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

A Distributed Learning Framework for Heterogeneous Data Sources

We present a probabilistic model-based framework for distributed learning that takes into account privacy restrictions and is applicable to scenarios where the different sites have diverse, possibly overlapping subsets of features. Our framework decouples data privacy issues from knowledge integration issues by requiring the individual sites to share only privacy-safe probabilistic models of the local data, which are then integrated to obtain a global probabilistic model based on the union of the features available at all the sites. We provide a mathematical formulation of the model integration problem using the maximum likelihood and maximum entropy principles and describe iterative algorithms that are guaranteed to converge to the optimal solution. For certain commonly occurring special cases involving hierarchically ordered feature sets or conditional independence, we obtain closed form solutions and use these to propose an efficient alternative scheme by recursive decomposition of the model integration problem. To address interpretability concerns, we also present a modified formulation where the global model is assumed to belong to a specified parametric family. Finally, to highlight the generality of our framework, we provide empirical results for various learning tasks such as clustering and classification on different kinds of datasets consisting of continuous vector, categorical and directional attributes. The results show that high quality global models can be obtained without much loss of privacy