# Euclidean Contractivity of Neural Networks with Symmetric Weights

## TL;DR

This paper analyzes the stability of certain neural network models with symmetric weights using contraction theory, providing conditions for Euclidean contractivity and applying results to quadratic optimization.

## Contribution

It introduces new algebraic results and sufficient conditions for Euclidean contractivity in neural networks with symmetric weights, including non-smooth activations.

## Key findings

- Contraction rates are log-optimal for most symmetric matrices.
- Provided stability conditions for Hopfield and firing-rate networks.
- Applied contraction analysis to optimize quadratic problems with box constraints.

## Abstract

This paper investigates stability conditions of continuous-time Hopfield and firing-rate neural networks by leveraging contraction theory. First, we present a number of useful general algebraic results on matrix polytopes and products of symmetric matrices. Then, we give sufficient conditions for strong and weak Euclidean contractivity, i.e., contractivity with respect to the $\ell_2$ norm, of both models with symmetric weights and (possibly) non-smooth activation functions. Our contraction analysis leads to contraction rates which are log-optimal in almost all symmetric synaptic matrices. Finally, we use our results to propose a firing-rate neural network model to solve a quadratic optimization problem with box constraints.