  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 $|\tau(x)|$,
$x\in\mathbb{R}^d$.
