Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back

TL;DR

This paper introduces tangent bundle convolutional neural networks (TNNs) that operate on vector fields over manifolds, generalizing existing filters and neural networks to continuous and discrete settings with proven convergence.

Contribution

It presents a novel convolution operation on tangent bundles, defines tangent bundle neural networks, and establishes a discretization method with convergence guarantees, linking to sheaf neural networks.

Findings

Effective on synthetic data

Successful on real-world tasks

Converges to continuous tangent bundle neural networks

Abstract

In this work we introduce a convolution operation over the tangent bundle of Riemann manifolds in terms of exponentials of the Connection Laplacian operator. We define tangent bundle filters and tangent bundle neural networks (TNNs) based on this convolution operation, which are novel continuous architectures operating on tangent bundle signals, i.e. vector fields over the manifolds. Tangent bundle filters admit a spectral representation that generalizes the ones of scalar manifold filters, graph filters and standard convolutional filters in continuous time. We then introduce a discretization procedure, both in the space and time domains, to make TNNs implementable, showing that their discrete counterpart is a novel principled variant of the very recently introduced sheaf neural networks. We formally prove that this discretized architecture converges to the underlying continuous TNN. Finally, we numerically evaluate the effectiveness of the proposed architecture on various learning tasks, both on synthetic and real data.