On the inconsistency of $\ell_1$-penalised sparse precision matrix estimation
This work highlights a critical limitation for researchers and practitioners using ℓ₁-based methods in fields like genomics, where sparse latent variable models are common, revealing that these methods are inconsistent in practical scenarios.
The paper investigates the consistency of ℓ₁-penalized methods like graphical lasso for sparse precision matrix estimation, showing they fail dramatically in models with nearly linear dependencies and become unreliable for larger gene networks, with assumptions for consistency not holding even in modest-sized real-world applications.
Various $\ell_1$-penalised estimation methods such as graphical lasso and CLIME are widely used for sparse precision matrix estimation. 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. In this paper, we explore the consistency of $\ell_1$-based methods for a class of sparse latent variable -like models, which are strongly motivated by several types of applications. We show that all $\ell_1$-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 $\ell_1$-based methods also become unreliable in practice for larger networks.