BackgroundVarious ℓ 1-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.ResultsWe explore the consistency of ℓ 1-based methods for a class of bipartite graphs motivated by the structure of models commonly used for gene regulatory networks. We show that all ℓ 1-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 ℓ 1-based methods also become unreliable in practice for larger networks.ConclusionsOur results demonstrate that ℓ 1-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.

Abstract. We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel matrices. Our results are new already in the general case of matrix convexity and our approach significantly simplifies the corresponding work on matrix monotonicity. We also obtain an extension of the original characterization for matrix convexity by Kraus, and tighten the relationship between monotonicity and convexity.

We give new characterizations for matrix monotonicity and convexity of fixed order which connects previous characterizations by Loewner, Dobsch, Donoghue, Kraus and Bendat-Sherman. The ideas introduced are then used to characterize matrix monotone functions of arbitrary order on general subsets of the real line.

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