Title: SHAP values via sparse Fourier representation
Venue: NeurIPS 2025
Accepted: yes
Keywords: SHAP

Interpretability

XAI

Fourier
Abstract: SHAP (SHapley Additive exPlanations) values are a widely used method for local feature attribution in interpretable and explainable AI. We propose an efficient two-stage algorithm for computing SHAP values in both black-box setting and tree-based models. Motivated by spectral bias in real-world predictors, we first approximate models using compact Fourier representations, exactly for trees and approximately for black-box models. In the second stage, we introduce a closed-form formula for {\em exactly} computing SHAP values using the Fourier representation, that ``linearizes'' the computation into a simple summation and is amenable to parallelization. As the Fourier approximation is computed only once, our method enables amortized SHAP value computation, achieving significant speedups over existing methods and a tunable trade-off between efficiency and precision.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces FourierSHAP, a novel method to approximate SHAP values for black-box models. First, a sparse Fourier approximation of the black-box is estimated. Then, thanks to a novel fast closed-form expression of SHAP values for Fourier representations of functions, the exact SHAP values (of the Fourier approximation) can be efficiently computed. The proposed method is specific to pseudoboolean functions.\n\nTo validate the FourierSHAP empirically, the authors consider four test cases with only binary or categorical features (transformed through one-hot encoding). For each test case, they first train a preliminary black-box model (MLP or tree ensemble) and compute a baseline with KernelSHAP. They show that FourierSHAP achieves good accuracy while offering several orders of magnitude improvement with respect to KernelSHAP.\n\nSTRENGTHS AND WEAKNESSES:\nI really enjoy the novelty of the paper, which shows that as long as a sparse Fourier representation is possible for a black-box model, it is possible to heavily reduce the computational burden of SHAP values. \nAs clearly motivated in the introduction, estimation of SHAP values is of great particular interest but usual running times prevent its broad application when the number of features is large. The new approach presented here is a new and promising point of view.\n\nThe paper is well-written and easy to follow, with novel (to the best of my knowledge) theoretical results fo Fourier representation. Sparse Fourier approximations of typical black-model models are validated in Figure 1, which is a very reassuring experiment. The running time improvements illustrated in all test cases are also impressive.\n\nHowever, I have a major concern about the limitation of the method on pseudoboolean functions. Of course categorical features can be transformed through one-hot encoding, and in Appendix F.1 the authors discuss quantization strategies to handle continuous features. To me, it is unfortunately still unclear:\n\n- how the SHAP value of the initial feature is recovered from the SHAP values of the new features after one-hot encoding (this may be trivial, but in this case this should be explicitly mentioned)\n- how running time improvement would change with respect to KernelSHAP used directly on the initial features (e.g., on SGEMM)\n- how continuous feature quantization would be handled in practice, and the impact of the increase in dimension\n\nI really think that the paper would benefit from at least one experiment with continuous feature to be more convincing.\n\nQUESTIONS:\n* As mentioned above, my main issue lies with the generalization to continuous features. Could the authors comment on the expected additional computational burden in such scenario?\n* Instead of reporting only relative improvement with respect to KernelSHAP, would it be possible to report also the running time of KernelSHAP in e.g., the Appendix? From a practical perspective a $10^3$ improvement is only interesting as long as KernelSHAP runs for more than seconds\n* Other sparse representations, given as low-order additive decompositions of multivariate functions, are also popular, and they would also be computationally efficient since they break (to some extent) the exponential complexity. How would the authors position their work with respect to such competitors?\n\nLIMITATIONS:\nThe authors discuss in Appendix F.1 the limitations related to continuous features, and in Appendix F.2 the extension of conditional SHAP values.\n\nI think Appendix F.1 should be expanded to give more practical details on how continuous features can be handled, and also contain at least one such illustration.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel approach to tackling the computational challenges associated with calculating Shapley Values (SV) for binary input vectors. The authors begin by reviewing the fundamentals of the Discrete Fourier Transform (DFT) applied to binary inputs, and they introduce key concepts such as k-sparsity and degree.\n\nThe authors provide a thoughtful discussion on the likelihood that many real-world prediction functions satisfy the sparsity requirement. They support this claim with educational examples, including modular functions and functions with only pairwise interactions, which effectively illustrate the practical relevance of their assumptions.\n\nBuilding on this foundation, the authors present their main result: surprisingly, the notoriously exponential summation inherent in SV computation can be expressed in a closed form when the input vector is represented in the binary Fourier space. This finding is a significant theoretical advancement.\n\nEmpirically, the authors demonstrate that their proposed method outperforms alternative approaches (FastSHAP, Kernel SHAP, DeepLIFT, and LigRegSHAP) in terms of SHAP value accuracy, further validating the practical utility of their contribution.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\nThe authors make a substantial contribution to the field of Explainable AI (XAI) and potentially to game theory as well. To the best of my knowledge, a closed-form solution for SV computation has not been previously established in the game theory literature, which underscores the high originality and novelty of this work.\n\nThe authors also provide a meticulous analysis of sparsity in the Fourier space, grounded in a deep understanding of spectral biases in neural networks. As a machine learning researcher in industry, I concur with their assertion that, in most real-world applications beyond benchmark problems in flagship domains (text, speech, vision, and molecular data), lower-order interactions often dominate the meaningful patterns worth extracting from data.\n\nThe manuscript is generally well-written and accessible. Additionally, the authors demonstrate a strong focus on the computational aspects of their method, leveraging JAX for efficient vectorization and parallelization, which enhances the practical applicability of their approach.\n\n**Weaknesses:**\nWhile the authors excel in explaining the implications of Propositions 1 and 2 with intuitive and straightforward examples, the main results in Section 4 are presented without sufficiently accessible explanations, which detracts from the paper’s readability. As a reviewer, I am providing a positive assessment under the assumption that the proofs are correct. However, if the authors have intentionally omitted detailed explanations to avoid scrutiny, I find this approach concerning and not conducive to transparent scientific discourse.\n\nQUESTIONS:\nI may add some later.\n\nLIMITATIONS:\nThe authors need to explain the main limitation upfront: The theory is only for binary vectors.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-30"}

comments: SUMMARY:
This paper introduces FOURIERSHAP, a novel two-stage algorithm for efficiently computing SHAP values for models with discrete or binary inputs. The core contribution is to first approximate a model using a sparse Fourier representation and then apply a newly derived closed-form formula that approximates SHAP values via a simple, parallelizable summation, bypassing the traditional method's exponential complexity. The authors demonstrate that this approach provides significant speedups over existing baselines while maintaining strong accuracy.

STRENGTHS AND WEAKNESSES:
## Strengths
* S1. The approach using Boolean Fourier analysis is theoretically novel and exciting.

* S2. The experimental validation is strong, with impressive and practical speedups.

## Weaknesses
* W1. The method is fundamentally designed for discrete inputs. However, the authors do suggest quantization as a workaround.

* W2. The method's effectiveness relies on the assumption of spectral sparsity. While the authors justify this for some models, it may not hold universally.

* W3. The justification for spectral bias in Section 3.1, which relies on the NTK, is not fully convincing. The author's arguments implicitly invoke the lazy regime, whereas there is now considerable work on more realistic "rich regimes". The motivation would be stronger with either more robust theoretical citations or direct empirical evidence of the claimed spectral sparsity.

QUESTIONS:
N/A

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 1
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper \"SHAP Values via Sparse Fourier Representation\" introduces FOURIERSHAP, a two-stage algorithm for efficiently computing SHAP values by leveraging the observation that many real-world models—especially neural networks and tree ensembles—exhibit spectral sparsity and can be approximated by sparse low-degree Fourier representations. In the first stage, the method approximates the black-box model using a sparse Fourier transform. In the second stage, it derives a closed-form, exact expression for the SHAP values of each Fourier basis function, avoiding the exponential summation over coalitions and enabling highly parallelizable, amortized computations. This approach results in orders-of-magnitude speedups over existing SHAP methods (including KERNELSHAP and FASTSHAP).\n\nSTRENGTHS AND WEAKNESSES:\nThe paper touches upon a computationally challenging problem of computing exact Shapley values, whose complexity grows exponentially with the number of features. The fact that the method can compute the exact Shapley value is the main strength of the paper. \n\nAlthough the paper seems a bit difficult to follow, FOURIERSHAP seems to approximate the trained model under explanation with a global, structured surrogate in the Fourier domain, enabling exact SHAP value computation without exponential cost. So, in principle, the authors put forward a global surrogate model that can approximate the trained model and able to compute Shapley values in closed form. Thus, the paper, IMHO, should be shared about providing a global surrogate model with Shapley value computation capability, not an approximation of the Shapley value itself for different models. As a result, the method is not comparable with amortizedSHAP (like FastSHAP which is studied in length in the paper). \n\nBased on this inception, the paper should discuss and study how this global surrogate model can approximate the functions, e.g., is it a universal approximation?  This approximation power would provide some perspective on how well the approximated Shapley values are, when the original model is replaced by the proposed surrogate model.\n\nIn the same line, the authors should discuss why this global surrogate model is preferred over TreeSHAP for instance, where we can compute the exact Shapley value. The current version of FourierSHAP seems to be focused on tabular data, and ensemble trees are the SOTA on tabular data, and we are able to compute exact Shapley values as well very efficiently. TreeSHAP is just an example, and there are a few other models that are capable of computing exact Shapley values.  \n\nThat being said, the authors ironically approximate the decision trees to apply FourierSHAP; why not using TreeSHAP?\n\nLast but not least, I did not why the authors use the term SHAP and FourierSHAP, while they try to approximate the Shapley value itself. So, I think the right terminology here is to use Shapley value and term the method as FourierShapley.\n\nQUESTIONS:\nWhy did you refer to the method as FourierSHAP and not FourierShapley?\n\nLIMITATIONS:\nSee above.", "TITLE": "Official Review", "RECOMMENDATION": 1, "REVIEWER_CONFIDENCE": 5, "CLARITY": 1, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-03"}