I know that we have truncated SVD that can compute the first say $k$ largest singular values (and corresponding singular vectors). However, I'd like to know if there is an algorithm that can find only the $k$-th largest singular value and the corresponding singular vectors without the need to find the larger singular values first.
The motivation is that I want to get the $k$-th largest singular value/vectors at computational/storage cost that is necessary for computing the principal singular vectors. I'm assuming that no good approximation for the singular values is available beforehand.