Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with fixed rank

TL;DR

This paper develops Riemannian Langevin Monte Carlo schemes for sampling fixed-rank positive semi-definite matrices, utilizing Euler-Maruyama discretization on manifolds with applications to Gibbs distributions.

Contribution

Introduces two explicit Riemannian Langevin schemes for sampling PSD matrices of fixed rank, with discretizations under two fundamental metrics and validation examples.

Findings

Schemes effectively sample from Gibbs distributions on PSD manifolds.

Numerical validation confirms the schemes' accuracy and applicability.

Applicable to energy functions with explicit Gibbs distributions.

Abstract

This paper introduces two explicit schemes to sample matrices from Gibbs distributions on $\mathcal S^{n,p}_+$, the manifold of real positive semi-definite (PSD) matrices of size $n\times n$ and rank $p$. Given an energy function $\mathcal E:\mathcal S^{n,p}_+\to \mathbb{R}$ and certain Riemannian metrics $g$ on $\mathcal S^{n,p}_+$, these schemes rely on an Euler-Maruyama discretization of the Riemannian Langevin equation (RLE) with Brownian motion on the manifold. We present numerical schemes for RLE under two fundamental metrics on $\mathcal S^{n,p}_+$: (a) the metric obtained from the embedding of $\mathcal S^{n,p}_+ \subset \mathbb{R}^{n\times n} $; and (b) the Bures-Wasserstein metric corresponding to quotient geometry. We also provide examples of energy functions with explicit Gibbs distributions that allow numerical validation of these schemes.