Unifying Single-Linkage and Max-Sum Clustering

To unify the motivating principles behind Max-Sum and Single-Linkage clustering, we consider an axiomatic approach to the theory of Clustering. We discuss abstract properties of clustering functions, following the framework of Kleinberg, [7]. By relaxing one of Kleinberg’s clustering axioms, we sidestep his impossibility result and arrive at a consistent set of axioms. We suggest to extend these axioms, aiming to provide an axiomatic taxonomy of clustering paradigms. Such a taxonomy should provide users some guidance concerning the choice of the appropriate clustering paradigm for a given task. The main result of this paper is a set of abstract properties that characterize the Max-Sum and Single-Linkage clustering functions. These functions have been traditionally treated separately, as the principles motivating their use have never been unified. In the current paper we provide alternative axiomatic definitions for Max-Sum and Single-Linkage which sheds light on their common underlying principles and will guide the user to decide which is appropriate for a particular task, if either of them.