Wide Gaps and Clustering Axioms

TL;DR

This paper introduces new clusterability properties and extensions of k-means to reconcile it with Kleinberg's axioms, enhancing theoretical understanding and practical dataset construction for clustering evaluation.

Contribution

It proposes variational and residual k-separability as new clusterability properties and extends k-means to satisfy Kleinberg's axioms, bridging theory and practice.

Findings

k-means violates Kleinberg's consistency axiom without clusterability assumptions

New clusterability properties ensure k-means aligns with axiomatic expectations

Method for constructing clusterable datasets with known global optima

Abstract

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.