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Value bounds and Convergence Analysis for Averages of LRP attributions

T0 review · 3 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Averaging LRP attributions converges with constants that ignore network weight size.

desk verdict Weight-independent convergence bound for LRP-beta is a new result worth engaging with, but the abstract overclaims, Theorem 3 is wrong, and Lemma 7 relies on an unstated denominator assumption that likely fails for ReLU networks. read the letter →

arxiv 2509.08963 v1 pith:PUYOVD6R submitted 2025-09-10 cs.LG cs.CV

Alexander Binder , Nastaran Takmil-Homayouni , Urun Dogan This is my paper · ORCID
classification cs.LGcs.CV
keywords Layer-wiserelevancepropagationLRP-betaattributionaveragingHoeffdinginequalityweight-normindependenceconvergenceboundstest-timeaugmentationSmoothGrad
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 proves that when you average Layer-wise relevance propagation (LRP) attributions over many randomly perturbed copies of an input—the recipe behind test-time augmentation and SmoothGrad-style smoothing—the average converges to its expectation at rate O(1/√m) with a constant that does not depend on the size of the network weights for the LRP-β rule. This contrasts with gradient-based attributions, whose convergence constant grows with the product of per-layer weight norms. The key step is a component-wise bound (Lemma 7) showing that after t layers the range of LRP-β attribution values is at most 2^{t-1}(1+β)^t for positive contributions and 2^{t-1}β(1+β)^{t-1} for negative ones, yielding the weight-independent total range 2^{n-1}(1+2β)(1+β)^{n-1} for β>0 and 1 for β=0. For LRP-γ the bound holds only under a condition relating γ to the ratio of positive and negative layer activations. Experiments on large vision models show empirically faster convergence for LRP than for gradients, even after ℓ2-normalization.

What carries the argument

The 'relevance-conserving' transition matrix M(g) for a layer collects the LRP attribution of each output neuron over its inputs; conservation means every column sums to 1, a generalization of stochasticity to non-square matrices that may contain negative entries. Lemma 7 propagates signed bounds on the attribution sums from layer to layer using only these column-sum identities and the separate normalization of positive and negative contributions in the β-rule, producing the weight-independent range bound (31). The singular-value analysis in Theorems 3 and 4 provides an earlier, looser bound and highlights the contrast with Jacobian-based gradient norms.

What would settle it

Construct or find a network with a dead ReLU unit (all pre-activations ≤ 0), run LRP-β, and check whether the attribution column sums still equal 1 and whether any attribution component exceeds the claimed range 2^{n-1}(1+2β)(1+β)^{n-1}. More directly, simulate a small random network where every neuron has nonzero positive and negative denominator sums and verify numerically that the range bound holds; a single counterexample would disprove the lemma.

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Extended reading notes

Core claim

The central claim is that LRP-type attribution maps can be represented as products of modified-gradient transition matrices, and for the relevance-conserving LRP-β rule the value range of every attribution component after backpropagating through t layers is bounded by 2^{t-1}(1+β)^t (positive part) and 2^{t-1}β(1+β)^{t-1} (negative part)—a bound that depends only on depth and the parameter β, not on weight magnitudes. Feeding this range into a standard concentration inequality for sums of bounded variables gives an O(1/√m) convergence rate for averages of LRP-β attributions whose multiplicative constant is free of weight norms, unlike the gradient bound which scales as 2L^{n-1}∏‖W_l‖_2. The

Load-bearing premise

The proof requires every column of each LRP-β transition matrix to sum exactly to 1, which needs both the positive and negative denominator sums to be nonzero at every neuron; for ReLU networks with dead or all-negative pre-activations the β-rule is undefined and the weight-independent bound can fail.

Editorial extensions

If this is right

  • Smoothing LRP-β attributions—via SmoothLRP or test-time augmentation—needs no extra samples to compensate for growing weight magnitudes, because the range bound in the concentration inequality does not involve weight norms.
  • The O(1/√m) rate matches gradient-based SmoothGrad, but the constant is fundamentally smaller for LRP-β, explaining prior empirical findings that LRP explanations are insensitive to top-down weight randomization.
  • For LRP-γ, convergence becomes weight-decoupled only when γ is chosen large enough to satisfy a per-layer condition on the relative scales of positive and negative activations, giving a practical guideline for setting γ.
  • The number of augmentations needed to keep a fixed deviation with probability 1−δ scales with the squared value range, so LRP-β's weight-independent range yields concrete computational savings in deep networks with large weights.
  • The bounds are stated for individual attribution components, so the same convergence guarantee applies component-wise to full attribution maps.

Reading between the lines

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

  • The component-wise bound suggests that LRP-β attribution values are effectively 'clipped' in a way that prevents the noise amplification seen in gradients; one could directly test this by comparing attribution variances across layer depths in a single network.
  • The transition-matrix formalism could be applied to other conservation-style attribution rules, such as DeepLIFT's rescale rule, to derive analogous weight-independent convergence constants.
  • For β=0 (LRP-0), the predicted per-component range lies in [0,1], implying an especially tight Hoeffding bound; experiments could compare the required sample sizes for β=0 versus β=1 in practice.
  • The LRP-γ condition hints at a principled way to choose γ during inference to guarantee fast averaging, moving beyond faithfulness-based tuning of attribution parameters.
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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

3 major / 6 minor

Summary. The paper analyzes LRP-β and LRP-γ attribution methods by representing them as products of modified gradient matrices. It derives singular-value bounds in §4 and component-wise range bounds in §5 (Lemmas 7 and 8). These bounds are combined with Hoeffding's inequality to obtain O(1/√m) convergence of averaged attribution maps, with constants independent of weight norms for LRP-β (Eq. (31)) and conditionally independent for LRP-γ. The theoretical results are complemented by an experimental comparison on ImageNet with pretrained networks under photometric and Gaussian augmentations.

Significance. If the range bounds are valid, the weight-independence result for LRP-β is a useful theoretical distinction from gradient-based and LRP-ε methods: it gives a concrete, weight-free sample-complexity estimate for SmoothLRP/test-time-averaged LRP attributions and provides a theoretical explanation for earlier observations about LRP's robustness to weight randomization. The derivations are self-contained, no parameters are fitted to the theory, and the appendix gives detailed proofs with the code in the supplement. The main risk is that the proof of Lemma 7 relies on every LRP-β denominator being nonzero, which is not guaranteed for the ReLU networks used in the experiments; this is load-bearing for Eq. (31).

major comments (3)
  1. [Theorem 3 / Eq. (17)] The proof uses Σ_b p_ab^+ = 1 and Σ_b p_ab^- = 1 for every neuron. These identities require both denominators in Eq. (7) to be nonzero. For ReLU networks, z_b ≥ 0, so a neuron whose incoming weight row satisfies w_ab ≤ 0 for all b has Σ_b (w_ab z_b)_+ = 0 for every input, making the positive LRP-β term 0/0. The paper states no fallback; if one adopts 0/0 := 0, the column sums are < 1 and the inductive steps in (85)-(86) fail. Consequently Eq. (31) is not established for the paper's own architectures, which may contain dead or all-negative-weight neurons. This is not a zero-measure event for such neurons. The authors should state a non-degeneracy assumption explicitly, or analyze a stable LRP variant (e.g., with an epsilon in the denominator) and prove the corresponding bounds.
  2. [Section 6.2 / Lemma 8] Theorem 3 is false as stated. For a column-stochastic matrix M, the computation (1/S) 1_S^T M M^T 1_S = R/S only gives a Rayleigh quotient, hence a lower bound σ_max ≥ √(R/S). It does not show that √(R/S) is a singular value, nor that 1_S/√S is a singular vector. Column stochasticity gives 1_S^T M = 1_R^T, not M 1_R ∝ 1_S. For example, M = [[0.8, 0.3], [0.2, 0.7]] is column-stochastic but M 1 = (1.1, 0.9), so 1 is not a singular vector. This theorem is not used in the main convergence argument, but it is a published mathematical claim and should be corrected to a lower bound or to a condition under which equality holds.
  3. [Section 5, Lemma 7] The experimental validation for LRP-γ is only partially covered by Lemma 8 because condition (29) is not verified for the networks and augmentations used; the paper acknowledges this in the text, but the tables present LRP-γ results without marking which cells are outside the theorem's scope. The central LRP-β claim is not affected, but the LRP-γ comparison should be explicitly labeled as exploratory in the main results, not only in the accompanying text.
minor comments (6)
  1. [Theorem 3] The typeset statement '√R√S' is ambiguous; it should be √(R/S) = √R / √S.
  2. [Lemma 7 / Eq. (28)] The notation P_{z_b ∈ Z_+} is confusing: it denotes a sum over all components b whose attribution is positive, but reads like a probability or a sum over a single component. Clarify with explicit Σ_{b: Y_+(b)} notation.
  3. [Definition 2] The sentence 'This property can be ensured for a non-zero attribution by normalization' is vague. Specify whether the authors propose renormalizing the LRP columns, adding an epsilon-rule, or restricting to architectures where Eq. (8) holds.
  4. [References] Reference [52] has a corrupted author field ('W. H. and'); it should cite Hoeffding (1963).
  5. [Tables 6.3-6.4] Several p-values are reported as 1.0 when the ratio is below 1; consider reporting exact values or 'n.s.' to avoid the appearance of a failed test with degenerate precision.
  6. [Section 6.1] The description of the experimental statistic says 'lower bound on the deviation' but the quantity in Eq. (34) is a norm of a difference of two independent averages; its expectation is not the same as the deviation from E[A(f,x)]. The text should be more explicit about why this is a lower-bound proxy, especially since the concentration theory applies to each component separately.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bounds are derived from the defining normalization of LRP-β; self-citations are motivational only.

full rationale

The paper's central claims—Lemma 7, Lemma 8, and equation (31)—are proved from the LRP-β/γ transition formulas themselves. Lemma 7's induction uses Σ_b p_ab^+ = 1 and Σ_b p_ab^- = 1, which are not fitted, imported, or predicted: they are the defining per-neuron normalizations in equations (7)–(8). Deriving consequences from the definition of the rule under analysis is not circularity; the paper does not claim to predict the weight-independent bound from data or from an external principle that already contains it. The self-citations (e.g., [8]) are used only to motivate why the result matters (empirically observed low sensitivity to top-down randomization) and to justify the ℓ2-normalization choice in experiments, not as load-bearing inputs to the proofs of Theorem 4 or Lemma 7. The paper also openly acknowledges that the LRP-γ condition (29) was not verified experimentally and that ℓ2-normalized results are not covered by Lemmata 7/8; these are honest limitations, not circular maneuvers. The unconditional validity of (31) does rest on a nonzero-denominator assumption—both sums in equation (7) being nonzero at every neuron—but this is a correctness/domain-of-definition caveat, not a circularity, and so is outside the scope of this pass. Overall, the derivation chain is self-contained: the bounds follow from the conservation properties embedded in the LRP rules, and the comparison to gradient bounds is made by an independent Hoeffding argument.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central LRP-beta claim rests on the LRP redistribution rule itself, the identity-activation backward assumption, and column-sum-to-1 denominators. No new physical or model entities are introduced. The only hand-chosen quantities are the rule parameters beta and gamma.

free parameters (3)
  • beta (LRP-beta rule parameter) = beta >= 0; experiments use beta = 0 and beta = 1
    The value-range bound and convergence constant in eq. (31) depend on beta. It is a method hyperparameter chosen by the user, not fitted to data.
  • gamma (LRP-gamma rule parameter) = gamma > 1; experiments use gamma = 100 and gamma = 1000
    The LRP-gamma bound in eq. (33) depends on gamma and on the condition (29) linking gamma to activation scales. Chosen by hand, not fitted.
  • noise variance for SmoothGrad experiments = sigma^2 = 1
    Used only in the empirical validation of SmoothGrad-type averaging, not in the theoretical bounds.
assumptions (7)
  • domain assumption LRP backward pass treats activation functions as identity maps even for non-piecewise-linear activations
    Stated in Section 3.1.2 and used to write the attribution map as a product of M-matrices in eq. (14).
  • domain assumption Batch normalization layers are fused into adjacent affine layers
    Stated in Section 3.1.2; needed for each layer to be a simple affine map g(z)=Wz+b.
  • domain assumption Each LRP-beta column sum is exactly 1, requiring nonzero positive and negative denominators in eq. (7)
    Definition 2 and proof of Lemma 7 (Appendix A.2) rely on sum_b p^+_ub = 1 and sum_b p^-_ub = 1.
  • domain assumption For LRP-gamma, condition (29) holds: gamma^{-1/2} * sum_{b:w_ab z_b<0} -w_ab z_b < sum_{b:w_ab z_b>0} w_ab z_b
    Lemma 8 requires this for all layers with positive connections; the bound in eq. (33) is conditional on it.
  • standard math Hoeffding's inequality applies to the bounded attribution components
    Known Result 6, used to convert value ranges into convergence probabilities.
  • domain assumption Activations are L-Lipschitz for the gradient-norm comparison in eq. (19)
    Used only to contrast the gradient bound with the LRP bound, not for the LRP result.
  • domain assumption Output initialization weights q_u are nonnegative and sum to 1
    Assumed in Lemmas 7 and 8 for the weighted output sum.

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Cite this review

Pith. "Pith review of Value bounds and Convergence Analysis for Averages of LRP attributions." pith.science (2026). https://pith.science/paper/PUYOVD6R

@misc{pith2026250908963,
  author       = {Pith},
  title        = {Pith review of: Value bounds and Convergence Analysis for Averages of LRP attributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUYOVD6R}},
  note         = {Machine review of arXiv:2509.08963}
}
read the original abstract

We analyze numerical properties of Layer-wise relevance propagation (LRP)-type attribution methods by representing them as a product of modified gradient matrices. This representation creates an analogy to matrix multiplications of Jacobi-matrices which arise from the chain rule of differentiation. In order to shed light on the distribution of attribution values, we derive upper bounds for singular values. Furthermore we derive component-wise bounds for attribution map values. As a main result, we apply these component-wise bounds to obtain multiplicative constants. These constants govern the convergence of empirical means of attributions to expectations of attribution maps. This finding has important implications for scenarios where multiple non-geometric data augmentations are applied to individual test samples, as well as for Smoothgrad-type attribution methods. In particular, our analysis reveals that the constants for LRP-beta remain independent of weight norms, a significant distinction from both gradient-based methods and LRP-epsilon.

Figures

Figures reproduced from arXiv: 2509.08963 by the authors.

Figure 1
Figure 1. Convergence statistics for EfficientNet-V2-S. Lower is better. First row: no normalization, photometric [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Convergence statistics for ResNet-50. Lower is better. First row: no normalization, photometric [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Convergence statistics for EfficientNet-V2-S. Lower is better. First row: no normalization, photometric [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Convergence statistics for ResNet-50. Lower is better. First row: no normalization, photometric [PITH_FULL_IMAGE:figures/full_fig_p035_4.png]
Figure 5
Figure 5. Figure 5: Convergence statistics for EfficientNet-V2-S. Lower is better. First row: no normalization, photometric [PITH_FULL_IMAGE:figures/full_fig_p036_5.png]
Figure 6
Figure 6. Figure 6: Convergence statistics for ResNet-50. Lower is better. First row: no normalization, photometric [PITH_FULL_IMAGE:figures/full_fig_p037_6.png]

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.