Although Artificial Neural Networks (ANNs) have achieved remarkable success across various tasks, they still suffer from limited generalization. We hypothesize that this limitation arises from the traditional sample-based (0--dimensionnal) regularization used in ANNs. To overcome this, we introduce Deep Sturm-Liouville (DSL), a novel function approximator that enables continuous 1D regularization along field lines in the input space by integrating the Sturm-Liouville Theorem (SLT) into the deep learning framework. DSL defines field lines traversing the input space, along which a Sturm-Liouville problem is solved to generate orthogonal basis functions, enforcing implicit regularization thanks to the desirable properties of SLT. These basis functions are linearly combined to construct the DSL approximator. Both the vector field and basis functions are parameterized by neural networks and learned jointly. We demonstrate that the DSL formulation naturally arises when solving a Rank-1 Parabolic Eigenvalue Problem. DSL is trained efficiently using stochastic gradient descent via implicit differentiation and achieves competitive performance on diverse multivariate datasets, including high-dimensional image datasets such as MNIST and CIFAR-10.

What is the shortest path between two data points lying in a high-dimensional space? While the answer is trivial in Euclidean geometry, it becomes significantly more complex when the data lies on a curved manifold—requiring a Riemannian metric to describe the space's local curvature. Estimating such a metric, however, remains a major challenge in high dimensions. In this work, we propose a method for deriving Riemannian metrics directly from pretrained Energy-Based Models (EBMs)—a class of generative models that assign low energy to high-density regions. These metrics define spatially varying distances, enabling the computation of geodesics—shortest paths that follow the data manifold’s intrinsic geometry. We introduce two novel metrics derived from EBMs and show that they produce geodesics that remain closer to the data manifold and exhibit lower curvature distortion, as measured by alignment with ground-truth trajectories. We evaluate our approach on increasingly complex datasets: synthetic datasets with known data density, rotated character images with interpretable geometry, and high-resolution natural images embedded in a pretrained VAE latent space. Our results show that EBM-derived metrics consistently outperform established baselines, especially in high-dimensional settings. Our work is the first to derive Riemannian metrics from EBMs, enabling data-aware geodesics and unlocking scalable, geometry-driven learning for generative modeling and simulation.

State-of-the-art approaches for training Differentially Private (DP) Deep Neural Networks (DNN) face difficulties to estimate tight bounds on the sensitivity of the network's layers, and instead rely on a process of per-sample gradient clipping. This clipping process not only biases the direction of gradients but also proves costly both in memory consumption and in computation. To provide sensitivity bounds and bypass the drawbacks of the clipping process, we propose to rely on Lipschitz constrained networks. Our theoretical analysis reveals an unexplored link between the Lipschitz constant with respect to their input and the one with respect to their parameters. By bounding the Lipschitz constant of each layer with respect to its parameters, we prove that we can train these networks with privacy guarantees. Our analysis not only allows the computation of the aforementioned sensitivities at scale, but also provides guidance on how to maximize the gradient-to-noise ratio for fixed privacy guarantees. To facilitate the application of Lipschitz networks and foster robust and certifiable learning under privacy guarantees, we provide a Python package that implements building blocks allowing the construction and private training of such networks.

A plethora of attribution methods have recently been developed to explain deep neural networks. These methods use different classes of perturbations (e.g, occlusion, blurring, masking, etc) to estimate the importance of individual image pixels to drive a model's decision. Nevertheless, the space of possible perturbations is vast and current attribution methods typically require significant computation time to accurately sample the space in order to achieve high-quality explanations. In this work, we introduce EVA (Explaining using Verified Perturbation Analysis) - the first explainability method which comes with guarantees that an entire set of possible perturbations has been exhaustively searched. We leverage recent progress in verified perturbation analysis methods to directly propagate bounds through a neural network to exhaustively probe a - potentially infinite-size - set of perturbations in a single forward pass. Our approach takes advantage of the beneficial properties of verified perturbation analysis, i.e., time efficiency and guaranteed complete - sampling agnostic - coverage of the perturbation space - to identify image pixels that drive a model's decision. We evaluate EVA systematically and demonstrate state-of-the-art results on multiple benchmarks. Our code isfreely available: github.com/deel-ai/formal-explainability

Attribution methods, which employ heatmaps to identify the most influential regions of an image that impact model decisions, have gained widespread popularity as a type of explainability method. However, recent research has exposed the limited practical value of these methods, attributed in part to their narrow focus on the most prominent regions of an image - revealing “where” the model looks, but failing to elucidate “what” the model sees in those areas. In this work, we try to fill in this gap with CRAFT - a novel approach to identify both “what” and “where” by generating concept-based explanations. We introduce 3 new ingredients to the automatic concept extraction literature: (i) a recursive strategy to detect and decompose concepts across layers, (ii) a novel method for a more faithful estimation of concept importance using Sobol indices, and (iii) the use of implicit differentiation to unlock Concept Attribution Maps. We conduct both human and computer vision experiments to demonstrate the benefits of the proposed approach. We show that the proposed concept importance estimation technique is more faithful to the model than previous methods. When evaluating the usefulness of the method for human experimenters on a human-centered utility benchmark, we find that our approach significantly improves on two of the three test scenarios.

Quantum walks on binary trees are used in many quantum algorithms to achieve important speedup over classical algorithms. The formulation of this kind of algorithms as quantum circuit presents the advantage of being easily readable, executable on circuit based quantum computers and simulators and optimal on the usage of resources. We propose a strategy to compose quantum circuit that performs quantum walk on binary trees following universal gate model quantum computation principles. We give a particular attention to NAND formula evaluation algorithm as it could have many applications in game theory and reinforcement learning. We therefore propose an application of this algorithm and show how it can be used to train a quantum reinforcement learning agent in a two player game environment.

This paper presents a new efficient black-box attribution method built on Hilbert-Schmidt Independence Criterion (HSIC). Based on Reproducing Kernel Hilbert Spaces (RKHS), HSIC measures the dependence between regions of an input image and the output of a model using the kernel embedding of their distributions. It thus provides explanations enriched by RKHS representation capabilities. HSIC can be estimated very efficiently, significantly reducing the computational cost compared to other black-box attribution methods.Our experiments show that HSIC is up to 8 times faster than the previous best black-box attribution methods while being as faithful.Indeed, we improve or match the state-of-the-art of both black-box and white-box attribution methods for several fidelity metrics on Imagenet with various recent model architectures.Importantly, we show that these advances can be transposed to efficiently and faithfully explain object detection models such as YOLOv4. Finally, we extend the traditional attribution methods by proposing a new kernel enabling an ANOVA-like orthogonal decomposition of importance scores based on HSIC, allowing us to evaluate not only the importance of each image patch but also the importance of their pairwise interactions. Our implementation is available at \url{https://github.com/paulnovello/HSIC-Attribution-Method}.

A plethora of methods have been proposed to explain how deep neural networks reach their decisions but comparatively, little effort has been made to ensure that the explanations produced by these methods are objectively relevant. While several desirable properties for trustworthy explanations have been formulated, objective measures have been harder to derive. Here, we propose two new measures to evaluate explanations borrowed from the field of algorithmic stability: mean generalizability MeGe and relative consistency ReCo. We conduct extensive experiments on different network architectures, common explainability methods, and several image datasets to demonstrate the benefits of the proposed measures. In comparison to ours, popular fidelity measures are not sufficient to guarantee trustworthy explanations. Finally, we found that 1-Lipschitz networks produce explanations with higher MeGe and ReCo than common neural networks while reaching similar accuracy. This suggests that 1-Lipschitz networks are a relevant direction towards predictors that are more explainable and trustworthy.

We describe a novel attribution method which is grounded in Sensitivity Analysis and uses Sobol indices. Beyond modeling the individual contributions of image regions, Sobol indices provide an efficient way to capture higher-order interactions between image regions and their contributions to a neural network's prediction through the lens of variance.We describe an approach that makes the computation of these indices efficient for high-dimensional problems by using perturbation masks coupled with efficient estimators to handle the high dimensionality of images.Importantly, we show that the proposed method leads to favorable scores on standard benchmarks for vision (and language models) while drastically reducing the computing time compared to other black-box methods -- even surpassing the accuracy of state-of-the-art white-box methods which require access to internal representations. Our code is freely available:github.com/fel-thomas/Sobol-Attribution-Method.