# Stability of the scattering transform for deformations with minimal regularity

arXiv: https://arxiv.org/abs/2205.11142 · HTML: https://mlsanity.com/paper/2205.11142/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 23 May 2022.

## Builds on (most cited first, 13 of 13)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [ADef: an Iterative Algorithm to Construct Adversarial Deformations](https://mlsanity.com/paper/1804.07729/) (2018)
- [On Lipschitz Bounds of General Convolutional Neural Networks](https://mlsanity.com/paper/1808.01415/) (2018)
- [Lipschitz Properties for Deep Convolutional Networks](https://mlsanity.com/paper/1701.05217/) (2017)
- [Deep Convolutional Neural Networks Based on Semi-Discrete Frames](https://mlsanity.com/paper/1504.05487/) (2015)

## Built on by (most cited first, 3 of 3)

- [Manifold Filter-Combine Networks](https://mlsanity.com/paper/2307.04056/) (2023)
- [InfoGain Wavelets: Furthering the Design of Graph Diffusion Wavelets](https://mlsanity.com/paper/2504.08802/) (2025)
- [Scattering Networks on Noncommutative Finite Groups](https://mlsanity.com/paper/2505.20950/) (2025)

## Abstract (arXiv)

Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by St'ephane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small C2 diffeomorphisms - a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the H"older regularity scale Cα, α >0. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class Cα, α>1, whereas instability phenomena can occur at lower regularity levels modelled by Cα, 0≤ α <1. While the behaviour at the threshold given by Lipschitz (or even C1) regularity remains beyond reach, we are able to prove a stability bound in that case, up to varepsilon losses.