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

In particular, the above approximation leads to important transferability results of graph neural In this work we introduce a convolution operation over the tangent networks (GNNs) [17,18], as well as to the introduction of Graphon bundle of Riemannian manifolds exploiting the Connection Laplacian and Manifold Neural Networks, continuous architectures shown to operator. We use the convolution to define tangent bundle filters be limit objects of GNNs [19, 20]. However, most of the previous and tangent bundle neural networks (TNNs), novel continuous works focus on scalar signals, e.g. one or more scalar values architectures operating on tangent bundle signals, i.e. vector fields attached to each node of graphs or point of manifolds; recent developments over manifolds. We discretize TNNs both in space and time domains, [21] show that processing vector data defined on tangent showing that their discrete counterpart is a principled variant bundles of manifolds or discrete vector bundles [22, 23] comes with of the recently introduced Sheaf Neural Networks.