Exact Functional ANOVA Decomposition for Categorical Inputs Models

Baptiste Ferrere, Nicolas Bousquet, Fabrice Gamboa, Jean-Michel Loubes, Joseph Muré
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:30879-30896, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-ferrere26a, title = {Exact Functional {ANOVA} Decomposition for Categorical Inputs Models}, author = {Ferrere, Baptiste and Bousquet, Nicolas and Gamboa, Fabrice and Loubes, Jean-Michel and Mur\'{e}, Joseph}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {30879--30896}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/ferrere26a/ferrere26a.pdf}, url = {https://proceedings.mlr.press/v306/ferrere26a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Exact Functional ANOVA Decomposition for Categorical Inputs Models %A Baptiste Ferrere %A Nicolas Bousquet %A Fabrice Gamboa %A Jean-Michel Loubes %A Joseph Muré %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-ferrere26a %I PMLR %P 30879--30896 %U https://proceedings.mlr.press/v306/ferrere26a.html %V 306 %X 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.
APA
Ferrere, B., Bousquet, N., Gamboa, F., Loubes, J. & Muré, J.. (2026). Exact Functional ANOVA Decomposition for Categorical Inputs Models. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:30879-30896 Available from https://proceedings.mlr.press/v306/ferrere26a.html.

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