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, learnt 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.

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, learnt 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.

Lipschitz constrained models have been used to solve specifics deep learning problems such as the estimation of Wasserstein distance for GAN, or the training of neural networks robust to adversarial attacks. Regardless the novel and effective algorithms to build such 1-Lipschitz networks, their usage remains marginal, and they are commonly considered as less expressive and less able to fit properly the data than their unconstrained counterpart. The goal of this paper is to demonstrate that, despite being empirically harder to train, 1-Lipschitz neural networks are theoretically better grounded than unconstrained ones when it comes to classification. We recall some results about 1-Lipschitz functions in the scope of deep learning and we extend and illustrate them to derive general properties for classification. We propose and demonstrate several new properties of 1-Lipschitz neural networks for classification. First, we show they can fit arbitrarily difficult frontiers, making them as expressive as classical ones, in addition to provide robustness certificates. We prove that when minimizing cross entropy loss the optimization problem under Lipschitz constraint is well posed and its solution generalizes well in the limit of big datasets, whereas regular neural networks can diverge even on remarkably simple situations. Then, we study the link between classification with 1-Lipschitz network and optimal transport thanks to regularized versions of Kantorovich-Rubinstein duality theory. Last, we derive preliminary bounds on their VC dimensions.