On uniqueness of the set of k-means

TL;DR

This paper establishes necessary and sufficient conditions for the uniqueness of k-means clustering for a probability distribution, analyzing the impact of the choice of k and providing statistical tools for testing uniqueness.

Contribution

It introduces a comprehensive framework for understanding k-means uniqueness, including asymptotic analysis, statistical characterizations, and a bootstrap test for practical assessment.

Findings

Conditions for k-means uniqueness are characterized.

A bootstrap test for assessing uniqueness is developed.

Simulation results demonstrate the effectiveness of the proposed methodology.

Abstract

We provide necessary and sufficient conditions for the uniqueness of the k-means set of a probability distribution. This uniqueness problem is related to the choice of k: depending on the underlying distribution, some values of this parameter could lead to multiple sets of k-means, which hampers the interpretation of the results and/or the stability of the algorithms. We give a general assessment on consistency of the empirical k-means adapted to the setting of non-uniqueness and determine the asymptotic distribution of the within cluster sum of squares (WCSS). We also provide statistical characterizations of k-means uniqueness in terms of the asymptotic behavior of the empirical WCSS. As a consequence, we derive a bootstrap test for uniqueness of the set of k-means. The results are illustrated with examples of different types of non-uniqueness and we check by simulations the performance of the proposed methodology.