Low Rank Convex Clustering for Matrix-Valued Observations

Common clustering methods, such as [Formula: see text]-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 [Formula: see text]-means method proposed by Z. Lyu, and D. Xia [ J. R. Stat. Soc. Ser. B. Methodol., 88 (2026), pp. 43–65]. 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.