On the inconsistency of $\ell_1$-penalised sparse precision matrix estimation

TL;DR

This paper investigates the limitations of $ ext{L}_1$-penalized methods like graphical lasso and CLIME in estimating sparse precision matrices, especially in models with near-linear dependencies and real gene expression data, revealing their practical unreliability.

Contribution

It demonstrates the failure of $ ext{L}_1$-based methods in certain latent variable models and real data scenarios, challenging assumptions about their consistency and reliability.

Findings

$ ext{L}_1$-methods fail with nearly linear dependencies

Inconsistency in gene expression data models

Practical unreliability for larger networks

Abstract

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.