# On the Consistency of Metric and Non-Metric K-medoids

arXiv: https://arxiv.org/abs/2010.09906 · HTML: https://mlsanity.com/paper/2010.09906/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2020.

## Built on by (most cited first, 3 of 3)

- [Asymptotic Theory of Geometric and Adaptive k-Means Clustering](https://mlsanity.com/paper/2202.13423/) (2022)
- [Consistency and inconsistency in k-means clustering](https://mlsanity.com/paper/2507.06226/) (2025)
- [The Condition-Number Principle for Prototype Clustering](https://mlsanity.com/paper/2604.07744/) (2026)

## Abstract (arXiv)

We establish the consistency of K-medoids in the context of metric spaces. We start by proving that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution under general conditions, including a wide selection of loss functions. This asymptotic equivalence, in turn, enables us to apply the work of Parna (1986) on the consistency of K-means. This general approach applies also to non-metric settings where only an ordering of the dissimilarities is available. We consider two types of ordinal information: one where all quadruple comparisons are available; and one where only triple comparisons are available. We provide some numerical experiments to illustrate our theory.