# Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian

arXiv: https://arxiv.org/abs/2303.11835 · HTML: https://mlsanity.com/paper/2303.11835/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 20 Mar 2023.

## Builds on (most cited first, 20 of 20)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [1D Convolutional Neural Networks and Applications: A Survey](https://mlsanity.com/paper/1905.03554/) (2019)
- [Orthogonalizing Convolutional Layers with the Cayley Transform](https://mlsanity.com/paper/2104.07167/) (2021)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Orthogonal Recurrent Neural Networks with Scaled Cayley Transform](https://mlsanity.com/paper/1707.09520/) (2017)
- [A Unified Algebraic Perspective on Lipschitz Neural Networks](https://mlsanity.com/paper/2303.03169/) (2023)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Lipschitz Bounded Equilibrium Networks](https://mlsanity.com/paper/2010.01732/) (2020)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Pay attention to your loss: understanding misconceptions about 1-Lipschitz neural networks](https://mlsanity.com/paper/2104.05097/) (2021)
- [Deep Neural Network Structures Solving Variational Inequalities](https://mlsanity.com/paper/1808.07526/) (2018)
- [Lipschitz constant estimation for 1D convolutional neural networks](https://mlsanity.com/paper/2211.15253/) (2022)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)

## Built on by (most cited first, 9 of 9)

- [Lipschitz constant estimation for general neural network architectures using control tools](https://mlsanity.com/paper/2405.01125/) (2024)
- [An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architectures](https://mlsanity.com/paper/2501.07930/) (2025)
- [Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations](https://mlsanity.com/paper/2401.14033/) (2024)
- [State space representations of the Roesser type for convolutional layers](https://mlsanity.com/paper/2403.11938/) (2024)
- [LipShiFT: A Certifiably Robust Shift-based Vision Transformer](https://mlsanity.com/paper/2503.14751/) (2025)
- [LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layers](https://mlsanity.com/paper/2410.22258/) (2024)
- [Learning Stable and Robust Linear Parameter-Varying State-Space Models](https://mlsanity.com/paper/2304.01828/) (2023)
- [Orthogonium : A Unified, Efficient Library of Orthogonal and 1-Lipschitz Building Blocks](https://mlsanity.com/paper/2601.13776/) (2026)
- [LipSSM: Structurally Lipschitz-Bounded Cascaded State-Space Model via Metric Transfer between Consecutive SSM Layers](https://mlsanity.com/paper/2609.30973/) (2026)

## Abstract (arXiv)

We establish a layer-wise parameterization for 1D convolutional neural networks (CNNs) with built-in end-to-end robustness guarantees. In doing so, we use the Lipschitz constant of the input-output mapping characterized by a CNN as a robustness measure. We base our parameterization on the Cayley transform that parameterizes orthogonal matrices and the controllability Gramian of the state space representation of the convolutional layers. The proposed parameterization by design fulfills linear matrix inequalities that are sufficient for Lipschitz continuity of the CNN, which further enables unconstrained training of Lipschitz-bounded 1D CNNs. Finally, we train Lipschitz-bounded 1D CNNs for the classification of heart arrythmia data and show their improved robustness.