Exact Sampling from Determinantal Point Processes

TL;DR

This paper introduces a method for exact sampling from determinantal point processes (DPPs) on continuous domains, highlighting its advantages over approximate methods in machine learning applications.

Contribution

It presents a novel approach for exact sampling from DPPs on continuous spaces, extending previous methods and demonstrating its benefits over approximate schemes.

Findings

Exact sampling is feasible on continuous domains.

Exact sampling can outperform approximate methods in precision-critical applications.

The method is generalized from univariate to multivariate spaces.

Abstract

Determinantal point processes (DPPs) are an important concept in random matrix theory and combinatorics. They have also recently attracted interest in the study of numerical methods for machine learning, as they offer an elegant "missing link" between independent Monte Carlo sampling and deterministic evaluation on regular grids, applicable to a general set of spaces. This is helpful whenever an algorithm explores to reduce uncertainty, such as in active learning, Bayesian optimization, reinforcement learning, and marginalization in graphical models. To draw samples from a DPP in practice, existing literature focuses on approximate schemes of low cost, or comparably inefficient exact algorithms like rejection sampling. We point out that, for many settings of relevance to machine learning, it is also possible to draw exact samples from DPPs on continuous domains. We start from an intuitive example on the real line, which is then generalized to multivariate real vector spaces. We also compare to previously studied approximations, showing that exact sampling, despite higher cost, can be preferable where precision is needed.