Dimension Reduction via Sum-of-Squares and Improved Clustering Algorithms for Non-Spherical Mixtures

We develop a new approach for clustering non-spherical (i.e., arbitrary component covariances) Gaussian mixture models via a subroutine based on the sum-of-squares method that finds a lowdimensional separation-preserving projection of the input data. Our method provides a non-spherical analog of the classical dimension reduction based on singular value decomposition that, among several other applications, forms a key component of the celebrated spherical clustering algorithm of Vempala and Wang (Vempala and Wang, 2004). As applications, we obtain an algorithm to (1) cluster an arbitrary total-variation separated mixture of k centered (i.e., zero-mean) Gaussians with n ≥ poly(d)f(w⁻¹_min) samples and poly(n) time, and (2) cluster an arbitrary total-variation separated mixture of k Gaussians with identical but arbitrary unknown covariance with n ≥ d^O(log w⁻¹_min)f(w⁻¹_min) samples and n^O(log w⁻¹_min) time. Here, wmin is the minimum mixing weight of the input mixture, and f does not depend on the dimension d. Our algorithms naturally extend to tolerate a dimension-independent fraction of arbitrary outliers. Before this work, the techniques in the state-of-the-art non-spherical clustering algorithms needed d^O(k)f(w⁻¹_min) samples and time for clustering such mixtures. Our results may come as a surprise in the context of the d^Ω(k) statistical query and sum-of-squares lower bounds (Diakonikolas et al., 2017, 2024) for clustering non-spherical Gaussian mixtures. While these results are usually thought to rule out d^o(k) cost algorithms for the problem, our results show that the lower bounds can, in fact, be circumvented for a remarkably general class of Gaussian mixtures.