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

This paper introduces two explicit schemes to sample matrices from Gibbs distributions on mathcal Sn,p_+, the manifold of real positive semi-definite (PSD) matrices of size n× n and rank p. Given an energy function mathcal E:mathcal Sn,p_+→ ℝ and certain Riemannian metrics g on mathcal Sn,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 Sn,p_+: (a) the metric obtained from the embedding of mathcal Sn,p_+ subset ℝn× 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.