[Paper Review] Consistent clustering using an $\ell_1$ fusion penalty

Peter Radchenko, Gourab Mukherjee|arXiv (Cornell University)|Dec 2, 2014

Statistical Methods and Inference40 references4 citations

TL;DR

This paper proposes a consistent convex clustering method using an ℓ₁ fusion penalty to minimize within-cluster sum of squares, proving that the sample clustering procedure asymptotically estimates its population counterpart. By reinterpreting clustering as a sequence of split decisions via maximization problems, the authors establish uniform convergence through empirical process theory and propose a post-processing refinement that improves cluster number and modality detection in simulations and single-cell virology data.

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, 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.

Motivation & Objective

• To establish the large sample behavior of a convex clustering framework with an ℓ₁ fusion penalty on cluster centroids.
• To resolve the lack of theoretical understanding regarding the asymptotic properties of this increasingly popular clustering method.
• To develop a population-level clustering procedure that serves as a theoretical benchmark for the sample clustering method.
• To propose a post-processing modification that enhances the identification of the true number of clusters and cluster modality.
• To validate the method’s performance on simulated data and real single-cell virology data for detecting cellular subpopulations.

Proposed method

• Represents the sample clustering procedure as a sequence of cluster splits determined by solving a series of maximization problems.
• Introduces a novel population-level clustering formulation derived from the sample procedure’s sequential split structure.
• Uses empirical process theory to prove uniform convergence of sample criterion functions to their population counterparts.
• Applies simultaneous analysis of a growing number of M-estimation problems to handle the increasing complexity of the clustering process.
• Proposes a post-processing step that refines cluster assignments by leveraging the sequential split structure and consistency results.
• Employs simulated data to compare the method with existing approaches for cluster number and modality assessment.

Experimental results

Research questions

• RQ1 Does the sample clustering procedure using an ℓ₁ fusion penalty consistently estimate the true underlying population clustering structure?
• RQ2 Can the sequential split representation of clustering be used to define a coherent and interpretable population-level clustering procedure?
• RQ3 How does the proposed post-processing step improve the detection of the number of clusters and cluster modality compared to existing methods?
• RQ4 What is the theoretical justification for the consistency of the clustering procedure under increasing sample size?
• RQ5 Can the method effectively detect biologically meaningful subpopulations in single-cell virology data?

Key findings

• The sample clustering procedure is proven to be consistent, meaning it asymptotically recovers the true population clustering structure.
• The population clustering procedure is formally defined through the sequential split representation, providing a clear theoretical foundation.
• The proposed post-processing modification improves performance in identifying the correct number of clusters and detecting cluster modality in simulated data.
• The method successfully detects biologically relevant cellular subpopulations in a single-cell virology dataset, demonstrating practical utility.
• Uniform convergence of the sample criterion to its population counterpart is established via a novel simultaneous analysis of M-estimation problems.
• The theoretical framework provides a new perspective on convex clustering as a sequence of optimization steps, enhancing interpretability and consistency.