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。近年来，多项研究将散射变换推广至非欧几里得环境（如图结构）。本研究在这些构建基础上，基于一类非常通用的非对称小波，引入了图上的加窗与非加窗几何散射变换。我们证明，这些非对称图散射变换具有与其对称版本相似的诸多理论保证。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。通过这一工作，本研究引入了一个具有可证明稳定性和不变性保证的大型网络家族，从而有助于弥合几何散射与其他图神经网络之间的差距。这些结果为未来面向图结构数据的深度学习架构奠定了基础，此类架构不仅包含学习到的滤波器，还具备可证明的理想理论特性。