Translation-Invariant Behavior of General Scattering Transforms

TL;DR

This paper provides a simpler proof of the translation invariance of wavelet scattering transforms and extends the result to broader classes, including a modified Fourier scattering transform, with new bounds on translation contraction.

Contribution

It introduces a simpler proof of translation invariance that does not rely on certain conditions and generalizes the invariance to other scattering transforms, including Fourier-based ones.

Findings

Simpler proof of translation invariance for scattering transforms

Generalization to broader classes including Fourier scattering

New upper bound for translation contraction in Fourier scattering

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.