Stability of the scattering transform for deformations with minimal regularity

The wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of harmonic and multiscale analysis can be ingeniously exploited to build a signal representation with provable geometric stability properties, such as the 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 Hölder 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 modeled by Cα, 0 ≤ α < 1. While the analysis 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 ε losses