Robustly Learning Mixtures of k Arbitrary Gaussians

We give a polynomial-time algorithm for the problem of robustly estimating a mixture of k arbitrary Gaussians in R-d, for any fixed k, in the presence of a constant fraction of arbitrary corruptions. This resolves the main open problem in several previous works on algorithmic robust statistics, which addressed the special cases of robustly estimating (a) a single Gaussian, (b) a mixture of TV-distance separated Gaussians, and (c) a uniform mixture of two Gaussians. Our main tools are an efficient partial clustering algorithm that relies on the sum-of-squares method, and a novel tensor decomposition algorithm that allows errors in both Frobenius norm and low-rank terms.