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.

这篇研究论文介绍了对 k-means 算法的扩展，以符合 Kleinberg 的公理系统，并提出了新的聚类性质，进一步解释了算法的特点和适用性。

广泛间隔与聚类公理

Wide Gaps and Clustering Axioms

The paper points at the grieving problems implied by the richness axiom in
the Kleinberg's axiomatic system and suggests resolutions. The richness induces
learnability problem in general and leads to conflicts with consistency axiom.
As a resolution, learnability constraints and usage of centric consistency or
restriction of the domain of considered clusterings to super-ball-clusterings
is proposed.

该研究论文指出 Kleinberg 的公理系统中富度公理所暗示的哀悼问题并提出解决方案，即提出了可学性约束，使用核心一致性或限制所考虑的簇的域到超球体簇。