Mixture models, robustness, and sum of squares proofs

We use the Sum of Squares method to develop new efficient algorithms for learning well-separated mixtures of Gaussians and robust mean estimation, both in high dimensions, that substantially improve upon the statistical guarantees achieved by previous efficient algorithms. Our contributions are: Mixture models with separated means: We study mixtures of poly(k)-many k-dimensional distributions where the means of every pair of distributions are separated by at least kε. In the special case of spherical Gaussian mixtures, we give a kO(1/ε)-time algorithm that learns the means assuming separation at least kε, for any ε> 0. This is the first algorithm to improve on greedy (“single-linkage”) and spectral clustering, breaking a long-standing barrier for efficient algorithms at separation k1/4. Robust estimation: When an unknown (1−ε)-fraction of X1,…,Xn are chosen from a sub-Gaussian distribution with mean µ but the remaining points are chosen adversarially, we give an algorithm recovering µ to error ε1−1/t in time kO(t), so long as sub-Gaussian-ness up to O(t) moments can be certified by a Sum of Squares proof. This is the first polynomial-time algorithm with guarantees approaching the information-theoretic limit for non-Gaussian distributions. Previous algorithms could not achieve error better than ε1/2. As a corollary, we achieve similar results for robust covariance estimation. Both of these results are based on a unified technique. Inspired by recent algorithms of Diakonikolas et al. in robust statistics, we devise an SDP based on the Sum of Squares method for the following setting: given X1,…,Xn ∈ ℝk for large k and n = poly(k) with the promise that a subset of X1,…,Xn were sampled from a probability distribution with bounded moments, recover some information about that distribution.