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REVIEW 4 major objections 3 minor 76 references

On the Complexity-Faithfulness Trade-off of Gradient-Based Explanations

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that noisy, unfaithful gradient explanations are two views of one spectral fact: the tail of a network's power spectrum. It proposes Expected Frequency and the explanation gap as a parameter-free way to measure and trade o

desk verdict Solid empirical story about smooth activations, but the spectral trade-off framework rests on an unproven equivalence that fails even in simple linear cases. read the letter →

arxiv 2508.10490 v1 pith:WFIQ3VFC submitted 2025-08-14 cs.LG cs.AIcs.CV

Amir Mehrpanah , Matteo Gamba , Kevin Smith , Hossein Azizpour This is my paper
classification cs.LGcs.AIcs.CV
keywords gradient-basedexplanationsexplanationcomplexityfaithfulnessexpectedfrequencygappowerspectrumReLUnetworksspectralanalysis
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

The paper tries to establish that the two standard complaints about gradient-based explanations — vanilla saliency maps look noisy, and smoothing methods fix the noise by betraying the model — are one phenomenon seen through a single lens: the tail of the network's power spectrum. The authors claim that on image-like data, the high-frequency tail of the network function is directly proportional to the high-frequency tail of its input gradient, so a summary statistic of that tail, the Expected Frequency, measures explanation complexity, and its change under a surrogate method, the 'explanation gap,' measures unfaithfulness. They further claim that smoothing the ReLU activation by convolving it with a Gaussian (approximated by SoftPlus) shortens the tail, producing simpler explanations and letting a designer choose where to sit on the complexity–faithfulness curve. If the claims hold, explainability stops being a matter of taste or heuristics: the trade-off becomes a measurable, adjustable spectral property of the architecture itself.

What carries the argument

The load-bearing objects are two spectral tails and the bridge between them: the Tail of the Power Spectrum (TPS) of the network $f$ and the Tail of the Spatial Power Spectrum (TSPS) of its input gradient $\nabla f$, whose proportional decay under high spatial autocorrelation is Theorem 1. On top of these sit two summary statistics: Expected Frequency $\mathrm{EF}(\mathbf{e}_f)=\int \omega S_{\mathbf{e}_f}(\omega)\,d\omega$ (Eq. 1), a one-number measure of explanation complexity, and the explanation gap, $G(f,\tilde f)\sim\Delta\mathrm{EF}:=|\mathrm{EF}(\nabla f)-\mathrm{EF}(\mathbf{e}_f)|$ (Eq. 7), a faithfulness proxy obtained by applying Parseval's theorem so the $L^2$ gap between origina

What would settle it

Train identical architectures on an image dataset with pixels randomly permuted within each image (destroying spatial autocorrelation) and compare VanillaGrad's expected frequency against the network's power-spectrum tail across the SoftPlus-to-ReLU parameter range. Theorem 1 predicts the two tails move together; if the proportionality breaks on such inputs, the autocorrelation assumption is doing the load-bearing work. A complementary check: on natural images, craft an explained sample whose pixel-value trajectory provably avoids all training trajectories and see whether EF still tracks the n

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a proportionality of tails: for a trained network $f$ on image-like data, the tail behavior of the power spectrum of $f$ is directly proportional to the tail of the spatial power spectrum of $\nabla f$ (Theorem 1). Because a network's Fourier transform is inaccessible, the gradient's spatial spectrum is read as the aperture onto the network's spectrum. From this the paper derives two quantities in one framework: Expected Frequency $\mathrm{EF}(\mathbf{e}_f)=\int \omega S_{\mathbf{e}_f}(\omega)\,d\omega$ for explanation complexity, and the explanation gap $G(f,\tilde f)\sim\Delta\mathrm{EF}=|\mathrm{EF}(\nabla f)-\mathrm{EF}(\mathbf{e}_f)|$ f

Load-bearing premise

That the input data and the features learned on it are highly autocorrelated in space (neighboring pixels vary slowly), and that the explained sample's pixel-value trajectory crosses a training sample's; the authors state in Appendix G that the autocorrelation property 'has not been theoretically proven yet,' so if either condition fails, the tail-proportionality theorem and the explanation-gap proxy built on it lose their foundation.

Editorial extensions

If this is right

  • Explanation complexity becomes measurable as a single number, EF, and faithfulness as its change, ΔEF, so post-hoc methods can be compared on both axes at once without extra hyperparameters.
  • Smoothing the activation function (SoftPlus parameterization of ReLU) demonstrably reduces EF; model designers can trade a little accuracy for substantially cleaner VanillaGrad explanations.
  • Surrogate-based methods act as low-pass filters: SmoothGrad and GradCAM reduce EF at a measurable faithfulness cost, with GradCAM showing the largest explanation gap across architectures.
  • The trade-off is continuous rather than a post-hoc versus ante-hoc dichotomy: one can achieve zero explanation gap at reduced complexity (VanillaGrad on a smooth network), making the two goals orthogonal axes rather than a binary choice.
  • The framework is shift- and rotation-invariant (Fourier basis plus power spectrum) and insensitive to gradient magnitude after rank normalization, addressing known confounds in pixel-removal faithfulness metrics.

Reading between the lines

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

  • My extension: if Theorem 1 survives closer scrutiny, the spectral tail becomes a candidate optimization objective for architecture search and activation design — networks could be selected or regularized directly for explanation complexity, which the paper only gestures at in its limitations.
  • My extension: the ΔEF proxy presumes surrogate methods act as low-pass filters; for methods that add or redistribute high-frequency content (e.g., Integrated Gradients' path dependence), ΔEF may understate the true explanation gap, so the faithfulness axis is only reliable within the low-pass family.
  • My extension: the framework predicts that any architectural choice known to flatten the learned function's spectrum — such as batch normalization, which the paper's ablation shows amplifies the tail — will shift EF in a predictable direction, giving a cheap testable fingerprint for new architectures, including transformers, where the paper notes its Laplace-kernel assumption breaks.
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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 / 3 minor

Summary. The paper proposes a spectral framework for analyzing gradient-based explanations. It defines Expected Frequency (EF), a summary statistic of the spatial power spectrum of input gradients, and argues that the tail of a network's power spectrum (TPS) is directly reflected in the tail of the spatial power spectrum of its gradient (TSPS). On this basis it introduces a measure of faithfulness, the 'explanation gap' G, and claims that G can be approximated by the absolute change in EF, ΔEF, induced by a surrogate explanation method. The authors further study a smooth parameterization of ReLU (SoftPlus, viewed as ReLU convolved with a Gaussian) and provide experiments across Fashion-MNIST, CIFAR-10, Imagenette, and ImageNet showing that smoother activations reduce EF and that post-hoc methods such as GradCAM and SmoothGrad have larger ΔEF. The central advertised contribution is a unified, parameter-free framework in which complexity and faithfulness can be measured and traded off.

Significance. If the framework were sound, it would provide a useful unifying spectral perspective on explanation complexity and faithfulness, and the empirical observation that smoother activations reduce gradient noise is plausible and well demonstrated across several datasets and ablations. The authors deserve credit for a broad experimental study (Figs. 2, 3, 7-9, 11; Tabs. 1-4), including ablations on depth, learning rate, input size, skip connections, and batch normalization, and for releasing code. However, the central theoretical bridge is not established. Specifically, the claimed equivalence G ~ ΔEF in Eq. (7) is not derived, is false in general, and is close to a definitional construct because ΔEF is defined as a difference of EF values. The main results therefore do not support the advertised trade-off framework.

major comments (4)
  1. [Sec. 4.4, Eq. (7) and Appendix E] Eq. (7) asserts G(f, f̃) ~ ΔEF(ef) := |EF(∇f) − EF(ef)|, but G and ΔEF are different functionals. By Parseval, G = ∫ ω²|F∇f − F ef|² dω, while ΔEF = |∫ω(|F∇f|² − |F ef|²)dω|. The former depends on the cross-spectrum (phase alignment) of ∇f and ef; the latter only on their marginal power spectra. Two explanations with identical power spectra but different phase (e.g., ef = −∇f, or a spatial mirror) give ΔEF = 0 while G > 0. Appendix E does not close this gap: it only shows that for a Laplace kernel both quantities scale as O(1/b) or O(b). Same leading-order asymptotics does not imply proportionality, and the appendix never computes G from the actual difference ∇f − ef. Thus the load-bearing bridge of the paper is unsupported.
  2. [Sec. 4.1 and Sec. 4.4] The claimed complexity-faithfulness trade-off is, to a large extent, definitional. Complexity is defined as EF in Eq. (1), and the faithfulness measure is then defined as ΔEF = |EF(∇f) − EF(ef)| in Eq. (7). Tables 1 and 2 therefore report the difference between two EF values as an 'explanation gap'; this is not an independent faithfulness metric. The paper does not validate ΔEF against any established faithfulness measure (e.g., pixel removal or sanity checks), so the trade-off is a relation between a quantity and its own variation rather than an empirical finding about faithfulness.
  3. [Theorem 1, Remark 6, Eq. (23)-(24), Appendix G] The proof of Theorem 1 rests on two assumptions that are acknowledged in the paper to be unproven or only heuristic: high spatial autocorrelation of inputs and learned features (Remark 6, Eq. (23)) and the existence of an intersection between the explained sample and a training sample in pixel-value space (Eq. (24)). Appendix G explicitly states that the autocorrelation property 'has not been theoretically proven yet.' Without these assumptions, Lemma 3 and the proportionality between the network TPS and the gradient TSPS collapse. The theorem is also stated informally and proved only in a drastically simplified setting (n=1, α=1, kernel expansion with the NTK replaced by a Laplace kernel), so it does not establish the claim for general trained networks.
  4. [Appendix F, Eq. (44)] The proof of Lemma 1 contains a mathematical error. The authors define gβ with precision β, i.e., variance 1/β. After convolving the activation with gβ, the covariance of the τ-transform inputs should be σ² + 1/β on the diagonal, not σ² + β as written in Eq. (44). The subsequent claim that increasing β makes the matrix 'closer to identity' is a consequence of this erroneous variance. With the correct variance, increasing β makes the distribution closer to the original ReLU case, not closer to identity. The empirical spectrum in Fig. 11 supports the intended conclusion, but the provided proof is not valid.
minor comments (3)
  1. [Throughout] There are several typos: 'whith' (Fig. 7 caption), 'hyperparamters' (Appendix B), 'an comprehensive' (Sec. 6), 'T est' (Fig. 5), 'hypothetize' (Sec. 3.2).
  2. [Sec. 4.3, Eq. (6)] The approximation that the low-frequency integral vanishes after surrogate smoothing is not justified. Low-pass surrogates do not generally leave low frequencies unchanged, and the threshold ω* is a free parameter. This step should be clarified or removed.
  3. [Sec. 5] The validation accuracy cap is described as a way to compare under similar training budgets, but it also means that the compared networks have different accuracy levels. The ablation in Fig. 5 addresses this, but the main text should state more explicitly that the EF reductions come with accuracy trade-offs.

Circularity Check

1 steps flagged · score 6.0 of 10

The faithfulness axis is defined as a difference of the complexity metric, making the advertised trade-off a mathematical identity rather than an empirical finding; the central G∼ΔEF proxy is asserted, not derived.

  1. self definitional [Sec. 4.4, Eq. (7); Tables 1 and 2]
    "More formally, we can use the absolute change in EF caused by an explanation method as a proxy for explanation gap, expressed as: G(f, ˜f ) ∼ ∆ EF(ef ) := | EF(∇f ) − EF(ef )| (7) ... This table shows the evaluation of expected frequency in Eq. (1) and explanation gap in Eq. (7)"

    EF is the paper's own complexity measure (Eq. 1). Eq. (7) defines the faithfulness proxy ΔEF as the absolute difference of two EF values, so for a fixed original model ΔEF is a deterministic function of the surrogate's EF. Consequently, any method that lowers EF automatically has a positive 'explanation gap'; the trade-off displayed in Fig. 1 and Tables 1–2 is a consequence of the definition, not an empirical discovery. The conclusion's claim that EF and ΔEF are 'two orthogonal axes' is contradicted by construction, since ΔEF is derived from the same expected-frequency statistic. The true gap G defined in Eq. (2) involves the L2 distance between gradients, which depends on cross-spectra/phase, and is not equal to a difference of marginal power-spectrum averages; the paper does not derive G

full rationale

The paper contains substantial independent content in Theorem 1 and Lemma 1: the tail of the spatial power spectrum of the gradient is related to the tail of the network's power spectrum under explicit assumptions (Eqs. 23–24), and smoothing ReLU by Gaussian convolution is shown to affect kernel spectral decay. These results are not circular. The main circularity is in the complexity–faithfulness framework itself. The faithfulness measure used for all empirical claims is ΔEF, which is defined as a difference of EF values. Since EF is the complexity measure, any surrogate that reduces EF necessarily increases ΔEF; the reported 'trade-off' is therefore a definitional identity rather than evidence about faithfulness. The paper also states in Appendix G that the key autocorrelation assumption 'has not been theoretically proven yet,' which is an unsupported premise, and Appendix E's derivation of Eq. (7) only analyzes kernel power spectra, never computing G from the actual gradient difference ∇f−∇f̃. These issues make the central trade-off claim only partially supported, but they are not self-citation circularity. The self-citation to [41] for the low-pass-filter view of explanation methods is a normal external reference and is not load-bearing in a circular way.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The framework rests on a kernel view of trained networks, high spatial autocorrelation of images, and an unproven replacement of NTK by the Laplace kernel. The metrics themselves involve hand-chosen parameters (β, accuracy cap, frequency split) and normalization choices. No new physical or ontological entities are postulated; EF, ΔEF and the smooth parameterization are definitions and design choices rather than entities.

free parameters (3)
  • Gaussian precision β (SoftPlus smoothness) = varied over 0.1 to 50 depending on dataset
    Hand-chosen hyperparameter interpolating between SoftPlus-smoothed ReLU and standard ReLU; all frequency-complexity results are reported as a function of β.
  • Validation accuracy cap = 60% Imagenette, 70% CIFAR-10, 80% Fashion-MNIST
    Chosen by hand to equalize training budgets and prevent initialization bias; the ablation without the cap shows the spectral trend is less clear (Fig. 6).
  • Frequency threshold ω* = not specified
    Introduced in Eq. (6) to split the explanation gap into low and high frequencies; the low-frequency integral is then assumed to vanish, but no value or sensitivity analysis is given.
assumptions (4)
  • domain assumption Input and learned features have high spatial autocorrelation (concentration of spatial power spectrum near zero).
    Explicitly assumed in Remark 6 (Eq. 23) and Appendix G; authors state this is intuitive for images but not theoretically proven. Without it, Lemma 3 and Theorem 1 lose the connection between network TPS and gradient TSPS.
  • domain assumption There exists at least one intersection between the explained sample and a training sample in pixel-value space (Eq. 24).
    Needed for Lemma 3 to linearize Δ(τ) and obtain the O(ω²|k̂|²) tail; the authors argue absence is improbable but provide no proof.
  • ad hoc to paper The trained network can be represented by a kernel expansion with its NTK, and the NTK can be replaced by the Laplace kernel without loss of generality for tail analysis.
    This substitution is asserted in Appendix C and used in Lemma 1 and the Eq. (7) derivation; only qualitative similarity between NTK and Laplace kernel is cited.
  • standard math Continuity and absolute integrability of the network function so that Fourier representations exist.
    Standard Fourier analysis assumption stated in Sec. 3.1; appropriate for continuous piecewise-linear ReLU networks.

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Pith. "Pith review of On the Complexity-Faithfulness Trade-off of Gradient-Based Explanations." pith.science (2026). https://pith.science/paper/WFIQ3VFC

@misc{pith2026250810490,
  author       = {Pith},
  title        = {Pith review of: On the Complexity-Faithfulness Trade-off of Gradient-Based Explanations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WFIQ3VFC}},
  note         = {Machine review of arXiv:2508.10490}
}
read the original abstract

ReLU networks, while prevalent for visual data, have sharp transitions, sometimes relying on individual pixels for predictions, making vanilla gradient-based explanations noisy and difficult to interpret. Existing methods, such as GradCAM, smooth these explanations by producing surrogate models at the cost of faithfulness. We introduce a unifying spectral framework to systematically analyze and quantify smoothness, faithfulness, and their trade-off in explanations. Using this framework, we quantify and regularize the contribution of ReLU networks to high-frequency information, providing a principled approach to identifying this trade-off. Our analysis characterizes how surrogate-based smoothing distorts explanations, leading to an ``explanation gap'' that we formally define and measure for different post-hoc methods. Finally, we validate our theoretical findings across different design choices, datasets, and ablations.

Figures

Figures reproduced from arXiv: 2508.10490 by the authors.

Figure 1
Figure 1. Exploring the Faithfulness-Complexity Trade-Off. This figure shows the trade-off typically made between faithfulness and complexity in post-hoc explanation methods. (b) presents VanillaGrad (VG) explanations for a ReLU network, which appear grainy due to the network’s reliance on high-frequency information. To mitigate this, post-hoc methods like SmoothGrad (SG), shown in (c), remove high-frequency components by cre… view at source ↗
Figure 2
Figure 2. ReLU Affects Gradient TSPS via Network’s TPS. This figure presents a practical implication of our theory on the tail behavior of the spatial power spectrum in relation to function sharpness, as formalized in Lemma 1. The plots depict the TSPS of input-gradients of networks trained with identical architectures and hyperparameters, varying only in smoothness parameter of ReLU (indicated in the legend) and datasets: (a… view at source ↗
Figure 3
Figure 3. Impact of Smoothness Parameter on Expected Frequency. This figure shows EF, defined in Eq. (1), used as a summary statistic of the frequency content of explanations in the spatial domain with respect to the smoothness parameter β in a (log) x-axis. To obtain the 95% CI, we repeated experiments using 10 seeds for each dataset: (a) Fashion MNIST, (b) CIFAR-10, (c) Imagenette (122×122), and (d) Imagenette (224 × 224). … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Overview of Key Theoretical Connections. A graphical overview of our contributions, which is based on prior works on spectral analysis of gradient-based explanation methods. The diagram illustrates the conceptual flow of the narrative presented in this paper. From TPS …
Figure 5
Figure 5. Figure 5: Ablation Study: Impact of Smoothness Parameter on Validation Accuracy. This figure presents an ablation study of our decision to impose an accuracy cap as an early stopping mech￾anism on Imagenette (224 × 224). By relaxing this constraint, we train smooth parameterizat…
Figure 6
Figure 6. Figure 6: Ablation Study: Impact of Validation Accuracy Cap and Learning Rate on the Spatial Power Spectrum. This figure presents an ablation study on our decision to impose a validation accuracy cap as an early stopping mechanism on Imagenette (224 × 224) across different learn…
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Ablation Study: Impact of Input Size on the Spatial Power Spectrum. This figure presents an ablation study, isolating the effect of input size by training multiple models on different versions of the Imagenette dataset with varying input resolutions. The input sizes ra…
Figure 9
Figure 9. Figure 9: Ablation Study: Effect of Skip Connections and Batch Normalization on the Spatial Power Spectrum Tail. This figure illustrates the impact of skip connections and batch normal￾ization on the tail of the spatial power spectrum on Imagenette (224 × 224). In (a) the power …
Figure 10
Figure 10. Figure 10: Illustration of Kernel Sharpness on the Spatial Power Spectrum Tail. A cartoon illustration depicts the theory regarding the effect of sharpness of the learned features on the tail of the power spectrum. (a) and (b): The x-axis represents a simplified version of spati…
Figure 11
Figure 11. Figure 11: Spectral Decay of the Empirical NTK Across Smooth Parameterizations of ReLU. This figure depicts the spectral decay of the empirical neural tangent kernel NTK, plotting the eigenvalues (x-axis) against their normalized magnitudes (y￾axis). This empirical evaluation su…
Figure 12
Figure 12. Figure 12: Impact of Low Spatial Autocorrelation on Spatial Power Spectrum of Gradient. This figure illustrates the impact of low spatial autocorrelation in the input and, consequently, in the learned features on the tail behavior of the spatial spectral density of the gradient.…
Figure 13
Figure 13. Figure 13: Sample Visualization of Explanations Outputs After Inverse Transformation Normalization. This figure presents a visualization of the outputs from various explanation methods after applying the inverse transformation method. This normalization technique aligns differen…
Figure 14
Figure 14. Figure 14: Sample Visualization of VanillaGrad Explanations Across Smooth ReLU Parameterizations. This figure presents addi￾tional samples from trained networks with different smooth parameterizations of ReLU, denoted by SP(β). The goal is to illustrate how the reliance on high-…

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