Solving the inverse Ising problem by mean-field methods in a clustered phase space with many states

In this work we explain how to properly use mean-field methods to solve the inverse Ising problem when thephase space is clustered, that is, many states are present. The clustering of the phase space can occur for manyreasons, e.g., when a system undergoes a phase transition, but also when data are collected in different regimes(e.g., quiescent and spiking regimes in neural networks). Mean-field methods for the inverse Ising problem aretypically used without taking into account the eventual clustered structure of the input configurations and maylead to very poor inference (e.g., in the low-temperature phase of the Curie-Weiss model). In this work we explainhow to modify mean-field approaches when the phase space is clustered and we illustrate the effectiveness of ourmethod on different clustered structures (low-temperature phases of Curie-Weiss and Hopfield models)