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

Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields $\tau\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$, to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces $U_s$ at scale $s$, stability in $L^2$ holds in the regime $\|\tau\|_{L^\infty}/s\ll 1$ - essentially as an effect of the uncertainty principle. Instability occurs when $\|\tau\|_{L^\infty}/s\gg 1$, 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 identically distributed variables $|\tau(x)|$, $x\in\mathbb{R}^d$.