Ising Model Selection Using $\ell_{1}$-Regularized Linear Regression: A Statistical Mechanics Analysis

We theoretically analyze the typical learning performance of ℓ 1-regularized linear regression (ℓ 1-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 ℓ 1-LinR is obtained. Remarkably, despite the model misspecification, ℓ 1-LinR is model selection consistent with the same order of sample complexity as ℓ 1-regularized logistic regression (ℓ 1-LogR), i.e. M=OlogN , 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 ℓ 1-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 ℓ 1-LinR, our method is readily applicable for precisely characterizing the typical learning performances of a wide class of ℓ 1-regularized M-estimators including ℓ 1-LogR and interaction screening.