Beyond Parallel Pancakes: Quasi-Polynomial Time Guarantees for Non-Spherical Gaussian Mixtures

We consider mixtures of $k\geq 2$ Gaussian components with unknown means and unknown covariance (identical for all components) that are well-separated, i.e., distinct components have statistical overlap at most $k^{-C}$ for a large enough constant $C\ge 1$. Previous statistical-query lower bounds [DKS17] give formal evidence that even distinguishing such mixtures from (pure) Gaussians may be exponentially hard (in $k$). We show that this kind of hardness can only appear if mixing weights are allowed to be exponentially small, and that for polynomially lower bounded mixing weights non-trivial algorithmic guarantees are possible in quasi-polynomial time. Concretely, we develop an algorithm based on the sum-of-squares method with running time quasi-polynomial in the minimum mixing weight. The algorithm can reliably distinguish between a mixture of $k\ge 2$ well-separated Gaussian components and a (pure) Gaussian distribution. As a certificate, the algorithm computes a bipartition of the input sample that separates a pair of mixture components, i.e., both sides of the bipartition contain most of the sample points of at least one component. For the special case of colinear means, our algorithm outputs a $k$ clustering of the input sample that is approximately consistent with the components of the mixture. A significant challenge for our results is that they appear to be inherently sensitive to small fractions of adversarial outliers unlike most previous results for Gaussian mixtures. The reason is that such outliers can simulate exponentially small mixing weights even for mixtures with polynomially lower bounded mixing weights. A key technical ingredient is a characterization of separating directions for well-separated Gaussian components in terms of ratios of polynomials that correspond to moments of two carefully chosen orders logarithmic in the minimum mixing weight.