Exponential Series Approaches for Nonparametric Graphical Models

Abstract

This thesis studies high-dimensional, continuous-valued pairwise Markov Random Fields. We are particularly interested in approximating pairwise densities whose logarithm belongs to a Sobolev space. For this problem we propose the method of exponential series [Crain, 1974; Barron and Sheu, 1991], which approximates the log density by a finite- dimensional exponential family with the number of sufficient statistics increasing with the sample size. We consider two approaches to estimating these models. The first is regularized maximum likelihood. This involves optimizing the sum of the log-likelihood of the data and a sparsity-inducing regularizer. We provide consistency and edge selection guarantees for this method. We then propose a variational approximation to the likelihood based on tree- reweighted, nonparametric message passing. We then consider estimation using regularized score matching. This approach uses an alternative scoring rule to the log-likelihood, which obviates the need to compute the normalizing constant of the distribution. For general continuous-valued exponential families, we provide parameter and edge consistency results. We then describe results for model selection in the nonparametric pairwise model using exponential series. The regularized score matching problem is shown to be a convex program; we provide scalable algorithms based on consensus Alternating Direction Method of Multipliers (ADMM, [Boyd et al., 2011]) and Coordinate-wise Descent. We compare our method to others in the literature as well as the aforementioned TRW estimator using simulated data.