On the inconsistency of ℓ1-penalised sparse precision matrix estimation

Various ℓ1-penalised estimation methods such as graphical lasso and CLIME are widely used for sparse precision matrix estimation and learning of undirected network structure from data. Many of these methods have been shown to be consistent under various quantitative assumptions about the underlying true covariance matrix. Intuitively, these conditions are related to situations where the penalty term will dominate the optimisation. We explore the consistency of ℓ1-based methods for a class of bipartite graphs motivated by the structure of models commonly used for gene regulatory networks. We show that all ℓ1-based methods fail dramatically for models with nearly linear dependencies between the variables. We also study the consistency on models derived from real gene expression data and note that the assumptions needed for consistency never hold even for modest sized gene networks and ℓ1-based methods also become unreliable in practice for larger networks. Our results demonstrate that ℓ1-penalised undirected network structure learning methods are unable to reliably learn many sparse bipartite graph structures, which arise often in gene expression data. Users of such methods should be aware of the consistency criteria of the methods and check if they are likely to be met in their application of interest.