Uniform distribution on Stiefel manifold

Question

I want to implement the method of sampling (uniformly) points on Stiefel manifold but I'm failing to find any kind of research/article/work that can give some info about the methods and techniques of doing it.

I found an old paper, but it is really hard to follow (no background info is given, and overall it is not very accessible).

• K. V. Mardia, C. G. Khatri, Uniform distribution on a Stiefel manifold, Journal of Multivariate Analysis, Volume 7, Issue 3, September 1977.

Is there any work that can help me to tackle the problem? I would be really grateful if you could share anything.

Answer 1

A simple method of generating such samples is as follows. Draw $n m$ random samples from $N(0,1)$ and arrange them into an $n\times m$ matrix $X$. Then $X(X^{\top}X)^{-1/2}$ is a random matrix that follows the uniform distribution on the Stiefel manifold $V_m(\mathbb{R}^n)$ (e.g., Theorem 2.2.1 in Chikuse, Y. (2003). Statistics on Special Manifolds).