Modeling and estimation in Gaussian graphical models : maximum-entropy methods and walk-sum analysis

Abstract

Graphical models provide a powerful formalism for statistical signal processing. Due to their sophisticated modeling capabilities, they have found applications in a variety of fields such as computer vision, image processing, and distributed sensor networks. In this thesis we study two central signal processing problems involving Gaussian graphical models, namely modeling and estimation. The modeling problem involves learning a sparse graphical model approximation to a specified distribution. The estimation problem in turn exploits this graph structure to solve high-dimensional estimation problems very efficiently. We propose a new approach for learning a thin graphical model approximation to a specified multivariate probability distribution (e.g., the empirical distribution from sample data). The selection of sparse graph structure arises naturally in our approach through the solution of a convex optimization problem, which differentiates our procedure from standard combinatorial methods. In our approach, we seek the maximum entropy relaxation (MER) within an exponential family, which maximizes entropy subject to constraints that marginal distributions on small subsets of variables are close to the prescribed marginals in relative entropy. We also present a primal-dual interior point method that is scalable and tractable provided the level of relaxation is sufficient to obtain a thin graph. A crucial element of this algorithm is that we exploit sparsity of the Fisher information matrix in models defined on chordal graphs. The merits of this approach are investigated by recovering the graphical structure of some simple graphical models from sample data. Next, we present a general class of algorithms for estimation in Gaussian graphical models with arbitrary structure.

(cont.) These algorithms involve a sequence of inference problems on tractable subgraphs over subsets of variables. This framework includes parallel iterations such as Embedded Trees, serial iterations such as block Gauss-Seidel, and hybrid versions of these iterations. We also discuss a method that uses local memory at each node to overcome temporary communication failures that may arise in distributed sensor network applications. We analyze these algorithms based on the recently developed walk-sum interpretation of Gaussian inference. We describe the walks "computed" by the algorithms using walk-sum diagrams, and show that for non-stationary iterations based on a very large and flexible set of sequences of subgraphs, convergence is achieved in walk-summable models. Consequently, we are free to choose spanning trees and subsets of variables adaptively at each iteration. This leads to efficient methods for optimizing the next iteration step to achieve maximum reduction in error. Simulation results demonstrate that these non-stationary algorithms provide a significant speedup in convergence over traditional one-tree and two-tree iterations.