A Density Consistency approach to the inverse Ising problem

TL;DR

This paper introduces a new Density Consistency method for the inverse Ising problem, providing closed-form parameter estimates and demonstrating competitive accuracy across various network configurations and data regimes.

Contribution

The work presents a novel DC-based approach with explicit formulas for parameters, improving inference accuracy especially with non-zero magnetization and external fields.

Findings

DC method offers closed-form solutions for model parameters.

Compared to existing methods, DC performs reliably across different regimes.

No single method outperforms others in all scenarios, but DC is among the most accurate.

Abstract

We propose a novel approach to the inverse Ising problem which employs the recently introduced Density Consistency approximation (DC) to determine the model parameters (couplings and external fields) maximizing the likelihood of given empirical data. This method allows for closed-form expressions of the inferred parameters as a function of the first and second empirical moments. Such expressions have a similar structure to the small-correlation expansion derived by Sessak and Monasson, of which they provide an improvement in the case of non-zero magnetization at low temperatures, as well as in presence of random external fields. The present work provides an extensive comparison with most common inference methods used to reconstruct the model parameters in several regimes, i.e. by varying both the network topology and the distribution of fields and couplings. The comparison shows that no method is uniformly better than every other one, but DC appears nevertheless as one of the most accurate and reliable approaches to infer couplings and fields from first and second moments in a significant range of parameters.