Euclidean Contractivity of Neural Networks With Symmetric Weights

This letter 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.

Automated summary

This letter investigates the stability conditions of continuous-time Hopfield and firing-rate neural networks using contraction theory. It presents useful algebraic results on matrix polytopes and products of symmetric matrices. Sufficient conditions for strong and weak Euclidean contractivity of both models with symmetric weights and non-smooth activation functions are given. The contraction analysis leads to log-optimal contraction rates in almost all symmetric synaptic matrices. Finally, a firing-rate neural network model is proposed to solve a quadratic optimization problem with box constraints.