Robert A. Kłopotek

Mieczysław A. Kłopotek, Robert A. Kłopotek

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.

Mieczyslaw A. Klopotek, Robert A. Klopotek

Development of new algorithms in the area of machine learning, especially clustering, comparative studies of such algorithms as well as testing according to software engineering principles requires availability of labeled data sets. While standard benchmarks are made available, a broader range of such data sets is necessary in order to avoid the problem of overfitting. In this context, theoretical works on axiomatization of clustering algorithms, especially axioms on clustering preserving transformations are quite a cheap way to produce labeled data sets from existing ones. However, the frequently cited axiomatic system of Kleinberg:2002, as we show in this paper, is not applicable for finite dimensional Euclidean spaces, in which many algorithms like $k$-means, operate. In particular, the so-called outer-consistency axiom fails upon making small changes in datapoint positions and inner-consistency axiom is valid only for identity transformation in general settings. Hence we propose an alternative axiomatic system, in which Kleinberg's inner consistency axiom is replaced by a centric consistency axiom and outer consistency axiom is replaced by motion consistency axiom. We demonstrate that the new system is satisfiable for a hierarchical version of $k$-means with auto-adjusted $k$, hence it is not contradictory. Additionally, as $k$-means creates convex clusters only, we demonstrate that it is possible to create a version detecting concave clusters and still the axiomatic system can be satisfied. The practical application area of such an axiomatic system may be the generation of new labeled test data from existent ones for clustering algorithm testing. %We propose the gravitational consistency as a replacement which does not have this deficiency.

Robert Kłopotek, Mieczysław Kłopotek

This paper investigates the validity of Kleinberg's axioms for clustering functions with respect to the quite popular clustering algorithm called $k$-means. While Kleinberg's axioms have been discussed heavily in the past, we concentrate here on the case predominantly relevant for $k$-means algorithm, that is behavior embedded in Euclidean space. We point at some contradictions and counter intuitiveness aspects of this axiomatic set within $\mathbb{R}^m$ that were evidently not discussed so far. Our results suggest that apparently without defining clearly what kind of clusters we expect we will not be able to construct a valid axiomatic system. In particular we look at the shape and the gaps between the clusters. Finally we demonstrate that there exist several ways to reconcile the formulation of the axioms with their intended meaning and that under this reformulation the axioms stop to be contradictory and the real-world $k$-means algorithm conforms to this axiomatic system.