# SHAP values through General Fourier Representations: Theory and Applications

arXiv: https://arxiv.org/abs/2511.00185 · HTML: https://mlsanity.com/paper/2511.00185/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 31 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Non-Gaussian processes and neural networks at finite widths](https://mlsanity.com/paper/1910.00019/) (2019)
- [Theory of the Frequency Principle for General Deep Neural Networks](https://mlsanity.com/paper/1906.09235/) (2019)
- [Wide Neural Networks with Bottlenecks are Deep Gaussian Processes](https://mlsanity.com/paper/2001.00921/) (2020)
- [A Scalable Walsh-Hadamard Regularizer to Overcome the Low-degree Spectral Bias of Neural Networks](https://mlsanity.com/paper/2305.09779/) (2023)
- [SHAP values via sparse Fourier representation](https://mlsanity.com/paper/2410.06300/) (2024)
- [Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform](https://mlsanity.com/paper/1504.07648/) (2015)
- [Random ReLU Neural Networks as Non-Gaussian Processes](https://mlsanity.com/paper/2405.10229/) (2024)
- [Finite Neural Networks as Mixtures of Gaussian Processes: From Provable Error Bounds to Prior Selection](https://mlsanity.com/paper/2407.18707/) (2024)

## Built on by (most cited first, 1 of 1)

- [Fair feature attribution for multi-output prediction: a Shapley-based perspective](https://mlsanity.com/paper/2602.22882/) (2026)

## Abstract (arXiv)

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.