An Impossibility Theorem for Clustering

Part of Advances in Neural Information Processing Systems 15 (NIPS 2002)

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Jon Kleinberg


Although the study of clustering is centered around an intuitively compelling goal, it has been very di(cid:14)cult to develop a uni(cid:12)ed framework for reasoning about it at a technical level, and pro- foundly diverse approaches to clustering abound in the research community. Here we suggest a formal perspective on the di(cid:14)culty in (cid:12)nding such a uni(cid:12)cation, in the form of an impossibility theo- rem: for a set of three simple properties, we show that there is no clustering function satisfying all three. Relaxations of these prop- erties expose some of the interesting (and unavoidable) trade-o(cid:11)s at work in well-studied clustering techniques such as single-linkage, sum-of-pairs, k-means, and k-median.