  We study the large sample behavior of a convex clustering framework, which
minimizes the sample within cluster sum of squares under an~$\ell_1$ 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. We then
demonstrate that the sample procedure consistently estimates its population
analog, and derive the corresponding rates of convergence. The proof conducts a
careful simultaneous analysis of a collection of M-estimation problems, whose
cardinality grows together with the sample size. Based on the new perspectives
gained from the asymptotic investigation, we propose a key post-processing
modification of the original clustering framework. We show, both theoretically
and empirically, that the resulting approach can be successfully used to
estimate the number of clusters in the population. 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.
