{"ID":2858817,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.07088","arxiv_id":"2510.07088","title":"Fourier Analysis on the Boolean Hypercube via Hoeffding Functional Decomposition","abstract":"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.","short_abstract":"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 arbit...","url_abs":"https://arxiv.org/abs/2510.07088","url_pdf":"https://arxiv.org/pdf/2510.07088v4","authors":"[\"Baptiste Ferrere\",\"Nicolas Bousquet\",\"Fabrice Gamboa\",\"Jean-Michel Loubes\",\"Joseph Muré\"]","published":"2025-10-08T14:46:20Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}