# Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators

arXiv: https://arxiv.org/abs/1903.01014 · HTML: https://mlsanity.com/paper/1903.01014/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2019.

## Built on by (most cited first, 30 of 55)

- [A Dynamical System Perspective for Lipschitz Neural Networks](https://mlsanity.com/paper/2110.12690/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [CLIP: Cheap Lipschitz Training of Neural Networks](https://mlsanity.com/paper/2103.12531/) (2021)
- [Certified Robustness via Dynamic Margin Maximization and Improved Lipschitz Regularization](https://mlsanity.com/paper/2310.00116/) (2023)
- [Robust Implicit Networks via Non-Euclidean Contractions](https://mlsanity.com/paper/2106.03194/) (2021)
- [On the expressivity of bi-Lipschitz normalizing flows](https://mlsanity.com/paper/2107.07232/) (2021)
- [Convergence of Deep ReLU Networks](https://mlsanity.com/paper/2107.12530/) (2021)
- [Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks](https://mlsanity.com/paper/2204.00846/) (2022)
- [Explainable AI via Learning to Optimize](https://mlsanity.com/paper/2204.14174/) (2022)
- [Convergence of Deep Convolutional Neural Networks](https://mlsanity.com/paper/2109.13542/) (2021)
- [Lipschitz constant estimation for 1D convolutional neural networks](https://mlsanity.com/paper/2211.15253/) (2022)
- [Adversarial robustness of sparse local Lipschitz predictors](https://mlsanity.com/paper/2202.13216/) (2022)
- [Connections between Numerical Algorithms for PDEs and Neural Networks](https://mlsanity.com/paper/2107.14742/) (2021)
- [A path-norm toolkit for modern networks: consequences, promises and challenges](https://mlsanity.com/paper/2310.01225/) (2023)
- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [Provably Tightest Linear Approximation for Robustness Verification of Sigmoid-like Neural Networks](https://mlsanity.com/paper/2208.09872/) (2022)
- [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)
- [Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity](https://mlsanity.com/paper/2111.12906/) (2021)
- [On Adversarial Robustness of Deep Image Deblurring](https://mlsanity.com/paper/2210.02502/) (2022)
- [Unfolded proximal neural networks for robust image Gaussian denoising](https://mlsanity.com/paper/2308.03139/) (2023)
- [Positive concave deep equilibrium models](https://mlsanity.com/paper/2402.04029/) (2024)
- [CertViT: Certified Robustness of Pre-Trained Vision Transformers](https://mlsanity.com/paper/2302.10287/) (2023)
- [What training reveals about neural network complexity](https://mlsanity.com/paper/2106.04186/) (2021)
- [Fixed points of nonnegative neural networks](https://mlsanity.com/paper/2106.16239/) (2021)
- [NESTANets: Stable, accurate and efficient neural networks for analysis-sparse inverse problems](https://mlsanity.com/paper/2203.00804/) (2022)
- [Uniform Convergence of Deep Neural Networks with Lipschitz Continuous Activation Functions and Variable Widths](https://mlsanity.com/paper/2306.01692/) (2023)
- [Bridging Dimensions: Confident Reachability for High-Dimensional Controllers](https://mlsanity.com/paper/2311.04843/) (2023)
- [Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificates](https://mlsanity.com/paper/2406.00826/) (2024)

## Abstract (arXiv)

Obtaining sharp Lipschitz constants for feed-forward neural networks is essential to assess their robustness in the face of perturbations of their inputs. We derive such constants in the context of a general layered network model involving compositions of nonexpansive averaged operators and affine operators. By exploiting this architecture, our analysis finely captures the interactions between the layers, yielding tighter Lipschitz constants than those resulting from the product of individual bounds for groups of layers. The proposed framework is shown to cover in particular many practical instances encountered in feed-forward neural networks. Our Lipschitz constant estimates are further improved in the case of structures employing scalar nonlinear functions, which include standard convolutional networks as special cases.