{"id":"aad87c52-4eca-49a0-8a04-c01d61ec5546","arxiv_id":"2501.06261","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new framework derives GradCAM and HiResCAM from Shapley values via Taylor expansion and introduces ShapleyCAM, a Hessian-based explanation method.","lead":"This paper introduces a game-theoretic framework, the Content Reserved Game-theoretic (CRG) Explainer, that recasts GradCAM, HiResCAM, and a new method called ShapleyCAM as Shapley value computations under Taylor-expanded utility functions. It also proposes a ReST utility function that combines pre- and post-softmax scores, with ImageNet experiments across 12 networks showing small gains on most metrics.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'exact Shapley value' claim is exact only for a second-order Taylor surrogate, whose fidelity across the coalition space is unvalidated and empirically fails on self-attention layers; this is the load-bearing assumption.","rationale":"The reader correctly identifies the Taylor approximation as the weakest assumption; I agree. The paper's theoretical derivation is internally consistent: Eq. 14 follows from the quadratic surrogate, and the Shapley value of a quadratic game is exactly the gradient term plus half the interaction terms. The issue is not internal inconsistency but external validity: the surrogate game may be a poor proxy for the actual network behavior over the subset distribution. This is load-bearing because the paper's contribution is framed as bridging heuristic CAMs and Shapley-based methods—if the quadratic surrogate is inaccurate, ShapleyCAM is just another heuristic tuned by a Hessian-vector product. The empirical evidence is too weak to rescue the claim: differences in ADCC/IC/ADD between ShapleyCAM and GradCAM are on the order of 0.1 points (Tables 1 and 3), no significance testing is reported, and on Swin Transformers the method does not clearly beat RandomCAM (Section 4.4). The proposed test directly measures the approximation error in the quantities that Shapley values average, so it would settle the concern. Since this concern matches the reader's weakest assumption and the conditional verdict already reflects it, no verdict change is needed.","tokens_in":24019,"tokens_out":16646,"duration_ms":144567,"concrete_test":"On a small sample (e.g., 50 images) from ImageNet, use ResNet-18 with the target layer preceding GAP. Sample 1,000 random coalitions S (drawn from the Shapley permutation distribution on the d spatial positions) and, for each S and a random player j, compute the true marginal contribution Δ_true = UReST(X_{S∪{j}}) − UReST(X_S) and the Taylor-approximated marginal Δ_approx from the quadratic surrogate in Theorem 2. Compute the Spearman correlation (or R²) between Δ_true and Δ_approx. If this correlation is low (<0.5) or no better than a first-order baseline, the second-order Taylor approximation is not faithful across coalitions, directly undermining the claim that ShapleyCAM computes exact Shapley values of a meaningful game.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical result (Theorem 2, Eq. 14) is a correct closed-form Shapley value for the quadratic game obtained by a second-order Taylor expansion of U(X_S) centered at the full activation tensor X_D. However, the Shapley value averages marginal contributions over all coalitions S⊆D, and for typical S, X_S is far from X_D (especially small coalitions, where most entries are zeroed). The paper provides no bound on the Taylor remainder, no diagnostic of approximation quality, and Section 4.4 acknowledges that for self-attention layers in Swin Transformers the approximation fails—most CAM methods, including ShapleyCAM, do not beat RandomCAM. For ReLU networks, the logits are piecewise linear in the target-layer activations, so the Hessian is zero almost everywhere; the only nonzero second-order contribution comes from the softmax/ReST nonlinearity, which is not representative of the global curvature of U over the subset space. Thus the main load-bearing assumption—that the second-order Taylor polynomial accurately represents U(X_S) for all S in the Shapley expectation—is unverified and likely false in general. The paper's own experiments (Tables 2 and 4) show the method's performance degrades markedly when this assumption is violated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Content Reserved Game-theoretic (CRG) Explainer, a cooperative-game framework in which each pixel of the downsampled explanation map is a player whose utility is the network output as a function of the target-layer activations. Within this framework, the paper proves that HiResCAM and GradCAM are first-order Taylor approximations of Shapley values, and it proposes ShapleyCAM, ShapleyCAM-H, and ShapleyCAM-E, which use a second-order Taylor expansion to obtain closed-form weights involving the gradient and Hessian of a proposed ReST utility function. The paper also gives a theoretical relation between pre-softmax and post-softmax GradCAM heatmaps and evaluates the methods on 12 ImageNet backbones with 6 metrics. The central closed-form Shapley derivation in Theorem 2 is mathematically correct for the quadratic surrogate, but the paper does not validate the fidelity of that surrogate across the coalition space, and the empirical margins over existing CAM methods are generally very small.","tokens_in":24270,"tokens_out":13259,"duration_ms":128101,"significance":"If the Taylor-surrogate assumption held uniformly over all coalitions, the paper would provide a valuable bridge between heuristic CAM methods and Shapley-based attribution: a one-extra-backward-pass method that produces exact Shapley values for an explicit quadratic approximation. The paper's main strengths are its identification of GradCAM and HiResCAM as first-order Shapley approximations, the correct closed-form second-order correction in Eq. (14), the ReST utility analysis, the full-ImageNet evaluation, and the public code release. However, the significance is currently limited by three concerns: the exact-Shapley claim is conditional on an unvalidated Taylor expansion, the empirical advantages over GradCAM are often fractions of a point in ADCC, and the proof of Theorem 3 contains an internal sign inconsistency. These issues are fixable but require additional analysis and experiments.","major_comments":[{"comment":"The chain-rule identity in the proof of Theorem 3 is stated with the wrong sign. The correct identity is ∂p_c/∂A_i = p_c Σ_{k≠c} p_k (∂y_c/∂A_i − ∂y_k/∂A_i), not p_c Σ_{k≠c} p_k (∂y_k/∂A_i − ∂y_c/∂A_i). The subsequent displayed equation g(∂p_c/∂A_i) = p_c Σ_{k≠c} p_k (g(∂y_k/∂A_i) − g(∂y_c/∂A_i)) inherits this sign error, yet Eq. (15) uses Epre_c − Epre_k, which would follow from the correct sign. As written, the derivation is internally inconsistent: the final formula is correct only after a second, unstated sign change. Because ReST is motivated by this relationship, the proof must be corrected before publication.","section":"Section 3.4, proof of Theorem 3 and Eq. (15)"},{"comment":"The claim that ShapleyCAM and ShapleyCAM-H 'consistently outperform' other CAM methods in ADCC, IC, and ADD is based on point estimates whose differences are very small. For example, in Table 1 ResNet-18, ShapleyCAM versus GradCAM gives ADCC 77.22 versus 77.10, IC 33.92 versus 33.77, and ADD 51.90 versus 51.89; in Table 3 ResNet-18 last-convolutional-layer, ShapleyCAM-H versus HiResCAM gives ADCC 77.21 versus 77.12, IC 30.50 versus 30.48, and ADD 49.37 versus 49.39. No confidence intervals, repeated runs, or significance tests are reported anywhere in the paper. Given the magnitude of these differences, the empirical support for the central claim of improved explainability is not yet established and needs a more rigorous reporting of variability or statistical testing.","section":"Tables 1 and 3, Section 4.4"},{"comment":"The exact-Shapley interpretation of ShapleyCAM is conditional on a second-order Taylor surrogate for U(X_S), but the paper provides no diagnostic that this surrogate is accurate across the coalition distribution. For typical small coalitions, X_S is far from the expansion point X_D, and the paper gives no bound on the Taylor remainder and no comparison against exact or Monte Carlo Shapley values on any configuration. The load-bearing nature of this assumption is visible in the paper's own Table 4: on the first normalization layer of Swin Transformer blocks, ShapleyCAM-H achieves ADCC 39.48 versus RandomCAM 49.21 for Swin-S, and most CAM methods fail to beat RandomCAM; Section 4.4 attributes this to the polynomial approximation of self-attention but leaves the hypothesis untested. A concrete test, such as computing Monte Carlo Shapley values for a reduced target layer and reporting correlation or error against ShapleyCAM weights, is needed to support the central claim.","section":"Sections 3.2–3.3 and 4.4, Eq. (14), Table 4"},{"comment":"The ReST ablation does not support the statement that ReST 'outperforms pre-softmax and post-softmax scores, except in Complexity.' Under the paper's primary composite metric ADCC, pre-softmax achieves 77.80 while ReST achieves 77.22; ReST is better on AD, Coh, IC, and ADD, but worse on Complexity and, overall, worse on ADCC. The text as written is misleading because the 'except in Complexity' exception is precisely what makes ReST lose the primary metric. The claim should be qualified, or the choice of primary metric should be defended in light of this result.","section":"Section 4.6, Table 5"}],"minor_comments":[{"comment":"The text contains a typo: 'CIF AR-10' should be 'CIFAR-10'.","section":"Section 2.2"},{"comment":"The word 'proccess' should be 'process'.","section":"Section 5"},{"comment":"The notation for row and column vectors is inconsistent: U'(X_D) is written both as a row vector (e.g., U'(X_D)X_j) and as a quantity involving transposes (e.g., X_D^T H_D). Defining all vectors as column vectors and using explicit transposes throughout would remove ambiguity.","section":"Equations (6) and (10)–(14)"},{"comment":"The justification for placing ReLU outside the summation relies on 'Shapley values greater than 0 indicating positive contribution,' but this does not by itself imply that negative Shapley values should be discarded; the paper should acknowledge that clipping is a visualization choice that removes potentially informative negative attributions.","section":"Section 3.5"}],"recommendation":"major_revision","confidential_remarks":"The paper's central derivation in Theorem 2 is sound and the code release is a strength. However, the proof of Theorem 3 contains a sign error, the empirical improvements are often within noise, and the Taylor-surrogate assumption is not tested against exact Shapley values. These issues are addressable in a major revision; I would not reject the paper, but I do not think the current evidence supports the advertised 'exact Shapley value' claim without a validation of the surrogate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The theoretical core is genuinely new and, as far as I can tell, correct. The paper shows that under a first-order Taylor approximation of the utility, the Shapley value per pixel reduces to the HiResCAM weighting (gradient times activation), and GradCAM is the channel-averaged version; under a second-order expansion you get the closed form W = grad - 1/2 X_D^T H_D. That is a clean result and I checked the algebra in Theorem 2 — it works. The CRG Explainer taxonomy (Type-I vs Type-II) is a useful way to organize CAM variants, and Theorem 3 (post-softmax GradCAM is a probability-weighted ensemble of pre-softmax heatmaps) is elegant. The ReST utility, y_c + ln softmax(y)_c, makes practical sense as a compromise between vanishing gradients of post-softmax and irrelevant highlights of pre-softmax.\n\nThe soft spots are in the load-bearing assumption and the empirics. The 'exact Shapley value' claim is exact only for the quadratic surrogate obtained by the Taylor expansion, not for the network itself. The Shapley expectation averages over coalitions of all sizes, and for small coalitions the masked activation vector X_S is far from the expansion point X_D. The paper gives no bound on the Taylor remainder, no diagnostic of approximation quality, and its own Swin Transformer results (Section 4.4) show the approximation collapses when the target layer is a self-attention layer — most methods, including ShapleyCAM, fall to RandomCAM levels. I think that is a real weakness, not a nitpick. The fact that for ReLU networks the logits are piecewise linear almost everywhere means the only Hessian contribution comes from the softmax/ReST nonlinearity, which may not represent the actual curvature over the subset space. That said, the first-order result for GradCAM/HiResCAM is exact for the pre-ReLU-to-logit map when no nonlinearity intervenes, so the framework is not empty.\n\nThe experiments are thorough (full ImageNet val, 12 architectures, 6 metrics) but the reported advantages are small — often 0.1-0.3 points in ADCC — and there is no significance testing. The paper's own discussion concedes the evaluation metrics are imperfect. I would not hang a strong empirical claim on these numbers as they stand.\n\nWho should read it: anyone working on the theoretical foundations of CAM or Shapley-based attribution. It deserves a serious referee, but I'd send it back with a request to validate the Taylor surrogate (e.g., measure the remainder on real subsets, or restrict the claim to layers where the approximation is faithful) and to report statistical significance or confidence intervals.","headline":"A clean game-theoretic reframing of CAMs whose load-bearing Taylor-surrogate assumption is unvalidated and whose empirical wins are small; worth a serious referee but needs revision.","tokens_in":24796,"tokens_out":3843,"would_cite":true,"duration_ms":34495,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A12","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that standard Class Activation Mapping heatmaps are Shapley-value attributions of a Taylor-expanded network score, and that adding a Hessian correction yields ShapleyCAM, a closed-form second-order explainer computable in…","keywords":["class activation mapping","Shapley value","cooperative game theory","Hessian matrix","explainable artificial intelligence","gradient-based explanations","ReST utility","feature attribution"],"falsifier":"Enumerate all subsets of a small activation tensor (say a $3\\times3$ spatial grid over a few channels), compute the exact Shapley value of the second-order Taylor-expanded utility using the definition, and compare each value with Equation (14); exact agreement is required if the derivation is correct. A second check is to evaluate the Taylor remainder on random subsets for a Swin Transformer's self-attention layer, where the paper itself reports that the approximation stops being predictive.","tokens_in":23813,"feed_emoji":"🎯","tokens_out":9921,"duration_ms":93106,"temperature":0.7,"pith_summary":"This paper tries to show that familiar CAM visual explanations are not ad hoc heuristics: they are Shapley values of a cooperative game played by spatial positions in the network's activation maps, computed under a Taylor approximation of the network score. By expanding the utility function to second order, it derives a closed-form weight for each activation map, making per-pixel Shapley values computable in one extra backward pass. On this view GradCAM and HiResCAM are first-order truncations of the same Shapley identity, and the new ShapleyCAM family is the second-order version. The paper also proposes a residual softmax utility that avoids off-target highlighting and gradient vanishing in confident predictions. If the derivation holds, CAM heatmaps inherit the fairness axioms of Shapley values at essentially the cost of a gradient-based method.","feed_headline":"One formula makes CAM heatmaps exact Shapley values","feed_subtitle":"GradCAM and HiResCAM are first-order cases; a Hessian term upgrades them at one extra backward pass.","key_machinery":"The central object is the Content Reserved Game-theoretic (CRG) Explainer, which defines a cooperative game at each pixel position: the $d$ spatial positions of the downsampled activation maps are players, and the utility $U$ is the network output. The explanation is $E = \\sum_i g(W^i) \\odot A^i$; Type-I keeps pixel-wise weights ($g(W^i)=W^i$, game-theoretic) and Type-II keeps only the channel mean ($g(W^i)=\\bar{W}^i \\mathbf{1}_d$, content-reserving). The proof machinery is the Taylor expansion of $U$ at $X_D$: first order gives $U'(X_D)X_j$, and the second-order contribution collapses through the subset-sum identity to $-\\frac{1}{2} X_D^\\top H_D X_j$. Only a Hessian-vector product is needed, so the computation costs one extra backward pass relative to GradCAM. The ReST utility $U_{\\mathrm{ReST}} = y_c + \\ln(\\mathrm{softmax}(y)_c)$ ties together the pre- and post-softmax analyses.","core_discovery":"Under the second-order Taylor expansion of the utility at the full activation tensor $X_D$, the Shapley value of the $j$-th spatial position is $\\phi(j;U) = U'(X_D)X_j - \\frac{1}{2} X_D^\\top H_D X_j$, where $X_j$ keeps only the $j$-th column across all activation maps and $H_D$ is the Hessian of the utility at $X_D$. Because this expression is linear in the entries of $X_j$, it decomposes as $\\sum_i W^i_j A^i_j$ with $W^i = [U'(X_D) - \\frac{1}{2} X_D^\\top H_D]^i$, exactly the weighting ShapleyCAM uses. The paper proves that ShapleyCAM-H is a Type-I CRG Explainer (per-pixel game-theoretic) and ShapleyCAM is a Type-II CRG Explainer (content-reserving); at first order the same theorem identifies HiResCAM and GradCAM as two valid truncations of one Shapley formula. It further proves that a post-softmax GradCAM heatmap is a probability-weighted sum of differences of pre-softmax heatmaps, which motivates adding the pre-softmax term back to form the ReST utility.","pith_inferences":["If the identity holds generally, any linear CAM weighting scheme can be read as a Taylor order of a Shapley expansion; an order-$k$ expansion would use $k$-th order derivative information and yield a hierarchy of CAM explainers trading precision against cost.","The proof technique transfers naturally to other cooperative-game solution concepts that differ from Shapley only in how subset marginals are averaged, so the same second-order closed form may produce new attribution methods beyond ShapleyCAM.","The ReST construction suggests a one-parameter family $U_\\lambda = y_c + \\lambda \\ln p_c$; tuning $\\lambda$ could interpolate between localization behavior and gradient stability for different confidence regimes.","The paper's explainability-versus-localization argument implies that current metrics such as ADCC reward foreground localization, so a target-class-specific evaluation, measuring whether the heatmap changes when the target class changes, would be a sharper test for CAM explainers."],"forward_implications":["ShapleyCAM, ShapleyCAM-H, and ShapleyCAM-E compute exact Shapley values for the second-order Taylor game, with only one additional backward pass beyond GradCAM.","GradCAM and HiResCAM inherit a Shapley-value interpretation as first-order approximations, and the placement of ReLU is settled: apply it after summing the weighted maps, not before.","The ReST utility removes both failure modes of the two standard score choices: off-target highlighting from pre-softmax and vanishing gradients from post-softmax.","Across the twelve tested networks, the second-order correction typically improves the ADCC, IC, and ADD explanation metrics over the corresponding first-order methods, especially for CNN architectures.","For Swin Transformers at deep self-attention target layers, the Taylor approximation is not reliable: most CAM methods, including ShapleyCAM, do not beat random-weight heatmaps, identifying a boundary of the theory."],"supporting_citations":[{"why":"Defines the Shapley value and its four axioms, the fairness notion the framework claims CAM heatmaps implement.","marker":"[13]"},{"why":"Provides the second-order Taylor-expansion trick for closed-form Shapley values that ShapleyCAM adapts from data valuation to feature attribution.","marker":"[17]"},{"why":"Introduces GradCAM and supplies the proof that gradients at the layer before GAP give identical weights, the Optimal CRG case.","marker":"[5]"},{"why":"Introduces HiResCAM, the element-wise weighting the paper identifies as the Type-I first-order explainer.","marker":"[4]"},{"why":"Proves equivalence relations among gradient-based CAM weights that the paper uses to identify first-order cases.","marker":"[9]"},{"why":"Shows that the needed Hessian-vector product costs one extra backward pass, making ShapleyCAM scalable.","marker":"[22]"},{"why":"Analyzes pre- versus post-softmax scores, the failure modes that ReST is designed to fix.","marker":"[23]"},{"why":"Supplies GradCAM-E and the ReLU-placement discussion that the paper generalizes and answers via Shapley values.","marker":"[12]"}],"fun_headline_variants":["CAM heatmaps get exact Shapley values via one Hessian term","GradCAM and HiResCAM are Shapley approximations at first order","ShapleyCAM gives exact Shapley heatmaps at one extra backward pass","New utility function makes CAM heatmaps provably Shapley","Bridge heuristic CAMs and Shapley values with one formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes the second-order Taylor expansion of the utility at the full activation tensor is a good approximation for every subset of pixel positions, which the paper's own Swin Transformer experiments show does not always hold.","fun_headline_variants_meta":{"raw":{"variants":["CAM heatmaps get exact Shapley values via one Hessian term","GradCAM and HiResCAM are Shapley approximations at first order","ShapleyCAM gives exact Shapley heatmaps at one extra backward pass","New utility function makes CAM heatmaps provably Shapley","Bridge heuristic CAMs and Shapley values with one formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000625,"raw_usage":{"total_tokens":2942,"prompt_tokens":1045,"completion_tokens":1897,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":1802}},"tokens_in":661,"tokens_out":1897,"duration_ms":12006,"temperature":1.0,"reasoning_tokens":1802,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:13:21.611463+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all subsets of a small activation tensor (say a $3\\times3$ spatial grid over a few channels), compute the exact Shapley value of the second-order Taylor-expanded utility using the definition, and compare each value with Equation (14); exact agreement is required if the derivation is correct. A second check is to evaluate the Taylor remainder on random subsets for a Swin Transformer's self-attention layer, where the paper itself reports that the approximation stops being predictive.","supporting_citations":[{"cited_title":"Contributions to the Theory of Games 2, 307–317 (1953)","cited_arxiv_id":null,"evidence_quote":"Defines the Shapley value and its four axioms, the fairness notion the framework claims CAM heatmaps implement."},{"cited_title":"In: International Joint Conference on Artificial Intelligence, pp","cited_arxiv_id":null,"evidence_quote":"Introduces GradCAM and supplies the proof that gradients at the layer before GAP give identical weights, the Optimal CRG case."},{"cited_title":"https://iclr-blogposts.github.io/2024/blog/ bench-hvp/ [Accessed: (Nov","cited_arxiv_id":null,"evidence_quote":"Shows that the needed Hessian-vector product costs one extra backward pass, making ShapleyCAM scalable."},{"cited_title":"1–4 (2023)","cited_arxiv_id":null,"evidence_quote":"Analyzes pre- versus post-softmax scores, the failure modes that ReST is designed to fix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies GradCAM-E and the ReLU-placement discussion that the paper generalizes and answers via Shapley values."}],"review_version":1}