High-dimensional Ising model selection using ${\ell_1}$-regularized logistic regression

TL;DR

This paper introduces a method using $ ext{l}_1$-regularized logistic regression for high-dimensional Ising model selection, providing theoretical guarantees for consistent graph recovery under certain conditions.

Contribution

The paper develops a novel high-dimensional neighborhood selection method for Ising models with theoretical analysis and improved sample complexity bounds.

Findings

Consistent neighborhood estimation with $n=\Omega(d^3\log p)$ samples.

Reduced sample size $n=\Omega(d^2\log p)$ suffices under certain conditions.

Method extends to general discrete Markov random fields.

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 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 $n=\Omega(d^3\log p)$ 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.