# Ising Model Selection Using ℓ1-Regularized Linear Regression: A Statistical Mechanics Analysis

arXiv: https://arxiv.org/abs/2102.03988 · HTML: https://mlsanity.com/paper/2102.03988/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 60 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 12 of 12)

- [Generalisation error in learning with random features and the hidden manifold model](https://mlsanity.com/paper/2002.09339/) (2020)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models](https://mlsanity.com/paper/1605.07252/) (2016)
- [Inverse statistical problems: from the inverse Ising problem to data science](https://mlsanity.com/paper/1702.01522/) (2017)
- [Optimal structure and parameter learning of Ising models](https://mlsanity.com/paper/1612.05024/) (2016)
- [Sparse Logistic Regression Learns All Discrete Pairwise Graphical Models](https://mlsanity.com/paper/1810.11905/) (2018)
- [Statistical Mechanics of High-Dimensional Inference](https://mlsanity.com/paper/1601.04650/) (2016)
- [Asymptotic errors for convex penalized linear regression beyond Gaussian matrices](https://mlsanity.com/paper/2002.04372/) (2020)
- [A statistical physics approach to learning curves for the Inverse Ising problem](https://mlsanity.com/paper/1705.05403/) (2017)
- [Learning performance in inverse Ising problems with sparse teacher couplings](https://mlsanity.com/paper/1912.11591/) (2019)
- [Structure Learning in Inverse Ising Problems Using ℓ2-Regularized Linear Estimator](https://mlsanity.com/paper/2008.08342/) (2020)
- [Statistical mechanics of the inverse Ising problem and the optimal objective function](https://mlsanity.com/paper/1611.04281/) (2016)

## Built on by (most cited first, 4 of 4)

- [Average case analysis of Lasso under ultra-sparse conditions](https://mlsanity.com/paper/2302.13093/) (2023)
- [Zero-Flow Encoders](https://mlsanity.com/paper/2602.00797/) (2026)
- [On Model Selection Consistency of Lasso for High-Dimensional Ising Models](https://mlsanity.com/paper/2110.08500/) (2021)
- [Transfer Learning in High-dimensional Ising Models](https://mlsanity.com/paper/2607.03005/) (2026)

## Abstract (arXiv)

We theoretically analyze the typical learning performance of ℓ1-regularized linear regression (ℓ1-LinR) for Ising model selection using the replica method from statistical mechanics. For typical random regular graphs in the paramagnetic phase, an accurate estimate of the typical sample complexity of ℓ1-LinR is obtained. Remarkably, despite the model misspecification, ℓ1-LinR is model selection consistent with the same order of sample complexity as ℓ1-regularized logistic regression (ℓ1-LogR), i.e., M=Oleft(log Nright), where N is the number of variables of the Ising model. Moreover, we provide an efficient method to accurately predict the non-asymptotic behavior of ℓ1-LinR for moderate M, N, such as precision and recall. Simulations show a fairly good agreement between theoretical predictions and experimental results, even for graphs with many loops, which supports our findings. Although this paper mainly focuses on ℓ1-LinR, our method is readily applicable for precisely characterizing the typical learning performances of a wide class of ℓ1-regularized M-estimators including ℓ1-LogR and interaction screening.