Title :

Monte Carlo with Determinantal Point Processes

Author(s) :

Bardenet, Remi [Auteur]
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Centre National de la Recherche Scientifique [CNRS]
Hardy, Adrien [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

The Annals of Applied Probability

Publisher :

Institute of Mathematical Statistics (IMS)

Publication date :

2020

ISSN :

1050-5164

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Analyse classique [math.CA]
Statistiques [stat]/Méthodologie [stat.ME]
Statistiques [stat]/Calcul [stat.CO]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Inférence bayésienne à ressources limitées - données massives et modèles coûteux

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL