{"id":"6a735bea-1dc3-4be9-96d5-ac569e7f36e4","arxiv_id":"2412.14509","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Averaging saliency values within super-pixel groups reduces the variance and improves the stability and generalizability of gradient-based interpretation maps.","lead":"This paper proposes grouping image pixels into super-pixels before computing gradient-based saliency maps, which reduces randomness in the maps and improves their stability and generalization. The method is simple, works with several interpretation techniques, and could make AI explanations more reliable for scientific discovery and medical imaging.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1 does not follow from Proposition 3 and Theorem 1: a smaller stability constant for g-SG only gives a smaller upper bound on epsilon_gen, not a smaller actual epsilon_gen than SG's.","rationale":"The reader's weakest_assumption focused on the semantic homogeneity of super-pixels, which is an empirical/fidelity concern. While that concern is legitimate, the more load-bearing issue is internal to the theoretical argument: Corollary 1 does not logically follow from the stated propositions. The central claim of the paper is that grouping pixels provably reduces the generalization error of saliency maps. If the proof of this corollary is invalid, the theoretical contribution collapses, even if the empirical results remain suggestive. The paper could still be conditionally accepted after either correcting the corollary to a true statement (e.g., 'the stability-based upper bound on epsilon_gen is not increased') or providing a valid proof. The empirical results on SSIM, MeGe, fidelity, and ROAD/ROAR are valuable and generally consistent with the practical recommendation, but they do not rescue an unsound theoretical claim. Therefore, the verdict remains CONDITIONAL, but for a different and more fundamental reason than the reader identified.","tokens_in":10294,"tokens_out":5727,"duration_ms":52843,"concrete_test":"Inspect the appendix proof of Corollary 1: if it derives epsilon_gen(g-SG) <= epsilon' and epsilon' <= epsilon but never proves epsilon_gen(g-SG) <= epsilon_gen(SG) directly, the corollary is a non-sequitur. Additionally, construct a minimal counterexample satisfying the paper's assumptions—e.g., a two-pixel linear model with population gradient (1,0) and a small training-induced perturbation—and compute epsilon_gen(SG) and epsilon_gen(g-SG) numerically. If any such setting yields epsilon_gen(g-SG) > epsilon_gen(SG) while the stability constants are ordered epsilon' <= epsilon, the corollary is false as a mathematical statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central theoretical claim, Corollary 1, is invalid as stated. Proposition 3 shows that if SG is epsilon-uniformly stable, then g-SG is epsilon'-uniformly stable for some epsilon' <= epsilon. Theorem 1 states that any epsilon-uniformly stable interpretation scheme satisfies epsilon_gen(I) <= epsilon. Combining these yields only epsilon_gen(g-SG) <= epsilon' <= epsilon and epsilon_gen(SG) <= epsilon. This does not imply epsilon_gen(g-SG) <= epsilon_gen(SG): both quantities share an upper bound, but the upper bound on SG is not a lower bound on its actual error. The corollary commits the fallacy of bounding a bound. To establish the claimed reduction, the authors would need either a lower bound on epsilon_gen(SG) in terms of its stability constant, or a direct argument comparing the two generalization errors (e.g., via the bias-variance decomposition of the grouped saliency). No such argument appears in the main text, and the appendix proof is not provided in this preprint. The empirical SSIM/MeGe improvements are consistent with the weaker statement that the stability-based worst-case guarantee improves, but they do not establish the stronger corollary. Since this corollary is the stated theoretical basis for 'the grouping of pixels reduces the generalization error,' the central theoretical contribution is unsupported as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a super-pixel-based grouping strategy to stabilize gradient-based saliency maps. The authors define an interpretation generalization error, prove a stability-based bound (Theorem 1), and claim that grouping pixels into super-pixels reduces this error (Corollary 1). They support the claim with Propositions 1–3 on the variance-reducing effect of averaging within groups, and they report experiments on CIFAR-10 and ImageNet measuring SSIM, MeGe, fidelity, and interpretability, which show consistent improvements over pixel-based maps.","tokens_in":10539,"tokens_out":10570,"duration_ms":90866,"significance":"If the theoretical claim were valid, the paper would contribute a simple, computationally inexpensive, and broadly applicable method for producing more stable and generalizable saliency maps, backed by a formal guarantee. The empirical study is extensive, includes multiple interpretation methods and metrics, and makes the code available. The averaging/grouping insight is natural and the propositions on variance reduction via Jensen's inequality are correct. However, the central theoretical corollary does not follow from the stated results, and the proof of the main theorem is not included in the preprint. As written, the paper establishes an improvement in the stability bound, not a reduction of the actual generalization error, which materially weakens the theoretical contribution.","major_comments":[{"comment":"The deduction 'εgen(g-SG) ≤ εgen(SG)' does not follow from Proposition 3 and Theorem 1. Proposition 3 states that if SG is ε-uniformly stable, then g-SG is ε'-uniformly stable for some ε' ≤ ε. Theorem 1 gives εgen(SG) ≤ ε and εgen(g-SG) ≤ ε'. These inequalities imply that both generalization errors are at most ε, with g-SG having a tighter upper bound, but they do not imply that the actual value of εgen(g-SG) is no larger than the actual value of εgen(SG). This is the fallacy of bounding a bound. To establish the corollary, the authors would need either a lower bound on εgen(SG) in terms of its stability constant or a direct comparison of the two generalization errors, for example via the bias-variance decomposition expressed in Proposition 2. Since Corollary 1 is the stated theoretical basis for the central claim that 'the grouping of pixels reduces the generalization error,' this is a load-bearing gap.","section":"Section 3.2, Corollary 1"},{"comment":"Theorem 1 is stated as a general bound 'εgen(I) ≤ ε' without specifying the conditions under which this stability-based generalization bound holds. In the standard algorithmic-stability framework, such bounds require assumptions such as symmetry of the algorithm and, for the absolute-value form used in Definition 2, careful handling of the expectation. The proof is deferred to an appendix that is not included in this preprint, so the correctness of the theorem cannot be verified from the manuscript. Because Theorem 1 is the foundation for the corollary, the authors should either provide the proof in the main text or appendix and state the precise assumptions (e.g., boundedness of the interpretation loss, if needed).","section":"Section 3.1, Theorem 1"}],"minor_comments":[{"comment":"The formula 'g-SG( f̂θ ,x) := AT ∇g f̂θ (x + A Wg)' appears to have a dimension error: A is d×p, so A^T is p×d, and A^T ∇g is not a well-defined d-dimensional vector. The subsequent simplification to κ(x)∇x f suggests that the intended broadcasting operator is A, not A^T (i.e., g-SG = A ∇g f = A W A^T ∇x f). Please correct this definition.","section":"Section 3.2, Definition of g-SG"},{"comment":"The sentence 'We let g = 0p to be the p-dimensional zero vector' is confusing because g is later used as a variable in the gradient. It would be clearer to say that g is a p-dimensional group-level input variable, and the gradient is evaluated at g = 0.","section":"Section 3.2, Notation"},{"comment":"The assumption that pixels within a super-pixel have essentially the same effect on the prediction is stated qualitatively and is crucial for the fidelity of the resulting maps. The main text defers validation to the appendix; a direct measurement of within-super-pixel gradient variance, or a comparison of grouped maps against ground-truth object boundaries, would strengthen the justification.","section":"Section 3.3, Super-pixel assumption"},{"comment":"The abbreviations 'MeGe', 'pixel', 'super', 'sup.', and 'rand' are used in figures and tables without full expansion in the captions; please define them for the reader.","section":"Figure 2 and Table 1"}],"recommendation":"major_revision","confidential_remarks":"The central theoretical claim in Corollary 1 is logically unsupported as written, which is a serious issue for a paper whose headline contribution is theoretical. However, the empirical results and the simpler stability-bound statement (ε' ≤ ε) provide a credible and publishable core if the authors revise the theoretical claim accordingly. The missing appendix with proofs of Theorem 1 and the corollary should also be supplied. I would not recommend rejection because the error is fixable within the manuscript's scope by weakening the corollary to a statement about stability bounds or by providing a genuine direct comparison of generalization errors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the method works empirically, and the stability framing is a reasonable way to think about the problem, but the paper's headline theoretical claim—Corollary 1—doesn't follow from their own results. That gap is real and load-bearing, because the corollary is what the title promises.\n\nThe idea is simple and useful: partition the image into super-pixels, average the gradient-based saliency within each group, and you get maps that are more stable across training runs. The experiments on CIFAR-10 and ImageNet back this up across several gradient methods, and the fidelity trade-off is modest. The comparison against square patches and random partitions is a nice control. Code is released. That is a solid empirical contribution.\n\nThe theory is where it falls down. Propositions 1 and 2 are fine applications of Jensen and variance decomposition. Proposition 3 presumably holds, since averaging with an orthogonal projection is non-expansive. But Corollary 1 is not a consequence of Proposition 3 plus Theorem 1. Theorem 1 gives an upper bound on generalization error in terms of the stability constant. A smaller stability constant for g-SG gives a smaller upper bound, not a smaller actual error. To conclude ε_gen(g-SG) ≤ ε_gen(SG), you'd need either a lower bound on SG's error or a direct bias–variance argument. Neither is in the main text, and the appendix is not in this preprint. The pointwise domination in Proposition 1 doesn't save it either, because the absolute gap between population and empirical means can behave differently. So the central theoretical claim is unsupported as written.\n\nThere is also the load-bearing assumption that pixels within a super-pixel share the same true importance. That is fine when super-pixels respect object boundaries, but it will blur real detail when they do not. The empirically chosen sizes (1000–2000 groups) seem to keep this in check, but it is a limitation worth stating more sharply.\n\nWho is this for? Anyone working on interpretability stability or applying saliency maps to high-stakes domains. The method is cheap, easy to implement, and likely to be useful even if the theory is reframed as “the worst-case bound improves” rather than “the generalization error decreases.”\n\nRecommendation: send to peer review. A good referee can catch the corollary issue, and the empirical contribution plus the correct propositions are worth publishing. I would suggest the authors either prove a lower bound for SG's error or explicitly reframe the claim as a stability improvement with a tighter bound.","headline":"Useful empirical method, but the main theoretical claim about reducing generalization error does not follow from the paper's own stability bound.","tokens_in":11054,"tokens_out":3204,"would_cite":true,"duration_ms":27281,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Averaging gradient-based saliency scores within super-pixels reduces the run-to-run variance and generalization error of neural-network explanations, with little or no loss in fidelity.","keywords":["saliency map","super-pixel","interpretation stability","algorithmic stability","generalization error","gradient-based interpretation","model interpretability"],"falsifier":"Train one architecture on two disjoint halves of a dataset and, for images where thin or high-frequency structures deliberately cross super-pixel boundaries, compute both pixel and super-pixel saliency maps. If the super-pixel map's MeGe generalization error does not fall below the pixel map's, or if its deletion curve removes little predicted class probability until regions far from the true object are erased, the within-group homogeneity assumption is violated and the variance-reduction claim fails for that partition.","tokens_in":10090,"feed_emoji":"🧩","tokens_out":7706,"duration_ms":58008,"temperature":0.7,"pith_summary":"Saliency maps produced by gradient-based interpretability methods change noticeably when a neural network is retrained on different data or with different random seeds, making them unreliable for understanding what a model is actually looking at. This paper argues that much of that instability can be removed by a simple preprocessing step: partition the image into super-pixels (perceptually homogeneous regions) and replace each pixel's importance score with the average score over its super-pixel. The paper proves that this grouping cannot increase the interpretation generalization error, defined as the expected gap between the saliency map computed on finite training data and the saliency map of the population-optimal network, and it demonstrates on CIFAR-10 and ImageNet that super-pixel maps are more stable across networks, more reproducible under data removal, and equally or more interpretable, with only small and controllable losses in fidelity. The reason the trick works is that averaging inside each group suppresses the high-dimensional gradient noise that stochastic training introduces, while the semantic alignment of super-pixels preserves the signal a human would want the explanation to carry.","feed_headline":"Super-pixel averaging tames noisy saliency maps","feed_subtitle":"Averaging gradients within super-pixels reduces run-to-run variance in explanations while preserving fidelity.","key_machinery":"The central object is the grouped-gradient operator g-SG( f, x) = κ(x)∇x f (x), where κ(x) is a d×d matrix that replaces the gradient at each pixel by the mean gradient over its super-pixel (κ(x)_{i,j} = 1/|S_k| when pixels i and j lie in the same group S_k, and 0 otherwise). This operator carries the argument: it is the device that converts a pixel-level explanation into a region-level one without retraining the network. The theoretical machinery is the algorithmic-stability framework: Theorem 1 converts ε-uniform stability of an interpretation scheme into a bound on its interpretation generalization error, Proposition 3 shows grouping preserves or improves the stability constant, and Proposition 2 quantifies the variance that the averaging removes. The super-pixel algorithms (SLIC, Quickshift, graph-based methods) supply the partition; they matter because the variance-reduction benefit is exactly the within-group variance of the gradient difference, which is small when groups are semantically homogeneous.","core_discovery":"The central claim is that for any neural network classifier, replacing the pixel-wise Simple Gradient map with a grouped map built from super-pixels reduces the expected deviation of the empirical explanation from the population explanation. Formally, the paper defines the interpretation loss as the L2 distance between the empirical map and the map of the population-trained network, and the interpretation generalization error as the expected gap between that loss on test data and on training data. It proves, using Jensen's inequality, that the grouped loss is pointwise no larger than the pixel-wise loss (Proposition 1), and that the difference equals the sum, over super-pixels, of the super-pixel size times the within-group variance of the gradient difference (Proposition 2). Extending the algorithmic-stability framework to explanations, it shows that if the pixel-wise map is ε-uniformly stable, the grouped map is stable with a no-worse constant (Proposition 3), and therefore εgen(g-SG) ≤ εgen(SG) (Corollary 1). The paper reads this as variance reduction by averaging: grouping lowers the effective dimension of the input, and the noise that stochastic training leaves in the gradient is averaged out within semantically meaningful regions.","pith_inferences":["The same averaging argument should apply to any high-dimensional input with a known grouping structure — audio frequency bands, gene modules, or time windows — as long as the groups align with the true decision-relevant features, suggesting a generic recipe for stabilizing explanations beyond images.","The paper's theory predicts the stability gain is exactly the within-group variance of the gradient difference (Proposition 2). A direct test of the mechanism would be to measure that variance at object boundaries and verify that the SSIM improvement tracks its magnitude; if the improvement persists even where the variance is high, the explanation would need to be revised.","An adaptive partition — small super-pixels near boundaries and large ones in uniform regions — is a natural next step that could push the fidelity–stability frontier beyond the fixed-size partitions used here."],"forward_implications":["Any gradient-based saliency method (Simple Gradient, Integrated Gradients, SmoothGrad, Sparsified SmoothGrad) can be made more reproducible across training runs by averaging the map within super-pixels, at the cost of a tunable and usually small fidelity loss.","Small super-pixels (about 1000–2000 groups on ImageNet) give most of the stability gain with negligible or even positive fidelity changes; overly large super-pixels degrade both stability and fidelity.","The grouping strategy is orthogonal to SmoothGrad-style noise averaging, so the two mechanisms can be combined to push stability further.","Because the partition is static and semantic, the method is computationally cheap relative to region-based approaches such as RISE, XRAI, and Score-CAM, and it can be applied to any gradient-based interpretation scheme."],"supporting_citations":[{"why":"Supplies the algorithmic-stability framework used to define ε-uniform stability and to derive the generalization bound in Theorem 1.","marker":"[5]"},{"why":"Provides the MeGe metric used to measure the generalizability of interpretation maps in the experiments.","marker":"[7]"},{"why":"Supplies the directional-convergence theory invoked to justify the existence of a unique population saliency map.","marker":"[16]"},{"why":"Introduces SmoothGrad, the noise-averaging baseline that the paper's grouping strategy is compared against and combined with.","marker":"[31]"},{"why":"Documents the initialization sensitivity of saliency maps, the core instability the paper addresses.","marker":"[38]"},{"why":"Provides the SLIC super-pixel algorithm used to generate the partitions in the CIFAR-10 experiments.","marker":"[1]"},{"why":"Provides the Quickshift super-pixel algorithm used in the ImageNet experiments.","marker":"[35]"},{"why":"Provides the deletion and insertion fidelity metrics used to evaluate the trade-off.","marker":"[26]"},{"why":"Provides the µFidelity metric used to quantify explanation fidelity.","marker":"[3]"},{"why":"Provides the ROAR metric used to measure interpretability via retraining.","marker":"[15]"}],"fun_headline_variants":["Super-pixel grouping stabilizes neural net saliency maps","Grouping pixels cuts variance in gradient explanations","Stable interpretation via super-pixel averaging","Super-pixels reduce noise in saliency maps","Averaging gradients within super-pixels stabilizes explanations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the assumption that every pixel inside a super-pixel has the same true importance to the network's decision; if a super-pixel crosses an object boundary or contains mixed textures, averaging will blur the very detail the explanation is meant to show.","fun_headline_variants_meta":{"raw":{"variants":["Super-pixel grouping stabilizes neural net saliency maps","Grouping pixels cuts variance in gradient explanations","Stable interpretation via super-pixel averaging","Super-pixels reduce noise in saliency maps","Averaging gradients within super-pixels stabilizes explanations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1250,"prompt_tokens":941,"completion_tokens":309,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":236}},"tokens_in":557,"tokens_out":309,"duration_ms":3218,"temperature":1.0,"reasoning_tokens":236,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:10:01.696575+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train one architecture on two disjoint halves of a dataset and, for images where thin or high-frequency structures deliberately cross super-pixel boundaries, compute both pixel and super-pixel saliency maps. If the super-pixel map's MeGe generalization error does not fall below the pixel map's, or if its deletion curve removes little predicted class probability until regions far from the true object are erased, the within-group homogeneity assumption is violated and the variance-reduction claim fails for that partition.","supporting_citations":[{"cited_title":"Stability and generalization","cited_arxiv_id":null,"evidence_quote":"Supplies the algorithmic-stability framework used to define ε-uniform stability and to derive the generalization bound in Theorem 1."},{"cited_title":"How good is your ex- planation? algorithmic stability measures to assess the quality of explanations for deep neural networks","cited_arxiv_id":null,"evidence_quote":"Provides the MeGe metric used to measure the generalizability of interpretation maps in the experiments."},{"cited_title":"Directional convergence and alignment in deep learning","cited_arxiv_id":null,"evidence_quote":"Supplies the directional-convergence theory invoked to justify the existence of a unique population saliency map."},{"cited_title":"Initialization noise in image gradients and saliency maps","cited_arxiv_id":null,"evidence_quote":"Documents the initialization sensitivity of saliency maps, the core instability the paper addresses."},{"cited_title":"Slic superpixels compared to state-of-the-art superpixel methods","cited_arxiv_id":null,"evidence_quote":"Provides the SLIC super-pixel algorithm used to generate the partitions in the CIFAR-10 experiments."},{"cited_title":"Quick shift and kernel methods for mode seek- ing","cited_arxiv_id":null,"evidence_quote":"Provides the Quickshift super-pixel algorithm used in the ImageNet experiments."}],"review_version":1}