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We study the clustering task under anisotropic Gaussian Mixture Models wherethe covariance matrices from different clusters are unknown and are notnecessarily the identical matrix. We characterize the dependence ofsignal-to-noise ratios on the cluster centers and covariance matrices andobtain the minimax lower bound for the clustering problem. In addition, wepropose a computationally feasible procedure and prove it achieves the optimalrate within a few iterations. The proposed procedure is a hard EM typealgorithm, and it can also be seen as a variant of the Lloyd's algorithm thatis adjusted to the anisotropic covariance matrices.

Optimal Clustering in Anisotropic Gaussian Mixture Models