将散射网络拓展到非交换有限群，提升CNN对称性建模能力。
Scattering Networks on Noncommutative Finite Groups
- 基于有限群构造类小波函数，支持非交换群上的信号分析。
- 散射变换具稳定、保能、非扩张等性质，深度增加时对群平移更鲁棒。
- 适用于阿贝尔与非阿贝尔群的数据分类任务，适合对称性建模研究者。
散射网络最初用于解释欧几里得空间上卷积神经网络（CNN）早期层的行为，其基础是小波。本文在群等变卷积神经网络（G-CNNs）框架下，为任意有限群（未必阿贝尔）引入了散射变换。我们构建了有限群上的小波，并分析其与经典小波的相似性。在特定小波系数条件下，证明了散射变换具有非扩张性、对形变的稳定性、能量保持性，且对左右群平移保持等变性；随着深度增加，散射系数对信号群平移的敏感性降低，这些性质均符合卷积神经网络的理想特性。此外，通过实例展示了该变换在阿贝尔与非阿贝尔群域数据分类中的应用。
原文摘要 · Abstract (English)
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.
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