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

• URL: http://arxiv.org/abs/2309.04072v1
• Date: Fri, 8 Sep 2023 02:09:40 GMT
• Title: Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with fixed rank
• Authors: Tianmin Yu and Shixin Zheng and Jianfeng Lu and Govind Menon and Xiangxiong Zhang
• Abstract summary: We present two explicit schemes to sample matrices from Gibbs distributions on $mathcal Sn,p_+$. We also provide examples of energy functions with explicit Gibbs distributions that allow numerical validation of these schemes.
• Score: 5.0397419406319095
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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