High-Dimensional Graphical 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 discrete Markov random field graph structure estimation, providing theoretical guarantees on sample size requirements for consistent neighborhood recovery.

Contribution

It develops a novel high-dimensional framework for graph estimation with theoretical analysis of sample complexity and consistency guarantees.

Findings

Consistent neighborhood selection achievable with $n = ext{Omega}(d^3 ext{log} p)$ samples.

Error probability decays exponentially with sample size.

Method applicable when both number of nodes and maximum neighborhood size grow with data.

Abstract

We consider the problem of estimating the graph structure associated with a discrete 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. 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 $n = \Omega(d^3 \log p)$, with the error decaying as $\order(\exp(-C n/d^3))$ for some constant $C$. If these same assumptions are imposed directly on the sample matrices, we show that $n = \Omega(d^2 \log p)$ samples are sufficient.