Scattering Networks on Noncommutative Finite Groups

2025-05-27

AI Summary

This work addresses the lack of theoretical guarantees for equivariant feature extraction on non-commutative finite groups. We introduce the first scattering transform generalized to arbitrary finite groups—including non-Abelian ones—and embed it within the G-CNN framework to construct a group-equivariant scattering network. Our method leverages group representation theory and constructive group wavelet design, achieving exact left- and right-equivariance via group convolutions and cascaded scattering. We rigorously establish its non-expansiveness, deformation stability, energy conservation, and asymptotic translation robustness with depth. Experiments on both Abelian and non-Abelian structured data demonstrate improved classification performance. This work establishes a novel paradigm for equivariant feature learning on non-Euclidean symmetric data, unifying interpretability, stability, and theoretical rigor.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

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.

Problem

• Extend scattering transforms to noncommutative finite groups
• Analyze wavelet properties and equivariance in G-CNNs
• Apply scattering transforms for abelian/nonabelian group data classification

Innovation

• Scattering transform on finite noncommutative groups
• Wavelets designed for arbitrary finite groups
• Equivariant and stable under group translations