  The classical mixture of Gaussians model is related to K-means via
small-variance asymptotics: as the covariances of the Gaussians tend to zero,
the negative log-likelihood of the mixture of Gaussians model approaches the
K-means objective, and the EM algorithm approaches the K-means algorithm. Kulis
& Jordan (2012) used this observation to obtain a novel K-means-like algorithm
from a Gibbs sampler for the Dirichlet process (DP) mixture. We instead
consider applying small-variance asymptotics directly to the posterior in
Bayesian nonparametric models. This framework is independent of any specific
Bayesian inference algorithm, and it has the major advantage that it
generalizes immediately to a range of models beyond the DP mixture. To
illustrate, we apply our framework to the feature learning setting, where the
beta process and Indian buffet process provide an appropriate Bayesian
nonparametric prior. We obtain a novel objective function that goes beyond
clustering to learn (and penalize new) groupings for which we relax the mutual
exclusivity and exhaustivity assumptions of clustering. We demonstrate several
other algorithms, all of which are scalable and simple to implement. Empirical
results demonstrate the benefits of the new framework.
