Abstract
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.
Highlights
Various 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
For instance, are nearly bipartite graphs with a small set of transcription factors regulating all the other genes. This structure has been successfully incorporated in gene regulatory network inference, often assuming a linear dependence between the regulators and targets, in both static (e.g. [1, 2]) as well as dynamic (e.g. [3, 4]) models
We explore the inconsistency of 1-penalised methods on models derived from real gene expression data and find the methods poorly suited for such applications
Summary
Various 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. For instance, are nearly bipartite graphs with a small set of transcription factors regulating all the other genes. This structure has been successfully incorporated in gene regulatory network inference, often assuming a linear dependence between the regulators and targets, in both static The motivation for the approach stems from the fact that for a Gaussian Markov random field model, zeros in the precision matrix translate exactly to absent edges in the corresponding undirected Gaussian graphical model, being informative about the marginal and conditional independence relationships among the variables
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