Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back
This work addresses a foundational challenge in geometric deep learning by extending neural networks to tangent bundles, with potential applications in fields like physics and computer graphics, though it appears incremental as it builds on existing sheaf neural networks.
The authors tackled the problem of defining convolution operations on Riemannian manifolds by introducing tangent bundle filters and neural networks (TNNs) that operate on vector fields, and they demonstrated effectiveness with a denoising task on the unit 2-sphere.
In this work we introduce a convolution operation over the tangent bundle of Riemannian manifolds exploiting the Connection Laplacian operator. We use the convolution to define tangent bundle filters and tangent bundle neural networks (TNNs), novel continuous architectures operating on tangent bundle signals, i.e. vector fields over manifolds. We discretize TNNs both in space and time domains, showing that their discrete counterpart is a principled variant of the recently introduced Sheaf Neural Networks. We formally prove that this discrete architecture converges to the underlying continuous TNN. We numerically evaluate the effectiveness of the proposed architecture on a denoising task of a tangent vector field over the unit 2-sphere.