Asymmetric Hopfield neural network and twisted tetrahedron equation

TL;DR

This paper extends Hopfield neural network models on a triangular lattice, revealing that their dynamics follow a Gibbs distribution and satisfy a deformed tetrahedron equation, linking neural networks with 3D statistical physics models.

Contribution

It introduces a novel generalization of Hopfield networks incorporating the twisted tetrahedron equation, connecting neural dynamics with integrable models in statistical physics.

Findings

Trajectory probabilities follow Gibbs distribution

Weight matrix satisfies the twisted tetrahedron equation

Model links neural networks to 3D statistical physics

Abstract

We generalize the approach of arXiv:1805.04138 for the case of the Hopfield neural network in the recall stage on a triangular lattice with isotropic weights. It appears that some properties of this model, in particular the probability of passing a trajectory in time dynamics, obeys the Gibbs distribution with a partition function having a vertex realization. Moreover the corresponding weight matrix satisfies the TTE - some deformation of the Zamolodchikov tetrahedron equation, the latter playing the role analogous to the Yang-Baxter equation in 3-dimensional statistical models.