. We demonstrate that regularization is generally beneficial for ensuring a proper solution. To solve the resulting optimization problem, we propose a proximal algorithm which minimizes a penalized criterion. Through simulations, we compare our algorithm with previously proposed approaches, illustrating their differing behaviors and providing empirical support for our theoretical findings.