Learning Graphical Models With Hubs

TL;DR

This paper introduces a convex framework with a row-column overlap norm penalty for learning high-dimensional graphical models that explicitly account for hub nodes, outperforming traditional methods that assume uniform edge likelihood.

Contribution

The authors propose a novel convex formulation and an efficient algorithm to model hub nodes in graphical models, extending beyond traditional l1-penalized approaches.

Findings

Outperforms existing methods on synthetic data

Effective in modeling hub nodes in real datasets

Applicable to Gaussian, covariance, and Ising models

Abstract

We consider the problem of learning a high-dimensional graphical model in which certain hub nodes are highly-connected to many other nodes. Many authors have studied the use of an l1 penalty in order to learn a sparse graph in high-dimensional setting. However, the l1 penalty implicitly assumes that each edge is equally likely and independent of all other edges. We propose a general framework to accommodate more realistic networks with hub nodes, using a convex formulation that involves a row-column overlap norm penalty. We apply this general framework to three widely-used probabilistic graphical models: the Gaussian graphical model, the covariance graph model, and the binary Ising model. An alternating direction method of multipliers algorithm is used to solve the corresponding convex optimization problems. On synthetic data, we demonstrate that our proposed framework outperforms competitors that do not explicitly model hub nodes. We illustrate our proposal on a webpage data set and a gene expression data set.