A sum-of-squares based dimension reduction subroutine gives d^{O(1)}-time clustering for centered non-spherical Gaussian mixtures and d^{O(log w_min^{-1})} for identical-covariance mixtures, improving on d^{O(k)} algorithms.
On spectral learning of mixtures of distributions
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Dimension Reduction via Sum-of-Squares and Improved Clustering Algorithms for Non-Spherical Mixtures
A sum-of-squares based dimension reduction subroutine gives d^{O(1)}-time clustering for centered non-spherical Gaussian mixtures and d^{O(log w_min^{-1})} for identical-covariance mixtures, improving on d^{O(k)} algorithms.