Computing singular vectors with random noise

Computing the first few singular vectors of a large matrix is an operation of great practical importance. For instance, it is at the core of PCA (principal component analysis), which is probably one of the first few things one might do when one has to study a large matrix.

In this talk, I would like to raise the following question: How accurate is the result when the original matrix is slightly perturbed by random noise? There are some surprises here, and a considerable amount of open questions