  Assignment methods are at the heart of many algorithms for unsupervised
learning and clustering - in particular, the well-known K-means and
Expectation-Maximization (EM) algorithms. In this work, we study several
different methods of assignment, including the "hard" assignments used by
K-means and the ?soft' assignments used by EM. While it is known that K-means
minimizes the distortion on the data and EM maximizes the likelihood, little is
known 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 expected
distortion, showing that K-means (and its extension for inferring general
parametric densities from unlabeled sample data) must implicitly manage a
trade-off between how similar the data assigned to each cluster are, and how
the data are balanced among the clusters. How well the data are balanced is
measured by the entropy of the partition defined by the hard assignments. In
addition to letting us predict and verify systematic differences between
K-means and EM on specific examples, the decomposition allows us to give a
rather general argument showing that K ?means will consistently find densities
with less "overlap" than EM. We also study a third natural assignment method
that we call posterior assignment, that is close in spirit to the soft
assignments of EM, but leads to a surprisingly different algorithm.
