Scattering Networks on Noncommutative Finite Groups

TL;DR

This work extends the scattering transform to signals defined on finite groups $G$, including noncommutative ones, by introducing $G$-wavelets built from group representations and establishing a Parseval-frame framework under a Calderón condition. The resulting $G$-scattering transform is non-expansive, stable to input perturbations, energy-preserving under admissibility, and equivariant to left and right translations with increasing approximate invariance at depth, enabling robust, symmetry-aware representations. The theory connects to Hammond et al.'s graph-wavelets in the abelian case and provides a scalable finite-feature architecture for G-CNNs on non-Euclidean domains. Numerical results on MNIST, bark-vs-meow on $Aff(\mathbb{F}_p)$, and symmetric-group function classification demonstrate strong performance and illustrate the method's ability to exploit noncommutative symmetries for improved discrimination.

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.

Figures (17)

• Figure 1: The structure of the scattering transform of a signal $f$.
• Figure 2: Representation of the Cayley graph of $(\mathbb{Z}_p^*, *)$
• Figure 3: G-Scattering transform
• Figure 4: Example of data
• Figure 5: Characters of $\mathbb{Z}/28\mathbb{Z}\times \mathbb{Z}/28\mathbb{Z}$

Theorems & Definitions (40)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Theorem 2.5: Inverse formula
• proof
• Theorem 2.6
• proof
• Remark 2.7
• Definition 2.8