Relative Layer-Wise Relevance Propagation: a more Robust Neural Networks eXplaination

Eric Nyiri; Olivier Gibaru

REVIEW 3 major objections 4 minor 41 references

Relative Layer-Wise Relevance Propagation: a more Robust Neural Networks eXplaination

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a new relative layer-wise relevance propagation rule, R-LRP, explains dense, convolutional, and residual image classifiers without dividing by near-zero values or tuning hyperparameters, and that it outperforms…

desk verdict R-LRP is a simple, hyperparameter-free attribution rule that empirically looks strong, but the paper's central conservation-law claim is false: for a linear layer the relevance sum is y^2/N, not c*y, so the residual normalization rests on a broken premise. read the letter →

arxiv 2501.14322 v1 pith:PRTT37Y7 submitted 2025-01-24 cs.LG cs.AI

Eric Nyiri , Olivier Gibaru This is my paper · ORCID

classification cs.LGcs.AI

keywords Layer-WiseRelevancePropagationeXplainableMachineLearninginterpretabilityneuralnetworkattributionsaliencymapsimageclassificationresidualnetworkshyperparameter-free

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 proposes a new layer-wise relevance propagation rule, R-LRP, that assigns each input pixel a share of a chosen network output without ever dividing by the neuron pre-activations that make classic LRP0 unstable. The authors claim R-LRP needs no hyperparameters and, when the lowest-relevance inputs are masked out, it preserves classification accuracy better than LRP0, LRP-epsilon, LRP-gamma, and LRP-alpha-beta on MNIST, a cat-versus-dog benchmark, and ImageNet-scale data with VGG16, VGG19, and ResNet50. If the claim holds, explainability for deep image classifiers becomes simpler and more reliable because the same rule works across dense, convolutional, and residual architectures. The paper also argues that negative relevance values carry useful signal, so taking absolute values of the contributions improves object localization.

What carries the argument

The central object is the relative contribution formula $z_{i,j,k}^{(l)} = \frac{1}{M_j^{(l+1)}} w_{i,j}^{(l+1)} x_i^{(l)} z_{j,k}^{(l+1)}$, followed by a layer-wise averaging step $z_{i,k}^{(l)} = \frac{\mathrm{Card}(J)}{N^{(l+1)}} \sum_J z_{i,j,k}^{(l)}$. It replaces the ratio-based LRP rule (which divides by the pre-activation sum over inputs) with a multiplication-only recurrence, so near-zero denominators never appear. The factor $\mathrm{Card}(J)/N^{(l+1)}$ rescales the sum of contributions so that the conservation law is preserved up to a global multiplicative constant, and the same machinery is extended to convolutions (via transposed convolutions), pooling, and residual blocks with a normalization of skip-connection and residual-path contributions.

What would settle it

A controlled benchmark with known discriminative features (for example, a synthetic dataset where a specific image region determines the class) would settle the claim: if R-LRP's top-k pixels do not localize that region with higher precision than random selection or LRP0, the reported superiority of R-LRP would be an artifact of the masking protocol.

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Extended reading notes

Core claim

R-LRP computes the contribution of neuron i in layer l to output k through a backward product of weights, activations, and already-computed upper-layer relevance, normalized by the number of inputs to each neuron and averaged over connected outputs, as in equations (6)–(7). Because no denominator is formed from the network's weighted sums, the rule does not blow up when a neuron's pre-activation is near zero. The authors state that the sum of all input contributions equals the selected output up to a single multiplicative factor, so the classical LRP conservation law holds in relative form. On their masked-image accuracy benchmarks, R-LRP outperforms the standard LRP variants at small keep-fractions, and it produces more object-focused maps in pointing-game and distance-to-mask tests.

Load-bearing premise

The evaluation assumes that a method is a better explanation if keeping only its highest-relevance inputs and zeroing the rest preserves classification accuracy.

Editorial extensions

If this is right

R-LRP can be dropped into existing dense, convolutional, and residual image classifiers without tuning epsilon, gamma, alpha, or beta, removing a common barrier to adopting LRP.
Explanations from R-LRP should remain stable in very deep networks because no stage of the backward pass divides by a quantity that can approach zero.
The masked-image accuracy protocol used in the paper, if it is accepted, provides a quantitative ranking of attribution methods that does not depend on a human looking at heatmaps.
Cross-network comparisons become feasible: the paper uses R-LRP to show that VGG16 and VGG19 rely on similar pixels while ResNet50 relies on different, more conceptual regions.
Absolute-value relevance can be reported as an additional map that highlights negative contributors, which the paper shows improves object localization.

Reading between the lines

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

The top-k masking protocol may reward concentrated maps even when they are not faithful to the network's actual decision; a fairer test would compare against a random pixel baseline and use multiple masking strategies.
Because R-LRP's conservation law holds only up to a global factor, the raw relevance values are not directly comparable across different output classes; a calibration step would be needed for cross-class comparisons.
The normalization step for residual blocks is heuristic, and its validity for very deep networks with many branches could be tested on architectures that are even deeper than ResNet50.
A natural and testable extension is applying R-LRP to transformer-based architectures, which the authors explicitly list as future work; whether the relative-relevance rule survives attention layers is an open empirical question.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's my take on the R-LRP paper. The core idea is genuinely simple: propagate the output back through the network by multiplying by weights and dividing by the layer width, never by the layer's weighted sum. That avoids the numerical blowups that plague LRP-0 and eliminates the need for epsilon/gamma/alpha-beta tuning. The authors ship code, test on MNIST, cat-vs-dog, and ImageNet-style data with multiple architectures, and report consistent improvements in top-k masking accuracy and pointing-game scores. That empirical work is real and the method is easy to implement, so as a practical baseline it has some value.

But the central theoretical claim does not survive contact with the math. In Definition 9, for a single linear layer with one output y = w^T x, the relevance sum over inputs is (1/N) * y^2 when the bias is zero, not a constant multiple of y. So Equation (2) fails even "up to a factor", because the factor y/N depends on the input. The paper asserts the conservation law holds up to a multiplicative factor and then builds the residual-block normalization (Proposition 14) on that exact equality. Since the premise is false, the residual handling is a self-referential fix, not a derivation. This is an internal inconsistency, not just a missing proof.

There's a second issue: the recursion is essentially Input*Gradient with a per-layer scaling constant and a specific seed. For linear layers the relative contribution ranking is identical to gradient*input. The paper doesn't cite or benchmark against Integrated Gradients or simple gradient*input, which overstates novelty. The ImageNet evaluation is also thin: two images per category, no error bars, assumed baseline accuracies of 90%/92%.

The masking protocol itself is a known evaluation style, so I don't fault the authors for using it; it's the interpretation that's fragile. The positive results are likely real in the sense that concentration on the object helps top-k accuracy, but that doesn't establish that the explanations are faithful.

Bottom line: the paper is a mixed bag. The method is simple and reproducible, the empirical comparisons are broad, but the load-bearing conservation claim is false and the novelty relative to Input*Gradient is undeclared. I'd send it to peer review, because a referee can pinpoint the algebra and the authors might salvage it as an empirical heuristic. But as it stands, I would not accept it as a novel scientific contribution, and I wouldn't cite it as a principled attribution method.

Referee Report

3 major / 4 minor

Summary. The paper proposes a new attribution method, Relative Layer-Wise Relevance Propagation (R-LRP), defined by Eqs. (6)-(7). The rule propagates a selected output backward through dense, convolutional, and residual layers without dividing by pre-activations, and it is claimed to involve no hyperparameters and to satisfy the classical LRP conservation law up to a multiplicative factor. The empirical sections compare R-LRP with LRP0, LRP-epsilon, LRP-gamma, and LRP-alpha-beta on a modified MNIST task, a cat-vs-dog dataset, and a subset of Synthetic ImageNet-1K with VGG16, VGG19, and ResNet50, using a top-k pixel/input masking accuracy protocol and two mask-based qualitative measures.

Significance. The proposal is attractive in spirit: the base R-LRP rule is simple, contains no learned parameters beyond the network weights, and the authors release source code. The experimental study covers several architectures and includes both quantitative and qualitative evaluations. If the central conservation-law claim were correct, R-LRP would be a valuable parameter-free alternative to existing LRP variants. However, the central theoretical claim is not correct: the sum of input relevances is not proportional to the selected output by a constant factor. The empirical superiority claim is also weakened by the chosen masking protocol and limited sampling, so the main contributions of the paper do not currently hold.

major comments (3)

[Section 2.1, Definition 9, Eqs. (6)-(7)] The claimed conservation law "up to a multiplicative factor" is false. Consider a single fully connected layer with one output neuron y = sum_i w_i x_i and bias zero. With z^{(q+1)} = y, Eq. (6) gives z_i = (1/N^(0)) w_i x_i y and Eq. (7) gives sum_i z_i = (1/N^(0)) y sum_i w_i x_i = y^2 / N^(0). The ratio (sum_i z_i)/y = y/N^(0) depends on the input through y, so there is no constant c satisfying sum_i z_i = c y for all inputs. Thus Definition 8 is not satisfied even up to a global factor, contradicting the abstract and the statement in Section 2.1 that "the sum of the inputs' contributions is equal to the selected output (up to a factor for our case)."
[Section 2.3, Proposition 14, Eq. (13)] The residual-block normalization in Eq. (13) is derived from the equality sum_N(l) z_p = sum_N(m) z_p, which is the conservation law shown above to fail for R-LRP. The rescaling factors in Eq. (13) are therefore data-dependent corrections rather than consequences of a valid conservation property. In addition, Eq. (13) divides by sums of relevances, which can be zero or arbitrarily small; this contradicts the paper's advertised property of avoiding division by small values, and the "except for Resnet skip connection" qualification does not cover these normalization denominators. The residual-block handling is therefore not soundly justified.
[Sections 3-5, evaluation protocol] The main empirical evidence for superiority is the top-k masking protocol in which all but the most relevant inputs/pixels are set to zero and the network accuracy is measured. This protocol rewards methods that produce concentrated maps, regardless of whether the selected regions correspond to the actual decision process, and it does not compare against any ground-truth attribution. The ImageNet-1K evaluation in Section 5 uses only two random images per category, and no error bars or per-image variance are reported. These limitations are partially acknowledged only for the qualitative mask-based evaluation in Section 6, but they also affect the quantitative claims of "better results" in Section 7. As presented, the empirical evidence does not support the claim that R-LRP is superior to other LRP methods in general.

minor comments (4)

[Title and Abstract] The word "eXplaination" in the title should be "Explanation."
[Authors' affiliations] The affiliation text "partial secondment at Buawei" appears to contain a typo; it should likely read "Huawei."
[Section 4.1] The dataset attribution "Charm Myae Zaw and all" is unclear; a formal Kaggle citation or a URL should be provided.
[Definition 9, Eq. (7)] The factor Card(J)/N^{(l+1)} in Eq. (7) is redundant when J is the full set of neurons in layer l+1, and its meaning for boundary positions in padded convolutions is not defined; the relationship between J and N^{(l+1)} should be clarified.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-definitional normalization in the residual-block rule; the core R-LRP definition and empirical comparisons are independent.

self definitional [Section 2.3, Proposition 14, Eq. (13)]
"From the conservation law (equation 2), we can state that, X_N(l) z (l) p = X_N(m) z (m) p , ∀(l, m) ∈ {0, ..., q} × {0, ..., q} Thus, the normalization consists in having the same value for sums at the end and the beginning of each part of the residual building block. ... Based on the conservation law property, we then define the contribution of the inputs of a learning block by z(l) i,j = z1(l) i,k P_N(m) z1(m) p / P_N(l) z1(l) p + z2(l) i,k P_N(m) z2(m) p / P_N(l) z2(l) p (13)"

Equation (13) is not derived from Eq. (6); it is constructed by multiplying each branch's start-of-block relevance by the ratio of that branch's end-of-block sum to its start-of-block sum. Therefore the block-level conservation equality is guaranteed by algebraic identity, i.e. the proposition's 'conservation law' is inserted as the normalization rule rather than obtained from the R-LRP propagation rule. The wording 'Based on the conservation law property' makes the target property an input to the definition. This is a self-definitional patch confined to residual blocks; the dense/convolutional R-LRP formula and the benchmark comparisons do not reduce to it.

full rationale

The core R-LRP rule (Eqs. 6-8) is a closed-form backpropagation formula using only network weights, activations, and a backward relevance variable; no parameter is fitted to the data whose explanations are evaluated, and no external self-cited result is needed to define it. The empirical claims (Sections 3-5) are tested against external datasets with a uniform masking protocol for all methods, and Section 6 even reports categories where R-LRP does not localize well, so the claimed superiority is not enforced by construction. The one exhibitable circular element is Proposition 14/Eq. (13), where residual-branch contributions are rescaled by their own sum ratios, making that block's conservation property definitional rather than derived; this is a secondary, disclosed normalization and does not drive the main empirical comparisons. The skeptic's algebraic objection to the 'up to a factor' claim for dense layers is a correctness concern about a false assertion, not a circularity: a false theorem is still non-circular. Overall, the central R-LRP contribution remains independently testable, so the score is 2 rather than higher.

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

R-LRP introduces no free parameters and no invented entities. The method rests on the DAG model of feed-forward networks, the (unproven and false) conservation law used in residual normalization, and an implicit treatment of ReLU as identity in the backward pass, since Equation (6) multiplies by the input activation x_i without including the activation derivative.

assumptions (3)

standard math The DAG representation of feed-forward neural networks (Definition 1) supports a unique backward propagation path for relevance.
Used throughout Section 1.1 to define network architectures and the R-LRP recursion.
domain assumption The conservation law (Definition 8) holds for LRP methods and is used to justify normalization in R-LRP.
Invoked in Section 2.3 (Proposition 14) to normalize residual and skip connection contributions; R-LRP itself is claimed to satisfy it up to a factor, which is not proven and is in fact false.
domain assumption Activation functions are piecewise linear (ReLU) or can be treated as identity in backward relevance computation.
The R-LRP formula in Equation (6) uses x_i^(l) directly without including the derivative of the activation function, effectively treating ReLU as identity.

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Cite this review

Pith. "Pith review of Relative Layer-Wise Relevance Propagation: a more Robust Neural Networks eXplaination." pith.science (2026). https://pith.science/paper/PRTT37Y7

@misc{pith2026250114322,
  author       = {Pith},
  title        = {Pith review of: Relative Layer-Wise Relevance Propagation: a more Robust Neural Networks eXplaination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRTT37Y7}},
  note         = {Machine review of arXiv:2501.14322}
}

read the original abstract

Machine learning methods are solving very successfully a plethora of tasks, but they have the disadvantage of not providing any information about their decision. Consequently, estimating the reasoning of the system provides additional information. For this, Layer-Wise Relevance Propagation (LRP) is one of the methods in eXplainable Machine Learning (XML). Its purpose is to provide contributions of any neural network output in the domain of its input. The main drawback of current methods is mainly due to division by small values. To overcome this problem, we provide a new definition called Relative LRP where the classical conservation law is satisfied up to a multiplicative factor but without divisions by small values except for Resnet skip connection. In this article, we will focus on image classification. This allows us to visualize the contributions of a pixel to the predictions of a multi-layer neural network. Pixel contributions provide a focus to further analysis on regions of potential interest. R-LRP can be applied for any dense, CNN or residual neural networks. Moreover, R-LRP doesn't need any hyperparameters to tune contrary to other LRP methods. We then compare the R-LRP method on different datasets with simple CNN, VGG16, VGG19 and Resnet50 networks.

Figures

Figures reproduced from arXiv: 2501.14322 by the authors.

**Figure 1.** Softmax implementation example in 1D. Proposition 2 (Feed-forward neural network output) Let us given a feed-forward neural network N (see Definition 1) and x ∈ R d with d = card(I) then SN (x) := {xv}v∈O is a function mapping R d into R p where p = card(O), called the output of N . The performance of these architectures is evaluated on new draws of data in the sense of probability or expectation of accuracy on sai… view at source ↗

**Figure 2.** Convolution part implementation example in 1D [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** A residual learning building block. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (28 more)

**Figure 4.** Figure 4: Images showing percentage of the most relevant pixels using LRP0 on a modified MNIST dataset using a simple dense network. Mask images are in black and white Image 1% 5% 10% 20% 50% [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Images showing percentage of the most relevant inputs using LRP0 on two images from Imagenet dataset with VGG16. 1.4 LRP modified formulas In order to avoid this problem, some LRP-methods use modified formulas to finally produce post processed heatmaps. Usually, these …

**Figure 6.** Figure 6: R-LRP problems with non-normalized sums for residual block contributions [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Contribution calculus for residual learning building block. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Images showing percentage of the most relevant pixels using LRP methods on a modified MNIST dataset using a simple CNN network.Positive contributions are in white while negative ones are in grey The evaluation gives the accuracy of the test set based on the percentage …

**Figure 9.** Figure 9: Accuracies of LRP methods over modified MNIST test dataset in terms of percentage of the most relevant pixels Although all these methods achieve quickly more than 95% of success, we naturally want the contributions of pixels to focus on the object in the input image a…

**Figure 10.** Figure 10: Images showing percentage of the most relevant pixels in absolute value using some LRP methods on a modified MNIST dataset using a simple CNN network [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: Accuracy of LRP methods over modified MNIST test dataset in terms of percentage of the most relevant pixels in absolute value 4 R-LRP validation with cat vs dog dataset In order to validate our new formula 6, we decided to use a more complex CNN than the one used for…

**Figure 12.** Figure 12: Two images, from the cat vs dog dataset, showing percentage of the most relevant inputs using some LRP methods with VGG16 based CNN network. using masked images in order to evaluate the LRP methods (figure 13). R-LRP outperforms all the other methods with an accuracy …

**Figure 13.** Figure 13: Accuracy of LRP methods over cat vs dog test dataset in terms of percentage of the most relevant inputs calculated mask images that contain a percentage of the highest contributions in absolute value, like for previous evaluation (see [PITH_FULL_IMAGE:figures/full_fi…

**Figure 14.** Figure 14: Two images, from the cat vs dog dataset, showing percentage of the most relevant inputs in absolute value using some LRP methods with VGG16 based CNN network. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

**Figure 15.** Figure 15: Accuracies of LRP methods over cat Vs dog test dataset in terms of percentage of the most relevant inputs in absolute value 4.5 Evaluations with pixels In the previous tests, we have used input values, which means that, for one pixel, we masked separately each input c…

**Figure 16.** Figure 16: Two images, from the cat Vs dog dataset, showing percentage of the most relevant pixels in absolute value using some LRP methods with VGG16 based CNN network [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]

**Figure 17.** Figure 17: Accuracy of LRP methods over cat Vs dog test dataset in terms of percentage of the most relevant pixels in absolute value images for each category from the dataset. We keep the same images for all our tests. For an image, the category taken into account for the tests …

**Figure 18.** Figure 18: percentage of the most relevant pixels in absolute value using some LRP methods with VGG16 network on some challenging images from Synthetic ImageNet-1K dataset. accuracy for these different percentage is better with our method than any other LRP method (see figure 19…

**Figure 19.** Figure 19: Accuracy of LRP methods in terms of percentage of the most relevant pixels in absolute value for VGG16 and VGG19 networks with Synthetic ImageNet-1K Because VGG16 and VGG19 networks are not so different, the test results are similar. 5.2 ResNet50 When we try to evalua…

**Figure 22.** Figure 22: figure 22.b, we can see that VGG19 and VGG16 use, in general, the same pixels to class [PITH_FULL_IMAGE:figures/full_fig_p021_22.png]

**Figure 20.** Figure 20: percentage of the most relevant pixels in absolute value using some LRP methods with ResNet50 network on some challenging images from Synthetic ImageNet-1K dataset. Evaluating the 2000 images used for the tests with the three networks, it shows that 1464 images get th…

**Figure 21.** Figure 21: Accuracy of LRP methods over ImageNet-1K dataset in terms of percentage of the most relevant pixels in absolute value using ResNet50 (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]

**Figure 22.** Figure 22: Cross comparison of networks using R-LRP. Relevant pixels for one network on ImageNet-1K are used to evaluate accuracy with another one (from left to right, VGG16, VGG19 and ResNet50) each folder. Thus, to be usable for qualitative evaluations of LRP methods, we only …

**Figure 23.** Figure 23: Mask and images with 20% of most relevant pixels using R-LRP and LRP-ab0505 for image 1040 (category 479) averaging the values over all images in a category, we get a score of the method for that category. The higher the score for a small percentage of pixels, the bet…

**Figure 24.** Figure 24: "Best" (cat 393) and "worst" (cat 928) average pixels in the mask for VGG16 [PITH_FULL_IMAGE:figures/full_fig_p024_24.png]

**Figure 25.** Figure 25: "Best" (cat 393) and "worst" (cat 928) average pixels in the mask for ResNet50 6.2 Average distances The average distance between relevant pixels and the object mask measures the dispersion of pixels in the image, and thus indicates whether relevant pixels are grouped…

**Figure 26.** Figure 26: Mask and images with 20% of most relevant pixels using R-LRP and LRP-ab0505 for image 1006 (category 277) In figure 27, we can see that R-LRP method gives better results than LRP-αβ and LRP-γ for categories 393 and 928 for VGG16 in terms of average distance from mask.…

**Figure 27.** Figure 27: "Best" (cat 393) and "worst" (cat 928) R-LRP average distance for VGG16 [PITH_FULL_IMAGE:figures/full_fig_p026_27.png]

**Figure 28.** Figure 28: "Best" (cat 393) and "worst" (cat 928) R-LRP average distance for ResNet50 [PITH_FULL_IMAGE:figures/full_fig_p026_28.png]

**Figure 29.** Figure 29: Mask and images with 20% of most relevant pixels using R-LRP and LRP-ab0505 for Image 21098 (category 486) Thus, as it was indicated previously, quality evaluations provide just indicative elements on the pixels defined as relevant by a method, with a human view. A me…

**Figure 30.** Figure 30: Mask and images with 20% of most relevant pixels using R-LRP and LRP-ab0505 for Image 1245 (category 928) for ResNet50 one (see figures 19 and 21). It seems that in terms of quality, R-LRP gives the best results for VGG16 (resp. VGG19) and good results for ResNet50. 7…

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