title: The graphical lasso: New insights and alternatives.
abstract: The graphical lasso [5] is an algorithm for learning the structure in an undirected Gaussian graphical model, using ℓ1regularization to control the number of zeros in the precision matrixΘ=Σ-1[2, 11]. The R package GLASSO [5] is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of GLASSO can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform GLASSO. By studying the "normal equations" we see that, GLASSO is solving thedualof the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in [2]. In this dual, the target of estimation isΣ, the covariance matrix, rather than the precision matrixΘ. We propose similar primal algorithms P-GLASSO and DP-GLASSO, that also operate by block-coordinate descent, whereΘis the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that DP-GLASSO is superior from several points of view.
authors.affiliation: Massachusetts Institute of Technology, Cambridge, MA 02139.
authors.forename: Rahul
authors.initials: R
authors.lastname: Mazumder
authors.affiliation: Departments of Statistics and Health Research and Policy, Stanford University, Stanford, CA 94305.
authors.forename: Trevor
authors.initials: T
authors.lastname: Hastie
keywords: Graphical lasso; convex analysis/optimization; positive definite matrices; precision matrix; semidefinite programming; sparse inverse covariance selection; sparsity