Monte Carlo with Determinantal Point Processes

Rémi Bardenet (1, 2) , Adrien Hardy (3)

1 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
2 CNRS - Centre National de la Recherche Scientifique
3 LPP - Laboratoire Paul Painlevé - UMR 8524

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.

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]; Méthodologie [stat.ME]; Calcul [stat.CO]

Rémi Bardenet, Adrien Hardy. Monte Carlo with Determinantal Point Processes. The Annals of Applied Probability, 2020. ⟨hal-01311263⟩