Scattering Networks on Noncommutative Finite Groups

TL;DR

This paper extends scattering networks to finite noncommutative groups, establishing their stability, equivariance, and energy preservation, and demonstrates their effectiveness in classifying data on complex group domains.

Contribution

It introduces a novel scattering transform on arbitrary finite groups, including nonabelian, with theoretical analysis and practical classification applications.

Findings

Scattering transform is non-expansive and stable under deformations.

The transform preserves energy and is equivariant to group translations.

Applications include classifying data on abelian and nonabelian group domains.

Abstract

Scattering Networks were initially designed to elucidate the behavior of early layers in Convolutional Neural Networks (CNNs) over Euclidean spaces and are grounded in wavelets. In this work, we introduce a scattering transform on an arbitrary finite group (not necessarily abelian) within the context of group-equivariant convolutional neural networks (G-CNNs). We present wavelets on finite groups and analyze their similarity to classical wavelets. We demonstrate that, under certain conditions in the wavelet coefficients, the scattering transform is non-expansive, stable under deformations, preserves energy, equivariant with respect to left and right group translations, and, as depth increases, the scattering coefficients are less sensitive to group translations of the signal, all desirable properties of convolutional neural networks. Furthermore, we provide examples illustrating the application of the scattering transform to classify data with domains involving abelian and nonabelian groups.