Stochastic processes on graphs with cycles : geometric and variational approaches

Abstract

Stochastic processes defined on graphs arise in a tremendous variety of fields, including statistical physics, signal processing, computer vision, artificial intelligence, and information theory. The formalism of graphical models provides a useful language with which to formulate fundamental problems common to all of these fields, including estimation, model fitting, and sampling. For graphs without cycles, known as trees, all of these problems are relatively well-understood, and can be solved efficiently with algorithms whose complexity scales in a tractable manner with problem size. In contrast, these same problems present considerable challenges in general graphs with cycles. The focus of this thesis is the development and analysis of methods, both exact and approximate, for problems on graphs with cycles. Our contributions are in developing and analyzing techniques for estimation, as well as methods for computing upper and lower bounds on quantities of interest (e.g., marginal probabilities; partition functions). In order to do so, we make use of exponential representations of distributions, as well as insight from the associated information geometry and Legendre duality. Our results demonstrate the power of exponential representations for graphical models, as well as the utility of studying collections of modified problems defined on trees embedded within the original graph with cycles. The specific contributions of this thesis include the following. We develop a method for performing exact estimation of Gaussian processes on graphs with cycles by solving a sequence of modified problems on embedded spanning trees.

(cont.) We introduce the tree-based reparameterization framework for approximate estimation of discrete processes. This perspective leads to a number of theoretical results on belief propagation and related algorithms, including characterizations of their fixed points and the associated approximation error. Next we extend the notion of reparameterization to a much broader class of methods for approximate inference, including Kikuchi methods, and present results on their fixed points and accuracy. Finally, we develop and analyze a novel class of upper bounds on the log partition function based on convex combinations of distributions in the exponential domain. In the special case of combining tree-structured distributions, the associated dual function gives an interesting perspective on the Bethe free energy.