Ising Model Selection Using _1-Regularized Linear Regression: A Statistical Mechanics Analysis

Abstract

We theoretically analyze the typical learning performance of _1-regularized linear regression (_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 _1-LinR is obtained. Remarkably, despite the model misspecification, _1-LinR is model selection consistent with the same order of sample complexity as _1-regularized logistic regression (_1-LogR), i.e., M=O( N), 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 _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 _1-LinR, our method is readily applicable for precisely characterizing the typical learning performances of a wide class of _1-regularized M-estimators including _1-LogR and interaction screening.

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Reproductions