# High-Dimensional Graphical Model Selection Using $ell 1$-Regularized Logistic Regression - Mathematics > Statistics Theory

High-Dimensional Graphical Model Selection Using $ell 1$-Regularized Logistic Regression - Mathematics > Statistics Theory - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: We consider the problem of estimating the graph structure associated with adiscrete Markov random field. We describe a method based on$\ell 1$-regularized logistic regression, in which the neighborhood of anygiven node is estimated by performing logistic regression subject to an$\ell 1$-constraint. Our framework applies to the high-dimensional setting, inwhich both the number of nodes $p$ and maximum neighborhood sizes $d$ areallowed to grow as a function of the number of observations $n$. Our mainresults provide sufficient conditions on the triple $n, p, d$ for the methodto succeed in consistently estimating the neighborhood of every node in thegraph simultaneously. Under certain assumptions on the population Fisherinformation matrix, we prove that consistent neighborhood selection can beobtained for sample sizes $n = \Omegad^3 \log p$, with the error decaying as$\order\exp-C n-d^3$ for some constant $C$. If these same assumptions areimposed directly on the sample matrices, we show that $n = \Omegad^2 \log p$samples are sufficient.

Autor: Pradeep Ravikumar, Martin J. Wainwright, John D. Lafferty

Fuente: https://arxiv.org/