Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back

TL;DR

This paper introduces tangent bundle neural networks (TNNs) that operate on vector fields over manifolds using a novel convolution based on the Connection Laplacian, bridging continuous and discrete geometric architectures.

Contribution

It proposes a new convolution operation for tangent bundle signals, defines TNNs, and demonstrates their convergence to continuous models, connecting manifold-based neural networks with sheaf neural networks.

Findings

TNNs effectively denoise tangent vector fields on the sphere.

The discrete TNN architecture converges to the continuous model.

The approach unifies manifold neural networks with sheaf neural networks.

Abstract

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.