Highdimensional Ising model selection using ${\ell_1}$regularized logistic regression
Abstract
We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on $\ell_1$regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an $\ell_1$constraint. The method is analyzed under highdimensional 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 $n=\Omega(d^3\log p)$ with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of $n=\Omega(d^2\log p)$ 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2010
 DOI:
 10.48550/arXiv.1010.0311
 arXiv:
 arXiv:1010.0311
 Bibcode:
 2010arXiv1010.0311R
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published in at http://dx.doi.org/10.1214/09AOS691 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)