# High-Dimensional Graphical Model Selection Using $\\ell\_1$-Regularized Logistic Regression

Source: [High-Dimensional Graphical Model Selection Using $\\ell\_1$-Regularized Logistic Regression](<https://www.sota2.com/research/paper/68fa4dd51e209131ce957baf>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:55:07.849Z

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

Published: 2008-04-26T03:47:39.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Classification on mushroom (Feature Attribute Evaluation)](<https://www.sota2.com/research/sota/classification-on-mushroom-feature-attribute-evaluation.md>): Classification; Mushroom; Poison Attribute Accuracy: 100.

- [Classification on adults (Fairness Metrics)](<https://www.sota2.com/research/sota/classification-on-adults-fairness-metrics.md>): Classification; adults; Performance (Income): 85.6.

- [Classification on breast cancer (recurrence, malignance)](<https://www.sota2.com/research/sota/classification-on-breast-cancer-recurrence-malignance.md>): Classification; Breast cancer; Recurrence Accuracy: 71.3.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/0804.4202>)

[PDF](<https://arxiv.org/pdf/0804.4202.pdf>)