{"id":"d6564f5b-44cd-447a-8ebc-4681afdbdb84","arxiv_id":"2505.24729","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Under linearity and a continuity assumption, every feature attribution method is an integral against a signed measure, so the design choice reduces to picking the measure.","lead":"This paper proposes a measure-theoretic framework for feature attribution, in which choosing an explanation method is equivalent to choosing a family of measures over the input space. It recovers known methods such as partial dependence plots as special cases and derives formulas for deep ReLU networks.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"FSC is the load-bearing assumption, and it excludes canonical gradient-based methods like Integrated Gradients; the framework is narrower than the paper suggests.","rationale":"The reader's verdict already identifies FSC as the weakest assumption; this stress-test confirms and sharpens that concern. The theorem itself is not invalid: any linear FSC functional is an integral, and the step-function construction is standard. But the FSC condition is not merely technical; it has real exclusionary power. Integrated Gradients is a canonical axiomatic attribution method (Sundararajan et al., 2017), and it violates FSC, as the f_ε example shows. More broadly, any method that depends on derivative information or on non-sup-norm functionals falls outside the framework. The paper should either explicitly restrict its scope to sup-norm-continuous methods or relax the topology (e.g., Sobolev spaces) if gradient-based methods are to be included. There is also an under-specified domain issue: the indicator functions used as atomic models are not in C([0,1]^d), so the statement of Thm. 3.2 silently expands the model class; this is fixable but needs clarification. Overall, the CONDITIONAL verdict remains appropriate, and no verdict change is needed.","tokens_in":27070,"tokens_out":20752,"duration_ms":278956,"concrete_test":"Compute Integrated Gradients for the sequence f_ε(x,y)=ε sin((x-y)/ε) with baseline (0,0) and input (1,1): IG_x=1 for every ε, although ||f_ε||∞→0. Then verify that no finite signed Borel measure μ can satisfy IG_x(f)=∫f dμ: if such μ existed, |IG_x(f_ε)|≤ε·TV(μ)→0, a contradiction. This settles whether IG lies in the FSC class of Thm. 3.2; it does not, so the framework's coverage of existing methods must be scoped to integral/expectation-based methods.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central representation theorem (Thm. 3.2) is correct as a conditional statement: if φ is linear and FSC, Riesz-Markov gives φ(x,f)=∫f dμ. The load-bearing issue is that FSC (Def. 3.1) is not a mild regularity condition: it is sup-norm continuity of the linear functional, and it fails for Integrated Gradients (IG). In d=2, take f_ε(x,y)=ε sin((x-y)/ε) on [0,1]^2, baseline 0, input (1,1). Then ||f_ε||∞=ε→0, but along the straight path γ(t)=(t,t), ∂_x f_ε=cos(0)=1, so IG_x(f_ε)=1 for every ε. Any attribution of the form f↦∫f dμ for a finite signed Borel measure would send this sequence to 0. Hence IG cannot be represented in this framework despite being linear and complete. The paper's Table 1 only recovers expectation/marginalization methods (Covert et al., PDP), not gradient-based saliency. If the paper claims a general first-principles framework for feature attribution, this is a substantial scope limitation; if the claim is only about FSC methods, it is correct but far less general than the title suggests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a measure-theoretic framework for feature attribution. Rather than listing axiomatic properties, the authors define an 'atomic' attribution on indicator functions of hyperrectangles and extend it to continuous models by requiring linearity and a continuity property called Functional Supremum Continuity (FSC). The main representation result (Theorem 3.2) states that, under these assumptions, every coordinate of the attribution of a continuous model is the integral of the model with respect to a finite signed Borel measure. The paper also proves a limitation result (Theorem 2.1) claiming that attribution methods satisfying completeness, sensitivity, linearity, and a Lipschitz condition resemble Gradient×Input up to a remainder bounded by the Hessian; derives closed-form attributions for piecewise-affine functions such as ReLU networks (Corollary 4.1); recovers several expectation-based attribution methods (Table 1); and formulates an optimization problem for feature-attribution measures with respect to recall and precision metrics (Section 4.3, Theorem 4.1).","tokens_in":27330,"tokens_out":12943,"duration_ms":162243,"significance":"The representation theorem is mathematically correct as a conditional statement: for a linear functional on C([0,1]^d) that is continuous in the supremum norm, the Riesz–Markov theorem yields an integral representation by a signed measure. The paper gives credit for connecting attribution methods to measure theory and for providing detailed appendix proofs, a public implementation, and closed-form expressions for ReLU networks within the assumed class. However, the framework is much narrower than the title and abstract suggest: FSC excludes standard gradient-based methods such as Integrated Gradients, and the table of recovered methods contains only expectation/marginalization methods. The paper also contains a formal gap in the domain over which FSC is applied. These issues are fixable by reframing and by making the domain explicit, but they materially affect the paper's central claim of being a general 'first principles' framework.","major_comments":[{"comment":"The central theorem is not well-posed as stated. FSC (Definition 3.1) only constrains phi on sequences inside a fixed function class F, and Theorem 3.1 takes F = C^0([0,1]^d). But the proof of Theorem 3.2, via Theorem B.2 and Equations (9)-(10), applies Linearity to the piecewise-constant approximations f_n of Theorem B.1, which are not continuous and therefore are not elements of F. If F is enlarged to include indicators and their finite sums, the Riesz-Markov representation on C^0 no longer determines the extension, and the 'atomic attribution' mu(R) is an additional, non-derived datum. The paper needs an explicit domain construction, such as a sup-norm continuous extension from C^0 to the algebra generated by rectangles, or a statement that F is that algebra, before Theorem 3.2 can be evaluated as a theorem rather than an implicit definition.","section":"Theorem 3.2 and Appendix B.3.1"},{"comment":"The scope of the framework is substantially narrower than the title and abstract suggest. FSC is continuity of each coordinate of the attribution in the sup norm, and the representation theorem covers exactly methods of the form f -> integral f d mu. Many standard linear attribution methods are not of this form. For example, set d=2, f_epsilon(x,y)=epsilon sin((x-y)/epsilon), baseline 0, and input x=(1,1). Then ||f_epsilon||_infinity = epsilon -> 0, but Integrated Gradients along the straight path gives partial_x f_epsilon(t,t)=cos(0)=1, so IG_x(f_epsilon)=1 for every epsilon. No finite signed measure mu can reproduce this while also reproducing IG on the zero function, since |integral f_epsilon d mu| <= epsilon * TV(mu) -> 0. Consequently Integrated Gradients, DeepLIFT, and related gradient-based saliency methods lie outside the framework, and Table 1 recovers only expectation/marginalization methods. The abstract's claim that the framework is built 'from the ground up' for feature attribution should be qualified to FSC-linear methods, otherwise the headline claim is misleading.","section":"Definition 3.1 / Section 3.2 / Table 1"},{"comment":"The claimed extension of Theorem 2.1 to deep ReLU networks via mollifiers is not justified. For the smoothed network f_n = phi_n * f, Theorem 2.1 bounds the remainder by L_x d/2 M_n with M_n = max ||nabla^2 f_n||_op. For a ReLU network with a nondifferentiable kink, M_n typically grows like the inverse mollifier width, because the smoothed second derivative contains a mollified Dirac mass; hence the bound diverges as n -> infinity. Uniform convergence of f_n to f gives, under FSC, convergence of the attributions, but it does not give a uniform smallness bound on the remainders. Therefore the conclusion that the issues of Gradient×Input transfer to all axiom-satisfying methods does not follow. The paper should either prove a uniform bound on M_n for mollified ReLU networks or restrict the claim to functions with uniformly bounded Hessian.","section":"Section 2.2, final paragraph"},{"comment":"The abstract and Section 2 claim that any attribution method satisfying Completeness, Sensitivity, and Linearity resembles Gradient×Input, but Theorem 2.1 additionally assumes the Lipschitz continuity condition of Definition 2.5. This is not a mild regularity assumption: it is a uniform sup-norm Lipschitz bound on the attribution as a function of the model, and it fails for the derivative-based methods that the Gradient×Input discussion concerns. The paper should state this condition in the abstract and conclusion, and it should not present the result as applying to 'any' method satisfying the three classical axioms.","section":"Theorem 2.1 / abstract"}],"minor_comments":[{"comment":"The paper says 'Rather than imposing axioms,' but the framework relies on Linearity, FSC, and a chosen atomic attribution for indicators; these are also modeling assumptions. The presentation would be clearer if the paper acknowledged that it replaces one axiomatic system with a different set of conditions.","section":"Section 1"},{"comment":"The formal statement of Sensitivity requires that a feature has no impact on the model's output for all inputs before the attribution is forced to zero; this is much stronger than the usual local sensitivity notion and should be discussed explicitly.","section":"Definition 2.3"},{"comment":"Recall is undefined when D_{1,w} is empty, for example when w=0; the paper should add a convention for this case.","section":"Section 4.3, Definition 4.2"},{"comment":"The first row writes P_X(dy | X_j = x_j), but for continuous X this is a regular conditional distribution rather than a pointwise-defined measure; a clarifying remark would avoid measure-theoretic ambiguity.","section":"Table 1"},{"comment":"The phrase 'This can be casted' should be 'This can be cast.'","section":"Section 4.3"},{"comment":"The proof of Proposition 2.1 uses the word 'Completude'; this should be 'Completeness.'","section":"Appendix D.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's core theorem is correct as a conditional statement, and the measure-theoretic reformulation could be a useful contribution if the scope is stated honestly. The main concerns are the domain gap in the statement of Theorem 3.2, the exclusion of canonical gradient-based methods from the framework, and the unsupported mollifier extension in Section 2.2. These are fixable within the manuscript's scope by clarifying assumptions and reframing the claims; hence major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the core result is correct but narrower than the title suggests. Any linear attribution that is continuous in sup norm is an integral against signed measures—that's Riesz-Markov, cleanly applied. The constructivist path through indicator functions is a nice way to present it, and the recovery of expectation-based methods (Covert et al., PDP) by choosing measures is a useful unifying view. That is the real contribution: not new mathematics, but a framing that may help design and compare methods.\n\nThe soft spot is scope. FSC is a strong assumption, not a mild one. The stress-test example is right: on a sequence f_epsilon = epsilon sin((x-y)/epsilon), sup norm goes to 0 but Integrated Gradients along the diagonal stays at 1. So IG is not FSC, and it cannot be represented by any finite signed measure. The paper's Table 1 only recovers integral/expectation methods. If the claim is restricted to FSC methods, fine; but the abstract and title say 'from first principles,' which overclaims. Relatedly, Theorem 2.1's 'any axiom-satisfying method resembles Gradient times Input' only holds under Lipschitz continuity, which is not in the abstract.\n\nThe ReLU part is formally elegant but limited: Corollary 4.1 gives a closed form for piecewise affine functions, but you need the polytope partition, which is exponential for deep nets. The mollifier extension in Section 2.2 is asserted, not proven, and it doesn't rescue IG since IG lacks FSC anyway. The metric optimization section is a small first step; it's honest about the projection losing information.\n\nThe proofs I checked are fine. Appendix Proposition A.2 has a slightly sloppy step, but the conclusion seems defensible. There are no experiments, and the code link is missing from the manuscript, which is a small reproducibility issue.\n\nWho this is for: someone working on expectation-based attributions or looking for a unifying framework. Not someone who needs to explain gradient-based saliency. It deserves a serious referee, but the revision should temper the claims in the abstract and explicitly discuss the FSC limitation.","headline":"A correct but narrow measure-theoretic reframing of attribution; the 'first principles' claim overreaches because FSC excludes gradient-based methods like Integrated Gradients.","tokens_in":27844,"tokens_out":2981,"would_cite":false,"duration_ms":35803,"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":"This paper argues that choosing a feature attribution method is equivalent to choosing a family of measures.","keywords":["feature attribution","explainable AI","measure-theoretic framework","Riesz-Markov theorem","signed Borel measures","indicator functions","ReLU networks","attribution axioms"],"falsifier":"Compute Integrated Gradients on a sequence of smooth mollified approximations of a fixed ReLU network: if the attributions do not converge as the approximations approach the network in sup norm, then Integrated Gradients fails functional supremum continuity, and no family of signed measures from Theorem 3.2 can represent it, marking the framework's boundary.","tokens_in":26823,"feed_emoji":"🎯","tokens_out":11673,"duration_ms":136192,"temperature":0.7,"pith_summary":"Feature attribution methods assign importance scores to input features, but there is no consensus on what a correct attribution is. This paper proposes to build attributions from the ground up: first decide what the attribution should be for the simplest models, the indicator functions of rectangles, and then extend that decision to all continuous models. The main theorem shows that, under two conditions—linearity and continuity of the attribution in the model under uniform convergence—this extension is forced: the attribution of any continuous model is the Lebesgue-Stieltjes integral of the model against the measures that defined the atomic attributions. The paper uses this to recover existing methods such as partial dependence plots and feature-removal explanations, to derive closed-form attributions for deep ReLU networks, and to turn evaluation-metric optimization into a search over measures. The reader should care because the framework replaces a sprawling set of axioms with a single constructive principle: choose the measures, and the attribution method is determined.","feed_headline":"Feature attribution reduces to choosing a measure","feed_subtitle":"Paper shows: fix attributions on simple rectangles and every continuous model's explanation becomes an integral.","key_machinery":"The central object is the family of signed Borel measures $\\{\\mu_{j,x}\\}$ on $[0,1]^d$, indexed by feature $j$ and explained input $x$. The construction assigns atomic attributions to rectangle indicators as $\\phi(x, \\mathbf{1}_R)_j = \\mu_{j,x}(R)$, then extends them by linearity to step-function approximations of a continuous model, and by functional supremum continuity—uniformly close models must have uniformly close attributions—passes to the limit. The Riesz-Markov theorem guarantees that the resulting Lebesgue-Stieltjes integral representation is unique in this class. A second workhorse is the piecewise affine decomposition of deep ReLU networks, which turns the integral in Corollary 4.1 into a finite sum over linear regions weighted by the measure of each region and evaluated at the region's center of mass.","core_discovery":"The paper's central claim is Theorem 3.2: for any linear and functionally supremum continuous attribution method, once the atomic attribution for an indicator function is fixed as $\\phi(x, \\mathbf{1}_R)_j = \\mu_{j,x}(R)$, the attribution of every continuous model $f$ on $[0,1]^d$ is $\\phi(x, f)_j = \\int f \\, d\\mu_{j,x}$. Riesz-Markov (Theorem 3.1) supplies the converse direction: every attribution method that is linear and continuous under uniform convergence of models is representable by a unique family of signed regular Borel measures. In the paper's reading, doing feature attribution is therefore equivalent to choosing a family of measures on the input space, one per feature and per explained point, and the atomic attribution of an indicator function simply reports the measure of the rectangle.","pith_inferences":["The functional supremum continuity assumption is not satisfied by popular gradient-based methods such as Integrated Gradients, since their outputs depend on derivatives rather than on sup-norm function values; if true, this means the measure-based representation covers a principled but limited class of attribution methods.","The framework suggests a direct empirical test for whether a given attribution method belongs to the class: evaluate it on many rectangle indicators to estimate an implied measure, then check whether its output on more complex models matches the corresponding integral; methods that fail are outside the representation.","Because Theorem 4.1 characterizes optima only through centers of mass, and many measures share the same center of mass, optimal attributions for a single model can be highly degenerate; evaluation metrics may need to be defined over distributions of models, as the paper's intersection construction starts to do.","The extension to unbounded input spaces or discontinuous models is not automatic, since the step-function approximation and the FSC topology are tied to compact domains and uniform convergence; the Stieltjes-integral formulation in the appendix is the paper's route, but it still requires bounded-variation integrands."],"forward_implications":["Any attribution method in this class is fully characterized by its measures: $\\phi(x, f)_j = \\int f \\, d\\mu_{j,x}$, so designing an attribution method is designing the measures.","Particular measure choices recover existing methods: conditional feature removal, independent-marginal feature removal, and partial dependence plots all appear in Table 1, and a delta-measure construction recovers the coefficients of a linear model.","For deep ReLU networks, which are piecewise affine continuous functions, the closed form of Corollary 4.1 computes attributions as a sum over activation regions of $\\mu_{j,x}(P)$ times the local linear model at the region's center of mass, and it also applies to CART trees and random forests.","Optimizing an evaluation metric such as Recall becomes a finite-dimensional geometric problem for linear models: an attribution is optimal exactly when each feature's measure has its center of mass in the half-space set described by Theorem 4.1, which also provides a sanity check for candidate attribution methods.","The Section 2 result gives a new rigidity argument: any attribution satisfying Completeness, Sensitivity, Linearity, and a Lipschitz condition must be close to Gradient×Input, so those axioms are too restrictive for a general foundation."],"supporting_citations":[{"why":"Supplies the Riesz-Markov theorem used to represent any Linear FSC attribution as an integral against signed measures.","marker":"[26]"},{"why":"Provides the uniform approximation of continuous functions on a compact by sums of indicator functions, the starting point of the construction.","marker":"[38]"},{"why":"Gives the multidimensional Riemann-Stieltjes integration theory used to take the limit of the step-function approximations.","marker":"[23]"},{"why":"Justifies identifying the Riemann-Stieltjes integral with the Lebesgue-Stieltjes integral in the proof of Theorem 3.2.","marker":"[14]"},{"why":"Establishes the bijection between signed Borel measures and functions of bounded variation used to pass from integrands to measures.","marker":"[3]"},{"why":"Provides the details for the measure-variation correspondence and for completely monotone functions yielding positive measures.","marker":"[9]"},{"why":"Gives the closed-form linear representation of deep ReLU networks on their activation regions used in Corollary 4.1.","marker":"[25]"},{"why":"Defines the Completeness, Sensitivity, and Linearity axioms that the paper argues are too restrictive in Section 2.","marker":"[41]"},{"why":"Defines partial dependence plots, one of the existing methods recovered by a particular choice of measures in Table 1.","marker":"[15]"},{"why":"Frames feature attribution through feature removal; the first two rows of Table 1 recover these explanations as measure choices.","marker":"[12]"}],"fun_headline_variants":["Attribution is choosing a measure per feature","Explaining models: from indicator boxes to integrals","Feature attribution = integral against a signed measure","Fix attributions on boxes, then integrate any model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on the assumption that a reasonable attribution method must be continuous in the model: if two models are uniformly close everywhere, their attributions must be close.","fun_headline_variants_meta":{"raw":{"variants":["Attribution is choosing a measure per feature","Explaining models: from indicator boxes to integrals","Feature attribution = integral against a signed measure","Fix attributions on boxes, then integrate any model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1637,"prompt_tokens":862,"completion_tokens":775,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":717}},"tokens_in":478,"tokens_out":775,"duration_ms":10866,"temperature":1.0,"reasoning_tokens":717,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:15:23.945688+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute Integrated Gradients on a sequence of smooth mollified approximations of a fixed ReLU network: if the attributions do not converge as the approximations approach the network in sup norm, then Integrated Gradients fails functional supremum continuity, and no family of signed measures from Theorem 3.2 can represent it, marking the framework's boundary.","supporting_citations":[{"cited_title":"Concrete Representation of Abstract (M)-Spaces (A characterization of the Space of Continuous Functions)","cited_arxiv_id":null,"evidence_quote":"Supplies the Riesz-Markov theorem used to represent any Linear FSC attribution as an integral against signed measures."},{"cited_title":"Principles of Mathematical Analysis","cited_arxiv_id":null,"evidence_quote":"Provides the uniform approximation of continuous functions on a compact by sums of indicator functions, the starting point of the construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the multidimensional Riemann-Stieltjes integration theory used to take the limit of the step-function approximations."},{"cited_title":"Functions of bounded variation, signed measures, and a general Koksma-Hlawka inequality","cited_arxiv_id":null,"evidence_quote":"Establishes the bijection between signed Borel measures and functions of bounded variation used to pass from integrands to measures."},{"cited_title":"Functions of bounded variation in one and multiple dimensions","cited_arxiv_id":null,"evidence_quote":"Provides the details for the measure-variation correspondence and for completely monotone functions yielding positive measures."},{"cited_title":"SplineCam: Exact Visualization and Characterization of Deep Network Geometry and Decision Boundaries","cited_arxiv_id":null,"evidence_quote":"Gives the closed-form linear representation of deep ReLU networks on their activation regions used in Corollary 4.1."},{"cited_title":"Axiomatic Attribution for Deep Networks","cited_arxiv_id":null,"evidence_quote":"Defines the Completeness, Sensitivity, and Linearity axioms that the paper argues are too restrictive in Section 2."},{"cited_title":"Greedy Function Approximation: A Gradient Boosting Machine","cited_arxiv_id":null,"evidence_quote":"Defines partial dependence plots, one of the existing methods recovered by a particular choice of measures in Table 1."},{"cited_title":"Explaining by Removing: A Unified Framework for Model Explanation","cited_arxiv_id":null,"evidence_quote":"Frames feature attribution through feature removal; the first two rows of Table 1 recover these explanations as measure choices."}],"review_version":1}