# On the explainable properties of 1-Lipschitz Neural Networks: An Optimal Transport Perspective

arXiv: https://arxiv.org/abs/2206.06854 · HTML: https://mlsanity.com/paper/2206.06854/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 71 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 30 of 54)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Sanity Checks for Saliency Maps](https://mlsanity.com/paper/1810.03292/) (2018)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [RISE: Randomized Input Sampling for Explanation of Black-box Models](https://mlsanity.com/paper/1806.07421/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Interpretation of Neural Networks is Fragile](https://mlsanity.com/paper/1710.10547/) (2017)
- [The (Un)reliability of saliency methods](https://mlsanity.com/paper/1711.00867/) (2017)
- [Counterfactual Visual Explanations](https://mlsanity.com/paper/1904.07451/) (2019)
- [On the (In)fidelity and Sensitivity for Explanations](https://mlsanity.com/paper/1901.09392/) (2019)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Evaluating and Aggregating Feature-based Model Explanations](https://mlsanity.com/paper/2005.00631/) (2020)
- [Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation](https://mlsanity.com/paper/1705.08475/) (2017)
- [Sanity Checks for Saliency Metrics](https://mlsanity.com/paper/1912.01451/) (2019)
- [Robust Large Margin Deep Neural Networks](https://mlsanity.com/paper/1605.08254/) (2016)
- [Fooling Neural Network Interpretations via Adversarial Model Manipulation](https://mlsanity.com/paper/1902.02041/) (2019)

## Built on by (most cited first, 6 of 6)

- [An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architectures](https://mlsanity.com/paper/2501.07930/) (2025)
- [Universal Neural Optimal Transport](https://mlsanity.com/paper/2212.00133/) (2022)
- [Efficient Robust Conformal Prediction via Lipschitz-Bounded Networks](https://mlsanity.com/paper/2506.05434/) (2025)
- [Wasserstein Distances Made Explainable: Insights Into Dataset Shifts and Transport Phenomena](https://mlsanity.com/paper/2505.06123/) (2025)
- [Informative Post-Hoc Explanations Only Exist for Simple Functions](https://mlsanity.com/paper/2508.11441/) (2025)
- [Mutual Equilibrium: Multimodal Representation Learning through Reciprocal Feedback](https://mlsanity.com/paper/2609.39456/) (2026)

## Abstract (arXiv)

Input gradients have a pivotal role in a variety of applications, including adversarial attack algorithms for evaluating model robustness, explainable AI techniques for generating Saliency Maps, and counterfactual explanations.However, Saliency Maps generated by traditional neural networks are often noisy and provide limited insights. In this paper, we demonstrate that, on the contrary, the Saliency Maps of 1-Lipschitz neural networks, learned with the dual loss of an optimal transportation problem, exhibit desirable XAI properties:They are highly concentrated on the essential parts of the image with low noise, significantly outperforming state-of-the-art explanation approaches across various models and metrics. We also prove that these maps align unprecedentedly well with human explanations on ImageNet.To explain the particularly beneficial properties of the Saliency Map for such models, we prove this gradient encodes both the direction of the transportation plan and the direction towards the nearest adversarial attack. Following the gradient down to the decision boundary is no longer considered an adversarial attack, but rather a counterfactual explanation that explicitly transports the input from one class to another. Thus, Learning with such a loss jointly optimizes the classification objective and the alignment of the gradient, i.e. the Saliency Map, to the transportation plan direction.These networks were previously known to be certifiably robust by design, and we demonstrate that they scale well for large problems and models, and are tailored for explainability using a fast and straightforward method.