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REVIEW 4 major objections 5 minor 62 references

On Spectral Properties of Gradient-based Explanation Methods

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Gradient-based explanations act as frequency-band selectors, so the perturbation scale — not per-pixel importance — decides which features get credited.

desk verdict A genuinely new spectral formalism for gradient explanations, but the σ-dependent band-pass result is derived under an assumption that the paper itself says only holds in the small-noise regime, where the predicted σ-dependence should vanish. read the letter →

arxiv 2508.10595 v1 pith:3CJWZY2T submitted 2025-08-14 cs.LG cs.AIcs.CV

Amir Mehrpanah , Erik Englesson , Hossein Azizpour This is my paper
classification cs.LGcs.AIcs.CV
keywords explainabilitygradient-basedexplanationsspectralanalysisSmoothGradband-passfilterperturbationscalesquaredgradientsLens
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

This paper tries to establish that gradient-based explanation methods, despite appearing to score individual pixels, actually act as frequency-selective filters over the model's decision boundary. The claim is that the gradient operator amplifies high frequencies (high-pass) while the input perturbation smooths them away (low-pass), so every gradient-plus-perturbation explanation is shaped by a band-pass filter whose center and width are controlled by the perturbation hyperparameters. That is why SmoothGrad-style explanations change radically with the noise scale: each scale reports a different frequency band, not a stable per-pixel ranking. The paper then derives practical consequences: squared gradients are a principled choice because they recover the power spectrum of the model rather than a signed, phase-sensitive quantity; a cosine-similarity criterion fixes a standard perturbation scale; and an aggregation method, SpectralLens, sums attributions over the full spectrum. If correct, this gives a formal explanation for the widely observed hyperparameter sensitivity of these methods and a way to reduce it.

What carries the argument

The central object is the spectral representation of an explanation: writing the expected gradient E[∇_{x̃}f(x̃)] and the expected squared gradient E[(∇_{x̃}f(x̃))^2] in the Fourier domain via the convolution theorem and Plancherel identity. This converts each explanation into the pointwise product of three spectra: the classifier's decision-boundary spectrum, the gradient operator's frequency response, and the perturbation kernel's spectrum. That product is the band-pass filter; the paper's propositions about inconsistency, its optimal perturbation scale, and its SpectralLens aggregation are all read off this expression.

What would settle it

Train a small network with a known decision boundary containing one high-frequency and one low-frequency feature of equal total contribution. Compute SmoothGrad-squared at several Gaussian noise scales. If the attribution ranking does not cross near the predicted threshold derived from the two frequencies and the Gaussian variance—or if the measured Fourier spectrum of the explanation does not show the predicted band-pass shape—the central claim is wrong.

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Extended reading notes

Core claim

The paper's central claim is that any explanation built from a gradient evaluated at perturbed inputs is a frequency-band selector, not a direct measurement of per-pixel contribution. In the Fourier domain, the expected gradient and expected squared gradient factor into the classifier's spectrum, a term from the gradient operator that grows with frequency, and the perturbation kernel's spectrum, which decays with frequency; their product is a band-pass filter whose peak location and width are set by the perturbation hyperparameters. The Fourier transform here is taken per pixel along the pixel-value axis, not across spatial dimensions. For SmoothGrad-squared with a Gaussian kernel, the attri

Load-bearing premise

The whole spectral derivation assumes the model's response to perturbed inputs is linear and convolutional, E[f(x+ξ)] = (f * κ)(x), so Fourier identities apply; the paper itself notes this is only valid when the perturbation stays near the sample, yet the band-pass formulas are applied to SmoothGrad at noise scales large enough that the concentration assumption fails.

Editorial extensions

If this is right

  • SmoothGrad and similar methods do not rank pixels by contribution; each explanation is the decision boundary's content within one frequency band selected by the noise scale.
  • Changing the noise scale changes which band is selected, so the same pixel can appear globally important at one scale and irrelevant at another; the paper formalizes this as a Rashomon-style inconsistency result.
  • Squared gradients are theoretically safer than gradients: they are nonnegative, carry phase-free power-spectrum information, and avoid destructive interference when averaging over hyperparameters.
  • Cosine similarity between the kernel's spectrum and the classifier's spectrum identifies a canonical perturbation scale, restoring reproducibility to SmoothGrad-squared.
  • Aggregating attribution across noise scales (SpectralLens) and reporting the scale of maximal attribution (ArgLens) gives a more complete picture than any single hyperparameter setting.

Reading between the lines

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

  • One testable extension is to use the band-pass view as a design tool: choosing a perturbation kernel whose spectrum matches a desired frequency band would let users deliberately expose high- or low-frequency features instead of searching noise scales by trial and error.
  • The same spectral decomposition suggests that evaluation metrics themselves are frequency-filtered: patch-based pixel removal acts as a spatial low-pass filter, so current benchmarks may systematically favor explanations of a certain bandwidth; a fair comparison would need to control for this.
  • If the linear-response assumption is relaxed, the band-pass picture likely becomes approximate; measuring how the explanation spectrum deviates from the predicted product at large noise scales could quantify when the formalism stops being predictive.
  • ArgLens might serve as a per-pixel robustness diagnostic: since high-frequency decision-boundary components emerge late in training and are unstable, the scale of maximal attribution could indicate which pixels rely on fragile features.
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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 / 5 minor

Summary. The paper proposes a probabilistic and spectral framework for gradient-based explanation methods. It models explanations as expectations of (powers of) gradients over input perturbations, derives spectral identities for gradient and squared-gradient explainers, and argues that the gradient acts as a high-pass filter while the perturbation acts as a low-pass filter, so their combination produces a band-pass filter controlled by the perturbation scale. This is used to explain hyperparameter-induced inconsistencies in SmoothGrad/SmoothGrad-Squared, to propose an optimal perturbation scale via cosine similarity between the perturbation kernel and the classifier PSD, and to introduce an aggregation method called SpectralLens. The claims are evaluated with insertion/deletion scores on ImageNet and Food101 using ResNet-50 and ViT-B/16.

Significance. The framework is original and, if the spectral identities were correct, would give a useful formal vocabulary to a largely heuristic area. The non-squared identity E[∇f(x+ξ)] = ∇(f*℘) is exact under mild regularity, and the high-pass/low-pass separation is intuitive and well connected to known phenomena. The paper also provides code and a reproducible evaluation pipeline, which is a strength. However, the squared-gradient identity, which drives the band-pass claim, the Rashomon-effect proposition, and the optimal-scale proposal, is not established: the derivation in Appendix A.3 computes a different quantity from the pointwise expectation used by SmoothGrad-Squared. Until that mismatch is corrected, the theoretical contributions are conditional; the empirical results remain interesting but do not by themselves validate the spectral mechanism.

major comments (4)
  1. [§3.1 and Appendix A.3, Eq. (3)] Eq. (3) is derived in Appendix A.3 as ∫ |(f' * φ)|² dx, the total L2 energy of the gradient of the smoothed response. The explanation method defined in Sec. 2 is the pointwise expectation E[(∇f(x+ξ))²] at a fixed input x. These differ: E[(∇f(x+ξ))²] = (∇(f*φ)(x))² + Var(∇f(x+ξ)), and the spectral formula averages away the phase needed for pixel localization. For f(v)=sin(ω0v), the true SG2 attribution is ω0²/2, independent of σ, whereas Eq. (4) predicts e^{-σ²ω0²} attenuation. Thus Eq. (4) does not describe SmoothGrad-Squared as defined and used in Sec. 5. The band-pass interpretation, Proposition 1, and the optimal-scale criterion (Eq. (6)) all rest on Eq. (4), so this is load-bearing.
  2. [§3.3, Eq. (4) and Appendix A.1] The stated validity regime is in tension with the predicted σ-dependence. Appendix A.1 says the linear-response assumption is valid when the perturbation distribution is concentrated near the sample. In that regime ∇f(x+ξ) ≈ ∇f(x), so E[(∇f(x+ξ))²] ≈ (∇f(x))², independent of σ; the e^{-σ²ω²} factor in Eq. (4) vanishes. When σ is large enough for the gradient to vary appreciably, the convolution/linear-response model is no longer a controlled approximation. Since Sec. 5 and Appendix E apply Eq. (4) at noise scales where this concentration is strained, the experiments do not isolate this gap.
  3. [§3.3, Proposition 1 and Appendix B] The Rashomon-effect proof assumes equal total contribution across pixels and single-frequency per-pixel spectra. These are not properties of trained networks; the proof only exhibits a toy model. Together with the Eq. (3) issue, it does not support the text's claim that this explains inconsistencies 'encountered in SG2 explanations.' The proposition should either be reframed explicitly as a minimal existence example or be supplemented with a perturbation-stability argument around those assumptions.
  4. [§4.1, Eq. (6)] The proposed optimal scale maximizes the overlap between the Gaussian kernel PSD and the classifier's own PSD. This selects the frequency band where the classifier's spectral energy concentrates, not a scale that maximizes explanation faithfulness. Since the criterion is built on Eq. (3), which is not the correct SG2 expression, the theoretical basis for the standardized scale is missing. The empirical gains in Table 1 do not isolate this mechanism, because SG2_opt changes the σ selection relative to the baseline heuristic, and the comparisons do not vary σ while holding all else fixed.
minor comments (5)
  1. [§2] The notation φ_E(x̃) is used before the perturbation distribution and explainer are fully defined; please make the three components (distribution, explainer, summary statistic) explicit in the notation.
  2. [§4.1, Eqs. (7)-(8)] The computation of the kernel norm is terse. Please state the normalization convention explicitly and explain why the L1 norm of flattened squared gradients is used for the classifier term.
  3. [Fig. 4 caption] The phrase 'When using a Dirac explanation prior in Eq. (9), where a perturbation level is assigned per image or dataset' is confusing because Eq. (9) defines SpectralLens. Clarify the connection between the explanation prior and the similarity computation.
  4. [Appendix C] The translation of RISE and Occlusion Sensitivity into the framework omits random shifts and spatial dependencies. Please state explicitly that this is a simplification and discuss whether those details affect the derived PSD.
  5. [§5] The statement that CIFAR is omitted because its resolution restricts the potential gap between high- and low-frequency features is asserted without evidence. Either support it with a small experiment or soften the claim.

Circularity Check

1 steps flagged · score 4.0 of 10

Central spectral band-pass derivation is self-contained; one secondary 'optimal perturbation scale' result is definitional (in-sample fit to the classifier PSD) and contributes partial circularity.

  1. self definitional [Sec. 4.1, Eq. (6), Remark 6]
    "Assuming unimodality, a nonzero contribution, and that the norm || |f̂|² ||₂ exists, then the cosine similarity is maximized at a certain perturbation scale that has the highest similarity with f. A higher cosine similarity guarantees recovering more information about the classifier."

    Eq. (6) defines the 'optimal' scale as the maximizer of cosine similarity between the perturbation-kernel PSD and the classifier PSD. The squared-gradient explanation in Eq. (3) is precisely this inner product (up to normalization), so Proposition 5's claim that the maximizer 'guarantees recovering more information' is an identity by construction, not a derived consequence. The scale is then fit on the same images used for evaluation (Fig. 4/Tab. 1), making the recommended 'standard' an in-sample optimization of the very quantity it is said to maximize. This is a secondary remedy; the band-pass spectral analysis in Sec. 3 is independent of this step.

full rationale

The paper's main theoretical result — that gradient-based explanations act as high-pass filters and perturbation kernels as low-pass filters, forming a band-pass with scale-dependent mode — is derived from Fourier identities (derivative operator, convolution theorem, Plancherel/Wiener-Khinchin) under an explicitly stated linearity-of-response assumption (Appendix A.1). That assumption is a modeling premise, not an output of the derivation, and the paper flags its validity is coupled to perturbation concentration. The spectral representations (Eqs. (1)-(4)) are self-contained given that premise and are not circular. No load-bearing self-citation is present: the only author self-reference ([25]) is a forward-looking connection to double descent, not a justification of the main results. The one circular element is the proposed 'optimal perturbation scale' in Sec. 4.1: the optimality criterion is defined as the cosine similarity between the kernel PSD and the classifier PSD, which is exactly the quantity the squared-gradient explanation computes, and the scale is tuned/evaluated on the same datasets. This makes the optimal-scale recommendation definitional rather than independently predictive, but it does not infect the central band-pass derivation. Hence partial circularity, score 4.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivations rest on the linear-response convolution model, standard Fourier analysis, and several simplifying assumptions specific to the proofs. The proposed optimal scale and SpectralLens involve design choices (uniform prior, empirical norm constancy) rather than derived necessities. No new physical entities are introduced.

free parameters (3)
  • Gaussian perturbation scale sigma (SmoothGrad/SG2) = Per-image or per-dataset argmax of cosine similarity; heuristic 0.1-0.3 for baseline
    Central hyperparameter. The paper proposes selecting it by maximizing Eq. (6), a data-dependent fit to the classifier's PSD, and evaluates SG2_opt at that scale.
  • Maximum perturbation scale cutoff = Set just before the anomalous predictive entropy trend (Appendix E.1)
    Chosen by hand from entropy curves; defines the range of noise levels used in experiments.
  • Explanation prior pi(sigma) for SpectralLens = Uniform
    Chosen for simplicity in Sec. 4.2; a different prior would change SL2 and ArgLens outputs.
assumptions (6)
  • domain assumption Linearity of response: model output under perturbation can be written as a convolution (f * xi)(x) with the perturbation kernel.
    Appendix A.1, Eq. (11)-(13). Needed for all Fourier results; acknowledged as a strong assumption, but used at large perturbation scales in Section 5.
  • standard math Fourier transforms exist and Plancherel theorem applies to f and the kernels.
    Appendix A.1, citing Plancherel theorem. Standard regularity condition.
  • domain assumption The cosine similarity objective is unimodal and the kernel norm exists, so a unique optimal perturbation scale exists.
    Section 4.1, before Eq. (6). Not proven; needed to justify the proposed scale selection.
  • ad hoc to paper For Proposition 1 (Rashomon effect), all pixels contribute equally in total and each feature contributes at a single frequency.
    Appendix B, Proof of Proposition 1. Toy assumption used to exhibit inconsistency; not a general proof of pervasiveness.
  • ad hoc to paper For Proposition 2 (sign of gradient), contributions are equal and even, with a specific spectral form.
    Appendix B, Proof of Proposition 2. Used to show sign flips can come from the kernel.
  • domain assumption The norm N_f stays constant when using a convex interpolation with noise.
    Appendix E.2. Empirically supported by observing dependence on input scale; needed to compute cosine similarity in Eq. (6).

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Cite this review

Pith. "Pith review of On Spectral Properties of Gradient-based Explanation Methods." pith.science (2026). https://pith.science/paper/3CJWZY2T

@misc{pith2026250810595,
  author       = {Pith},
  title        = {Pith review of: On Spectral Properties of Gradient-based Explanation Methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CJWZY2T}},
  note         = {Machine review of arXiv:2508.10595}
}
read the original abstract

Understanding the behavior of deep networks is crucial to increase our confidence in their results. Despite an extensive body of work for explaining their predictions, researchers have faced reliability issues, which can be attributed to insufficient formalism. In our research, we adopt novel probabilistic and spectral perspectives to formally analyze explanation methods. Our study reveals a pervasive spectral bias stemming from the use of gradient, and sheds light on some common design choices that have been discovered experimentally, in particular, the use of squared gradient and input perturbation. We further characterize how the choice of perturbation hyperparameters in explanation methods, such as SmoothGrad, can lead to inconsistent explanations and introduce two remedies based on our proposed formalism: (i) a mechanism to determine a standard perturbation scale, and (ii) an aggregation method which we call SpectralLens. Finally, we substantiate our theoretical results through quantitative evaluations.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.