Fast sampling from $\beta$-ensembles

Abstract

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 $\mathcal{O}(\log (N))$.

arXiv comment37 pages, 8 figures, code at https://github.com/guilgautier/DPPy

References in corpus (12)

• Determinantal Processes and Independence
• Matrix Models for Beta Ensembles
• On the Numerical Evaluation of Fredholm Determinants
• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Orthogonal polynomial ensembles in probability theory
• How many eigenvalues of a Gaussian random matrix are positive?
• Universality of the Stochastic Airy Operator
• Canonical moments and random spectral measures
• Polynomial Ensembles and Recurrence Coefficients
• Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms
• Numerical solution of Dyson Brownian motion and a sampling scheme for invariant matrix ensembles
• Distributions on unbounded moment spaces and random moment sequences

Cited by in corpus (1)

• A note on the exact simulation of a random eigenvalue of a GUE matrix