A Uniqueness Theorem for Clustering

Despite the widespread use of Clustering, there is distressingly little general theory of clustering available. Questions like "What distinguishes a clustering of data from other data partitioning?", "Are there any principles governing all clustering paradigms?", "How should a user choose an appropriate clustering algorithm for a particular task?", etc. are almost completely unanswered by the existing body of clustering literature. We consider an axiomatic approach to the theory of Clustering. We adopt the framework of Kleinberg, [Kle03]. 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 Single-Linkage clustering function. This characterization result provides new insight into the properties of desired data groupings that make Single-Linkage the appropriate choice. We conclude by considering a taxonomy of clustering functions based on abstract properties that each satisfies.