Exact Functional ANOVA Decomposition for Categorical Inputs Models
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.
Keywords
Cite
@article{arxiv.2603.02673,
title = {Exact Functional ANOVA Decomposition for Categorical Inputs Models},
author = {Baptiste Ferrere and Nicolas Bousquet and Fabrice Gamboa and Jean-Michel Loubes and Joseph Muré},
journal= {arXiv preprint arXiv:2603.02673},
year = {2026}
}