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 $C^2$ 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.

On the stability of deep convolutional neural networks under irregular or random deformations

The problem of robustness under location deformations for deep convolutional neural networks (DCNNs) is of great theoretical and practical interest. This issue has been studied in pioneering works, especially for scattering-type architectures, for deformation vector fields $\tau(x)$ with some regularity - at least $C^1$. Here we address this issue for any field $\tau\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$, without any additional regularity assumption, hence including the case of wild irregular deformations such as a noise on the pixel location of an image. We prove that for signals in multiresolution approximation spaces $U_s$ at scale $s$, whenever the network is Lipschitz continuous (regardless of its architecture), stability in $L^2$ holds in the regime $\|\tau\|_{L^\infty}/s\ll 1$, essentially as a consequence of the uncertainty principle. When $\|\tau\|_{L^\infty}/s\gg 1$ instability can occur even for well-structured DCNNs such as the wavelet scattering networks, and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space $B^{d/2}_{2,1}$ tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when $\tau(x)$ is modeled as a random field (not bounded, in general) with with identically distributed variables $|\tau(x)|$, $x\in\mathbb{R}^d$.