# Achieving robustness in classification using optimal transport with hinge regularization

arXiv: https://arxiv.org/abs/2006.06520 · HTML: https://mlsanity.com/paper/2006.06520/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2020.

## Built on by (most cited first, 27 of 27)

- [Pay attention to your loss: understanding misconceptions about 1-Lipschitz neural networks](https://mlsanity.com/paper/2104.05097/) (2021)
- [Learning to Align Sequential Actions in the Wild](https://mlsanity.com/paper/2111.09301/) (2021)
- [CC-Cert: A Probabilistic Approach to Certify General Robustness of Neural Networks](https://mlsanity.com/paper/2109.10696/) (2021)
- [Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/2108.05623/) (2021)
- [DP-SGD Without Clipping: The Lipschitz Neural Network Way](https://mlsanity.com/paper/2305.16202/) (2023)
- [Robust One-Class Classification with Signed Distance Function using 1-Lipschitz Neural Networks](https://mlsanity.com/paper/2303.01978/) (2023)
- [Detecting Brittle Decisions for Free: Leveraging Margin Consistency in Deep Robust Classifiers](https://mlsanity.com/paper/2406.18451/) (2024)
- [1-Lipschitz Neural Distance Fields](https://mlsanity.com/paper/2407.09505/) (2024)
- [An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architectures](https://mlsanity.com/paper/2501.07930/) (2025)
- [On the explainable properties of 1-Lipschitz Neural Networks: An Optimal Transport Perspective](https://mlsanity.com/paper/2206.06854/) (2022)
- [CertViT: Certified Robustness of Pre-Trained Vision Transformers](https://mlsanity.com/paper/2302.10287/) (2023)
- [Adversarial Attacks and Defenses in Machine Learning-Powered Networks: A Contemporary Survey](https://mlsanity.com/paper/2303.06302/) (2023)
- [Robust-by-Design Classification via Unitary-Gradient Neural Networks](https://mlsanity.com/paper/2209.04293/) (2022)
- [Efficient Robust Conformal Prediction via Lipschitz-Bounded Networks](https://mlsanity.com/paper/2506.05434/) (2025)
- [FAME: Formal Abstract Minimal Explanation for Neural Networks](https://mlsanity.com/paper/2603.10661/) (2026)
- [Towards Robust Semantic Segmentation against Patch-based Attack via Attention Refinement](https://mlsanity.com/paper/2401.01750/) (2024)
- [Input Convex Lipschitz Recurrent Neural Networks for Robust and Efficient Process Modeling and Optimization](https://mlsanity.com/paper/2401.07494/) (2024)
- [Stability Bounds for the Unfolded Forward-Backward Algorithm](https://mlsanity.com/paper/2412.17888/) (2024)
- [Diffusion Models are Robust Pretrainers](https://mlsanity.com/paper/2511.02793/) (2025)
- [Fast and Flexible Robustness Certificates for Semantic Segmentation](https://mlsanity.com/paper/2512.06010/) (2025)
- [NeuralDP Differentially private neural networks by design](https://mlsanity.com/paper/2107.14582/) (2021)
- [Aortic root landmark localization with optimal transport loss for heatmap regression](https://mlsanity.com/paper/2407.04921/) (2024)
- [Fine-Tuning Adversarially-Robust Transformers for Single-Image Dehazing](https://mlsanity.com/paper/2504.17829/) (2025)
- [Orthogonium : A Unified, Efficient Library of Orthogonal and 1-Lipschitz Building Blocks](https://mlsanity.com/paper/2601.13776/) (2026)
- [LipSSD: Lipschitz-Constrained Single-Shot Detection for Adversarially Robust Object Detection](https://mlsanity.com/paper/2607.06592/) (2026)
- [LiST: Lipschitz Scaling Training for Robust and Calibrated Neural Networks](https://mlsanity.com/paper/2607.07745/) (2026)
- [Building Trust in Artificial Intelligence: A Necessity for Railway Applications](https://mlsanity.com/paper/2609.18278/) (2026)

## Abstract (arXiv)

Adversarial examples have pointed out Deep Neural Networks vulnerability to small local noise. It has been shown that constraining their Lipschitz constant should enhance robustness, but make them harder to learn with classical loss functions. We propose a new framework for binary classification, based on optimal transport, which integrates this Lipschitz constraint as a theoretical requirement. We propose to learn 1-Lipschitz networks using a new loss that is an hinge regularized version of the Kantorovich-Rubinstein dual formulation for the Wasserstein distance estimation. This loss function has a direct interpretation in terms of adversarial robustness together with certifiable robustness bound. We also prove that this hinge regularized version is still the dual formulation of an optimal transportation problem, and has a solution. We also establish several geometrical properties of this optimal solution, and extend the approach to multi-class problems. Experiments show that the proposed approach provides the expected guarantees in terms of robustness without any significant accuracy drop. The adversarial examples, on the proposed models, visibly and meaningfully change the input providing an explanation for the classification.