  The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of the success of convolutional neural
networks (ConvNets) in image data analysis and other tasks. Inspired by recent
interest in geometric deep learning, which aims to generalize ConvNets to
manifold and graph-structured domains, we generalize the scattering transform
to compact manifolds. Similar to the Euclidean scattering transform, our
geometric scattering transform is based on a cascade of designed filters and
pointwise nonlinearities, which enables rigorous analysis of the feature
extraction provided by scattering layers. Our main focus here is on theoretical
understanding of this geometric scattering network, while setting aside
implementation aspects, although we remark that application of similar
transforms to graph data analysis has been studied recently in related work.
Our results establish conditions under which geometric scattering provides
localized isometry invariant descriptions of manifold signals, which are also
stable to families of diffeomorphisms formulated in intrinsic manifolds terms.
These results not only generalize the deformation stability and local
roto-translation invariance of Euclidean scattering, but also demonstrate the
importance of linking the used filter structures (e.g., in geometric deep
learning) to the underlying manifold geometry, or the data geometry it
represents.
