Translation-Invariant Behavior of General Scattering Transforms

Abstract: The main result of our paper offers an alternative, simpler, proof of Mallat's result on the translation invariance of the limiting behavior of sequences of Wavelet Scattering Transforms, which (unlike Mallat's proof) does not rely on the admissibility condition or on the density of a logarithmic Sobolev space in $L^2$. Furthermore, this result is generalized to a broader class of scattering transforms, including, for instance, a modification of the Fourier Scattering Transform. As a result, we also prove a new upper bound for the translation contraction for the Fourier Scattering Transform.