Stability of the scattering transform for deformations with minimal regularity
This work addresses the theoretical understanding of geometric stability in deep convolutional neural networks for researchers in mathematical analysis, though it is incremental as it builds on prior results by exploring specific regularity conditions.
The paper investigates the stability of the wavelet scattering transform under deformations with minimal regularity, proving that stability is achievable for deformations of class C^α with α>1, while instability can occur for α<1, and providing a stability bound up to ε losses for the Lipschitz or C^1 threshold.
Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by Stéphane 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 Hölder regularity scale $C^α$, $α>0$. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class $C^α$, $α>1$, whereas instability phenomena can occur at lower regularity levels modelled by $C^α$, $0\le α<1$. 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.