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The problem of robustness under location deformations for deep convolutionalneural networks (DCNNs) is of great theoretical and practical interest. Thisissue has been studied in pioneering works, especially for scattering-typearchitectures, for deformation vector fields $\tau(x)$ with some regularity -at least $C^1$. Here we address this issue for any field $\tau\inL^\infty(\mathbb{R}^d;\mathbb{R}^d)$, without any additional regularityassumption, hence including the case of wild irregular deformations such as anoise on the pixel location of an image. We prove that for signals inmultiresolution approximation spaces $U_s$ at scale $s$, whenever the networkis Lipschitz continuous (regardless of its architecture), stability in $L^2$holds in the regime $\|\tau\|_{L^\infty}/s\ll 1$, essentially as a consequenceof the uncertainty principle. When $\|\tau\|_{L^\infty}/s\gg 1$ instability canoccur even for well-structured DCNNs such as the wavelet scattering networks,and we provide a sharp upper bound for the asymptotic growth rate. Thestability results are then extended to signals in the Besov space$B^{d/2}_{2,1}$ tailored to the given multiresolution approximation. We alsoconsider the case of more general time-frequency deformations. Finally, weprovide stochastic versions of the aforementioned results, namely we study theissue of stability in mean when $\tau(x)$ is modeled as a random field (notbounded, in general) with with identically distributed variables $|\tau(x)|$,$x\in\mathbb{R}^d$.

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