Translation-Invariant Behavior of General Scattering Transforms
Wojciech Czaja, Brandon Kolstoe, David Koralov
TL;DR
The paper proves a simpler translation-invariance result for the limiting behavior of coherent sequences of scattering transforms, avoiding Mallat's admissibility condition and density arguments in $L^2$. It generalizes the framework to broader scattering transforms, including a variant of the Fourier Scattering Transform, and derives a concrete upper bound on translation contraction: $||\tilde{S}[P]f-\tilde{S}[P]T_c f||_{L^2l^2}^2\le(2\pi D|c|)^2||f||_{L^2}^2$ with $D=\sup(|\xi|: \hat{g}_0(\xi)\neq0)$. For coherent sequences, this bound implies translation-invariance in the limit $J\to\infty$, i.e., $\lim_{J\to\infty}||\tilde{S}_J[P_J]f-\tilde{S}_J[P_J]T_c f||_{L^2l^2}=0$, including the Fourier Scattering case under Uniform Covering Frames where norm-preservation holds without an admissibility condition. Overall, the work broadens the stability and invariance guarantees of scattering transforms, enabling robust translation handling in practical audio and image processing tasks without relying on stringent frame admissibility.
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.