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Motivated by the problem of robustness to deformations of the input for deepconvolutional neural networks, we identify signal classes which are inherentlystable to irregular deformations induced by distortion fields $\tau\inL^\infty(\mathbb{R}^d;\mathbb{R}^d)$, to be characterized in terms of ageneralized modulus of continuity associated with the deformation operator.Resorting to ideas of harmonic and multiscale analysis, we prove that forsignals in multiresolution approximation spaces $U_s$ at scale $s$, stabilityin $L^2$ holds in the regime $\|\tau\|_{L^\infty}/s\ll 1$ - essentially as aneffect of the uncertainty principle. Instability occurs when$\|\tau\|_{L^\infty}/s\gg 1$, and we provide a sharp upper bound for theasymptotic growth rate. The stability results are then extended to signals inthe Besov space $B^{d/2}_{2,1}$ tailored to the given multiresolutionapproximation. We also consider the case of more general time-frequencydeformations. Finally, we provide stochastic versions of the aforementionedresults, namely we study the issue of stability in mean when $\tau(x)$ ismodeled as a random field (not bounded, in general) with identicallydistributed variables $|\tau(x)|$, $x\in\mathbb{R}^d$.

Generalized moduli of continuity under irregular or random deformations via multiscale analysis