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Singular values and vectors under random perturbation

Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors?聽聽

Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank.聽

Time permitting, I will also discuss some applications of these bounds to community detection and matrix recovery type problems.聽 This talk is based on joint work with Van Vu and Ke Wang.