Identifying Network Hubs with the Partial Correlation Graphical LASSO

Abstract: The Partial Correlation Graphical LASSO (PCGLASSO) offers a scale-invariant alternative to the standard GLASSO. This paper provides the first comprehensive treatment of the PCGLASSO estimator. We introduce a novel and highly efficient algorithm. Our central theoretical contribution is the first scale-invariant irrepresentability criterion for PCGLASSO, which guarantees consistent model selection. We prove this condition is significantly weaker than its GLASSO counterpart, providing the first theoretical justification for PCGLASSO's superior empirical performance, especially in recovering networks with hub structures. Furthermore, we deliver the first analysis of the estimator's non-convex solution landscape, establishing new conditions for global uniqueness and guaranteeing the consistency of all minimizers.