Stability of the scattering transform for deformations with minimal regularity

• URL: http://arxiv.org/abs/2205.11142v1
• Date: Mon, 23 May 2022 09:08:21 GMT
• Title: Stability of the scattering transform for deformations with minimal regularity
• Authors: Fabio Nicola and S. Ivan Trapasso
• Abstract summary: We investigate the relationship between the scattering structure and the regularity of the deformation in the H"older regularity scale $Calpha$, $alpha >0$. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class $Calpha$, $alpha>1$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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