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REVIEW 4 major objections 6 minor 42 references

Feature Attribution from First Principles

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that choosing a feature attribution method is equivalent to choosing a family of measures.

desk verdict A correct but narrow measure-theoretic reframing of attribution; the 'first principles' claim overreaches because FSC excludes gradient-based methods like Integrated Gradients. read the letter →

arxiv 2505.24729 v1 pith:X7A7EPDV submitted 2025-05-30 cs.LG

Magamed Taimeskhanov , Damien Garreau This is my paper · ORCID
classification cs.LG
keywords featureattributionexplainableAImeasure-theoreticframeworkRiesz-MarkovtheoremsignedBorelmeasuresindicatorfunctionsReLUnetworksaxioms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

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).

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 (4)
  1. [Theorem 3.2 and Appendix B.3.1] 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.
  2. [Definition 3.1 / Section 3.2 / Table 1] 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.
  3. [Section 2.2, final paragraph] 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.
  4. [Theorem 2.1 / abstract] 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.
minor comments (6)
  1. [Section 1] 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.
  2. [Definition 2.3] 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.
  3. [Section 4.3, Definition 4.2] Recall is undefined when D_{1,w} is empty, for example when w=0; the paper should add a convention for this case.
  4. [Table 1] 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.
  5. [Section 4.3] The phrase 'This can be casted' should be 'This can be cast.'
  6. [Appendix D.1] The proof of Proposition 2.1 uses the word 'Completude'; this should be 'Completeness.'

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the representation theorem is a direct Riesz–Markov application; atomic-attribution choices are explicit design inputs, not fitted outputs.

full rationale

The paper's central claim (Theorem 3.2) is not circular. The authors explicitly assume that a Linear and FSC attribution method exists, invoke the Riesz–Markov theorem (Theorem 3.1) to obtain representing signed measures, and then define atomic attributions of indicator functions as the measure of the rectangle: "Assume that the atomic attributions of indicator functions are given by: ... ϕ(x,1_R)_j := μ_{j,x}(R)." The conclusion that the attribution of a continuous model is ∫ f dμ_{j,x} follows directly from Riesz–Markov once the atomic attribution is identified with the representing measure. This is a constructivist definition, not a hidden re-use of the conclusion. Crucially, no parameter is fitted to data and then renamed as a prediction; the choice of measure is the design input, and the paper explicitly states "doing feature attribution is equivalent to choosing a family of measures over the input space." Likewise, Table 1 recovers methods such as Covert et al. and PDP by deliberately selecting the corresponding conditional/marginal distributions as the measures; this is a transparent instantiation, not a circular derivation. The self-citations present ([16], [29]) appear only in related-work discussions and are not load-bearing for any theorem. The skeptical concern that Functional Supremum Continuity excludes gradient-based methods such as Integrated Gradients is a scope limitation, not a circularity: it does not mean the paper's conditional derivation assumes its own conclusion. Overall, the derivation chain is self-contained and the central representation theorem is imported from classical functional analysis rather than from the authors' prior work.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The framework's contribution is the reduction of attribution design to measure choice. The measures are free parameters, FSC is an ad hoc regularity axiom, and the atomic attribution is an invented primitive with no external evidence.

free parameters (2)
  • family of measures μ_{j,x} = user-specified (e.g., δ_{x_j}⊗L for PDP)
    The framework's output is fully determined by the chosen family of signed Borel measures; the paper does not derive these from data or first principles, leaving the central design choice free.
  • atomic attribution for indicator functions = not fixed
    The 'first principle' is the user's assignment of a measure to each rectangle; this is a free modeling choice, not derived.
assumptions (5)
  • domain assumption Linearity of the attribution in the model (Definition 2.4)
    Assumed as a reasonable property; not derived. It is required for the integral representation.
  • ad hoc to paper Functional supremum continuity (FSC, Definition 3.1)
    Introduced to make the step-function limit valid; it is a strong regularity assumption that excludes several popular gradient-based methods.
  • standard math Input functions are continuous on [0,1]^d and models are approximated by step functions (Theorem B.1)
    Uniform approximation of continuous functions by step functions is classical.
  • standard math Riesz-Markov representation theorem
    Basis for Theorem 3.1/3.2; every linear continuous functional on C(X) is an integral against a unique signed Borel measure.
  • ad hoc to paper Lipschitz continuity of the attribution in sup norm (Definition 2.5) for Theorem 2.1
    Needed to bound the remainder in Theorem 2.1 but not part of the cited axioms; fails for gradient-based methods.
invented entities (1)
  • atomic attribution on indicator functions as the primitive of the framework
    purpose: Defines the attribution for the simplest models, from which all other attributions are derived
    The paper postulates that the user first specifies attributions for indicator functions; this primitive has no independent empirical justification and fully determines the resulting method.

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Pith. "Pith review of Feature Attribution from First Principles." pith.science (2026). https://pith.science/paper/X7A7EPDV

@misc{pith2026250524729,
  author       = {Pith},
  title        = {Pith review of: Feature Attribution from First Principles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X7A7EPDV}},
  note         = {Machine review of arXiv:2505.24729}
}
read the original abstract

Feature attribution methods are a popular approach to explain the behavior of machine learning models. They assign importance scores to each input feature, quantifying their influence on the model's prediction. However, evaluating these methods empirically remains a significant challenge. To bypass this shortcoming, several prior works have proposed axiomatic frameworks that any feature attribution method should satisfy. In this work, we argue that such axioms are often too restrictive, and propose in response a new feature attribution framework, built from the ground up. Rather than imposing axioms, we start by defining attributions for the simplest possible models, i.e., indicator functions, and use these as building blocks for more complex models. We then show that one recovers several existing attribution methods, depending on the choice of atomic attribution. Subsequently, we derive closed-form expressions for attribution of deep ReLU networks, and take a step toward the optimization of evaluation metrics with respect to feature attributions.

Figures

Figures reproduced from arXiv: 2505.24729 by the authors.

Figure 1
Figure 1. Illustration of Eq. (1). In red the function f to be approximated; in blue the approximation f10. In other words, any FSC and Linear feature attribution method for models in C 0 pr0, 1s d q is just an integral. In the setting of Section 3.1 with d “ 2, the measures tµj,xujPJ2K from Th. 3.1 take the form dµ1,xpx, yq :“ dδx1 pxqdy and dµ2,xpx, yq :“ dxdδx2 pyq, thereby recov￾ering the attribution of Equation (4). Here… view at source ↗
Figure 2
Figure 2. Description of the solution to (6) for fw with d “ 2 and w P R 2 . The green and blue areas correspond to the solution set Sj (j P J2K) of Th. 4.1 when j P D1,w. The red hatched area repre￾sent the input space. All measures such that their center of mass belong to the green or blue areas are optimal for (5). Informally, Recall measures the proportion of features retrieved by ϕ that are actually used by the model fw.… view at source ↗
Figure 3
Figure 3. Illustration of the increment operator ∆p¨; ¨q in the 2D setting. Given a rectangle R :“ ra, bs ˆ rc, ds Ă r0, 1s 2 , the 2D increment for a function g : R 2 Ñ R is defined as ∆pg; Rq :“ `gpa, cq ´ gpb, cq ´ gpa, dq ` gpb, dq. B Appendix for Section 3 B.1 Appendix for Section 3.1 In this section, we present the approximation result which underpins our constructivist approach to feature attribution. Theorem B.1 (Appr… view at source ↗

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    #´ś p‰j ş r0,1s 1 dp1ypą0q ¯ş r0,1s yk dp2ykq if k“ j ,ş r0,1s yk dp1yką0q ş r0,1s 1 dp2yjqś p‰j,k ş r0,1s 1 dp1ypą0q else. (Fubini’s Theorem) “

    Isabella Verdinelli and Larry Wasserman. Feature Importance: A Closer Look at Shapley Values and LOCO. Statistical Science, 2024. 12 Appendix Contents of the Appendix A Appendix for Section 2 13 A.1 Appendix for Section 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . ....

    2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.