Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SHAP values through General Fourier Representations: Theory and Applications

Roberto Morales

TL;DR

This work develops a generalized Fourier framework for SHAP values on discrete, multi-valued input spaces by constructing an orthonormal tensor-product basis under product measures. It proves that SHAP attributions are linear functionals of the model’s Fourier coefficients and provides both deterministic truncation stability bounds and probabilistic convergence results to Gaussian-process limits, including infinite-width neural networks. A GP-based analysis yields explicit mean and concentration bounds for SHAP truncation errors, while a neural-network experiment demonstrates substantial computational savings and strong agreement with Kernel-SHAP on a clinical dataset with significant class imbalance. The results bridge cooperative-game explanations with harmonic analysis, enabling scalable and robust SHAP interpretations for complex, non-binary domains and guiding sparse spectral recovery in practice.

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.

SHAP values through General Fourier Representations: Theory and Applications

TL;DR

This work develops a generalized Fourier framework for SHAP values on discrete, multi-valued input spaces by constructing an orthonormal tensor-product basis under product measures. It proves that SHAP attributions are linear functionals of the model’s Fourier coefficients and provides both deterministic truncation stability bounds and probabilistic convergence results to Gaussian-process limits, including infinite-width neural networks. A GP-based analysis yields explicit mean and concentration bounds for SHAP truncation errors, while a neural-network experiment demonstrates substantial computational savings and strong agreement with Kernel-SHAP on a clinical dataset with significant class imbalance. The results bridge cooperative-game explanations with harmonic analysis, enabling scalable and robust SHAP interpretations for complex, non-binary domains and guiding sparse spectral recovery in practice.

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.

Paper Structure

This paper contains 20 sections, 8 theorems, 63 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Methodology and main contributions
  4. Related works
  5. Organization of the paper
  6. Mathematical Setting and Orthonormal Basis Construction
  7. Main results
  8. Deterministic results: Spectral truncation and error analysis
  9. Probabilistic results: Asymptotic convergence and Gaussian limits
  10. Numerical experiments
  11. Setting, training and sparse representation
  12. Results
  13. Time and Peak memory
  14. Conclusions and open problems
  15. Spectral Stability of SHAP for Neural Networks
  16. ...and 5 more sections

Key Result

Proposition 2.2

The family $\{\Psi_k\}_{k\in \mathcal{I}}$ is an orthonormal basis of $L^2(\mu)$.

Figures (3)

  • Figure 1: ROC curve of the trained classifier on the test set. The curve illustrates the trade-off between the true positive rate and the false positive rate across different classification thresholds. An AUC of approximately 0.83 indicates strong discriminative ability, showing that the model ranks positive samples above negatives in about 83% of the cases.
  • Figure 2: Mean absolute SHAP values on the logit scale across age bins. Each panel displays bar plots for the same set of clinical and demographic covariates within a specific age bin; higher bars indicate greater average contribution to the model's output magnitude. The log scale highlights both dominant and secondary drivers of risk across bins, facilitating cross-age comparisons of feature importance.
  • Figure 3: Barplots of the logarithmic runtime and peak memory for the Kernel and Fourier SHAP values across data bins.

Theorems & Definitions (24)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 3.1
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4: Expected $L^2$ error under a Gaussian-process prior
  • ...and 14 more