This is a follow-up question to this one about eigenvalues of matrix sums. Suppose you have matrices $A$ and $B$, and know their singular values. What can you say about the singular values of $A+B$?

For Hermitian matrices and eigenvalues, this question was answered by a famous theorem of Knutson and Tao, but I don't know of anything similar for the more general case of singular values. This result would have come in useful for an estimate that I needed. I was able to obtain the estimate in a differenr way, but now I'm curious about the question.