Year: 2012
Type: article
Abstract: 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.
Source: Electronic Journal of Statistics
DOI: 10.1214/12-ejs740
Authors: Rahul Mazumder, Trevor Hastie
Institutions: Massachusetts Institute of Technology, Stanford University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W1486916140