We study the large sample behavior of a convex clustering framework, which minimizes the sample within cluster sum of squares under an fusion constraint on the cluster centroids. This recently proposed approach has been gaining in popularity, however, its asymptotic properties have remained mostly unknown. Our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems. We use this representation to provide a simple and intuitive formulation for the population clustering procedure, and demonstrate that the sample procedure consistently estimates its population analog. The proof conducts a careful simultaneous analysis of a growing number of M-estimation problems, taking advantage of results from the empirical process theory to establish uniform convergence of the sample criterion functions to their population counterparts. Based on the new perspectives gained from the asymptotic investigation, we propose a key post-processing modification of the original clustering approach. Using simulated data, we compare the proposed method with existing number of clusters and modality assessment approaches, and obtain encouraging results. We also demonstrate the applicability of our clustering method for the detection of cellular subpopulations in a single-cell virology study.
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