Sparse inverse covariance estimation with the graphical lasso - PubMed (original) (raw)
Sparse inverse covariance estimation with the graphical lasso
Jerome Friedman et al. Biostatistics. 2008 Jul.
Abstract
We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate descent procedure for the lasso, we develop a simple algorithm--the graphical lasso--that is remarkably fast: It solves a 1000-node problem ( approximately 500,000 parameters) in at most a minute and is 30-4000 times faster than competing methods. It also provides a conceptual link between the exact problem and the approximation suggested by Meinshausen and Bühlmann (2006). We illustrate the method on some cell-signaling data from proteomics.
Figures
Fig. 1.
Number of CPU seconds required for the graphical lasso procedure.
Fig. 2.
Directed acylic graph from cell-signaling data, from Sachs and others (2003).
Fig. 3.
Cell-signaling data: Undirected graphs from graphical lasso with different values of the penalty parameter ρ.
Fig. 4.
Cell-signaling data: Profile of coefficients as the total_L_1norm of the coefficient vector increases, that is as_ρ_decreases. Profiles for the largest coefficients are labeled with the corresponding pair of proteins.
Fig. 5.
Cell-signaling data. Left panel shows 10-fold cross-validation using both regression and likelihood approaches (details in text). Right panel compares the regression sum of squares of the exact graphical lasso approach to the Meinhausen–Buhlmann approximation.
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