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

arXiv: https://arxiv.org/abs/2309.04072 · HTML: https://mlsanity.com/paper/2309.04072/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 8 Sep 2023.

## Builds on (most cited first, 6 of 6)

- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020)
- [Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion](https://mlsanity.com/paper/1804.04310/) (2018)
- [Efficient sampling from the Bingham distribution](https://mlsanity.com/paper/2010.00137/) (2020)
- [Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints: an extended version](https://mlsanity.com/paper/2204.07830/) (2022)

## Built on by (most cited first, 1 of 1)

- [On the implicit regularization of Langevin dynamics with projected noise](https://mlsanity.com/paper/2602.12257/) (2026)

## Abstract (arXiv)

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.