Stability of the scattering transform for deformations with minimal regularity

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.