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

2026-05-18

AI Summary

This work addresses the challenge of explicitly expressing and effectively estimating functional ANOVA decompositions in realistic settings where input variables exhibit dependencies. By integrating Hilbert space theory with generalized functional ANOVA, the authors present the first closed-form Riesz basis decomposition for continuous and correlated inputs, enabling computable representations of main effects and higher-order interactions. The proposed model-agnostic algorithm not only unifies classical results established under independence assumptions but also establishes theoretical connections to prominent interpretability methods such as SHAP values and generalized additive models. Empirical evaluations demonstrate that the approach outperforms several state-of-the-art explanation techniques in both accuracy and practical utility.

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.

Problem

• functional ANOVA
• dependent inputs
• additive explanations
• Hoeffding decomposition
• model interpretability

Innovation

• Generalized Functional ANOVA
• Riesz Basis
• Hilbert Space Methods
• Additive Explanations
• Model-Agnostic Interpretability