REVIEW 3 major objections 4 minor 12 references
Statistical Test for Saliency Maps of Graph Neural Networks via Selective Inference
T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read Conditioning on the selected subgraph turns GNN saliency maps into exact, finite-sample hypothesis tests, so a highlighted region can be called meaningful only when the difference between salient and non-salient nodes is unlikely under…
desk verdict Useful selective-inference application to GNN saliency maps, but the unconditional Type I error claim in Theorem 1 is not supported; the conditional guarantee is correct and sufficient. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the selective p-value, defined by conditioning the test statistic T(X) = η^T X / $\sqrt$(η^T Σ η) on the event that the same salient and non-salient subgraphs are selected and on the sufficient statistic Q_X that removes the nuisance parameter. Two structural facts make this computable: Lemma 1, that the CAM saliency map is a piecewise linear function of the input features for ReLU-based GCN and GIN architectures, and Lemma 2, that the conditional data space collapses to a one-dimensional line X(z) = a + b z along which the truncation set Z is a union of intervals. The intervals are found by parametric programming, solving systems of linear inequalities derived from the thresholding rules that define the salient and non-salient subgraphs. This turns a high-dimensional inverse problem into a one-dimensional search, and the final p-value is the tail probability of a truncated standard normal over Z.
What would settle it
Simulate many graphs under the null hypothesis with known covariance but non-Gaussian heavy-tailed noise, run the published algorithm at α = 0.05, and count the rejection rate: a clear departure from 0.05 beyond Monte Carlo error would falsify the exact claim. A second direct check is to train the GNN on the same data that are then tested; if the empirical Type I error exceeds the nominal level, the claim that the training process needs no assumptions fails in that regime.
Extended reading notes
Core claim
The paper's central claim is Theorem 1: under the null hypothesis H0 in equation (5), which says that the mean feature values of the salient and non-salient subgraphs are equal, the selective p-value in equation (13) satisfies P_H0(p_selective ≤ α) = α for every α in (0,1). This is an exact finite-sample statement, not an asymptotic approximation. The mechanism is conditioning on V_X, the selected subgraph pair, and on Q_X defined in equation (11); under H0 the conditional test statistic is a truncated standard normal, so the probability integral transform gives conditional uniformity. After marginalizing over the conditioning variables, the paper obtains the exact Type I error control. Because the GNN and its weights are treated as fixed after training, the paper states that Theorem 1 makes no additional assumption on the training data or training process. In experiments, the naive p-value over-rejects because it ignores selection, Bonferroni correction controls error but is too conservative, while the proposed selective p-value keeps error at the nominal level and has higher power; on the EEG dataset, positive samples containing P300 responses receive small selective p-values and negative samples receive large ones.
Load-bearing premise
The exact Type I error guarantee rests on the feature vector being generated as Gaussian noise with known covariance matrix and on the GNN being fixed independently of the test data: if the covariance is estimated from the same data or the network is trained or retrained on the test instance, Theorem 1 no longer applies and only empirical error control is reported.
Editorial extensions
If this is right
- Saliency maps from GNNs can be reported with exact p-values, so a highlighted subgraph is statistically certified only when the salient versus non-salient difference is extreme under the null.
- The Type I error guarantee holds at any significance level and for finite numbers of nodes and features, so small-sample applications such as EEG studies do not need asymptotic sample-size justifications.
- Any saliency method that is piecewise linear in the features, including Grad-CAM, Grad, and Grad-Input on GCN and GIN models, can use the same conditioning and one-dimensional search.
- The test is valid for any fixed pair of thresholds used to define salient and non-salient subgraphs, including after normalizing saliency values to the range [0,1].
Reading between the lines
- Beyond the paper: the exact guarantee should be read as conditional on the GNN being fixed before the test data are seen; the conditioning event covers only the subgraph selection, not any training or retraining on the test instance, so a fully exact statement for end-to-end learned explanations would need a model for the training process.
- Beyond the paper: because the framework handles any piecewise-linear saliency method, approximating softmax attention in graph transformers by piecewise-linear splines would open the same testing scheme to attention-based GNNs, though the paper does not make that claim.
- Beyond the paper: the paper leaves Lemma 1 unproved with a note that it follows from standard composition results; supplying the full proof would remove the only structural gap between the algorithmic recipe and the theorem.
- Beyond the paper: using the selective p-value as a gate before interpretability evaluation could reduce false discoveries in neuroscience pipelines without replacing existing explanation metrics, a synthesis the paper only sketches as future work.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a selective-inference (SI) framework for testing the statistical significance of GNN saliency maps. Under the model X = mu + epsilon with Gaussian noise and known covariance, the authors define salient and non-salient subgraphs by thresholding a CAM-based saliency map, introduce a data-driven null hypothesis comparing mean features of the two subgraphs, and compute a selective p-value conditioned on the observed subgraph selection and a sufficient statistic for the nuisance parameter. The p-value is computed exactly by exploiting piecewise linearity of the GNN-CAM map and reducing the conditional data space to a one-dimensional search via parametric programming. The authors claim unconditional Type I error control (Theorem 1) and support this with synthetic and EEG experiments.
Significance. If the central claims are correctly stated, the paper makes a useful contribution: it is among the first to apply exact selective inference to GNN saliency maps, and the conditional SI guarantee is obtained non-asymptotically for a broad class of piecewise-linear GNN architectures and saliency methods. The paper also ships reproducible code and reports extensive experiments, including robustness checks under non-Gaussian noise, estimated covariance, varied thresholds, and multiple GNN/saliency combinations. These are concrete strengths. However, the paper's headline validity statement in Theorem 1 is stated as an unconditional guarantee, and that statement is not supported by the proof; the conditional SI guarantee underlying the method is standard and likely sufficient for the paper's practical claims, but the unconditional claim needs correction.
major comments (3)
- [Section 4.2, Theorem 1, Appendix C.2] The unconditional Type I error claim in Theorem 1 is not established. The proof in Appendix C.2 first shows that, conditional on V_X = V_obs and Q_X = Q_obs, the selective p-value is uniform under eta(V_obs)^T mu = 0. This is the standard SI result. The subsequent marginalization over Q_X given V_X is valid. The final marginalization over V_X, however, uses probabilities P_H0(V_X = V_obs) for a hypothesis H0 that is not a single distribution but a family of constraints indexed by the selected subgraph. For a fixed nonzero mu, eta(V)^T mu = 0 holds only for those subgraphs V for which the observed selection is exactly the one satisfying the null; for other subgraphs the conditional rejection probability is a power, not alpha. Hence the equality P_H0(p_selective <= alpha) = alpha in Eq. (8) does not follow except under a global null such as mu = 0, or when the statement is understood conditionally on the observed selection. The fix is to restate Theorem 1 conditionally on V_X = V_obs (and Q_X = Q_obs), which is the accepted SI notion of Type I error control and is sufficient for the method's practical validity claim.
- [Section 4.2, paragraph beginning 'Furthermore, we emphasize'] The claim that the method makes 'no assumptions on the training data or training process' is too strong. Theorem 1 treats the GNN weights as fixed inputs: the distributional calculation in Section 3.1 concerns only the test feature vector X, and the selection event is defined through a fixed network. If the GNN is trained on data that include the test instance, or if the weights are random and dependent on X, the saliency map and the selection event are not exogenous and the stated guarantee does not follow. The method is valid for a fixed, pre-trained GNN used on an independent test instance. The sentence 'Our method guarantees control of the Type I error rate even when the GNN is trained on ill-conditioned data' should be qualified accordingly.
- [Section 4.3, Lemma 1] There is a dimension mismatch in Lemma 1. The lemma states that S_i(G_X) = C_i^(k) X + b_i^(k) with C_i^(k) in R^{n x nd} and b_i^(k) in R^n. Since S_i(G_X) is a scalar for each node i, C_i^(k) should be a row vector in R^{1 x nd} and b_i^(k) a scalar in R. As written, the expression C_i^(k) X + b_i^(k) would be n-dimensional, which contradicts the scalar nature of the saliency score. This is a local typo, but it occurs in a formally stated lemma and should be corrected.
minor comments (4)
- [Section 3.2, Eq. (7)] The paper first defines T(X) as the raw mean difference in Eq. (6) and then redefines it as a normalized version in Eq. (7) 'without loss of generality'. Please state explicitly that the statistical test and the selective p-value are defined for the normalized statistic, and clarify that eta in Eq. (7) is the same eta defined in Appendix B.
- [Appendix D.1, Eqs. (17)-(18)] The interval formulas divide by beta_i, but the beta_i = 0 case is not discussed. When the saliency score is constant in z for some node, the threshold inequality imposes either no constraint or an infeasible constraint, and the corresponding interval branch should be specified. Please add a sentence handling zero-slope cases.
- [Appendix E.2 and footnote 3] The exact validity statement in Theorem 1 assumes a known covariance Sigma. In Appendix E.2, Sigma is estimated from data and the Type I error is evaluated only empirically. The main text should explicitly note that the theoretical guarantee does not extend to the estimated-covariance experiment, which is a robustness check rather than an exact result.
- [Throughout] There are several typographical and minor language slips, for example 'the propery in(8)', 'satistifies', and 'wchich'. A careful proofreading pass is recommended before publication.
Circularity Check
No significant circularity: the selective p-value derivation is a self-contained application of standard selective inference, and the cited parametric-programming work is computational support rather than a load-bearing self-citation.
full rationale
The paper's central derivation (selective p-value in Eq. 13, conditional data space in Eq. 14, and truncation intervals in Lemma 2) is built directly on the standard selective-inference conditioning argument: after conditioning on the selected subgraph V_X and the nuisance sufficient statistic Q_X, the test statistic is a truncated normal under the conditional null, and the p-value is uniform by probability integral transform. No fitted parameter is renamed as a prediction; thresholds are user-set, the GNN weights are fixed inputs, and the null distribution is constructed from the model X = mu + epsilon with known Sigma. The self-citations to Duy & Takeuchi (2022) and related SI work are used for efficient computation of the truncation intervals, and the paper supplies its own proof of Lemma 2 plus an explicit interval-identification procedure in Appendix D; nothing in the validity proof depends on an unverified uniqueness theorem or on an ansatz imported from the authors' prior work. The unconditional phrasing of Theorem 1 is statistically debatable because H0 is data-dependent and marginalizing over V_X requires care, but that is a correctness or interpretation concern rather than circularity; the conditional SI guarantee is the standard, self-contained result. The paper also discloses the known-Sigma limitation and the need for a fixed GNN, so its own stated assumptions do not smuggle in the conclusion.
Assumptions & free parameters
free parameters (3)
- thresholds tau_u and tau_l =
0.7 and 0.3 in experiments
- estimated covariance matrix Sigma =
sample covariance from 520 negative EEG samples per participant
- GNN weights =
trained on 15 EEG participants in real data; unspecified in synthetic experiments
assumptions (4)
- domain assumption Feature vector X follows X = mu + epsilon with epsilon ~ N(0, Sigma), Sigma known (Eq. 4)
- domain assumption The GNN is fixed and independent of the test data X
- domain assumption The saliency map is a piecewise linear function of X (Lemma 1)
- standard math Conditioning on the selection event and the sufficient statistic yields a truncated normal null distribution
Cite this review
Pith. "Pith review of Statistical Test for Saliency Maps of Graph Neural Networks via Selective Inference." pith.science (2026). https://pith.science/paper/3JMBNJHL
@misc{pith2026250516893,
author = {Pith},
title = {Pith review of: Statistical Test for Saliency Maps of Graph Neural Networks via Selective Inference},
year = {2026},
howpublished = {\url{https://pith.science/paper/3JMBNJHL}},
note = {Machine review of arXiv:2505.16893}
}
abstract
Graph Neural Networks (GNNs) have gained prominence for their ability to process graph-structured data across various domains. However, interpreting GNN decisions remains a significant challenge, leading to the adoption of saliency maps for identifying salient subgraphs composed of influential nodes and edges. Despite their utility, the reliability of GNN saliency maps has been questioned, particularly in terms of their robustness to input noise. In this study, we propose a statistical testing framework to rigorously evaluate the significance of saliency maps. Our main contribution lies in addressing the inflation of the Type I error rate caused by double-dipping of data, leveraging the framework of Selective Inference. Our method provides statistically valid $p$-values while controlling the Type I error rate, ensuring that identified salient subgraphs contain meaningful information rather than random artifacts. The method is applicable to a variety of saliency methods with piecewise linearity (e.g., Class Activation Mapping). We validate our method on synthetic and real-world datasets, demonstrating its capability in assessing the reliability of GNN interpretations.
Figures
Figures from the paper (9 more)
Reference graph
Works this paper leans on
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[1]
was ∆ for each distribution family, with∆∈{ 0.01, 0.05, 0.1, 0.15}. Other settings were the same as the default settings in the Type I error rate evaluation in Section 6.2. The results are shown in Figure 6 demonstrating that our proposed method can effectively control the Type I error rate for non-Gaussian noise. 0.01 0.05 0.1 0.15 1-Wasserstein Distance...
work page 2022
-
[3]
Statistical Test for Diffusion-Based Anomaly Localization via Selective Inference
Teruyuki Katsuoka, Tomohiro Shiraishi, Daiki Miwa, Vo Nguyen Le Duy, and Ichiro Takeuchi. Statistical test on diffusion model-based anomaly detection by selective inference.arXiv preprint arXiv:2402.11789,
-
[9]
Daiki Miwa, Tomohiro Shiraishi, Vo Nguyen Le Duy, Teruyuki Katsuoka, and Ichiro Takeuchi. Statistical test for anomaly detections by variational auto-encoders.arXiv preprint arXiv:2402.03724,
-
[10]
Tomohiro Shiraishi, Tatsuya Matsukawa, Shuichi Nishino, and Ichiro Takeuchi. Statistical test for feature selection pipelines by selective inference.arXiv preprint arXiv:2406.18902, 2024a. Tomohiro Shiraishi, Daiki Miwa, Teruyuki Katsuoka, Vo Nguyen Le Duy, Kouichi Taji, and Ichiro Takeuchi. Statistical test for attention maps in vision transformers. In F...
-
[11]
Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps.arXiv preprint arXiv:1312.6034,
-
[1986]
Graph Neural Network Explanations are Fragile
Jiate Li, Meng Pang, Yun Dong, Jinyuan Jia, and Binghui Wang. Graph neural network explanations are fragile. arXiv preprint arXiv:2406.03193, 2024a. Zhong Li, Simon Geisler, Yuhang Wang, Stephan Günnemann, and Matthijs van Leeuwen. Explainable graph neural networks under fire.arXiv preprint arXiv:2406.06417, 2024b. Xuefen Lin, Jielin Chen, Weifeng Ma, Wei...
-
[2014]
Jelena Markovic, Lucy Xia, and Jonathan Taylor. Unifying approach to selective inference with applications to cross-validation.arXiv preprint arXiv:1703.06559,
-
[2017]
Statistical Test for Auto Feature Engineering by Selective Inference
Tatsuya Matsukawa, Tomohiro Shiraishi, Shuichi Nishino, Teruyuki Katsuoka, and Ichiro Takeuchi. Statis- tical test for auto feature engineering by selective inference.arXiv preprint arXiv:2410.19768,
Show all 12 references
-
[2021]
Graphsvx: Shapley value explanations for graph neural networks
Alexandre Duval and Fragkiskos D Malliaros. Graphsvx: Shapley value explanations for graph neural networks. In Machine Learning and Knowledge Discovery in Databases. Research Track: European Con- ference, ECML PKDD 2021, Bilbao, Spain, September 13–17, 2021, Proceedings, Part ...
2021
-
[2022]
Optimal inference after model selection.arXiv preprint arXiv:1410.2597,
William Fithian, Dennis Sun, and Jonathan Taylor. Optimal inference after model selection.arXiv preprint arXiv:1410.2597,
-
[2023]
A significance test for forward stepwise model selection.arXiv preprint arXiv:1405.3920,
15 Joshua R Loftus and Jonathan E Taylor. A significance test for forward stepwise model selection.arXiv preprint arXiv:1405.3920,
-
[2024]
si4onnx: A python package for selective inference in deep learning models.arXiv preprint arXiv:2501.17415,
Teruyuki Katsuoka, Tomohiro Shiraishi, Daiki Miwa, Shuichi Nishino, and Ichiro Takeuchi. si4onnx: A python package for selective inference in deep learning models.arXiv preprint arXiv:2501.17415,
Reviewed August 7, 2026 · model on record in the stance chip above.
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