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

Rewiring with Positional Encodings for Graph Neural Networks

Several recent works use positional encodings to extend the receptive fields of graph neural network (GNN) layers equipped with attention mechanisms. These techniques, however, extend receptive fields to the complete graph, at substantial computational cost and risking a change in the inductive biases of conventional GNNs, or require complex architecture adjustments. As a conservative alternative, we use positional encodings to expand receptive fields to $r$-hop neighborhoods. More specifically, our method augments the input graph with additional nodes/edges and uses positional encodings as node and/or edge features. We thus modify graphs before inputting them to a downstream GNN model, instead of modifying the model itself. This makes our method model-agnostic, i.e. compatible with any existing GNN architectures. We also provide examples of positional encodings that are lossless with a one-to-one map between the original and the modified graphs. We demonstrate that extending receptive fields via positional encodings and a virtual fully-connected node significantly improves GNN performance and alleviates over-squashing using small $r$. We obtain improvements on a variety of models and datasets, and reach state-of-the-art performance using traditional GNNs or graph Transformers