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.

翻译：本文中，我们引入了一种基于联络拉普拉斯算子指数的黎曼流形切丛上的卷积运算。我们定义了基于该卷积运算的切丛滤波器和切丛神经网络（TNNs），这是一种作用于切丛信号（即流形上的向量场）的新型连续架构。切丛滤波器具有一种谱表示，该表示推广了标量流形滤波器、图滤波器以及连续时间标准卷积滤波器的谱表示。随后我们引入了空间域和时间域上的离散化程序，使TNNs得以实现，并证明了其离散对应物是近期提出的层神经网络的原理性新变体。我们从数学上严格证明了这种离散化架构收敛于原始的连续TNN。最后，我们通过多项合成数据与真实数据的机器学习任务，数值评估了所提架构的有效性。