Robustly Clustering a Mixture of Gaussians

We give an efficient algorithm for robustly clustering of a mixture of two arbitrary Gaussians, a central open problem in the theory of computationally efficient robust estimation, assuming only that the the means of the component Gaussians are well-separated or their covariances are well-separated. Our algorithm and analysis extend naturally to robustly clustering mixtures of well-separated strongly logconcave distributions. The mean separation required is close to the smallest possible to guarantee that most of the measure of each component can be separated by some hyperplane (for covariances, it is the same condition in the second degree polynomial kernel). We also show that for Gaussian mixtures, separation in total variation distance suffices to achieve robust clustering. Our main tools are a new identifiability criterion based on isotropic position and the Fisher discriminant, and a corresponding Sum-of-Squares convex programming relaxation, of fixed degree.