# High-Dimensional Graphical Model Selection Using ℓ_{1}-Regularized Logistic Regression

journal contribution

posted on 01.07.2000, 00:00 by Pradeep Ravikumar, Martin J. Wainwright, John LaffertyWe consider the problem of estimating the graph structure associated with a discrete
Markov random field. We describe a method based on ℓ

_{1}-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an ℓ_{1}-constraint. Our framework applies to the high-dimensional setting, in which both the number of nodes p and maximum neighborhood sizes 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) for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. Under certain assumptions on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes n = Ω(d^{3}log p), with the error decaying as O(exp(−Cn/d^{3})) for some constant C. If these same assumptions are imposed directly on the sample matrices, we show that n= Ω(d^{2}log p) samples are sufficient.