published: Oct. 3, 2011, recorded: September 2011, views: 3367
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In spite of the increasing interest into clustering research within the last decades, a unified clustering theory that is independent of a particular algorithm, or underlying the data structure and even the objective function has not be formulated so far. In the paper at hand, we take the first steps towards a theoretical foundation of clustering, by proposing a new notion of "clusterability" of data sets based on the density of the data within a specific region. Specifically, we give a formal definition of what we call "α-clusterable" set and we utilize this notion to prove that the principles proposed in Kleinberg’s impossibility theorem for clustering , are consistent. We further propose an unsupervised clustering algorithm which is based on the notion of α-clusterable set. The proposed algorithm exploits the ability of the well known and widely used particle swarm optimization  to maximize the recently proposed window density function . The obtained clustering quality is compared favorably to the corresponding clustering quality of various other well-known clustering algorithms.
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