Fourier Analysis on the Boolean Hypercube via Hoeffding Functional Decomposition

Abstract

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 \emph{explicit} decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any \emph{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 \emph{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, \textit{e.g.} when dealing with \emph{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.