Robust Sub-Gaussian Principal Component Analysis and Width-Independent Schatten Packing

We develop two methods for the following fundamental statistical task: given an \eps -corrupted set of n samples from a d -dimensional sub-Gaussian distribution, return an approximate top eigenvector of the covariance matrix. Our first robust PCA algorithm runs in polynomial time, returns a 1 - O(\eps\log\eps {-1}) -approximate top eigenvector, and is based on a simple iterative filtering approach. Our second, which attains a slightly worse approximation factor, runs in nearly-linear time and sample complexity under a mild spectral gap assumption. These are the first polynomial-time algorithms yielding non-trivial information about the covariance of a corrupted sub-Gaussian distribution without requiring additional algebraic structure of moments. As a key technical tool, we develop the first width-independent solvers for Schatten- p norm packing semidefinite programs, giving a (1 \eps) -approximate solution in O(p\log(\tfrac{nd}{\eps})\eps {-1}) input-sparsity time iterations (where n, d are problem dimensions).