# Reconstructing the Hopfield network as an inverse Ising problem

## TL;DR

This paper evaluates four mean field algorithms for reconstructing Hopfield networks as inverse Ising problems, revealing phase-dependent reconstruction success and highlighting the paramagnetic phase as optimal for network inference.

## Contribution

It provides the first detailed analysis of Hopfield network reconstruction using inverse Ising algorithms across different phases and conditions.

## Key findings

- Reconstruction algorithms fail in the retrieval phase during memory recall.
- Algorithms perform well in the paramagnetic phase for network reconstruction.
- Performance depends on system size, memory load, and temperature.

## Abstract

We test four fast mean field type algorithms on Hopfield networks as an inverse Ising problem. The equilibrium behavior of Hopfield networks is simulated through Glauber dynamics. In the low temperature regime, the simulated annealing technique is adopted. Although performances of these network reconstruction algorithms on the simulated network of spiking neurons are extensively studied recently, the analysis of Hopfield networks is lacking so far. For the Hopfield network, we found that, in the retrieval phase favored when the network wants to memory one of stored patterns, all the reconstruction algorithms fail to extract interactions within a desired accuracy, and the same failure occurs in the spin glass phase where spurious minima show up, while in the paramagnetic phase, albeit unfavored during the retrieval dynamics, the algorithms work well to reconstruct the network itself. This implies that, as a inverse problem, the paramagnetic phase is conversely useful for reconstructing the network while the retrieval phase loses all the information about interactions in the network except for the case where only one pattern is stored. The performances of algorithms are studied with respect to the system size, memory load and temperature, sample-to-sample fluctuations are also considered.