Scattering Networks on Noncommutative Finite Groups
This provides a theoretical foundation for scattering transforms in non-commutative settings, potentially benefiting researchers in geometric deep learning and group theory applications.
The authors extended scattering networks to arbitrary finite groups (including non-abelian ones) within group-equivariant CNNs, proving that under certain wavelet conditions the transform has desirable CNN properties like non-expansiveness, stability, and equivariance, and demonstrated its application to classifying data on both abelian and non-abelian groups.
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.