  Cluster analysis has attracted more and more attention in the field of
machine learning and data mining. Numerous clustering algorithms have been
proposed and are being developed due to diverse theories and various
requirements of emerging applications. Therefore, it is very worth establishing
an unified axiomatic framework for data clustering. In the literature, it is an
open problem and has been proved very challenging. In this paper, clustering
results are axiomatized by assuming that an proper clustering result should
satisfy categorization axioms. The proposed axioms not only introduce
classification of clustering results and inequalities of clustering results,
but also are consistent with prototype theory and exemplar theory of
categorization models in cognitive science. Moreover, the proposed axioms lead
to three principles of designing clustering algorithm and cluster validity
index, which follow many popular clustering algorithms and cluster validity
indices.
