A novel sampler for Gauss-Hermite determinantal point processes with application to Monte Carlo integration

TL;DR

This paper introduces a new sampling method for Gauss-Hermite determinantal point processes on 1, demonstrating improved Monte Carlo integration with Gaussian measures, relevant for machine learning applications.

Contribution

It develops a novel sampling scheme for Gauss-Hermite determinantal point processes on 1, extending existing methods to Rd and enhancing Monte Carlo integration techniques.

Findings

Samples effectively improve Monte Carlo integration accuracy.

The new sampler outperforms existing methods in Gaussian measure contexts.

Application potential in machine learning tasks involving high-dimensional integrals.

Abstract

Determinantal points processes are a promising but relatively under-developed tool in machine learning and statistical modelling, being the canonical statistical example of distributions with repulsion. While their mathematical formulation is elegant and appealing, their practical use, such as simply sampling from them, is far from straightforward.Recent work has shown how a particular type of determinantal point process defined on the compact multidimensional space $[-1, 1]^d$ can be practically sampled and further shown how such samples can be used to improve Monte Carlo integration.This work extends those results to a new determinantal point process on $\mathbb{R}^d$ by constructing a novel sampling scheme. Samples from this new process are shown to be useful in Monte Carlo integration against Gaussian measure, which is particularly relevant in machine learning applications.