  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.
