Optimal Clustering by Lloyd Algorithm for Low-Rank Mixture Model

This paper investigates the computational and statistical limits in clustering matrix-valued observations. We propose a low-rank mixture model (LrMM), adapted from the classical Gaussian mixture model (GMM) to treat matrix-valued observations, which assumes low-rankness for population center matrices. A computationally efficient clustering method is designed by integrating Lloyd algorithm and low-rank approximation. Once well-initialized, the algorithm converges fast and achieves an exponential-type clustering error rate that is minimax optimal. Meanwhile, we show that a tensor-based spectral method delivers a good initial clustering. Comparable to GMM, the minimax optimal clustering error rate is decided by the separation strength, i.e, the minimal distance between population center matrices. By exploiting low-rankness, the proposed algorithm is blessed with a weaker requirement on separation strength. Unlike GMM, however, the statistical and computational difficulty of LrMM is characterized by the signal strength, i.e, the smallest non-zero singular values of population center matrices. Evidences are provided showing that no polynomial-time algorithm is consistent if the signal strength is not strong enough, even though the separation strength is strong. The performance of our low-rank Lloyd algorithm is further demonstrated under sub-Gaussian noise. Intriguing differences between estimation and clustering under LrMM are discussed. The merits of low-rank Lloyd algorithm are confirmed by comprehensive simulation experiments. Finally, our method outperforms others in the literature on real-world datasets.