Exact Functional ANOVA Decomposition for Categorical Inputs Models

TL;DR

This paper introduces a closed-form, efficient method for functional ANOVA decomposition tailored for categorical inputs, extending interpretability tools like SHAP to dependent feature distributions without relying on sampling.

Contribution

It derives a novel closed-form functional ANOVA decomposition for categorical inputs with arbitrary dependence, bridging functional analysis and discrete Fourier analysis, and generalizes SHAP values.

Findings

Provides a computationally efficient closed-form solution

Extends ANOVA and SHAP to dependent categorical features

Recovers classical independent case as a special instance

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.