The scattering transform is a multilayered wavelet-based deep learning architecture that acts as a model of convolutional neural networks. Recently, several works have introduced generalizations of the scattering transform for non-Euclidean settings such as graphs. Our work builds upon these constructions by introducing windowed and non-windowed geometric scattering transforms for graphs based upon a very general class of asymmetric wavelets. We show that these asymmetric graph scattering transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. In doing so, this work helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties.

翻译：散射变换是一种基于小波的多层深度学习架构，可作为卷积神经网络的模型。近期，多项研究将散射变换推广至非欧几里得环境（如图数据）。本文在上述基础上，基于一类非常广泛的非对称小波，引入了图数据的窗口化与非窗口化几何散射变换。我们证明，这些非对称图散射变换具有与其对称对应物相同的理论保证。因此，所提出的构建统一并扩展了现有多种图散射架构的已知理论结果。通过此举，本研究引入了一类具有可证明稳定性和不变性保证的大规模网络，有助于弥合几何散射与其他图神经网络之间的鸿沟。这些结果为未来基于图结构数据的学习型滤波器深度学习架构奠定了基础，并使这类架构具备可证明的理想理论特性。