Conditional Matrix Flows for Gaussian Graphical Models

TL;DR

This paper introduces a unified variational inference framework using matrix-variate Normalizing Flows for Gaussian Graphical Models, enabling efficient exploration of regularization parameters and norms, including non-convex cases.

Contribution

It proposes a novel approach that trains a single flow model to represent a continuum of sparse models across all regularization parameters and norms, combining Bayesian and frequentist advantages.

Findings

Unified model for all regularization parameters and norms

Efficient posterior inference for multiple models

Access to solution paths and model selection metrics

Abstract

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.