Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with
  fixed rank
Tianmin Yu Shixin Zheng Jianfeng Lu Govind Menon Xiangxiong Zhang
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.