  Bayesian nonparametrics are a class of probabilistic models in which the
model size is inferred from data. A recently developed methodology in this
field is small-variance asymptotic analysis, a mathematical technique for
deriving learning algorithms that capture much of the flexibility of Bayesian
nonparametric inference algorithms, but are simpler to implement and less
computationally expensive. Past work on small-variance analysis of Bayesian
nonparametric inference algorithms has exclusively considered batch models
trained on a single, static dataset, which are incapable of capturing time
evolution in the latent structure of the data. This work presents a
small-variance analysis of the maximum a posteriori filtering problem for a
temporally varying mixture model with a Markov dependence structure, which
captures temporally evolving clusters within a dataset. Two clustering
algorithms result from the analysis: D-Means, an iterative clustering algorithm
for linearly separable, spherical clusters; and SD-Means, a spectral clustering
algorithm derived from a kernelized, relaxed version of the clustering problem.
Empirical results from experiments demonstrate the advantages of using D-Means
and SD-Means over contemporary clustering algorithms, in terms of both
computational cost and clustering accuracy.
