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.

翻译：布尔超立方体上的傅里叶分析，其基本定义为伪布尔函数空间关于均匀概率测度的正交分解。本文提出一种基于方差分析（ANOVA）的广义傅里叶分解方法，适用于赋予任意概率测度的布尔超立方体。我们给出了一个显式分解基，该基在布尔超立方体上的任意概率测度下，推广了沃尔什-哈达玛基（或称奇偶函数基）。我们将整个函数分解的计算表述为一个最小二乘问题，并提供了一种应对经典维度灾难挑战的方法。我们全面推广了布尔超立方体上的傅里叶分析，使其能够处理现实世界机器学习任务中固有的非均匀配置空间，例如处理独热编码特征时的情况。最后，通过与SHAP、TreeHFD等特征归因方法进行比较研究，我们展示了该方法在可解释人工智能领域的实际影响。