On Generalizations of the Nonwindowed Scattering Transform

TL;DR

This paper extends wavelet scattering transforms by defining new norms, proving their mathematical properties, and developing operators invariant or equivariant to rotations, enhancing their applicability in analyzing signals.

Contribution

It introduces generalized wavelet scattering transforms with proven norms, Lipschitz continuity, and rotation invariance/equivariance, broadening the theoretical framework and potential applications.

Findings

Operators are well-defined and Lipschitz continuous.

Extended to rotation-invariant and equivariant operators.

Provided mathematical bounds and properties for the generalized transforms.

Abstract

In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as $\Lb^q(\mathbb{R}^n)$ norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then provide norms for these operators, prove that these operators are well-defined, and are Lipschitz continuous to the action of $C^2$ diffeomorphisms in specific cases. Lastly, we extend our results to formulate an operator invariant to the action of rotations $R \in \text{SO}(n)$ and an operator that is equivariant to the action of rotations of $R \in \text{SO}(n)$.