I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together.
However, I'm wondering how random this method is.
Since this paper claims that one can generate random unitary matrices uniformly from $U(n)$, which is the set of all the unitary matrices of order $n$.
Is there a method which can generate a random matrix with given singular values uniformly from all matrices with same set of singular values?
Thank you so much!