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

arXiv: https://arxiv.org/abs/2604.15238 · HTML: https://mlsanity.com/paper/2604.15238/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 49 references that are in the map.
- First posted on arXiv 16 Apr 2026.

## Builds on (most cited first, 17 of 17)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Lipschitz Bounded Equilibrium Networks](https://mlsanity.com/paper/2010.01732/) (2020)
- [Robust Implicit Networks via Non-Euclidean Contractions](https://mlsanity.com/paper/2106.03194/) (2021)
- [RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks](https://mlsanity.com/paper/2106.08928/) (2021)
- [Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach](https://mlsanity.com/paper/2305.15595/) (2023)
- [Online Optimization as a Feedback Controller: Stability and Tracking](https://mlsanity.com/paper/1805.09877/) (2018)
- [Euclidean Contractivity of Neural Networks with Symmetric Weights](https://mlsanity.com/paper/2302.13452/) (2023)
- [Contracting Neural Networks: Sharp LMI Conditions with Applications to Integral Control and Deep Learning](https://mlsanity.com/paper/2604.00119/) (2026)
- [Controller Design for Structured State-space Models via Contraction Theory](https://mlsanity.com/paper/2604.07069/) (2026)
- [Small-gain analysis of exponential incremental input/output-to-state stability for large-scale distributed systems](https://mlsanity.com/paper/2604.07081/) (2026)

## Abstract (arXiv)

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.