# Noise-adding Methods of Saliency Map as Series of Higher Order Partial Derivative

arXiv: https://arxiv.org/abs/1806.03000 · HTML: https://mlsanity.com/paper/1806.03000/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2018.

## Built on by (most cited first, 11 of 11)

- [Don't Lie to Me! Robust and Efficient Explainability with Verified Perturbation Analysis](https://mlsanity.com/paper/2202.07728/) (2022)
- [Gradient based Feature Attribution in Explainable AI: A Technical Review](https://mlsanity.com/paper/2403.10415/) (2024)
- [Xplique: A Deep Learning Explainability Toolbox](https://mlsanity.com/paper/2206.04394/) (2022)
- [Interpretability Needs a New Paradigm](https://mlsanity.com/paper/2405.05386/) (2024)
- [New Definitions and Evaluations for Saliency Methods: Staying Intrinsic, Complete and Sound](https://mlsanity.com/paper/2211.02912/) (2022)
- [Rationalizing Predictions by Adversarial Information Calibration](https://mlsanity.com/paper/2301.06009/) (2023)
- [Saliency strikes back: How filtering out high frequencies improves white-box explanations](https://mlsanity.com/paper/2307.09591/) (2023)
- [Looking deeper into interpretable deep learning in neuroimaging: a comprehensive survey](https://mlsanity.com/paper/2307.09615/) (2023)
- [Explainability as statistical inference](https://mlsanity.com/paper/2212.03131/) (2022)
- [COMET: Neural Cost Model Explanation Framework](https://mlsanity.com/paper/2302.06836/) (2023)
- [The Directed Prediction Change - Efficient and Trustworthy Fidelity Assessment for Local Feature Attribution Methods](https://mlsanity.com/paper/2511.21363/) (2025)

## Abstract (arXiv)

SmoothGrad and VarGrad are techniques that enhance the empirical quality of standard saliency maps by adding noise to input. However, there were few works that provide a rigorous theoretical interpretation of those methods. We analytically formalize the result of these noise-adding methods. As a result, we observe two interesting results from the existing noise-adding methods. First, SmoothGrad does not make the gradient of the score function smooth. Second, VarGrad is independent of the gradient of the score function. We believe that our findings provide a clue to reveal the relationship between local explanation methods of deep neural networks and higher-order partial derivatives of the score function.