Estimation of Graphical Models through Structured Norm Minimization

TL;DR

This paper introduces a convex optimization framework using a novel structured norm to estimate complex graphical models with sparse, structured sparse, and dense components from high-dimensional data, applicable in various scientific fields.

Contribution

It proposes a new structured norm-based method for estimating complex models with multiple components, solved efficiently via a linearized multi-block ADMM algorithm.

Findings

Superior performance on synthetic datasets.

Effective application to real-world datasets.

Handles complex structures beyond simple sparsity.

Abstract

Estimation of Markov Random Field and covariance models from high-dimensional data represents a canonical problem that has received a lot of attention in the literature. A key assumption, widely employed, is that of {\em sparsity} of the underlying model. In this paper, we study the problem of estimating such models exhibiting a more intricate structure comprising simultaneously of {\em sparse, structured sparse} and {\em dense} components. Such structures naturally arise in several scientific fields, including molecular biology, finance, and political science. We introduce a general framework based on a novel structured norm that enables us to estimate such complex structures from high-dimensional data. The resulting optimization problem is convex and we introduce a linearized multi-block alternating direction method of multipliers (ADMM) algorithm to solve it efficiently. We illustrate the superior performance of the proposed framework on a number of synthetic data sets generated from both random and structured networks. Further, we apply the method to a number of real data sets and discuss the results.