# Wide Gaps and Clustering Axioms

arXiv: https://arxiv.org/abs/2308.03464 · HTML: https://mlsanity.com/paper/2308.03464/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 7 Aug 2023.

## Builds on (most cited first, 9 of 9)

- [Heavy hitters via cluster-preserving clustering](https://mlsanity.com/paper/1604.01357/) (2016)
- [Axioms for graph clustering quality functions](https://mlsanity.com/paper/1308.3383/) (2013)
- [Possibility results for graph clustering: A novel consistency axiom](https://mlsanity.com/paper/1806.06142/) (2018)
- [Towards Continuous Consistency Axiom](https://mlsanity.com/paper/2202.06015/) (2022)
- [A Clustering Preserving Transformation for k-Means Algorithm Output](https://mlsanity.com/paper/2202.10455/) (2022)
- [An Aposteriorical Clusterability Criterion for k-Means++ and Simplicity of Clustering](https://mlsanity.com/paper/1704.07139/) (2017)
- [Computational Feasibility of Clustering under Clusterability Assumptions](https://mlsanity.com/paper/1501.00437/) (2015)
- [An Effective and Efficient Approach for Clusterability Evaluation](https://mlsanity.com/paper/1602.06687/) (2016)
- [Privacy Preserving K-Means Clustering: A Secure Multi-Party Computation Approach](https://mlsanity.com/paper/2009.10453/) (2020)

## Abstract (arXiv)

The widely applied k-means algorithm produces clusterings that violate our expectations with respect to high/low similarity/density and is in conflict with Kleinberg's axiomatic system for distance based clustering algorithms that formalizes those expectations in a natural way. k-means violates in particular the consistency axiom. We hypothesise that this clash is due to the not explicated expectation that the data themselves should have the property of being clusterable in order to expect the algorithm clustering hem to fit a clustering axiomatic system. To demonstrate this, we introduce two new clusterability properties, variational k-separability and residual k-separability and show that then the Kleinberg's consistency axiom holds for k-means operating in the Euclidean or non-Euclidean space. Furthermore, we propose extensions of k-means algorithm that fit approximately the Kleinberg's richness axiom that does not hold for k-means. In this way, we reconcile k-means with Kleinberg's axiomatic framework in Euclidean and non-Euclidean settings. Besides contribution to the theory of axiomatic frameworks of clustering and for clusterability theory, practical contribution is the possibility to construct {datasets for testing purposes of algorithms optimizing k-means cost function. This includes a method of construction of {clusterable data with known in advance global optimum.