Matrix perturbation inequalities, such as Weyl's theorem (concerning the
singular values) and the Davis-Kahan theorem (concerning the singular vectors),
play essential roles in quantitative science; in particular, these bounds have
found application in data analysis as well as related areas of engineering and
computer science. In many situations, the perturbation is assumed to be random,
and the
... [Show full abstract] original matrix (data) has certain structural properties (such as
having low rank). We show that, in this scenario, classical perturbation
results, such as Weyl and Davis-Kahan, can be improved significantly. We
believe many of our new bounds are close to optimal.
As an application, we give a uniform and simple analysis of many matrix
reconstruction problems including hidden clique, hidden coloring, clustering,
and Netflix-type problems. In certain cases, our method generalizes and
improves existing results.