Node-Based Learning of Multiple Gaussian Graphical Models

TL;DR

This paper introduces a node-based approach for estimating multiple high-dimensional Gaussian graphical models with structured differences, using convex optimization and scalable algorithms, demonstrated on biological and synthetic data.

Contribution

It proposes a novel node-based framework with convex penalties for modeling structured differences across multiple Gaussian graphical models, scalable to high dimensions.

Findings

Effective in recovering network structures in synthetic data.

Successfully applied to biological gene expression data.

Scalable algorithm with proven decomposition properties.

Abstract

We consider the problem of estimating high-dimensional Gaussian graphical models corresponding to a single set of variables under several distinct conditions. This problem is motivated by the task of recovering transcriptional regulatory networks on the basis of gene expression data {containing heterogeneous samples, such as different disease states, multiple species, or different developmental stages}. We assume that most aspects of the conditional dependence networks are shared, but that there are some structured differences between them. Rather than assuming that similarities and differences between networks are driven by individual edges, we take a node-based approach, which in many cases provides a more intuitive interpretation of the network differences. We consider estimation under two distinct assumptions: (1) differences between the K networks are due to individual nodes that are perturbed across conditions, or (2) similarities among the K networks are due to the presence of common hub nodes that are shared across all K networks. Using a row-column overlap norm penalty function, we formulate two convex optimization problems that correspond to these two assumptions. We solve these problems using an alternating direction method of multipliers algorithm, and we derive a set of necessary and sufficient conditions that allows us to decompose the problem into independent subproblems so that our algorithm can be scaled to high-dimensional settings. Our proposal is illustrated on synthetic data, a webpage data set, and a brain cancer gene expression data set.