Identifying Network Hubs with the Partial Correlation Graphical LASSO

TL;DR

This paper studies the Partial Correlation Graphical LASSO (PCGLASSO), introducing a scale-invariant irrepresentability condition that ensures consistent model selection and explains its empirical advantages over traditional GLASSO.

Contribution

The paper introduces a new scale-invariant irrepresentability condition for PCGLASSO, proving its sufficiency for model selection consistency and demonstrating its weaker nature compared to GLASSO's condition.

Findings

The irrepresentability condition for PCGLASSO is weaker than for GLASSO.

PCGLASSO exhibits improved empirical behavior in hub-structured graphs.

Algorithms for computing PCGLASSO are developed and analyzed for global optimality.

Abstract

Graphical LASSO (GLASSO) is a widely used method for estimating sparse precision matrices and learning undirected graphical models in high-dimensional settings. Because GLASSO penalizes entries of the precision matrix directly, however, it is not scale-invariant. Partial Correlation Graphical LASSO (PCGLASSO), introduced by Carter et al. (2024), addresses this limitation by penalizing partial correlations, which directly characterize conditional dependence. In this paper, we study both statistical and computational properties of the PCGLASSO estimator. Our main contribution is the introduction of a scale-invariant irrepresentability condition for PCGLASSO and the proof that this condition is sufficient for consistent model selection. We further show that this condition is weaker than the corresponding irrepresentability condition for GLASSO, helping to explain the improved empirical behavior of PCGLASSO in settings such as hub-structured graphs. In addition, we develop two efficient algorithms for computing the estimator and analyze the nonconvex optimization problem underlying PCGLASSO, deriving conditions for global uniqueness and showing consistency of all minimizers.