We introduce vector diffusion wavelets (VDWs), a novel family of wavelets inspired by the vector diffusion maps algorithm that was introduced to analyze data lying in the tangent bundle of a Riemannian manifold. We show that these wavelets may be effectively incorporated into a family of geometric graph neural networks, which we refer to as VDW-GNNs. We demonstrate that such networks are effective on synthetic point cloud data, as well as on real-world data derived from wind-field measurements and neural activity data. Theoretically, we prove that these new wavelets have desirable frame theoretic properties, similar to traditional diffusion wavelets. Additionally, we prove that these wavelets have desirable symmetries with respect to rotations and translations.

翻译：我们引入了向量扩散小波（VDWs），这是一种受向量扩散映射算法启发的新型小波族，该算法最初是为分析黎曼流形切丛上的数据而提出的。我们证明，这些小波可以有效地融入一系列几何图神经网络中，我们将其称为VDW-GNNs。我们通过实验表明，此类网络在合成点云数据、以及源自风场测量和神经活动数据的真实世界数据上均表现优异。理论上，我们证明了这些新小波具有与传统扩散小波类似的优良框架理论性质。此外，我们还证明了这些小波在旋转和平移变换下具有理想的对称性。