arXiv subjects

Baptiste Ferrere

Publications and source records attributed to Baptiste Ferrere.

4 recordsLinked to original sources

arXiv2026

Minimax Additive Regression under Unknown Dependent Designs

We study additive regression under a potentially non-product random design on $[0,1]^d$, allowing the dimension $d$ to grow with the sample size $n$. We introduce coupled smoothness classes that separately control the regularity of the marginal densities and the density-weighted additive components. To handle dependence, we adapt a Riesz-basis construction for functional ANOVA models and establish compatibility bounds with constants independent of the dimension under uniform bounds on the joint density. We construct thresholded least-squares estimators and establish matching minimax upper and lower bounds for prediction with known or unknown marginal densities, under suitable dimension-growth conditions. When the marginal densities are at least as smooth as the weighted components, the unknown-density problem attains the known-density minimax rate. When the densities are less smooth, their regularity determines the minimax rate over the coupled class. Finally, we show that the centered additive components can be recovered at the same aggregate upper rate, without an additional order of error.

stat.ML↗

arXiv2026

Generalized Functional ANOVA: A Complete Theoretical Framework

The functional ANOVA provides a fundamental representation of square-integrable multivariate functions into main effects and higher-order interactions. For independent inputs, the components belong to mutually orthogonal Hilbert subspaces and admit an explicit representation. For dependent inputs, however, the components are only hierarchically orthogonal: although existence and uniqueness results are available, the Hilbert subspaces underlying the generalized decomposition have remained implicit. We resolve this representation problem for continuous inputs supported on a bounded hyperrectangle whose joint density is bounded above and away from zero. We introduce a distribution-adapted family of functions and prove that it forms a Riesz basis of the $L^2$ space, thereby guaranteeing a unique, stable, and unconditionally convergent representation. We then show that, for every coalition of variables, the corresponding block of this basis exactly characterizes the Hilbert subspace containing the functional ANOVA component. Our construction recovers the classical orthogonal decomposition under input independence. As a direct consequence, computing the generalized functional ANOVA reduces to estimating coefficients in an explicit, distribution-adapted basis. Finally, as a \emph{proof of concept}, we introduce an elementary, fast and model-agnostic estimator based on our theoretical results. Experiments on synthetic and real-world datasets illustrate its connections with established tabular explanation methods and show that low order components often capture most of the signal in the model output.

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arXiv2026

Exact Functional ANOVA Decomposition for Categorical Inputs Models

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.

stat.ML↗

arXiv2026

Fourier Analysis on the Boolean Hypercube via Hoeffding Functional Decomposition

Fourier analysis on the Boolean hypercube is fundamentally defined as the orthogonal decomposition of the space of pseudo-Boolean functions with respect to the uniform probability measure. In this work, we propose an ANOVA-based generalization of the Fourier decomposition on the Boolean hypercube endowed with any arbitrary probability measure. We provide an \emph{explicit} decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any \emph{arbitrary} probability measure on the Boolean hypercube. We formulate the computation of the entire functional decomposition as a least squares problem and also provide a method to address the classical \emph{curse of dimensionality} challenge. We provide a comprehensive generalization of Fourier analysis on the Boolean hypercube, enabling the handling of non-uniform configuration spaces inherent to real-world machine learning tasks, \textit{e.g.} when dealing with \emph{one-hot encoded} features. Finally, we demonstrate its practical impact in the field of explainable AI, by conducting comparative studies with feature attribution methods such as SHAP or TreeHFD.

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