Geometric Scattering on Manifolds

TL;DR

This paper extends the Euclidean scattering transform to compact manifolds, providing a mathematically rigorous framework for analyzing signals on manifolds with invariance and stability properties.

Contribution

It introduces a geometric scattering transform for manifolds, establishing theoretical conditions for invariance and stability, thus generalizing Euclidean scattering to non-Euclidean domains.

Findings

Provides localized isometry invariant descriptions of manifold signals

Demonstrates stability to diffeomorphisms on manifolds

Links filter structures to manifold geometry for better analysis

Abstract

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.