# Euclidean Contractivity of Neural Networks with Symmetric Weights

arXiv: https://arxiv.org/abs/2302.13452 · HTML: https://mlsanity.com/paper/2302.13452/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2023.

## Built on by (most cited first, 3 of 3)

- [Embedding of Low-Dimensional Sensory Dynamics in Recurrent Networks: Implications for the Geometry of Neural Representation](https://mlsanity.com/paper/2601.19019/) (2026)
- [A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning](https://mlsanity.com/paper/2604.15238/) (2026)
- [Predictive Coding with Bayesian Priors via Proximal Gradients](https://mlsanity.com/paper/2606.08374/) (2026)

## Abstract (arXiv)

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 ℓ2 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.