On Convex Clustering Solutions

TL;DR

This paper investigates the properties of convex clustering, proving it can only learn convex clusters and characterizing its solutions, regularization, and limitations, thus deepening understanding of its capabilities and constraints.

Contribution

It provides the first theoretical analysis showing convex clustering only learns convex clusters and describes the geometric structure of its solutions.

Findings

Convex clustering can only learn convex clusters.

Clusters have disjoint bounding balls with significant gaps.

The paper characterizes solutions, hyperparameters, and limitations.

Abstract

Convex clustering is an attractive clustering algorithm with favorable properties such as efficiency and optimality owing to its convex formulation. It is thought to generalize both k-means clustering and agglomerative clustering. However, it is not known whether convex clustering preserves desirable properties of these algorithms. A common expectation is that convex clustering may learn difficult cluster types such as non-convex ones. Current understanding of convex clustering is limited to only consistency results on well-separated clusters. We show new understanding of its solutions. We prove that convex clustering can only learn convex clusters. We then show that the clusters have disjoint bounding balls with significant gaps. We further characterize the solutions, regularization hyperparameters, inclusterable cases and consistency.