Conditional Matrix Flows for Gaussian Graphical Models

Studying conditional independence among many variables with few observations is a challenging task.Gaussian Graphical Models (GGMs) tackle this problem by encouraging sparsity in the precision matrix through l_q regularization with q\leq1 .However, most GMMs rely on the l_1 norm because the objective is highly non-convex for sub- l_1 pseudo-norms.In the frequentist formulation, the l_1 norm relaxation provides the solution path as a function of the shrinkage parameter \lambda .In the Bayesian formulation, sparsity is instead encouraged through a Laplace prior, but posterior inference for different \lambda requires repeated runs of expensive Gibbs samplers.Here we propose a general framework for variational inference with matrix-variate Normalizing Flow in GGMs, which unifies the benefits of frequentist and Bayesian frameworks.As a key improvement on previous work, we train with one flow a continuum of sparse regression models jointly for all regularization parameters \lambda and all l_q norms, including non-convex sub- l_1 pseudo-norms.Within one model we thus have access to (i) the evolution of the posterior for any \lambda and any l_q (pseudo-) norm, (ii) the marginal log-likelihood for model selection, and (iii) the frequentist solution paths through simulated annealing in the MAP limit.