A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning

TL;DR

This paper introduces a nonlinear separation principle based on contraction theory, providing stability conditions and design methods for neural networks and control systems, with applications to RNNs and neural network design.

Contribution

It develops a contraction-based nonlinear separation principle, derives LMI conditions for neural network stability, and applies these to neural network design and control.

Findings

Established a global exponential stability guarantee for interconnected contracting systems.

Derived LMI conditions ensuring contractivity of neural network architectures.

Designed a neural network-based classifier with competitive accuracy and parameter efficiency.

Abstract

This paper establishes a nonlinear separation principle based on contraction theory and derives sharp stability conditions for recurrent neural networks (RNNs). First, we introduce a nonlinear separation principle that guarantees global exponential stability for the interconnection of a contracting state-feedback controller and a contracting observer, alongside parametric extensions for robustness and equilibrium tracking. Second, we derive sharp linear matrix inequality (LMI) conditions that guarantee the contractivity of both firing rate and Hopfield neural network architectures. We establish structural relationships among these certificates-demonstrating that continuous-time models with monotone non-decreasing activations maximize the admissible weight space-and extend these stability guarantees to interconnected systems and Graph RNNs. Third, we combine our separation principle and LMI framework to solve the output reference tracking problem for RNN-modeled plants. We provide LMI synthesis methods for feedback controllers and observers, and rigorously design a low-gain integral controller to eliminate steady-state error. Finally, we derive an exact, unconstrained algebraic parameterization of our contraction LMIs to design highly expressive implicit neural networks, achieving competitive accuracy and parameter efficiency on standard image classification benchmarks.