I am trying to understand the behavior of distributions over the Unitary group (i.e. the set of square matrices $P$ such that $P^tP = I_d$), or in general distribution over the Stiefel manifold (set of $D\times K$ matrices such that $P^tP = I_K$).

In particular, I need to understand how to compute any marginal distributions. Let's say for instance, the marginals of the uniform distribution of one the set defined above.

By Intuition I would like to say that spherical symmetries imply that the marginals are uniform on the corresponding Stiefel manifold, but I don't manage to write it properly...

Any clue?

  • $\begingroup$ The question of computing the marginals requires knowing in what form the original distribution is represented--otherwise only generic answers are possible (that is, you compute a marginal distribution they way you would in any situation). Could you elaborate on how your distributions are given to you? $\endgroup$ – whuber May 15 '19 at 23:16