Mean-field theory for the inverse Ising problem at low temperatures

TL;DR

This paper introduces a clustering-based mean-field approach to accurately reconstruct Ising model parameters from data at low temperatures, overcoming previous limitations due to multiple thermodynamic states.

Contribution

It proposes a novel method combining clustering with mean-field approximations to improve inverse Ising problem solutions at low temperatures.

Findings

Effective reconstruction of Ising models at low temperatures.

Clustering spin configurations captures multiple thermodynamic states.

Method outperforms traditional mean-field approaches in challenging regimes.

Abstract

The large amounts of data from molecular biology and neuroscience have lead to a renewed interest in the inverse Ising problem: how to reconstruct parameters of the Ising model (couplings between spins and external fields) from a number of spin configurations sampled from the Boltzmann measure. To invert the relationship between model parameters and observables (magnetisations and correlations) mean-field approximations are often used, allowing to determine model parameters from data. However, all known mean-field methods fail at low temperatures with the emergence of multiple thermodynamic states. Here we show how clustering spin configurations can approximate these thermodynamic states, and how mean-field methods applied to thermodynamic states allow an efficient reconstruction of Ising models also at low temperatures.