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 ๐ฎ^n,p_+, the manifold of real positive semi-definite (PSD) matrices of size nร n and rank p. Given an energy function โฐ:๐ฎ^n,p_+โโ and certain Riemannian metrics g on ๐ฎ^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 ๐ฎ^n,p_+: (a) the metric obtained from the embedding of ๐ฎ^n,p_+ โโ^nร 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.
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