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Biostatistics Advance Access originally published online on December 12, 2007
Biostatistics 2008 9(3):432-441; doi:10.1093/biostatistics/kxm045
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© The Author 2007. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Sparse inverse covariance estimation with the graphical lasso

Jerome Friedman

Department of Statistics, Stanford University, CA 94305, USA

Trevor Hastie

Department of Statistics and Department of Health Research & Policy, Stanford University, CA 94305, USA

Robert Tibshirani*

Department of Health Research & Policy and Department of Statistics, Stanford University, CA 94305, USA tibs{at}stanford.edu

* To whom correspondence should be addressed.

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 (~500000 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.

Keywords: Gaussian covariance; Graphical model; L1; Lasso

Received August 16, 2007; revised November 6, 2007; accepted for publication November 8, 2007.


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