Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
This work addresses the challenge of extending deep learning to non-Euclidean domains like manifolds, which is incremental as it builds on prior Euclidean scattering transforms.
The authors tackled the problem of generalizing convolutional neural networks to manifold-structured data by defining a geometric scattering transform on compact Riemannian manifolds, achieving invariance to local isometries and stability to certain diffeomorphisms with empirical utility on geometric learning tasks.
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.