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

09/08/2023

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by   Tianmin Yu, et al.

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