# Probabilistic Lipschitzness and the Stable Rank for Comparing Explanation Models

arXiv: https://arxiv.org/abs/2402.18863 · HTML: https://mlsanity.com/paper/2402.18863/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 36 references that are in the map.
- First posted on arXiv 29 Feb 2024.

## Builds on (most cited first, 28 of 28)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [A Survey on Explainable Artificial Intelligence (XAI): Towards Medical XAI](https://mlsanity.com/paper/1907.07374/) (2019)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [On the Robustness of Interpretability Methods](https://mlsanity.com/paper/1806.08049/) (2018)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [On the (In)fidelity and Sensitivity for Explanations](https://mlsanity.com/paper/1901.09392/) (2019)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Efficient Decision-based Black-box Adversarial Attacks on Face Recognition](https://mlsanity.com/paper/1904.04433/) (2019)
- [Summary of ChatGPT-Related Research and Perspective Towards the Future of Large Language Models](https://mlsanity.com/paper/2304.01852/) (2023)
- [Beyond Periodicity: Towards a Unifying Framework for Activations in Coordinate-MLPs](https://mlsanity.com/paper/2111.15135/) (2021)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [A Rigorous Study of Integrated Gradients Method and Extensions to Internal Neuron Attributions](https://mlsanity.com/paper/2202.11912/) (2022)
- [The Unreasonable Effectiveness of Deep Learning in Artificial Intelligence](https://mlsanity.com/paper/2002.04806/) (2020)
- [Towards the Unification and Robustness of Perturbation and Gradient Based Explanations](https://mlsanity.com/paper/2102.10618/) (2021)
- [Stable Rank Normalization for Improved Generalization in Neural Networks and GANs](https://mlsanity.com/paper/1906.04659/) (2019)
- [Smoothed Geometry for Robust Attribution](https://mlsanity.com/paper/2006.06643/) (2020)
- [Finding the right XAI method -- A Guide for the Evaluation and Ranking of Explainable AI Methods in Climate Science](https://mlsanity.com/paper/2303.00652/) (2023)
- [Explainable Artificial Intelligence Applications in Cyber Security: State-of-the-Art in Research](https://mlsanity.com/paper/2208.14937/) (2022)
- [Analytical bounds on the local Lipschitz constants of ReLU networks](https://mlsanity.com/paper/2104.14672/) (2021)
- [Analyzing Explainer Robustness via Probabilistic Lipschitzness of Prediction Functions](https://mlsanity.com/paper/2206.12481/) (2022)

## Built on by (most cited first, 4 of 4)

- [Algebraic Adversarial Attacks on Integrated Gradients](https://mlsanity.com/paper/2407.16233/) (2024)
- [Tangentially Aligned Integrated Gradients for User-Friendly Explanations](https://mlsanity.com/paper/2503.08240/) (2025)
- [Value bounds and Convergence Analysis for Averages of LRP attributions](https://mlsanity.com/paper/2509.08963/) (2025)
- [UbiQVision: Quantifying Uncertainty in XAI for Image Recognition](https://mlsanity.com/paper/2512.20288/) (2025)

## Abstract (arXiv)

Explainability models are now prevalent within machine learning to address the black-box nature of neural networks. The question now is which explainability model is most effective. Probabilistic Lipschitzness has demonstrated that the smoothness of a neural network is fundamentally linked to the quality of post hoc explanations. In this work, we prove theoretical lower bounds on the probabilistic Lipschitzness of Integrated Gradients, LIME and SmoothGrad. We propose a novel metric using probabilistic Lipschitzness, normalised astuteness, to compare the robustness of explainability models. Further, we prove a link between the local Lipschitz constant of a neural network and its stable rank. We then demonstrate that the stable rank of a neural network provides a heuristic for the robustness of explainability models.