# On uniqueness of the set of k-means

arXiv: https://arxiv.org/abs/2410.13495 · HTML: https://mlsanity.com/paper/2410.13495/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 31 references that are in the map.
- First posted on arXiv 17 Oct 2024.

## Builds on (most cited first, 1 of 1)

- [Moment-based Uniform Deviation Bounds for k-means and Friends](https://mlsanity.com/paper/1311.1903/) (2013)

## Built on by (most cited first, 1 of 1)

- [K-means with learned metrics](https://mlsanity.com/paper/2603.14601/) (2026)

## Abstract (arXiv)

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.