The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.

翻译：欧几里得散射变换于近十年前被提出，旨在提升对卷积神经网络的数学理解。受近期几何深度学习研究热潮的启发（该领域致力于将卷积神经网络推广至流形与图结构领域），本文定义了流形上的几何散射变换。与欧几里得散射变换类似，几何散射变换基于小波滤波器与逐点非线性函数的级联结构。该变换具备对局部等距变换的不变性，并对特定类型的微分同胚具有稳定性。实验结果表明了其在若干几何学习任务中的实用性。本文的研究成果将欧几里得散射的形变稳定性与局部平移不变性推广至更一般情形，并揭示了滤波器结构与数据底层几何特性之间的关联对于深度学习的关键作用。