Structure Learning in Inverse Ising Problems Using $\ell_2$-Regularized Linear Estimator

Inferring interaction parameters from observed data is a ubiquitous requirement in various fields of science and engineering. Recent studies have shown that the pseudolikelihood (PL) method is highly effective in meeting this requirement even though the maximum likelihood method is computationally intractable when used directly. To the best of our knowledge, most existing studies assume that the postulated model used in the inference stage covers the true model that generates the data. However, such an assumption does not necessarily hold in practical situations. From this perspective, we discuss the utility of the PL method in model mismatch cases. Specifically, we examine the inference performance of the PL method when $\ell_2$-regularized (ridge) linear regression is applied to data generated from sparse Boltzmann machines of Ising spins using methods of statistical mechanics. Our analysis indicates that despite the model mismatch, one can perfectly identify the network topology using naive linear regression without regularization when the dataset size $M$ is greater than the number of Ising spins, $N$. Further, even when $M < N$, perfect identification is possible using a two-stage estimator with much better quantitative performance compared to naive usage of the PL method. Results of extensive numerical experiments support our findings.