SHAP values through General Fourier Representations: Theory and Applications

2025-10-31

AI Summary

This paper addresses the weak interpretability, lack of mathematical unification, and absence of theoretical stability guarantees for SHAP values in discrete or multi-valued input spaces. To resolve these issues, we establish a rigorous theoretical framework grounded in generalized Fourier spectral analysis. We first embed the SHAP attribution system into the spectral domain, introducing a generalized Fourier expansion under a tensor-product orthogonal basis and revealing a linear mapping between SHAP values and Fourier coefficients. Under both deterministic and probabilistic settings, we derive stability estimates and convergence theorems with explicit error bounds. Theoretical analysis leverages Lipschitz continuity, Gaussian process limits, and concentration inequalities, and establishes convergence for both Fourier truncation and infinite-width neural networks. Numerical experiments validate the theoretical results on real-world clinical imbalanced datasets.

Technology Category

Machine Learning: Other Foundations of Machine LearningReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyNatural Language Processing: Interpretability, Analysis, and Evaluation of NLP Models

Application Category

Search and Retrieval-Augmented AI: Web query analysis, representation and understandingWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This article establishes a rigorous spectral framework for the mathematical analysis of SHAP values. We show that any predictive model defined on a discrete or multi-valued input space admits a generalized Fourier expansion with respect to an orthonormalisation tensor-product basis constructed under a product probability measure. Within this setting, each SHAP attribution can be represented as a linear functional of the model's Fourier coefficients. Two complementary regimes are studied. In the deterministic regime, we derive quantitative stability estimates for SHAP values under Fourier truncation, showing that the attribution map is Lipschitz continuous with respect to the distance between predictors. In the probabilistic regime, we consider neural networks in their infinite-width limit and prove convergence of SHAP values toward those induced by the corresponding Gaussian process prior, with explicit error bounds in expectation and with high probability based on concentration inequalities. We also provide a numerical experiment on a clinical unbalanced dataset to validate the theoretical findings.

Problem

• Establishing spectral framework for SHAP value analysis
• Studying SHAP stability under Fourier truncation regimes
• Proving SHAP convergence in infinite-width neural networks

Innovation

• Generalized Fourier expansion for SHAP values analysis
• Lipschitz continuity of SHAP under Fourier truncation
• Neural network SHAP convergence to Gaussian process