ResearchTrend.AI
  • Papers
  • Communities
  • Events
  • Blog
  • Pricing
Papers
Communities
Social Events
Terms and Conditions
Pricing
Parameter LabParameter LabTwitterGitHubLinkedInBlueskyYoutube

© 2025 ResearchTrend.AI, All rights reserved.

  1. Home
  2. Papers
  3. 2309.04072
8
0

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

8 September 2023
Tianmin Yu
Shixin Zheng
Jianfeng Lu
Govind Menon
Xiangxiong Zhang
ArXiv (abs)PDFHTML
Abstract

This paper introduces two explicit schemes to sample matrices from Gibbs distributions on S+n,p\mathcal S^{n,p}_+S+n,p​, the manifold of real positive semi-definite (PSD) matrices of size n×nn\times nn×n and rank ppp. Given an energy function E:S+n,p→R\mathcal E:\mathcal S^{n,p}_+\to \mathbb{R}E:S+n,p​→R and certain Riemannian metrics ggg on S+n,p\mathcal S^{n,p}_+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\mathcal S^{n,p}_+S+n,p​: (a) the metric obtained from the embedding of S+n,p⊂Rn×n\mathcal S^{n,p}_+ \subset \mathbb{R}^{n\times n} S+n,p​⊂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.

View on arXiv
Comments on this paper