Existence of the solution to the graphical lasso

TL;DR

The paper addresses whether the graphical lasso and its off-diagonal variant admit a finite maximiser when the sample covariance $S$ is only positive semidefinite. It provides dual-free proofs based on eigenvalue behavior to show that a diagonal $l_1$ penalty guarantees existence for any PSD $S$, while the off-diagonal version may fail unless all diagonals of $S$ are nonzero; Gaussian sampling ensures this condition with probability 1 under common settings. Key contributions include precise existence conditions for both glasso and odglasso, along with a uniqueness result from strict concavity and a generalization to broader penalty functions. The findings offer theoretical guarantees for high-dimensional Gaussian graphical modeling and inform penalty design to ensure estimators exist when $n\le p$.

Abstract

The graphical lasso (glasso) is an $l_1$ penalised likelihood estimator for a Gaussian precision matrix. A benefit of the glasso is that it exists even when the sample covariance matrix is not positive definite but only positive semidefinite. This note collects a number of results concerning the existence of the glasso both when the penalty is applied to all entries of the precision matrix and when the penalty is only applied to the off-diagonals. New proofs are provided for these results which give insight into how the $l_1$ penalty achieves these existence properties. These proofs extend to a much larger class of penalty functions allowing one to easily determine if new penalised likelihood estimates exist for positive semidefinite sample covariance.

Theorems & Definitions (13)

• Proposition 1
• proof
• Corollary 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof