Monte Carlo with Determinantal Point Processes

TL;DR

This paper introduces a Monte Carlo method using determinantal point processes that achieves faster convergence rates than traditional methods, especially in higher dimensions, by leveraging repulsive random variables and orthogonal polynomial structures.

Contribution

It develops a novel Monte Carlo quadrature based on determinantal point processes, proving a CLT for these processes, and demonstrates improved convergence rates over classical Monte Carlo methods.

Findings

Achieves convergence rate of N^{-(1+1/d)/2} in d dimensions.

Proves a CLT for linear statistics of determinantal point processes.

Introduces a general Monte Carlo method applicable to any measure with a smooth density.

Abstract

We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as $N^{-(1+1/d)/2}$, where $d$ is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition, a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which, at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term.