Categorical Hopfield Networks

TL;DR

This paper presents a categorical Hopfield network model that generalizes neural network resource assignments using category theory, linking resource dynamics with neural network weight updates.

Contribution

It introduces a categorical framework for Hopfield networks, connecting resource assignment dynamics with neural network weight updates via functorial methods.

Findings

Demonstrates how resource assignments can be modeled categorically.

Shows the categorical model reproduces neural network weight updates.

Provides a toy example illustrating the categorical Hopfield equations.

Abstract

This paper discusses a simple and explicit toy-model example of the categorical Hopfield equations introduced in previous work of Manin and the author. These describe dynamical assignments of resources to networks, where resources are objects in unital symmetric monoidal categories and assignments are realized by summing functors. The special case discussed here is based on computational resources (computational models of neurons) as objects in a category of DNNs, with a simple choice of the endofunctors defining the Hopfield equations that reproduce the usual updating of the weights in DNNs by gradient descent.