Low Rank Convex Clustering For Matrix-Valued Observations

TL;DR

This paper introduces a low rank convex clustering method specifically designed for matrix-valued data, providing theoretical guarantees and an efficient algorithm, with extensive experiments demonstrating its effectiveness.

Contribution

It extends convex clustering to matrix data with low rank structure, offers theoretical recovery guarantees, and develops a practical algorithm.

Findings

Exact cluster recovery for finite samples

Asymptotic cluster recovery as sample size grows

Finite sample prediction error bounds

Abstract

Common clustering methods, such as $k$-means and convex clustering, group similar vector-valued observations into clusters. However, with the increasing prevalence of matrix-valued observations, which often exhibit low rank characteristics, there is a growing need for specialized clustering techniques for these data types. In this paper, we propose a low rank convex clustering model tailored for matrix-valued observations. Our approach extends the convex clustering model originally designed for vector-valued data to classify matrix-valued observations. Additionally, it serves as a convex relaxation of the low rank $k$-means method proposed by Z. Lyu, and D. Xia (arXiv:2207.04600). Theoretically, we establish exact cluster recovery for finite samples and asymptotic cluster recovery as the sample size approaches infinity. We also give a finite sample bound on prediction error in terms of centroid estimation, and further establish the prediction consistency. To make the model practically useful, we develop an efficient double-loop algorithm for solving it. Extensive numerical experiments are conducted to show the effectiveness of our proposed model.