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

TL;DR

This work develops tangent bundle convolution to process vector-field signals on Riemannian manifolds by leveraging the Connection Laplacian Δ. It defines tangent bundle filters and Tangent Bundle Neural Networks (TNNs), derives their spectral representation, and provides a principled space-time discretization via orthogonal cellular sheaves, yielding DD-TNNs with a formal convergence guarantee to the continuous model. The approach unifies manifold, graph, and time-domain filtering for vector fields, and is validated on torus denoising, wind-field reconstruction/forecasting, and manifold classification, demonstrating clear benefits from incorporating tangent-bundle geometry. By linking tangent bundle processing to cellular sheaves and algebraic signal processing, the paper offers a theoretically grounded, scalable framework with potential implications for geometric deep learning and physics-informed 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.

Figures (7)

• Figure 1: An example of tangent vector
• Figure 2: Vector diffusion
• Figure 3: An example of tangent bundle signal
• Figure 4: Illustration of a lowpass, non-amplifying, Lipschitz continuous tangent bundle filter. The $x$-axis stands for the spectrum with each sample representing an eigenvalue. Here the eigenvalues increase at a logarithmic rate. The red dotted line is $\lambda_i^{-2}$ and the blue dotted line is the filter, obtained with impulse response $\widetilde{h}(t) = t^2/6$, thus ${\hat{h}}(\lambda) = \frac{1}{3}\lambda^{-3}$, from \ref{['frequency_filt']}.
• Figure 5: Pictorial view of discrete parallel transport.

Theorems & Definitions (23)

• Definition 1: Tangent Bundle
• Definition 2: Riemann Metric
• Remark 1
• Definition 3: Tangent Bundle Inner Product
• Definition 4: Tangent Bundle Filter
• Proposition 1: Parametric Filter
• Remark 2
• Proposition 2: Frequency Representation
• proof
• Definition 5: Frequency Response