Outlier-Robust Clustering of Non-Spherical Mixtures
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We give the first outlier-robust efficient algorithm for clustering a mixture of k statistically separated d-dimensional Gaussians (k-GMMs). Concretely, our algorithm takes input an ϵ-corrupted sample from a k-GMM and outputs an approximate clustering that misclassifies at most O(kϵ)+η fraction of the points whenever every pair of components is separated by 1-(-poly(k/η)) in total variation distance. This is the statistically weakest possible notion of separation and allows, for e.g., clustering of mixtures with components with the same mean with covariances differing in a single unknown direction or separated in Frobenius distance. The running time of our algorithm is d^O(log(κ)) poly(k/η) where κ is a measure of spread of the mixture in any direction. For k=2, our algorithms run in time and samples poly(d) with no dependence on the spread κ. Such a results were not known prior to our work, even for k=2. More generally, our algorithm succeeds for mixtures of any distribution that satisfies two well-studied analytic assumptions - certifiable hypercontractivity and anti-concentration. Thus, they extend to clustering mixtures of arbitrary affine transforms of the uniform distribution on the d-dimensional unit sphere. Even the information theoretic clusterability of distributions satisfying our analytic assumptions was not known and is likely to be of independent interest. Our algorithms build on the recent flurry of work relying on certifiable anti-concentration, first introduced in [KKK'19, RY'20]. Our techniques expand the sum-of-squares toolkit to show robust certifiability of TV-separated Gaussian clusters in data. This involves a low-degree sum-of-squares proof of statements that relate parameter distance to total variation distance simply relying on hypercontractivity and anti-concentration.
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