Open Access
June 2010 High-dimensional Ising model selection using 1-regularized logistic regression
Pradeep Ravikumar, Martin J. Wainwright, John D. Lafferty
Ann. Statist. 38(3): 1287-1319 (June 2010). DOI: 10.1214/09-AOS691


We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on 1-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an 1-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes p and maximum neighborhood size d are allowed to grow as a function of the number of observations n. Our main results provide sufficient conditions on the triple (n, p, d) and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes n=Ω(d3log p) with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of n=Ω(d2log p) suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.


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Pradeep Ravikumar. Martin J. Wainwright. John D. Lafferty. "High-dimensional Ising model selection using 1-regularized logistic regression." Ann. Statist. 38 (3) 1287 - 1319, June 2010.


Published: June 2010
First available in Project Euclid: 8 March 2010

zbMATH: 1189.62115
MathSciNet: MR2662343
Digital Object Identifier: 10.1214/09-AOS691

Primary: 62F12
Secondary: 68T99

Keywords: convex risk minimization , graphical models , high-dimensional asymptotics , ℓ_1-regularization , Markov random fields , Model selection , structure learning

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 3 • June 2010
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