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.

翻译：我们研究的是不同组群的共变矩阵不为人知,不一定是相同的矩阵,在类集混合模型下进行分组任务研究;我们将信号对噪音比率对集集中心和共变矩阵的依赖性定性为对集集中心和共变矩阵的依赖性,并获得组群问题的最低约束值;此外,我们提出一个计算可行的程序,并证明它在几个迭代中达到最佳比率;拟议程序是一种硬式EM类型算法,也可以被视为劳埃德算法的一种变体,该算法被调整到厌同式变量。