# Fourier Analysis on the Boolean Hypercube via Hoeffding Functional Decomposition

arXiv: https://arxiv.org/abs/2510.07088 · HTML: https://mlsanity.com/paper/2510.07088/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 34 references that are in the map.
- First posted on arXiv 8 Oct 2025.

## Builds on (most cited first, 12 of 12)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [From Shapley Values to Generalized Additive Models and back](https://mlsanity.com/paper/2209.04012/) (2022)
- [Purifying Interaction Effects with the Functional ANOVA: An Efficient Algorithm for Recovering Identifiable Additive Models](https://mlsanity.com/paper/1911.04974/) (2019)
- [Global Sensitivity Analysis with Dependence Measures](https://mlsanity.com/paper/1311.2483/) (2013)
- [Statistical Aspects of SHAP: Functional ANOVA for Model Interpretation](https://mlsanity.com/paper/2208.09970/) (2022)
- [Reluctant Interaction Modeling](https://mlsanity.com/paper/1907.08414/) (2019)
- [SHAP values via sparse Fourier representation](https://mlsanity.com/paper/2410.06300/) (2024)
- [On the coalitional decomposition of parameters of interest](https://mlsanity.com/paper/2301.02539/) (2023)
- [Proportional marginal effects for global sensitivity analysis](https://mlsanity.com/paper/2210.13065/) (2022)

## Abstract (arXiv)

Fourier analysis on the Boolean hypercube is fundamentally defined as the orthogonal decomposition of the space of pseudo-Boolean functions with respect to the uniform probability measure. In this work, we propose an ANOVA-based generalization of the Fourier decomposition on the Boolean hypercube endowed with any arbitrary probability measure. We provide an explicit decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any arbitrary probability measure on the Boolean hypercube. We formulate the computation of the entire functional decomposition as a least squares problem and also provide a method to address the classical curse of dimensionality challenge. We provide a comprehensive generalization of Fourier analysis on the Boolean hypercube, enabling the handling of non-uniform configuration spaces inherent to real-world machine learning tasks, textite.g. when dealing with one-hot encoded features. Finally, we demonstrate its practical impact in the field of explainable AI, by conducting comparative studies with feature attribution methods such as SHAP or TreeHFD.