A fast Metropolis-Hastings method for generating random correlation matrices

TL;DR

This paper introduces a simple, fast, and theoretically sound Metropolis-Hastings algorithm for uniformly sampling correlation matrices, outperforming existing methods in speed and convergence.

Contribution

It presents a novel, intuitive MCMC method based on Cholesky factorization for sampling correlation matrices, with detailed convergence analysis and empirical validation.

Findings

Faster convergence compared to existing methods

Significantly improved sampling speed

Theoretically justified convergence properties

Abstract

We propose a novel Metropolis-Hastings algorithm to sample uniformly from the space of correlation matrices. Existing methods in the literature are based on elaborated representations of a correlation matrix, or on complex parametrizations of it. By contrast, our method is intuitive and simple, based the classical Cholesky factorization of a positive definite matrix and Markov chain Monte Carlo theory. We perform a detailed convergence analysis of the resulting Markov chain, and show how it benefits from fast convergence, both theoretically and empirically. Furthermore, in numerical experiments our algorithm is shown to be significantly faster than the current alternative approaches, thanks to its simple yet principled approach.