247 research outputs found
-MLE: A fast algorithm for learning statistical mixture models
We describe -MLE, a fast and efficient local search algorithm for learning
finite statistical mixtures of exponential families such as Gaussian mixture
models. Mixture models are traditionally learned using the
expectation-maximization (EM) soft clustering technique that monotonically
increases the incomplete (expected complete) likelihood. Given prescribed
mixture weights, the hard clustering -MLE algorithm iteratively assigns data
to the most likely weighted component and update the component models using
Maximum Likelihood Estimators (MLEs). Using the duality between exponential
families and Bregman divergences, we prove that the local convergence of the
complete likelihood of -MLE follows directly from the convergence of a dual
additively weighted Bregman hard clustering. The inner loop of -MLE can be
implemented using any -means heuristic like the celebrated Lloyd's batched
or Hartigan's greedy swap updates. We then show how to update the mixture
weights by minimizing a cross-entropy criterion that implies to update weights
by taking the relative proportion of cluster points, and reiterate the mixture
parameter update and mixture weight update processes until convergence. Hard EM
is interpreted as a special case of -MLE when both the component update and
the weight update are performed successively in the inner loop. To initialize
-MLE, we propose -MLE++, a careful initialization of -MLE guaranteeing
probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201
Geometric Inference in Bayesian Hierarchical Models with Applications to Topic Modeling
Unstructured data is available in abundance with the rapidly growing size of digital information. Labeling such data is expensive and impractical, making unsupervised learning an increasingly important field. Big data collections often have rich latent structure that statistical modeler is challenged to uncover. Bayesian hierarchical modeling is a particularly suitable approach for complex latent patterns. Graphical model formalism has been prominent in developing various procedures for inference in Bayesian models, however the corresponding computational limits often fall behind the demands of the modern data sizes. In this thesis we develop new approaches for scalable approximate Bayesian inference. In particular, our approaches are driven by the analysis of latent geometric structures induced by the models.
Our specific contributions include the following. We develop full geometric recipe of the Latent Dirichlet Allocation topic model. Next, we study several approaches for exploiting the latent geometry to first arrive at a fast weighted clustering procedure augmented with geometric corrections for topic inference, and then a nonparametric approach based on the analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Estimates produced by our methods are shown to be statistically consistent under some conditions. Finally, we develop a series of models for temporal dynamics of the latent geometric structures where inference can be performed in online and distributed fashion. All our algorithms are evaluated with extensive experiments on simulated and real datasets, culminating at a method several orders of magnitude faster than existing state-of-the-art topic modeling approaches, as demonstrated by experiments working with several million documents in a dozen minutes.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146051/1/moonfolk_1.pd
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