Generalized Functional ANOVA in Closed-Form: A Unified View of Additive Explanations

TL;DR

This paper introduces a unified, explicit framework for functional ANOVA decomposition applicable to dependent inputs, enabling easier computation and estimation of model explanations in a data-driven, model-agnostic manner.

Contribution

It develops a Hilbert space-based Riesz Basis for functional ANOVA, unifying independent and dependent input cases, and proposes an efficient algorithm for data-driven decomposition estimation.

Findings

The proposed method recovers classical ANOVA in independent cases.

It enables explicit decomposition computation for dependent inputs.

Empirical comparisons show competitive performance with state-of-the-art explanation methods.

Abstract

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.