Random perturbation of low-rank matrices and applications
摘要：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. It is naturable to understand the essential spectral parameters of this perturbed matrix, such as its spectral norm, the leading singular values, and vectors, or the subspace formed by the first few singular vectors. 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 the data matrix has low rank. I will also discuss some applications of our results. This talk is based on joint works with Sean O'Rourke and Van Vu.