{"id":"9a4e190b-5032-4d4c-91d8-63b2305ae0b8","arxiv_id":"2412.13623","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A unification framework for XAI attribution methods whose core canonical decomposition theorem fails because the components sum to the fully removed function rather than to the original function.","lead":"This paper proposes a unified mathematical framework that casts attribution-based explanation methods as removal-based methods tied to cooperative game theory. It claims all additive decompositions are canonical and derives behavioral axioms, but a central formula is inconsistent with its own definitions.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The CAD as defined fails completeness: summation property gives Σ_S g_S = P_[d], not f, and the proof of Theorem 19 misstates P_∅ as the full sum. The claimed equivalence between additive decompositions and CADs therefore does not hold.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the proof of Theorem 19 relies on the false equality f = P_∅(f) = Σ_{S⊆∅} g_S(f) = Σ_{S⊆[d]} g_S(f), and the CAD as defined does not satisfy Definition 14 completeness. My independent check of the summation property confirms that Σ_{S⊆[d]} g_S equals P_[d], not P_∅, and the d=1 averaging-removal example shows the CAD components sum to a constant rather than to f. This invalidates the central equivalence and the derived game-theoretic representation. I see no way to reinterpret the definitions that repairs the argument without changing the CAD's construction or the removal-operator axioms. The verdict of REJECT is therefore appropriate, and my read does not change the reader's assessment.","tokens_in":57512,"tokens_out":4219,"duration_ms":34983,"concrete_test":"Compute the CAD for d=1, f(x)=x, removal operators P_∅=I and P_1(f)=E_{U[0,1]}[f]. The CAD gives g_∅=f and g_1=E[f]-f, so g_∅+g_1=E[f]≠f, contradicting completeness. Independently, take the trivial additive decomposition g_[d]=f, g_S=0 otherwise and set P_S=Σ_{T⊆S}g_T; then P_∅=0, violating the removal-operator condition P_∅=I. Both checks settle the failure of Theorem 19.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 14 requires an additive functional decomposition to satisfy f = Σ_{S⊆[d]} g_S(f). Definition 18 constructs g_S from removal operators with P_∅ = I and derives the summation property P_T = Σ_{S⊆T} g_S(f). Taking T = [d] gives Σ_{S⊆[d]} g_S(f) = P_[d](f), and P_[d](f) is required to be independent of all features, i.e. constant in x. Thus for any non-constant f the CAD components do not sum to f, so the CAD is not an additive functional decomposition in general. The proof of Theorem 19 asserts f = P_∅(f) = Σ_{S⊆∅} g_S(f) = Σ_{S⊆[d]} g_S(f); the middle equality is false because Σ_{S⊆∅} g_S is g_∅, not the total sum. A minimal example is d=1, f(x)=x, removal P_1(f)=E[f]: then g_∅=f and g_1=E[f]-f, so g_∅+g_1=E[f]≠f. The converse direction of Theorem 19 also fails: defining P_S := Σ_{T⊆S} g_T from a valid additive decomposition gives P_∅=g_∅, which is not the identity operator required of removal operators. Since Theorem 43 (RBAM representation theorem) and the functional-axiom results all invoke the CAD as a valid decomposition, this is a load-bearing error, not a notational slip.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a unifying framework for removal-based attribution methods (RBAMs) in explainable AI. It formalizes RBAMs through removal operators, behaviour mappings, and aggregation coefficients; introduces the canonical additive decomposition (CAD); claims (Theorem 19) that every additive functional decomposition is a CAD and that every RBAM is a game-theoretic value or interaction index of a pointwise cooperative game (Theorem 43); derives sufficient conditions for functional axioms; and discusses efficient approximations. The central claimed equivalence is the foundation for the rest of the framework.","tokens_in":57789,"tokens_out":7867,"duration_ms":62292,"significance":"If correct, the framework would unify a broad class of attribution methods, enable rigorous method selection, and provide a game-theoretic interpretation of many heuristics. The paper also introduces functional axioms and a useful taxonomy of game-theoretic values and indices. However, the central mathematical claim is false under the paper's own definitions, so the proposed unification does not hold.","major_comments":[{"comment":"The proof of Theorem 19 asserts f = P_∅(f) = Σ_{S⊆∅} g_S(f) = Σ_{S⊆[d]} g_S(f). Under Definition 18 the summation property is P_T = Σ_{S⊆T} g_S, so P_∅ = g_∅, not the full sum. Consequently the CAD components sum to P_[d](f), which is required to be feature-independent and is generally not equal to f. For example, with d=1, f(x)=x, and removal P_1(f)=E[f] (so P_∅=I), the CAD gives g_∅=x and g_1=E[f]-x, and g_∅+g_1=E[f] ≠ f. Thus the CAD does not generally satisfy the completeness axiom of Definition 14, and the theorem's claim that every CAD is an additive functional decomposition is false.","section":"Theorem 19 / Appendix A"},{"comment":"The pointwise cooperative game is defined as v_Φ_G(f,x)(S) = Φ(P_S(f))(x) − Φ(P_[d](f))(x), and the equality with Φ(Σ_{T⊆S} g_T(f))(x) − Φ(g_∅(f))(x) is claimed to follow from the summation property. This substitution is incorrect: the summation property gives P_[d] = Σ_{T⊆[d]} g_T, not g_∅. The error propagates to Theorem 43, where the constant shift Φ(g_∅(f))(x) should be Φ(P_[d](f))(x); these differ for any non-constant P_[d](f).","section":"Definition 40 / Section 7.1"},{"comment":"In the converse direction, the proof defines P_S := Σ_{T⊆S} g_T for a given additive decomposition. Then P_∅ = g_∅, which is not generally the identity operator. Since removal operators are required to satisfy P_∅ = I, the constructed operators are not valid removal operators. Hence the claimed 'if and only if' fails in both directions.","section":"Theorem 19, converse / Appendix A"},{"comment":"The efficient-computation results and the functional-axiom conditions all rely on the corresponding functional decomposition being a valid additive decomposition of f. Because the CAD does not generally satisfy completeness, these results do not follow from the framework as stated. For instance, Proposition 47 expresses m(f,S) as a linear combination of components g_T(f); without completeness, those components do not sum to f, so the attribution method is not anchored in a decomposition of the explained function.","section":"Sections 7.3-7.4, Propositions 47-53"}],"minor_comments":[{"comment":"The manuscript switches between 'I' and 'we' (e.g., §4.2.2, §4.3.3); please harmonize.","section":"Throughout"},{"comment":"Table 2 lists PredDiff and CXPlain without references.","section":"Table 2"},{"comment":"The notation for the complement of a set S is defined but appears visually identical to S in the provided text; please ensure the typeset version distinguishes them (e.g., \\overline{S}).","section":"Section 3"},{"comment":"Definition 40 uses G for both the decomposition and the set of games; consider renaming one to avoid confusion.","section":"Definition 40"}],"recommendation":"reject","confidential_remarks":"The manuscript appears to be based on a dissertation chapter and contains a fundamental mathematical error that invalidates the main contribution. The authors may need to substantially revise the definitions (e.g., by imposing P_[d]=I or redefining the CAD) and reassess which of the claimed results survive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Arne,\n\nThis one is a genuine mixed bag. The ambition is real: the paper tries to unify a large class of removal-based attribution methods under one game-theoretic umbrella, and the survey material (Tables 1 and 2, the XBR comparison, the taxonomy of game-theoretic values) is genuinely useful. The claim that any removal-based method, not just Shapley-based ones, can be represented as a value or interaction index of a pointwise cooperative game is new relative to Bordt-von Luxburg and the XBR line, and the functional axioms are a sensible extension of the game-theoretic ones.\n\nThe problem is in the core construction. The CAD as defined in Definition 18, with removal operators satisfying P_∅ = I and P_T removing features in T, does not satisfy the completeness condition of Definition 14. The summation property P_T = Σ_{S⊆T} g_S gives, at T=∅, P_∅ = g_∅, and at T=[d] the total Σ_{S⊆[d]} g_S = P_[d], which is constant in x. So the components sum to the fully-removed function, not to f. The proof of Theorem 19 in the appendix asserts f = P_∅(f) = Σ_{S⊆∅} g_S = Σ_{S⊆[d]} g_S, and the middle equality is false. The d=1, f(x)=x example with P_1 = E[f] is a clean counterexample: g_∅=f, g_1=E[f]-f, and the sum is E[f] ≠ f. This is not a notational pimple; Theorem 19 is the load-bearing wall for Theorem 43 (the RBAM representation theorem), the pointwise game construction, and the sufficient conditions for the functional axioms.\n\nThe good news is the fix looks straightforward: re-index the removal operators so that P_T keeps features in T (P_∅ = constant baseline, P_[d] = identity). Then the Möbius inversion runs over kept sets and the completeness property returns. But as submitted, the paper's central theorem is contradicted by its own definitions.\n\nOther than that, the related-work handling is fair, the citations to the authors' own approximation papers are used appropriately as applications, and the writing is clear. The scope is right for a major venue; the math just isn't there yet in this form.\n\nI'd send it to a serious referee only after the authors fix the indexing—the idea is valuable enough to deserve a careful major revision. As it stands, though, it's a reject.","headline":"The CAD is indexed the wrong way around: under the paper's own definitions the components sum to P_[d], not f, so Theorem 19 and everything riding on it collapses—though the framework looks salvageable with a re-indexing.","tokens_in":58365,"tokens_out":5267,"would_cite":false,"duration_ms":48018,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A12","68T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that every removal-based attribution method—Shapley values, LIME, occlusion, LOCO and many others—is fully determined by a behaviour mapping, a set of removal operators, and aggregation coefficients, and that each such…","keywords":["attribution methods","removal-based attribution","functional decomposition","canonical additive decomposition","cooperative game theory","Shapley values","interaction indices","explainable AI"],"falsifier":"Compute the canonical additive decomposition for the one-variable function f(x) = x using the removal operator that averages over the feature, P_1(f) = E[f]. The recursion gives g_∅(f) = f and g_1(f) = E[f] − f, whose sum is E[f], not f; checking whether the components add back to the original function for this case settles the completeness claim.","tokens_in":57248,"feed_emoji":"🧩","tokens_out":8961,"duration_ms":72360,"temperature":0.7,"pith_summary":"The paper sets out to unify the many feature-attribution explanation methods in machine learning into a single formal framework. It defines removal-based attribution methods (RBAMs) by three choices: what behaviour of the model is explained, how features are removed, and how the removed-feature behaviours are aggregated. It then introduces the canonical additive decomposition (CAD), a recursive construction built from the removal operators, and claims that every valid additive functional decomposition is a CAD. From there it shows that any RBAM is a value or interaction index of a pointwise cooperative game determined by the CAD, so the entire field of attribution methods becomes one family of game-theoretic indices parameterized by three choices. The payoff of the claimed unification is a behaviour-based taxonomy: formal 'functional axioms'—null, dummy, symmetry, anonymity—that tell practitioners which method guarantees which behaviour, plus a route to fast approximations by decomposing functions additively.","feed_headline":"Attribution methods reduce to three choices and a game","feed_subtitle":"Shapley, LIME, occlusion and LOCO all become one additive decomposition plus a value.","key_machinery":"The canonical additive decomposition (CAD): given removal operators {P_T}, define components recursively as g_S(f) = P_S(f) − Σ_{T⊂S} g_T(f), equivalently g_S(f) = Σ_{T⊆S} (−1)^{|S|−|T|} P_T(f). Its summation property P_T = Σ_{S⊆T} g_S connects removal operators to function components, and the pointwise cooperative game v(S) = Φ(P_S(f))(x) − Φ(P_[d](f))(x) turns the decomposition into a cooperative game. The RBAM representation theorem then writes any RBAM's attribution as an aggregation of this game, making the CAD the bridge between removal-based explanations and game theory.","core_discovery":"The central claim is that the class of removal-based attribution methods, which includes Shapley-based methods, permutation importance, occlusion, LIME on tabular data, and many interaction indices, is exactly the class of linear functions of pointwise cooperative games. For any RBAM, the removal operators define a canonical additive decomposition of the explained function; the decomposition plus the behaviour mapping defines a cooperative game on feature subsets; and the aggregation coefficients define a value (for single features) or interaction index (for feature groups) on that game. The paper further claims that any additive functional decomposition—any way of writing f as a sum of components each depending on a subset of features—can be represented as a CAD for some removal operators, which would make the CAD a genuinely canonical construction. This equivalence turns questions about explanation methods into questions about additive decompositions and game-theoretic indices, and yields sufficient conditions under which methods satisfy intuitive guarantees about independent, additive, symmetric, or anonymous features.","pith_inferences":["The same decomposition machinery could be applied to non-removal explanations that still rely on feature subsets, such as counterfactual explanations, extending the unification beyond attribution scores (an extension the paper mentions as future work).","One can design new attribution methods by pairing any set of removal operators with any game-theoretic value or interaction index; the paper's sufficient conditions then serve as a testable recipe for creating methods with prescribed functional behaviours.","The framework's functional axioms could be turned into a method-selection tool: a practitioner picks the axiom they need (e.g., functional null) and the sufficient conditions tell them which removal operator and value combinations are compatible."],"forward_implications":["Every removal-based attribution method can be viewed as a value or interaction index for a specific pointwise cooperative game, so heuristics like occlusion and LOCO are unified with Shapley-based methods into one family.","If the corresponding decomposition is minimal, then probabilistic and MC attribution methods automatically satisfy the functional dummy and null axioms, giving provable behavioural guarantees.","Any internally consistent MC attribution method can be written as a weighted sum of functional components, and for cardinal-probabilistic methods the weights collapse to a linear-time computation.","The framework provides a taxonomy of attribution methods by pointwise game-theoretic axioms, clarifying which axioms a method actually guarantees once behaviour and removal choices are fixed."],"supporting_citations":[{"why":"Supplies the XBR framework of behaviour, removal, and aggregation that the paper formalizes as RBAMs.","marker":"(Covert et al., 2021)"},{"why":"Provides the general decomposition formula that the CAD generalizes; its linear separable operators are a special case.","marker":"(Kuo et al., 2010)"},{"why":"Gives the taxonomy of values (linear, marginal contribution, probabilistic, random-order) used for the pointwise axioms.","marker":"(Weber, 1988)"},{"why":"Supplies the interaction-index taxonomy and the MC/probabilistic interaction indices used to define pointwise values.","marker":"(Grabisch and Roubens, 1999)"},{"why":"Provides the Shapley-Taylor interaction index used as an example of a higher-order RBAM and an interaction-efficiency axiom.","marker":"(Sundararajan and Najmi, 2020)"},{"why":"Defines SHAP, the primary instance of a Shapley-based RBAM that the framework unifies.","marker":"(Lundberg and Lee, 2017)"}],"fun_headline_variants":["Attribution methods reduce to one game and a decomposition","Every attribution method is a game value or interaction index","Attribution methods are linear functions of cooperative games","One canonical decomposition unifies attribution methods"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim depends on the assumption that the canonical additive decomposition's components always sum back to the original function, which in turn requires the empty-set removal operator to recover the full sum of components.","fun_headline_variants_meta":{"raw":{"variants":["Attribution methods reduce to one game and a decomposition","Every attribution method is a game value or interaction index","Attribution methods are linear functions of cooperative games","One canonical decomposition unifies attribution methods"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001943,"raw_usage":{"total_tokens":7632,"prompt_tokens":1009,"completion_tokens":6623,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":6564}},"tokens_in":625,"tokens_out":6623,"duration_ms":41347,"temperature":1.0,"reasoning_tokens":6564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:59:49.611738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the canonical additive decomposition for the one-variable function f(x) = x using the removal operator that averages over the feature, P_1(f) = E[f]. The recursion gives g_∅(f) = f and g_1(f) = E[f] − f, whose sum is E[f], not f; checking whether the components add back to the original function for this case settles the completeness claim.","supporting_citations":[{"cited_title":"Explaining by removing: A unified framework for model explanation","cited_arxiv_id":null,"evidence_quote":"Supplies the XBR framework of behaviour, removal, and aggregation that the paper formalizes as RBAMs."},{"cited_title":"Probabilistic values for games","cited_arxiv_id":null,"evidence_quote":"Gives the taxonomy of values (linear, marginal contribution, probabilistic, random-order) used for the pointwise axioms."}],"review_version":1}