An Alternative Graphical Lasso Algorithm for Precision Matrices

The Graphical Lasso (GLasso) algorithm is fast and widely used for estimating sparse precision matrices (Friedman et al., 2008). Its central role in the literature of high-dimensional covariance estimation rivals that of Lasso regression for sparse estimation of the mean vector. Some mysteries regarding its optimization target, convergence, positive-definiteness and performance have been unearthed, resolved and presented in Mazumder and Hastie (2011), leading to a new/improved (dual-primal) DP-GLasso. Using a new and slightly different reparametriztion of the last column of a precision matrix we show that the regularized normal log-likelihood naturally decouples into a sum of two easy to minimize convex functions one of which is a Lasso regression problem. This decomposition is the key in developing a transparent, simple iterative block coordinate descent algorithm for computing the GLasso updates with performance comparable to DP-GLasso. In particular, our algorithm has the precision matrix as its optimization target right at the outset, and retains all the favorable properties of the DP-GLasso algorithm.