Fabrice Gamboa

Yohann De Castro, Fabrice Gamboa, Didier Henrion et al.

We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems to solve numerically the approximate optimal design problem. The geometry of the design is recovered via semidefinite programming duality theory. This article shows that the hierarchy converges to the approximate optimal design as the order of the hierarchy increases. Furthermore, we provide a dual certificate ensuring finite convergence of the hierarchy and showing that the approximate optimal design can be computed numerically with our method. As a byproduct, we revisit the equivalence theorem of the experimental design theory: it is linked to the Christoffel polynomial and it characterizes finite convergence of the moment-sum-of-square hierarchies.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

Functional ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions. For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability. However, the lack of an explicit closed-form expression for general dependent distributions has forced practitioners to rely on costly sampling-based approximations. We completely resolve this limitation for categorical inputs. By bridging functional analysis with the extension of discrete Fourier analysis, we derive a closed-form decomposition without any assumption. Our formulation is computationally very efficient. It seamlessly recovers the classical independent case and extends to arbitrary dependence structures, including distributions with non-rectangular support. Furthermore, leveraging the intrinsic link between SHAP and ANOVA under independence, our framework yields a natural generalization of SHAP values for the general categorical setting.

Alejandro Cholaquidis, Fabrice Gamboa, Leonardo Moreno

Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for non Euclidean data. Circular data is a classic example, but so is data in the space of covariance matrices, data on the Grassmannian manifold obtained as a result of principal component analysis, among many others. In this work we investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by $X$, lies in an Euclidean space. This extends the concepts delineated in \cite{waser14} to this novel context. Aligning with traditional principles in conformal inference, these prediction sets are distribution-free, indicating that no specific assumptions are imposed on the joint distribution of $(X,Y)$, and they maintain a non-parametric character. We prove the asymptotic almost sure convergence of the empirical version of these regions on the manifold to their population counterparts. The efficiency of this method is shown through a comprehensive simulation study and an analysis involving real-world data.

Justin Reverdi, Sixin Zhang, Fabrice Gamboa et al.

Feature maps associated with positive definite kernels play a central role in kernel methods and learning theory, where regularity properties such as Lipschitz continuity are closely related to robustness and stability guarantees. Despite their importance, explicit characterizations of the Lipschitz constant of kernel feature maps are available only in a limited number of cases. In this paper, we study the Lipschitz regularity of feature maps associated with integral kernels under differentiability assumptions. We first provide sufficient conditions ensuring Lipschitz continuity and derive explicit formulas for the corresponding Lipschitz constants. We then identify a condition under which the feature map fails to be Lipschitz continuous and apply these results to several important classes of kernels. For infinite width two-layer neural network with isotropic Gaussian weight distributions, we show that the Lipschitz constant of the associated kernel can be expressed as the supremum of a two-dimensional integral, leading to an explicit characterization for the Gaussian kernel and the ReLU random neural network kernel. We also study continuous and shift-invariant kernels such as Gaussian, Laplace, and Matérn kernels, which admit an interpretation as neural network with cosine activation function. In this setting, we prove that the feature map is Lipschitz continuous if and only if the weight distribution has a finite second-order moment, and we then derive its Lipschitz constant. Finally, we raise an open question concerning the asymptotic behavior of the convergence of the Lipschitz constant in finite width neural networks. Numerical experiments are provided to support this behavior.

Hai-Vy Nguyen, Fabrice Gamboa, Reda Chhaibi et al.

We measure the Out-of-domain uncertainty in the prediction of Neural Networks using a statistical notion called ``Lens Depth'' (LD) combined with Fermat Distance, which is able to capture precisely the ``depth'' of a point with respect to a distribution in feature space, without any assumption about the form of distribution. Our method has no trainable parameter. The method is applicable to any classification model as it is applied directly in feature space at test time and does not intervene in training process. As such, it does not impact the performance of the original model. The proposed method gives excellent qualitative result on toy datasets and can give competitive or better uncertainty estimation on standard deep learning datasets compared to strong baseline methods.

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa et al.

Understanding the behavior of predictive models with random inputs can be achieved through functional decompositions into sub-models that capture interpretable effects of input groups. Building on recent advances in uncertainty quantification, the existence and uniqueness of a generalized Hoeffding decomposition have been established for correlated input variables, using oblique projections onto suitable functional subspaces. This work focuses on the case of Bernoulli inputs and provides a complete analytical characterization of the decomposition. We show that, in this discrete setting, the associated subspaces are one-dimensional and that the decomposition admits a closed-form representation. One of the main contributions of this study is to generalize the classical Fourier--Walsh--Hadamard decomposition for pseudo-Boolean functions to the correlated case, yielding an oblique version when the underlying distribution is not a product measure, and recovering the standard orthogonal form when independence holds. This explicit structure offers a fully interpretable framework, clarifying the contribution of each input combination and theoretically enabling model reverse engineering. From this formulation, explicit sensitivity measures-such as Sobol' indices and Shapley effects-can be directly derived. Numerical experiments illustrate the practical interest of the approach for decision-support problems involving binary features. The paper concludes with perspectives on extending the methodology to high-dimensional settings and to models involving inputs with finite, non-binary support.

Hai-Vy Nguyen, Fabrice Gamboa, Sixin Zhang et al.

In this paper, we propose a method for transferring feature representation to lightweight student models from larger teacher models. We mathematically define a new notion called \textit{perception coherence}. Based on this notion, we propose a loss function, which takes into account the dissimilarities between data points in feature space through their ranking. At a high level, by minimizing this loss function, the student model learns to mimic how the teacher model \textit{perceives} inputs. More precisely, our method is motivated by the fact that the representational capacity of the student model is weaker than the teacher model. Hence, we aim to develop a new method allowing for a better relaxation. This means that, the student model does not need to preserve the absolute geometry of the teacher one, while preserving global coherence through dissimilarity ranking. Importantly, while rankings are defined only on finite sets, our notion of \textit{perception coherence} extends them into a probabilistic form. This formulation depends on the input distribution and applies to general dissimilarity metrics. Our theoretical insights provide a probabilistic perspective on the process of feature representation transfer. Our experiments results show that our method outperforms or achieves on-par performance compared to strong baseline methods for representation transferring.

Hai-Vy Nguyen, Fabrice Gamboa, Sixin Zhang et al.

In this paper, we introduce a novel spatial attention module that can be easily integrated to any convolutional network. This module guides the model to pay attention to the most discriminative part of an image. This enables the model to attain a better performance by an end-to-end training. In conventional approaches, a spatial attention map is typically generated in a position-wise manner. Thus, it is often resulting in irregular boundaries and so can hamper generalization to new samples. In our method, the attention region is constrained to be rectangular. This rectangle is parametrized by only 5 parameters, allowing for a better stability and generalization to new samples. In our experiments, our method systematically outperforms the position-wise counterpart. So that, we provide a novel useful spatial attention mechanism for convolutional models. Besides, our module also provides the interpretability regarding the \textit{where to look} question, as it helps to know the part of the input on which the model focuses to produce the prediction.

Reda Chhaibi, Fabrice Gamboa, Christophe Oger et al.

We propose an active sampling flow, with the use-case of simulating the impact of combined variations on analog circuits. In such a context, given the large number of parameters, it is difficult to fit a surrogate model and to efficiently explore the space of design features. By combining a drastic dimension reduction using sensitivity analysis and Bayesian surrogate modeling, we obtain a flexible active sampling flow. On synthetic and real datasets, this flow outperforms the usual Monte-Carlo sampling which often forms the foundation of design space exploration.

Virgile Foy, Fabrice Gamboa, Reda Chhaibi

In the field of computer vision, the numerical encoding of 3D surfaces is crucial. It is classical to represent surfaces with their Signed Distance Functions (SDFs) or Unsigned Distance Functions (UDFs). For tasks like representation learning, surface classification, or surface reconstruction, this function can be learned by a neural network, called Neural Distance Function. This network, and in particular its weights, may serve as a parametric and implicit representation for the surface. The network must represent the surface as accurately as possible. In this paper, we propose a method for learning UDFs that improves the fidelity of the obtained Neural UDF to the original 3D surface. The key idea of our method is to concentrate the learning effort of the Neural UDF on surface edges. More precisely, we show that sampling more training points around surface edges allows better local accuracy of the trained Neural UDF, and thus improves the global expressiveness of the Neural UDF in terms of Hausdorff distance. To detect surface edges, we propose a new statistical method based on the calculation of a $p$-value at each point on the surface. Our method is shown to detect surface edges more accurately than a commonly used local geometric descriptor.

François Bachoc, Fabrice Gamboa, Max Halford et al.

In this paper, we present a new explainability formalism designed to shed light on how each input variable of a test set impacts the predictions of machine learning models. Hence, we propose a group explainability formalism for trained machine learning decision rules, based on their response to the variability of the input variables distribution. In order to emphasize the impact of each input variable, this formalism uses an information theory framework that quantifies the influence of all input-output observations based on entropic projections. This is thus the first unified and model agnostic formalism enabling data scientists to interpret the dependence between the input variables, their impact on the prediction errors, and their influence on the output predictions. Convergence rates of the entropic projections are provided in the large sample case. Most importantly, we prove that computing an explanation in our framework has a low algorithmic complexity, making it scalable to real-life large datasets. We illustrate our strategy by explaining complex decision rules learned by using XGBoost, Random Forest or Deep Neural Network classifiers on various datasets such as Adult Income, MNIST, CelebA, Boston Housing, Iris, as well as synthetic ones. We finally make clear its differences with the explainability strategies LIME and SHAP, that are based on single observations. Results can be reproduced by using the freely distributed Python toolbox https://gems-ai.aniti.fr/.

François Bachoc, Baptiste Broto, Fabrice Gamboa et al.

In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we explore the less classical case where X is the non commutative finite group of permutations. In this setting, we propose and study an harmonic analysis of the covariance operators that enables to consider Gaussian processes models and forecasting issues. Our theory is motivated by statistical ranking problems.

Thomas Epelbaum, Fabrice Gamboa, Jean-Michel Loubes et al.

In this paper, we propose deep learning architectures (FNN, CNN and LSTM) to forecast a regression model for time dependent data. These algorithm's are designed to handle Floating Car Data (FCD) historic speeds to predict road traffic data. For this we aggregate the speeds into the network inputs in an innovative way. We compare the RMSE thus obtained with the results of a simpler physical model, and show that the latter achieves better RMSE accuracy. We also propose a new indicator, which evaluates the algorithms improvement when compared to a benchmark prediction. We conclude by questioning the interest of using deep learning methods for this specific regression task.

François Bachoc, Fabrice Gamboa, Jean-Michel Loubes et al.

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling.

Guillaume Allain, Fabrice Gamboa, Philippe Goudal et al.

We provide a model to understand how adverse weather conditions modify traffic flow dynamic. We first prove that the microscopic Free Flow Speed of the vehicles is changed and then provide a rule to model this change. For this, we consider a thresholded linear model, corresponding to an application of a MARS model to road trafficking. This model adapts itself locally to the whole road network and provides accurate unbiased forecasted speed using live or short term forecasted weather data information.