Convex clustering via $\ell_1$ fusion penalization

TL;DR

This paper analyzes the asymptotic behavior of a convex clustering method with an $$ fusion penalty, establishing its consistency and convergence rates, and proposes a post-processing step for better cluster number estimation, validated through simulations and real data.

Contribution

It provides the first thorough asymptotic analysis of the convex clustering with $$ fusion penalty and introduces a post-processing modification for improved cluster number estimation.

Findings

The sample clustering procedure consistently estimates the population clustering.

The convergence rates of the estimator are derived.

The proposed method effectively estimates the number of clusters in practice.

Abstract

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.