Towards a Principled Theory of Clustering

To answer the question “Which clustering function should one use?” for a given task, we consider an axiomatic approach to the theory of Clustering, with special focus on uniqueness theorems characterizing popular clustering functions. We argue that such theorems can be used to decide exactly when a particular clustering function should be used or avoided. We discuss abstract properties of clustering functions, following the framework of Kleinberg, [Kleinberg, 2003]. By altering one of Kleinberg’s axioms, we sidestep his impossibility result and arrive at a consistent set of axioms, aiming to provide an axiomatic taxonomy of clustering paradigms. 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. These uniqueness theorems will guide the user to decide which is appropriate for a particular task, if either of them. Our results also provide a theoretical foundation for empirical observations of clustering performed by humans in [Dry et al., 2009].