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.

翻译：本文提出了两种显式方案，用于从吉布斯分布中采样 $\mathcal S^{n,p}_+$ 上的矩阵，其中 $\mathcal S^{n,p}_+$ 是大小为 $n\times n$、秩为 $p$ 的实半正定矩阵流形。给定能量函数 $\mathcal E:\mathcal S^{n,p}_+\to \mathbb{R}$ 及 $\mathcal S^{n,p}_+$ 上的特定黎曼度量 $g$，这些方案基于流形上布朗运动的黎曼朗之万方程（RLE）的欧拉-丸山离散化。我们针对 $\mathcal S^{n,p}_+$ 上的两种基本度量提出了RLE的数值方案：（a）由 $\mathcal S^{n,p}_+ \subset \mathbb{R}^{n\times n}$ 嵌入诱导的度量；（b）对应商几何的Bures-Wasserstein度量。我们还提供了具有显式吉布斯分布的能量函数示例，以支持这些方案的数值验证。