  Bayesian models offer great flexibility for clustering
applications---Bayesian nonparametrics can be used for modeling infinite
mixtures, and hierarchical Bayesian models can be utilized for sharing clusters
across multiple data sets. For the most part, such flexibility is lacking in
classical clustering methods such as k-means. In this paper, we revisit the
k-means clustering algorithm from a Bayesian nonparametric viewpoint. Inspired
by the asymptotic connection between k-means and mixtures of Gaussians, we show
that a Gibbs sampling algorithm for the Dirichlet process mixture approaches a
hard clustering algorithm in the limit, and further that the resulting
algorithm monotonically minimizes an elegant underlying k-means-like clustering
objective that includes a penalty for the number of clusters. We generalize
this analysis to the case of clustering multiple data sets through a similar
asymptotic argument with the hierarchical Dirichlet process. We also discuss
further extensions that highlight the benefits of our analysis: i) a spectral
relaxation involving thresholded eigenvectors, and ii) a normalized cut graph
clustering algorithm that does not fix the number of clusters in the graph.
