Beyond inverse Ising model: structure of the analytical solution for a class of inverse problems

TL;DR

This paper explores an analytical approach to inverse problems in statistical models, extending beyond the inverse Ising model by identifying conditions for direct solutions based on entropy and correlation structures.

Contribution

It introduces a non-iterative method for solving inverse problems using entropy decomposition, applicable to models with factorized forms and sparse Fisher information.

Findings

Inverse problems are local with sparse Fisher information in factorized models.

Entropy can be expressed as a sum of small cluster contributions.

Numerical simulations support the analytical results.

Abstract

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.