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Assignment methods are at the heart of many algorithms for unsupervisedlearning and clustering - in particular, the well-known K-means andExpectation-Maximization (EM) algorithms. In this work, we study severaldifferent methods of assignment, including the "hard" assignments used byK-means and the ?soft' assignments used by EM. While it is known that K-meansminimizes the distortion on the data and EM maximizes the likelihood, little isknown about the systematic differences of behavior between the two algorithms.Here we shed light on these differences via an information-theoretic analysis.The cornerstone of our results is a simple decomposition of the expecteddistortion, showing that K-means (and its extension for inferring generalparametric densities from unlabeled sample data) must implicitly manage atrade-off between how similar the data assigned to each cluster are, and howthe data are balanced among the clusters. How well the data are balanced ismeasured by the entropy of the partition defined by the hard assignments. Inaddition to letting us predict and verify systematic differences betweenK-means and EM on specific examples, the decomposition allows us to give arather general argument showing that K ?means will consistently find densitieswith less "overlap" than EM. We also study a third natural assignment methodthat we call posterior assignment, that is close in spirit to the softassignments of EM, but leads to a surprisingly different algorithm.

An Information-Theoretic Analysis of Hard and Soft Assignment Methods for Clustering