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Value bounds and Convergence Analysis for Averages of LRP attributions

Alexander BinderNastaran Takmil-HomayouniUrun Dogan

article2025arXiv0 citations

Establishes formal value bounds and convergence guarantees for averaged Layer-wise Relevance Propagation attributions, demonstrating that LRP-beta uniquely avoids sensitivity to weight norms in smoothed and data-augmented model explanations.

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In critical domains such as healthcare and the sciences, understanding which input features drive deep learning predictions is vital for safety, accountability, and regulatory compliance. Many common explanation frameworks rely on averaging attribution maps across multiple perturbed versions of an image, such as in test-time data augmentation and noise-smoothing methods. However, standard gradient-based explanations often suffer from severe noise, requiring large numbers of perturbations that significantly increase runtime and computational costs.

The article provides theoretical value bounds and evaluates the convergence behavior of averaged Layer-wise Relevance Propagation explanations compared to gradient-based methods. It aims to determine how many augmented samples are needed to produce stable, faithful explanations across diverse neural network architectures.

To conduct this evaluation, the authors formulated a matrix-based theoretical framework that models relevance propagation in parallel to standard gradient calculus. They then validated these theoretical bounds through empirical experiments across three standard architectures: ResNet-50, EfficientNet-V2-S, and Swin-Transformer-V2-Tiny. The analysis evaluated convergence speeds using 1,000 validation images tested across sample sizes of 25, 50, and 100 under both photometric distortions and additive Gaussian noise.

The findings demonstrate three major insights. First, mathematical derivations prove that Layer-wise Relevance Propagation under the beta rule possesses value ranges that are entirely decoupled from model weight norms, behaving analogously to gradient clipping. In contrast, standard gradient value bounds scale directly with the dimensionality and scale of network weights. Second, empirical testing confirms that unnormalized relevance propagation explanations converge dramatically faster than gradient-based alternatives, exhibiting difference ratios hundreds to thousands of times smaller across all architectures. Third, even when controlling for numerical scale using vector length normalization, relevance propagation methods consistently maintained superior or comparable stability across the majority of testing conditions.

These results demonstrate that relevance propagation produces statistically stable explanations with far fewer sample iterations than standard gradient averaging. In practical deployments, this significantly reduces the latency and compute expenses associated with model interpretability, enabling real-time explanation pipelines in high-throughput workflows. Furthermore, the findings explain why relevance propagation exhibits robust resilience against model parameter variations where gradient-based methods falter.

Practitioners implementing test-time augmentation or smoothing techniques should consider adopting Layer-wise Relevance Propagation variants to optimize computational efficiency. While the core theoretical guarantees apply to unnormalized maps and high parameter values in the gamma rule, users should carefully tune hyper-parameters to avoid numerical instability from negative activations. Future research should develop formal convergence bounds specifically for normalized attribution spaces and explore parameter optimization criteria beyond traditional explanation fidelity metrics.

Leipzig UniversityMicrosoftOtto-von-Guericke-Universität Magdeburgdata augmentationmultiplicationnumerical solutionLRP attributionsconvergence analysisvalue bounds
arXiv: 2509.08963
  • Paper: Methods for interpreting and understanding deep neural networks, Grégoire Montavon et al. (2018). It introduces Layer-wise Relevance Propagation (LRP) and its propagation rules, providing the foundational attribution framework whose numerical bounds and convergence properties are rigorously analyzed in the source.
  • Paper: Evaluating the Visualization of What a Deep Neural Network Has Learned, Wojciech Samek et al. (2015). It details practical implementations and quantitative evaluations of Layer-wise Relevance Propagation relative to gradient-based heatmaps, which directly motivates the source's theoretical analysis of LRP attribution distributions.
  • Paper: Axiomatic Attribution for Deep Networks, Mukund Sundararajan et al. (2017). It establishes the foundational axiomatic framework and gradient-path formulations for feature attribution methods, serving as essential context for understanding how LRP compares to other attribution operators.
  • Paper: Learning Important Features Through Propagating Activation Differences, Avanti Shrikumar et al. (2017). It provides crucial background on backpropagation-style score decomposition algorithms that relate directly to the modified gradient matrix representations studied in the source.
  • Paper: An Introduction to Matrix Concentration Inequalities, Joel A. Tropp (2015). It provides essential mathematical tools and inequalities for bounding singular values and spectral norms of matrix products and sums, which the source adapts to establish bounds on LRP Jacobian matrices.

No sufficiently relevant recommendations were found.

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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.

Topics

data augmentationmultiplicationnumerical solutiontest samples

Keywords

LRP attributionsconvergence analysisvalue boundsgradient-based attribution mapslayer-wise relevance propagationcomponent-wise bounds

Table of Contents

  • 1 Introduction
  • 2 Related Work
  • 3 The problem setup: Convergence problems considered
  • 3.1 Base quantities and Notation
  • 3.1.1 Gradient
  • 3.1.2 LRP
  • 4 Analysis of Singular values
  • 4.1 Comparison to the norm of the gradient attribution map
  • 5 Analysis of Value Ranges and Convergence Speed for LRP-β\beta and LRP-γ\gamma
  • 6 Experimental validation of convergence speed
  • 6.1 Experimental details
  • 6.2 Experimental results
  • 6.3 Unnormalized case, covered by the theoretical results
  • 6.4 ℓ2\ell_{2}-normalized case, not covered by theoretical results
  • 7 Conclusion
  • References
  • A Technical Appendices and Supplementary Material
  • A.1 Proof of Theorem
  • A.2 Proof of Lemma
  • A.3 Proof of Lemma
  • B Convergence Statistics for for LRP-β\beta and the gradient
  • C Convergence Statistics for LRP-β\beta and the gradient times input
  • D Convergence Statistics for for LRP-γ\gamma and the gradient

Knowls

  1. Knowl 1 — Matrix Representation and Relevance Conservation for LRP

    model/method

    For an nn-layer neural network f(x)=g(n)∘σ(n−1)∘g(n−1)∘⋯∘σ(1)∘g(1)(x)f(x) = g^{(n)} \circ \sigma^{(n-1)} \circ g^{(n-1)} \circ \dots \circ \sigma^{(1)} \circ g^{(1)}(x) with affine layers g(r)(z)=W(r)z+b(r)g^{(r)}(z) = W^{(r)}z + b^{(r)} mapping RS→RR\mathbb{R}^S \to \mathbb{R}^R and activation functions σ(r)\sigma^{(r)}, Layer-wise Relevance Propagation (LRP) computes attributions by backward matrix multiplication analogous to the chain rule of differentiation. Assuming identity mappings for activation functions in the backward pass and fusing batch-normalization into adjacent layers, the attribution map for a weighted combination of network outputs ∑uqufu(x)\sum_u q_u f_u(x) with ∑uqu=1\sum_u q_u = 1 (qu≥0q_u \ge 0) is expressed as:

    ∑uquAtt(fu,x)=q⊤M⊤(g(n))M⊤(g(n−1))⋯M⊤(g(1))(x)\sum_u q_u \text{Att}(f_u, x) = q^\top M^\top(g^{(n)}) M^\top(g^{(n-1)}) \cdots M^\top(g^{(1)})(x)

    where M(g)∈RS×RM(g) \in \mathbb{R}^{S \times R} is the modified gradient matrix defined by components Mb,a(g)=Att(ga,zb)M_{b, a}(g) = \text{Att}(g_a, z_b), representing the scalar attribution of input neuron zbz_b to output neuron gag_a.

    A modified gradient method is defined as relevance conserving if every column of M(g)M(g) sums to 1:

    1S⊤M(g)=1R⊤\mathbf{1}_S^\top M(g) = \mathbf{1}_R^\top

    where 1S∈RS\mathbf{1}_S \in \mathbb{R}^S and 1R∈RR\mathbf{1}_R \in \mathbb{R}^R are all-ones vectors.

    For LRP-β\beta (with β≥0\beta \ge 0), the local attribution is defined as:

    Att(ga,zb)=(1+β)(wabzb)+∑b′(wab′zb′)+−β(wabzb)−∑b′(wab′zb′)−\text{Att}(g_a, z_b) = (1+\beta) \frac{(w_{ab}z_b)_+}{\sum_{b'} (w_{ab'}z_{b'})_+} - \beta \frac{(w_{ab}z_b)_-}{\sum_{b'} (w_{ab'}z_{b'})_-}

    where (z)+=max⁡(z,0)(z)_+ = \max(z, 0) and (z)−=min⁡(z,0)(z)_- = \min(z, 0), satisfying ∑bAtt(ga,zb)=1+β−β=1\sum_b \text{Att}(g_a, z_b) = 1 + \beta - \beta = 1.

    For LRP-γ\gamma (with γ≥0\gamma \ge 0), the local attribution is defined as:

    Att(ga,zb)=wabzb+γ(wabzb)+∑b′(wab′zb′+γ(wab′zb′)+)\text{Att}(g_a, z_b) = \frac{w_{ab}z_b + \gamma (w_{ab}z_b)_+}{\sum_{b'} (w_{ab'}z_{b'} + \gamma (w_{ab'}z_{b'})_+)}

    which also satisfies ∑bAtt(ga,zb)=1\sum_b \text{Att}(g_a, z_b) = 1.

  2. Knowl 2 — Component-Wise Bounds and Value Range for LRP-beta Attributions

    theoretical result

    For an nn-layer neural network with output initialization weights qu≥0q_u \ge 0 satisfying ∑uqu=1\sum_u q_u = 1, the range of backpropagated LRP-β\beta attribution scores at layer n−tn-t (1≤t≤n−11 \le t \le n-1) for each feature component zb(n−t)z_b^{(n-t)} is bounded. Let Z+(n−t)={zb(n−t):∑uquAtt(gu(n),zb(n−t))>0}Z^{(n-t)}_+ = \{z_b^{(n-t)} : \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) > 0\} and Z−(n−t)={zb(n−t):∑uquAtt(gu(n),zb(n−t))<0}Z^{(n-t)}_- = \{z_b^{(n-t)} : \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) < 0\}.

    For β>0\beta > 0:

    ∑zb(n−t)∈Z+(n−t)∑uquAtt(gu(n),zb(n−t))≤2t−1(1+β)t\sum_{z_b^{(n-t)} \in Z^{(n-t)}_+} \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) \le 2^{t-1}(1+\beta)^t

    ∑zb(n−t)∈Z−(n−t)∑uquAtt(gu(n),zb(n−t))≥−2t−1β(1+β)t−1\sum_{z_b^{(n-t)} \in Z^{(n-t)}_-} \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) \ge -2^{t-1}\beta(1+\beta)^{t-1}

    For β=0\beta = 0, all attribution values are non-negative and satisfy:

    ∑b∑uquAtt(gu(n),zb(n−t))∈[0,1]\sum_b \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) \in [0, 1]

    Consequently, the overall attribution value range zu−zlz_u - z_l at the input layer (t=n−1t = n-1) across the network is:

    zu−zl={2n−1(1+2β)(1+β)n−1if β>01if β=0z_u - z_l = \begin{cases} 2^{n-1}(1+2\beta)(1+\beta)^{n-1} & \text{if } \beta > 0 \\ 1 & \text{if } \beta = 0 \end{cases}

    Crucially, this bound is completely decoupled from the neural network weight norms ∥W(l)∥2\|W^{(l)}\|_2 and the layer dimensions RlR_l.

  3. Knowl 3 — Hoeffding Bound on Empirical Averages of Augmented Attributions

    theoretical result

    Consider the empirical mean 1m∑i=1mA(f,x(i))\frac{1}{m}\sum_{i=1}^m A(f, x^{(i)}) of attribution maps computed over mm conditionally independent variations x(i)x^{(i)} of an input xx (such as test-time photometric augmentations x(i)=Tci(x)x^{(i)} = T_{c_i}(x) with ci∼Qc_i \sim Q, or SmoothGrad/SmoothLRP noise additions x(i)=x+ϵix^{(i)} = x + \epsilon_i).

    By Hoeffding's inequality, for any attribution method whose component values lie almost surely in [zl,zu][z_l, z_u], the deviation of the empirical mean from its expectation E[A(f,x)]\mathbb{E}[A(f, x)] satisfies:

    P(∣1m∑i=1mA(f,x(i))−E[A(f,x)]∣≥t)≤2exp⁡(−2t2m(zu−zl)2)P\left( \left| \frac{1}{m}\sum_{i=1}^m A(f, x^{(i)}) - \mathbb{E}[A(f, x)] \right| \ge t \right) \le 2 \exp\left( -2 \frac{t^2 m}{(z_u - z_l)^2} \right)

    Equivalently, with probability at least 1−δ1-\delta, the deviation is bounded by:

    t(δ)=(zu−zl)−12ln⁡(δ2)1mt(\delta) = (z_u - z_l) \sqrt{-\frac{1}{2}\ln\left(\frac{\delta}{2}\right)} \frac{1}{\sqrt{m}}

    To guarantee a maximum deviation tt with probability at least 1−δ1-\delta, the required sample size is:

    m=(zu−zl)212ln⁡(2δ)1t2m = (z_u - z_l)^2 \frac{1}{2}\ln\left(\frac{2}{\delta}\right) \frac{1}{t^2}

    While both gradient-based and LRP attribution averages converge asymptotically at rate O(1/m)\mathcal{O}(1/\sqrt{m}), the constant factor zu−zlz_u - z_l for LRP-β\beta is independent of network weight norms. In contrast, for standard gradient attributions with LL-Lipschitz activation functions, the range scales with layer weight norms:

    zu−zl=2Ln−1∥Wn∥2∥Wn−1∥2⋯∥W1∥2z_u - z_l = 2 L^{n-1} \|W_n\|_2 \|W_{n-1}\|_2 \cdots \|W_1\|_2

    For Gaussian initialized weights wd∼N(0,σ2)w_d \sim \mathcal{N}(0, \sigma^2) where E[∥Wl∥2]=Rlσ\mathbb{E}[\|W_l\|_2] = \sqrt{R_l}\sigma, the gradient constant scales as ∏l=1nRl\prod_{l=1}^n \sqrt{R_l}.

  4. Knowl 4 — Component-Wise Bounds and Value Range for LRP-gamma Attributions

    theoretical result

    For an nn-layer neural network with output initialization weights qu≥0q_u \ge 0 satisfying ∑uqu=1\sum_u q_u = 1, assume γ>1\gamma > 1 is chosen large enough such that for all layers simultaneously with positive connections (∑b:wabzb>0wabzb>0\sum_{b: w_{ab}z_b > 0} w_{ab}z_b > 0), the following condition holds:

    γ−1/2∑b:wabzb<0−wabzb<∑b:wabzb>0wabzb\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

    Defining b(γ)=max⁡(1γ1/2−1,1+γ1+γ−γ1/2)b(\gamma) = \max\left(\frac{1}{\gamma^{1/2}-1}, \frac{1+\gamma}{1+\gamma-\gamma^{1/2}}\right) (which equals 1+γ1+γ−γ1/2\frac{1+\gamma}{1+\gamma-\gamma^{1/2}} for γ≥4\gamma \ge 4), the attribution values at layer n−tn-t satisfy:

    ∑zb(n−t)∈Z+(n−t)∑uquAtt(gu(n),zb(n−t))≤2t−11+γ1+γ−γ1/2b(γ)t−1\sum_{z_b^{(n-t)} \in Z^{(n-t)}_+} \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) \le 2^{t-1} \frac{1+\gamma}{1+\gamma-\gamma^{1/2}} b(\gamma)^{t-1}

    ∑zb(n−t)∈Z−(n−t)∑uquAtt(gu(n),zb(n−t))≥2t−1−1γ1/2−1b(γ)t−1\sum_{z_b^{(n-t)} \in Z^{(n-t)}_-} \sum_u q_u \text{Att}(g_u^{(n)}, z_b^{(n-t)}) \ge 2^{t-1} \frac{-1}{\gamma^{1/2}-1} b(\gamma)^{t-1}

    The overall value range zu−zlz_u - z_l for the input layer (t=n−1t = n-1) across the nn-layer network is:

    zu−zl=2n−1b(γ)n−1(1+γ1+γ−γ1/2+1γ1/2−1)z_u - z_l = 2^{n-1} b(\gamma)^{n-1} \left( \frac{1+\gamma}{1+\gamma-\gamma^{1/2}} + \frac{1}{\gamma^{1/2}-1} \right)

  5. Knowl 5 — Singular Value Bounds for LRP Transition Matrices

    theoretical result

    For an affine layer g:RS→RRg: \mathbb{R}^S \to \mathbb{R}^R with relevance-conserving modified gradient matrix M(g)∈RS×RM(g) \in \mathbb{R}^{S \times R}:

    1. The normalized vector of ones 1S1S\frac{1}{\sqrt{S}}\mathbf{1}_S is a singular vector of M(g)M(g) with singular value RS\sqrt{\frac{R}{S}}, satisfying:

    1S1S⊤MM⊤1S1S=RS\frac{1}{\sqrt{S}}\mathbf{1}_S^\top M M^\top \frac{1}{\sqrt{S}}\mathbf{1}_S = \frac{R}{S}

    1. For LRP-β\beta (β≥0\beta \ge 0), the largest singular value of the layer transition matrix M(g)M(g) is bounded by:

    sup⁡v:∥v∥2=1∥M⊤v∥2≤R(1+β)2+β2≤R(1+2β)\sup_{v: \|v\|_2=1} \|M^\top v\|_2 \le \sqrt{R}\sqrt{(1+\beta)^2 + \beta^2} \le \sqrt{R}(1+\sqrt{2}\beta)

    1. In the limit γ→∞\gamma \to \infty, the singular value upper bound for LRP-γ\gamma converges to R\sqrt{R}.

    Across an nn-layer network, the product of LRP-β\beta transition matrices satisfies:

    ∥M(g(n))⋯M(g(1))∥2≤(1+2β)n∏l=1nRl\|M(g^{(n)}) \cdots M(g^{(1)})\|_2 \le (1+\sqrt{2}\beta)^n \prod_{l=1}^n \sqrt{R_l}

    This bound is independent of the weight matrices W(l)W^{(l)}, acting as an analog of gradient clipping for modified gradients.

  6. Knowl 6 — Theoretical Decoupling from Weight Norms and Sanity Check Insensitivity

    theoretical result

    The theoretical finding that LRP-β\beta's component-wise value bounds and convergence constants are decoupled from layer weight norms ∥W(l)∥2\|W^{(l)}\|_2 provides a formal explanation for the low sensitivity of LRP to top-down model parameter randomization tests (cascading randomization sanity checks).

    In standard gradient-based attribution methods, randomizing upper network layers significantly changes the scale and distribution of attributions because the bound depends multiplicatively on ∏l=1n∥W(l)∥2\prod_{l=1}^n \|W^{(l)}\|_2. In contrast, LRP-β\beta's layer-wise normalization isolates attribution magnitude and value range from weight norms, rendering the attribution range invariant to weight scaling.

  7. Knowl 7 — Empirical Metrics for Evaluating Attribution Average Convergence

    experimental setup

    To evaluate empirical convergence toward the true expectation without computing analytical expectations, two sample averages of size mm are computed over two disjoint sets of transformed inputs {x(i,1)}i=1m\{x^{(i, 1)}\}_{i=1}^m and {x(k,2)}k=1m\{x^{(k, 2)}\}_{k=1}^m for each base image xx.

    1. The unnormalized difference statistic is defined as:

    s1,m(x)=∥1m∑i=1mA(f,x(i,1))−1m∑k=1mA(f,x(k,2))∥2s_{1,m}(x) = \left\| \frac{1}{m}\sum_{i=1}^m A(f, x^{(i, 1)}) - \frac{1}{m}\sum_{k=1}^m A(f, x^{(k, 2)}) \right\|_2

    1. The ℓ2\ell_2-normalized difference statistic (isolating directional convergence from scale differences) is defined as:

    s2,m(x)=∥1m∑i=1mA(f,x(i,1))∥1m∑i′=1mA(f,x(i′,1))∥2−1m∑k=1mA(f,x(k,2))∥1m∑k′=1mA(f,x(k′,2))∥2∥2s_{2,m}(x) = \left\| \frac{\frac{1}{m}\sum_{i=1}^m A(f, x^{(i, 1)})}{\left\|\frac{1}{m}\sum_{i'=1}^m A(f, x^{(i', 1)})\right\|_2} - \frac{\frac{1}{m}\sum_{k=1}^m A(f, x^{(k, 2)})}{\left\|\frac{1}{m}\sum_{k'=1}^m A(f, x^{(k', 2)})\right\|_2} \right\|_2

    As n→∞n \to \infty, the distance ∥1m∑i=1mA(f,x(i,1))−1n∑k=1nA(f,x(k,2))∥2\|\frac{1}{m}\sum_{i=1}^m A(f, x^{(i,1)}) - \frac{1}{n}\sum_{k=1}^n A(f, x^{(k,2)})\|_2 converges to ∥1m∑i=1mA(f,x(i,1))−E[A(f,x)]∥2\|\frac{1}{m}\sum_{i=1}^m A(f, x^{(i,1)}) - \mathbb{E}[A(f, x)]\|_2, making s1,m(x)s_{1,m}(x) a computable lower bound on deviation.

    Evaluation is performed on the first 1000 ImageNet validation images across three architectures: ResNet-50, EfficientNet-V2-S, and Swin-V2-Tiny. Two augmentation regimes are used:

    • Photometric distortions (RandomPhotometricDistort with brightness in [0.875,1.125][0.875, 1.125], contrast in [0.5,1.5][0.5, 1.5], saturation in [0.8,1.2][0.8, 1.2], hue in [−0.1,0.1][-0.1, 0.1]).
    • Gaussian noise N(0,1)\mathcal{N}(0, 1) (SmoothGrad setting).

    Sample sizes tested are m∈{25,50,100}m \in \{25, 50, 100\}. Statistical significance is evaluated using a one-sided paired Wilcoxon signed-rank test on the median ratio of gradient-based statistics versus LRP statistics.

  8. Knowl 8 — Empirical Convergence Rates for Unnormalized Attribution Averages

    empirical result

    For unnormalized attribution maps evaluated using s1,m(x)s_{1,m}(x) across 1000 ImageNet validation samples, empirical means of LRP attributions converge drastically faster than raw gradient (∇\nabla) and gradient-times-input (∇×x\nabla \times x) across all tested models (ResNet-50, EfficientNet-V2-S, Swin-V2-Tiny), sample sizes m∈{25,50,100}m \in \{25, 50, 100\}, and augmentations (Gaussian noise and photometric distortion), with p<10−60p < 10^{-60} (typically p≈1.7×10−165p \approx 1.7 \times 10^{-165}) in one-sided paired Wilcoxon signed-rank tests.

    Representative median ratios (Median(Gradient-based)/Median(LRP)\text{Median}(\text{Gradient-based}) / \text{Median}(\text{LRP})):

    • EfficientNet-V2-S:
      • Gradient vs. LRP-β=0\beta=0: 2377.52377.5 (m=25m=25, Gaussian) to 7671.27671.2 (m=100m=100, photometric).
      • Gradient vs. LRP-β=1\beta=1: 726.1726.1 (m=25m=25, Gaussian) to 2337.22337.2 (m=100m=100, photometric).
      • ∇×x\nabla \times x vs. LRP-β=0\beta=0: 1460.61460.6 to 2883.22883.2.
      • Gradient vs. LRP-γ=103\gamma=10^3: 2217.52217.5 to 7092.87092.8.
    • ResNet-50:
      • Gradient vs. LRP-β=0\beta=0: 447.1447.1 to 1796.21796.2.
      • ∇×x\nabla \times x vs. LRP-β=0\beta=0: 336.9336.9 to 835.7835.7.
    • Swin-V2-Tiny:
      • Gradient vs. LRP-β=0\beta=0: 14156.214156.2 (m=25m=25, Gaussian) to 78109.178109.1 (m=100m=100, photometric).
      • ∇×x\nabla \times x vs. LRP-β=0\beta=0: 11461.411461.4 to 38074.838074.8.

  9. Knowl 9 — Empirical Convergence Comparison for L2-Normalized Attribution Averages

    empirical result

    Under the scale-invariant ℓ2\ell_2-normalized statistic s2,m(x)s_{2,m}(x), LRP-β\beta maintains a convergence advantage over gradient methods across most settings:

    • LRP-β=0\beta = 0 converges faster than raw gradients and ∇×x\nabla \times x across all architectures and augmentations (p<10−115p < 10^{-115}).
      • Gradient vs. LRP-β=0\beta=0 median ratios: 2.12.1 to 3.13.1 on ResNet-50; 2.82.8 to 4.94.9 on EfficientNet-V2-S; 3.33.3 to 5.15.1 on Swin-V2-Tiny.
      • ∇×x\nabla \times x vs. LRP-β=0\beta=0 median ratios: 2.72.7 to 8.08.0 on ResNet-50; 2.32.3 to 8.78.7 on EfficientNet-V2-S; 3.03.0 to 8.08.0 on Swin-V2-Tiny.
    • LRP-β=1\beta = 1 outperforms gradient methods in most cases (ratios 1.11.1 to 3.53.5), with a few comparable cases under photometric augmentation on ResNet-50 and EfficientNet-V2-S (ratios ≈0.9\approx 0.9 to 1.21.2).
    • LRP-γ\gamma (γ∈{102,103}\gamma \in \{10^2, 10^3\}) converges faster on EfficientNet-V2-S and ResNet-50 (ratios 2.02.0 to 8.78.7). On Swin-V2-Tiny, however, LRP-γ\gamma exhibits larger normalized distances (ratios 0.20.2 to 0.80.8 with p=1.0p=1.0), where γ=100\gamma = 100 produces large discrepancies between mean and median due to outlier samples violating the positive connection condition.

  10. Knowl 10 — Sample Efficiency of LRP in Averaged and Smoothed Explanations

    empirical result

    Because the required sample size mm in statistical attribution averaging scales with (zu−zl)2(z_u - z_l)^2, the weight-independent value range of LRP-β\beta enables practical test-time data augmentation averaging and SmoothLRP using small sample sizes (m∈[25,100]m \in [25, 100]). In contrast, gradient-based methods (like SmoothGrad) suffer from large variance driven by weight norm accumulation across layers, necessitating substantially higher sample counts to achieve comparable convergence.

Coverage note — Derivations and algebraic steps in the proofs of Theorem 4, Lemma 7, and Lemma 8 in the appendix were omitted in accordance with the requirement to exclude proof-only material; full result statements and preconditions were retained.

References

  1. 1.K. Simonyan, A. Vedaldi, and A. Zisserman, “Deep inside convolutional networks: Visualising image classification models and saliency maps,” in 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Workshop Track Proceedings (Y. Bengio and Y. LeCun, eds.), 2014.
  2. 2.J. Springenberg, A. Dosovitskiy, T. Brox, and M. Riedmiller, “Striving for simplicity: The all convolutional net,” in ICLR (workshop track), 2015.
  3. 3.R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, “Grad-cam: Visual explanations from deep networks via gradient-based localization,” in 2017 IEEE International Conference on Computer Vision (ICCV), pp. 618–626, 2017.
  4. 4.A. Shrikumar, P. Greenside, and A. Kundaje, “Learning important features through propagating activation differences,” in Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017 (D. Precup and Y. W. Teh, eds.), vol. 70 of Proceedings of Machine Learning Research, pp. 3145–3153, PMLR, 2017.
  5. 5.D. Balduzzi, M. Frean, L. Leary, J. P. Lewis, K. W. Ma, and B. McWilliams, “The shattered gradients problem: If resnets are the answer, then what is the question?,” in International Conference on Machine Learning (ICML), vol. 70 of PMLR, pp. 342–350, PMLR, 2017.
  6. 6.S. Bach, A. Binder, G. Montavon, F. Klauschen, K. R. Müller, and W. Samek, “On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation,” PLoS ONE, vol. 10, no. 7, pp. 1–46, 2015.
  7. 7.S. Lapuschkin, S. Wäldchen, A. Binder, G. Montavon, W. Samek, and K.-R. Müller, “Unmasking clever hans predictors and assessing what machines really learn,” Nature Communications, vol. 10, p. 1096, Mar 2019.
  8. 8.A. Binder, L. Weber, S. Lapuschkin, G. Montavon, K.-R. Müller, and W. Samek, “Shortcomings of top-down randomization-based sanity checks for evaluations of deep neural network explanations,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 16143–16152, June 2023.
  9. 9.R. Achtibat, S. M. V. Hatefi, M. Dreyer, A. Jain, T. Wiegand, S. Lapuschkin, and W. Samek, “AttnLRP: Attention-aware layer-wise relevance propagation for transformers,” in Proceedings of the 41st International Conference on Machine Learning (R. Salakhutdinov, Z. Kolter, K. Heller, A. Weller, N. Oliver, J. Scarlett, and F. Berkenkamp, eds.), vol. 235 of Proceedings of Machine Learning Research, pp. 135–168, PMLR, 21–27 Jul 2024.
  10. 10.F. Rezaei Jafari, G. Montavon, K.-R. Müller, and O. Eberle, “Mambalrp: Explaining selective state space sequence models,” in Advances in Neural Information Processing Systems (A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, eds.), vol. 37, pp. 118540–118570, Curran Associates, Inc., 2024.
  11. 11.J. Adebayo, J. Gilmer, M. Muelly, I. J. Goodfellow, M. Hardt, and B. Kim, “Sanity checks for saliency maps,” in Advances in Neural Information Processing Systems 31, pp. 9525–9536, 2018.
  12. 12.M. Ancona, E. Ceolini, C. Öztireli, and M. Gross, “Towards better understanding of gradient-based attribution methods for deep neural networks,” in 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings, OpenReview.net, 2018.
  13. 13.M. Sundararajan, A. Taly, and Q. Yan, “Axiomatic attribution for deep networks,” in Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017 (D. Precup and Y. W. Teh, eds.), vol. 70 of Proceedings of Machine Learning Research, pp. 3319–3328, PMLR, 2017.
  14. 14.D. Smilkov, N. Thorat, B. Kim, F. B. Viégas, and M. Wattenberg, “Smoothgrad: removing noise by adding noise,” arXiv preprint arXiv:1706.03825, 2017.
  15. 15.A. P. Raulf, S. Däubener, B. Hack, A. Mosig, and A. Fischer, “Smoothlrp: Smoothing LRP by averaging over stochastic input variations,” in 29th European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning, ESANN 2021, Online event (Bruges, Belgium), October 6-8, 2021, 2021.
  16. 16.E. Štrumbelj and I. Kononenko, “Explaining prediction models and individual predictions with feature contributions,” Knowledge and Information Systems, vol. 41, pp. 647–665, Dec 2014.
  17. 17.S. M. Lundberg and S. Lee, “A unified approach to interpreting model predictions,” in Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA (I. Guyon, U. von Luxburg, S. Bengio, H. M. Wallach, R. Fergus, S. V. N. Vishwanathan, and R. Garnett, eds.), pp. 4765–4774, 2017.
  18. 18.V. Petsiuk, A. Das, and K. Saenko, “RISE: randomized input sampling for explanation of black-box models,” in British Machine Vision Conference 2018, BMVC 2018, Newcastle, UK, September 3-6, 2018, p. 151, BMVA Press, 2018.
  19. 19.R. C. Fong and A. Vedaldi, “Interpretable explanations of black boxes by meaningful perturbation,” in IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017, pp. 3449–3457, IEEE Computer Society, 2017.
  20. 20.C. Agarwal and A. Nguyen, “Explaining image classifiers by removing input features using generative models,” in Computer Vision - ACCV 2020 - 15th Asian Conference on Computer Vision, Kyoto, Japan, November 30 - December 4, 2020, Revised Selected Papers, Part VI (H. Ishikawa, C. Liu, T. Pajdla, and J. Shi, eds.), vol. 12627 of Lecture Notes in Computer Science, pp. 101–118, Springer, 2020.
  21. 21.D. Alvarez-Melis and T. S. Jaakkola, “Towards robust interpretability with self-explaining neural networks,” in Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada (S. Bengio, H. M. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, eds.), pp. 7786–7795, 2018.
  22. 22.W. Samek, A. Binder, G. Montavon, S. Lapuschkin, and K. Müller, “Evaluating the visualization of what a deep neural network has learned,” IEEE Trans. Neural Networks Learn. Syst., vol. 28, no. 11, pp. 2660–2673, 2017.
  23. 23.C. Yeh, C. Hsieh, A. S. Suggala, D. I. Inouye, and P. Ravikumar, “On the (in)fidelity and sensitivity of explanations,” in Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada (H. M. Wallach, H. Larochelle, A. Beygelzimer, F. d’Alché-Buc, E. B. Fox, and R. Garnett, eds.), pp. 10965–10976, 2019.
  24. 24.U. Bhatt, A. Weller, and J. M. F. Moura, “Evaluating and aggregating feature-based model explanations,” in Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI 2020 (C. Bessiere, ed.), pp. 3016–3022, ijcai.org, 2020.
  25. 25.L. Rieger and L. K. Hansen, “IROF: a low resource evaluation metric for explanation methods,” CoRR, vol. abs/2003.08747, 2020.
  26. 26.A. Nguyen and M. R. Martínez, “On quantitative aspects of model interpretability,” CoRR, vol. abs/2007.07584, 2020.
  27. 27.S. Dasgupta, N. Frost, and M. Moshkovitz, “Framework for evaluating faithfulness of local explanations,” in International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA (K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvári, G. Niu, and S. Sabato, eds.), vol. 162 of Proceedings of Machine Learning Research, pp. 4794–4815, PMLR, 2022.
  28. 28.C. Agarwal, N. Johnson, M. Pawelczyk, S. Krishna, E. Saxena, M. Zitnik, and H. Lakkaraju, “Rethinking stability for attribution-based explanations,” CoRR, vol. abs/2203.06877, 2022.
  29. 29.Y. Rong, T. Leemann, V. Borisov, G. Kasneci, and E. Kasneci, “A consistent and efficient evaluation strategy for attribution methods,” in International Conference on Machine Learning, ICML 2022, 17-23 July 2022, Baltimore, Maryland, USA (K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvári, G. Niu, and S. Sabato, eds.), vol. 162 of Proceedings of Machine Learning Research, pp. 18770–18795, PMLR, 2022.
  30. 30.A. Hedström, L. Weber, S. Lapuschkin, and M. M. Höhne, “Sanity checks revisited: An exploration to repair the model parameter randomisation test,” CoRR, vol. abs/2401.06465, 2024.
  31. 31.R. Hesse, S. Schaub-Meyer, and S. Roth, “Funnybirds: A synthetic vision dataset for a part-based analysis of explainable AI methods,” in IEEE/CVF International Conference on Computer Vision, ICCV 2023, Paris, France, October 1-6, 2023, pp. 3958–3968, IEEE, 2023.
  32. 32.R. Hesse, S. Schaub-Meyer, and S. Roth, “Benchmarking the attribution quality of vision models,” in Advances in Neural Information Processing Systems (A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, eds.), vol. 37, pp. 97928–97947, Curran Associates, Inc., 2024.
  33. 33.T. Han, S. Srinivas, and H. Lakkaraju, “Which explanation should i choose? a function approximation perspective to characterizing post hoc explanations,” in Advances in Neural Information Processing Systems (S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, eds.), vol. 35, pp. 5256–5268, Curran Associates, Inc., 2022.
  34. 34.A.-C. Woerl, J. Disselhoff, and M. Wand, “Initialization noise in image gradients and saliency maps,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1766–1775, June 2023.
  35. 35.S. Agarwal, S. Jabbari, C. Agarwal, S. Upadhyay, S. Wu, and H. Lakkaraju, “Towards the unification and robustness of perturbation and gradient based explanations,” in Proceedings of the 38th International Conference on Machine Learning (M. Meila and T. Zhang, eds.), vol. 139 of Proceedings of Machine Learning Research, pp. 110–119, PMLR, 18–24 Jul 2021.
  36. 36.L. Zhou, C. Ma, Z. Wang, and X. Shi, “Rethinking the principle of gradient smooth methods in model explanation,” 2024.
  37. 37.L. Simpson, K. Millar, A. Cheng, C.-C. Lim, and H. G Chew, “Probabilistic lipschitzness and the stable rank for comparing explanation models,” 2024.
  38. 38.M. Scalbert, M. Vakalopoulou, and F. Couzinié-Devy, “Test-time image-to-image translation ensembling improves out-of-distribution generalization in histopathology,” 2022.
  39. 39.J. Dippel, N. Prenißl, J. Hense, P. Liznerski, T. Winterhoff, S. Schallenberg, M. Kloft, O. Buchstab, D. Horst, M. Alber, L. Ruff, K.-R. Müller, and F. Klauschen, “Ai-based anomaly detection for clinical-grade histopathological diagnostics,” 2024.
  40. 40.C. Xu, Z. Wen, Z. Liu, and C. Ye, “Improved domain generalization for cell detection in histopathology images via test-time stain augmentation,” in Medical Image Computing and Computer Assisted Intervention – MICCAI 2022 (L. Wang, Q. Dou, P. T. Fletcher, S. Speidel, and S. Li, eds.), (Cham), pp. 150–159, Springer Nature Switzerland, 2022.
  41. 41.A. Vulli, P. N. Srinivasu, M. S. K. Sashank, J. Shafi, J. Choi, and M. F. Ijaz, “Fine-tuned densenet-169 for breast cancer metastasis prediction using fastai and 1-cycle policy,” Sensors, vol. 22, no. 8, 2022.
  42. 42.M. Gaillochet, C. Desrosiers, and H. Lombaert, “Taal: Test-time augmentation for active learning in medical image segmentation,” in MICCAI Workshop on Data Augmentation, Labelling, and Imperfections, pp. 43–53, Springer, 2022.
  43. 43.A. Mahbod, G. Dorffner, I. Ellinger, R. Woitek, and S. Hatamikia, “Improving generalization capability of deep learning-based nuclei instance segmentation by non-deterministic train time and deterministic test time stain normalization,” Computational and Structural Biotechnology Journal, vol. 23, pp. 669–678, 2024.
  44. 44.Y. Liu, S. J. Wagner, and T. Peng, “Multi-modality microscopy image style augmentation for nuclei segmentation,” Journal of Imaging, vol. 8, no. 3, 2022.
  45. 45.M. Jahanifar, A. Shephard, N. Z. Tajeddin, R. M. S. Bashir, M. Bilal, S. A. Khurram, F. Minhas, and N. Rajpoot, “Stain-robust mitotic figure detection for the mitosis domain generalization challenge,” 2021.
  46. 46.X. Ma, Y. Tao, Y. Zhang, Z. Ji, Y. Zhang, and Q. Chen, “Test-time generative augmentation for medical image segmentation,” 2024.
  47. 47.D. Shanmugam, D. Blalock, G. Balakrishnan, and J. Guttag, “Better aggregation in test-time augmentation,” 2021.
  48. 48.M. S. Ayhan and P. Berens, “Test-time data augmentation for estimation of heteroscedastic aleatoric uncertainty in deep neural networks,” in Medical Imaging with Deep Learning, 2018.
  49. 49.G. Montavon, A. Binder, S. Lapuschkin, W. Samek, and K.-R. Müller, “Layer-wise relevance propagation: an overview,” in Explainable AI: interpreting, explaining and visualizing deep learning, pp. 193–209, Springer, 2019.
  50. 50.H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung),” Mathematische Annalen, vol. 71, pp. 441–479, Dec 1912.
  51. 51.R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1990.
  52. 52.W. H. and, “Probability inequalities for sums of bounded random variables,” Journal of the American Statistical Association, vol. 58, no. 301, pp. 13–30, 1963.
  53. 53.K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778, 2016.
  54. 54.M. Tan and Q. V. Le, “EfficientNetV2: Smaller models and faster training,” in Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event (M. Meila and T. Zhang, eds.), vol. 139 of Proceedings of Machine Learning Research, pp. 10096–10106, PMLR, 2021.
  55. 55.A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. M. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Köpf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. J. Bai, and S. Chintala, “Pytorch: An imperative style, high-performance deep learning library,” Advances in Neural Information Processing Systems 32 (Nips 2019), vol. 32, 2019. Bp0id Times Cited:23510 Cited References Count:43 Advances in Neural Information Processing Systems.
  56. 56.Z. Liu, H. Hu, Y. Lin, Z. Yao, Z. Xie, Y. Wei, J. Ning, Y. Cao, Z. Zhang, L. Dong, F. Wei, and B. Guo, “Swin transformer V2: scaling up capacity and resolution,” in IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2022, New Orleans, LA, USA, June 18-24, 2022, pp. 11999–12009, IEEE, 2022.
  57. 57.J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255, IEEE, 2009.

Citation

MLA
Binder, A., et al. “Value Bounds and Convergence Analysis for Averages of LRP Attributions”. arXiv, 2025, http://arxiv.org/abs/2509.08963v1.
APA
Binder, A., Takmil-Homayouni, N., & Dogan, U. (2025). Value bounds and Convergence Analysis for Averages of LRP attributions. arXiv. http://arxiv.org/abs/2509.08963v1
Chicago
Binder, A., N. Takmil-Homayouni, and U. Dogan. 2025. “Value Bounds and Convergence Analysis for Averages of LRP Attributions”. arXiv. http://arxiv.org/abs/2509.08963v1.
Harvard
Binder, A., Takmil-Homayouni, N. and Dogan, U. (2025) “Value bounds and Convergence Analysis for Averages of LRP attributions”, arXiv [Preprint]. Available at: http://arxiv.org/abs/2509.08963v1.
Vancouver
1. Binder A, Takmil-Homayouni N, Dogan U (2025) Value bounds and Convergence Analysis for Averages of LRP attributions. arXiv

BibTeX

@article{binder2025value,
  title = {Value bounds and Convergence Analysis for Averages of LRP attributions},
  author = {Binder, Alexander and Takmil-Homayouni, Nastaran and Dogan, Urun},
  year = {2025},
  journal = {arXiv},
  url = {http://arxiv.org/abs/2509.08963v1},
  eprint = {2509.08963}
}
Metadata:arXiv

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