Fast sampling from beta-ensembles

Gautier, Guillaume; Bardenet, Remi; Valko, Michal

Document type :

Article dans une revue scientifique

DOI :

10.1007/s11222-020-09984-0

Title :

Fast sampling from beta-ensembles

Author(s) :

Gautier, Guillaume [Auteur]
Scool [Scool]
Bardenet, Remi [Auteur]

Scool [Scool]
Valko, Michal [Auteur]

DeepMind [Paris]

Journal title :

Statistics and Computing

Publisher :

Springer Verlag (Germany)

Publication date :

2021-01-12

ISSN :

0960-3174

English keyword(s) :

β-ensembles
Tridiagonal random matrices
Orthogonal polynomials
Gibbs sampling

HAL domain(s) :

Mathématiques [math]/Statistiques [math.ST]

English abstract : [en]

We study sampling algorithms for $\beta$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal ...
Show more >We study sampling algorithms for $\beta$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of $\beta$-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size $N$ mixes in $\mathcal{O}(\log(N))$.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

ANR Project :

Inférence bayésienne à ressources limitées - données massives et modèles coûteux

Comment :

37 pages, 8 figures, code at https://github.com/guilgautier/DPPy