# Value bounds and Convergence Analysis for Averages of LRP attributions

arXiv: https://arxiv.org/abs/2509.08963 · HTML: https://mlsanity.com/paper/2509.08963/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 57 references that are in the map.
- First posted on arXiv 10 Sep 2025.

## Builds on (most cited first, 30 of 41)

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- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
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- [RISE: Randomized Input Sampling for Explanation of Black-box Models](https://mlsanity.com/paper/1806.07421/) (2018)
- [Interpretable Explanations of Black Boxes by Meaningful Perturbation](https://mlsanity.com/paper/1704.03296/) (2017)
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- [Unmasking Clever Hans Predictors and Assessing What Machines Really Learn](https://mlsanity.com/paper/1902.10178/) (2019)
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- [AttnLRP: Attention-Aware Layer-Wise Relevance Propagation for Transformers](https://mlsanity.com/paper/2402.05602/) (2024)
- [A Consistent and Efficient Evaluation Strategy for Attribution Methods](https://mlsanity.com/paper/2202.00449/) (2022)
- [Which Explanation Should I Choose? A Function Approximation Perspective to Characterizing Post Hoc Explanations](https://mlsanity.com/paper/2206.01254/) (2022)
- [On quantitative aspects of model interpretability](https://mlsanity.com/paper/2007.07584/) (2020)
- [Framework for Evaluating Faithfulness of Local Explanations](https://mlsanity.com/paper/2202.00734/) (2022)
- [Rethinking Stability for Attribution-based Explanations](https://mlsanity.com/paper/2203.06877/) (2022)
- [FunnyBirds: A Synthetic Vision Dataset for a Part-Based Analysis of Explainable AI Methods](https://mlsanity.com/paper/2308.06248/) (2023)
- [Towards the Unification and Robustness of Perturbation and Gradient Based Explanations](https://mlsanity.com/paper/2102.10618/) (2021)
- [IROF: a low resource evaluation metric for explanation methods](https://mlsanity.com/paper/2003.08747/) (2020)

## Abstract (arXiv)

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.