Optimal Clustering by Lloyd Algorithm for Low-Rank Mixture Model

TL;DR

This paper introduces a low-rank mixture model for matrix clustering, proposing an efficient Lloyd-based algorithm that achieves minimax optimal error rates under certain conditions, with theoretical and empirical validation.

Contribution

It develops a low-rank adaptation of Lloyd's algorithm for matrix clustering, providing theoretical guarantees and demonstrating improved performance over existing methods.

Findings

Algorithm converges quickly with good initialization.

Achieves minimax optimal clustering error rate.

Outperforms existing methods on real datasets.

Abstract

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's 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 the separation strength. Unlike GMM, however, the computational difficulty of LrMM is characterized by the signal strength, i.e., the smallest non-zero singular values of population center matrices. Evidence is provided showing that no polynomial-time algorithm is consistent if the signal strength is not strong enough, even though the separation strength is strong. Intriguing differences between estimation and clustering under LrMM are discussed. The merits of low-rank Lloyd's algorithm are confirmed by comprehensive simulation experiments. Finally, our method outperforms others in the literature on real-world datasets.