Stability of the scattering transform for deformations with minimal regularity

TL;DR

This paper analyzes the stability of the wavelet scattering transform under deformations with minimal regularity, identifying thresholds for stability based on the deformation's Hölder regularity.

Contribution

It precisely characterizes the stability threshold of the scattering transform for deformations with different Hölder regularities, extending understanding of its robustness.

Findings

Stability is maintained for deformations with Hölder regularity greater than 1.

Instability can occur for deformations with Hölder regularity less than 1.

A stability bound is established for Lipschitz (C^1) deformations, up to small losses.

Abstract

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 $C^2$ 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 $C^\alpha$, $\alpha >0$. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class $C^{\alpha}$, $\alpha>1$, whereas instability phenomena can occur at lower regularity levels modelled by $C^\alpha$, $0\le \alpha <1$. While the behaviour at the threshold given by Lipschitz (or even $C^1$) regularity remains beyond reach, we are able to prove a stability bound in that case, up to $\varepsilon$ losses.