# Structure Learning in Inverse Ising Problems Using ℓ2-Regularized Linear Estimator

arXiv: https://arxiv.org/abs/2008.08342 · HTML: https://mlsanity.com/paper/2008.08342/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Aug 2020.

## Built on by (most cited first, 3 of 3)

- [Average case analysis of Lasso under ultra-sparse conditions](https://mlsanity.com/paper/2302.13093/) (2023)
- [Ising Model Selection Using ℓ1-Regularized Linear Regression: A Statistical Mechanics Analysis](https://mlsanity.com/paper/2102.03988/) (2021)
- [On Model Selection Consistency of Lasso for High-Dimensional Ising Models](https://mlsanity.com/paper/2110.08500/) (2021)

## Abstract (arXiv)

The inference performance of the pseudolikelihood method is discussed in the framework of the inverse Ising problem when the ℓ2-regularized (ridge) linear regression is adopted. This setup is introduced for theoretically investigating the situation where the data generation model is different from the inference one, namely the model mismatch situation. In the teacher-student scenario under the assumption that the teacher couplings are sparse, the analysis is conducted using the replica and cavity methods, with a special focus on whether the presence/absence of teacher couplings is correctly inferred or not. The result indicates that despite the model mismatch, one can perfectly identify the network structure using naive linear regression without regularization when the number of spins N is smaller than the dataset size M, in the thermodynamic limit N→ ∞. Further, to access the underdetermined region M < N, we examine the effect of the ℓ2 regularization, and find that biases appear in all the coupling estimates, preventing the perfect identification of the network structure. We, however, find that the biases are shown to decay exponentially fast as the distance from the center spin chosen in the pseudolikelihood method grows. Based on this finding, we propose a two-stage estimator: In the first stage, the ridge regression is used and the estimates are pruned by a relatively small threshold; in the second stage the naive linear regression is conducted only on the remaining couplings, and the resultant estimates are again pruned by another relatively large threshold. This estimator with the appropriate regularization coefficient and thresholds is shown to achieve the perfect identification of the network structure even in 0<M/N<1. Results of extensive numerical experiments support these findings.