Faster Algorithms for High-Dimensional Robust Covariance Estimation

We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem with near-optimal error guarantees for several natural structured distributions. Our main contribution is to develop faster algorithms for this problem whose running time nearly matches that of computing the empirical covariance. Given N = (Omega) over tilde (d(2)/epsilon(2)) samples from a d-dimensional Gaussian distribution, an epsilon-fraction of which may be arbitrarily corrupted, our algorithm runs in time (O) over tilde (d(3.26))= poly(epsilon) and approximates the unknown covariance matrix to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees all have runtimes (Omega) over tilde (d(2 omega)) when epsilon = Omega(1), where omega is the exponent of matrix multiplication. We also provide evidence that improving the running time of our algorithm may require new algorithmic techniques.