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2012 The graphical lasso: New insights and alternatives
Rahul Mazumder, Trevor Hastie
Electron. J. Statist. 6: 2125-2149 (2012). DOI: 10.1214/12-EJS740


The graphical lasso [5] is an algorithm for learning the structure in an undirected Gaussian graphical model, using $\ell_{1}$ regularization to control the number of zeros in the precision matrix $\boldsymbol{\Theta}=\boldsymbol{\Sigma}^{-1}$ [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 the dual of 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 is $\boldsymbol{\Sigma}$, the covariance matrix, rather than the precision matrix $\boldsymbol{\Theta}$. We propose similar primal algorithms P-GLASSO and DP-GLASSO, that also operate by block-coordinate descent, where $\boldsymbol{\Theta}$ is 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.


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Rahul Mazumder. Trevor Hastie. "The graphical lasso: New insights and alternatives." Electron. J. Statist. 6 2125 - 2149, 2012.


Published: 2012
First available in Project Euclid: 9 November 2012

zbMATH: 1295.62066
MathSciNet: MR3020259
Digital Object Identifier: 10.1214/12-EJS740

Primary: 62-09 , 62H99
Secondary: 62-04

Keywords: convex analysis/optimization , graphical lasso , positive definite matrices , precision matrix , semidefinite programming , sparse inverse covariance selection , Sparsity

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society


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