  We study the clustering task under anisotropic Gaussian Mixture Models where
the covariance matrices from different clusters are unknown and are not
necessarily the identical matrix. We characterize the dependence of
signal-to-noise ratios on the cluster centers and covariance matrices and
obtain the minimax lower bound for the clustering problem. In addition, we
propose a computationally feasible procedure and prove it achieves the optimal
rate within a few iterations. The proposed procedure is a hard EM type
algorithm, and it can also be seen as a variant of the Lloyd's algorithm that
is adjusted to the anisotropic covariance matrices.
