# Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity

arXiv: https://arxiv.org/abs/2111.12906 · HTML: https://mlsanity.com/paper/2111.12906/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 25 Nov 2021.

## Builds on (most cited first, 26 of 26)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [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)
- [Domain-Adversarial Training of Neural Networks](https://mlsanity.com/paper/1505.07818/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Ensemble Adversarial Training: Attacks and Defenses](https://mlsanity.com/paper/1705.07204/) (2017)
- [Distillation as a Defense to Adversarial Perturbations against Deep Neural Networks](https://mlsanity.com/paper/1511.04508/) (2015)
- [Adversarial Training for Free!](https://mlsanity.com/paper/1904.12843/) (2019)
- [One pixel attack for fooling deep neural networks](https://mlsanity.com/paper/1710.08864/) (2017)
- [Adversarial Examples Are Not Easily Detected: Bypassing Ten Detection Methods](https://mlsanity.com/paper/1705.07263/) (2017)
- [Empirical Evaluation of Rectified Activations in Convolutional Network](https://mlsanity.com/paper/1505.00853/) (2015)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [The Singular Values of Convolutional Layers](https://mlsanity.com/paper/1805.10408/) (2018)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators](https://mlsanity.com/paper/1903.01014/) (2019)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Adversarial Robustness with Semi-Infinite Constrained Learning](https://mlsanity.com/paper/2110.15767/) (2021)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Cross-Entropy Loss and Low-Rank Features Have Responsibility for Adversarial Examples](https://mlsanity.com/paper/1901.08360/) (2019)
- [Robust Design of Deep Neural Networks against Adversarial Attacks based on Lyapunov Theory](https://mlsanity.com/paper/1911.04636/) (2019)
- [On the Effectiveness of Defensive Distillation](https://mlsanity.com/paper/1607.05113/) (2016)

## Built on by (most cited first, 7 of 7)

- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Lipschitz constant estimation for 1D convolutional neural networks](https://mlsanity.com/paper/2211.15253/) (2022)
- [Improving Adversarial Robustness via Joint Classification and Multiple Explicit Detection Classes](https://mlsanity.com/paper/2210.14410/) (2022)
- [State space representations of the Roesser type for convolutional layers](https://mlsanity.com/paper/2403.11938/) (2024)
- [LipKernel: Lipschitz-Bounded Convolutional Neural Networks via Dissipative Layers](https://mlsanity.com/paper/2410.22258/) (2024)
- [Robust Convolution Neural ODEs via Contractivity-promoting regularization](https://mlsanity.com/paper/2508.11432/) (2025)

## Abstract (arXiv)

Adversarial examples can easily degrade the classification performance in neural networks. Empirical methods for promoting robustness to such examples have been proposed, but often lack both analytical insights and formal guarantees. Recently, some robustness certificates have appeared in the literature based on system theoretic notions. This work proposes an incremental dissipativity-based robustness certificate for neural networks in the form of a linear matrix inequality for each layer. We also propose an equivalent spectral norm bound for this certificate which is scalable to neural networks with multiple layers. We demonstrate the improved performance against adversarial attacks on a feed-forward neural network trained on MNIST and an Alexnet trained using CIFAR-10.