This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle's original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher-order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.

翻译：本文提出了一种在黎曼流形上进行等变消息传递的几何洞见。如前所述，黎曼流形上的数值特征被表示为流形上与坐标无关的特征场。对于流形上的任意与坐标无关的特征场，存在一个从主丛到数值特征空间的等变嵌入。我们认为，该嵌入在数值特征空间上诱导的度量应最优地保留主丛的原始度量。这一最优性准则导致需要最小化该嵌入图相对于数值特征空间的扭曲Polyakov作用量，从而在相关的向量丛上产生一个等变扩散过程。通过将扩散方程流在固定时间步长下离散化，我们得到了流形上的消息传递方案。我们提出了一种高阶等变扩散过程，该过程等价于底流形笛卡尔积上的扩散过程。该高阶扩散过程在图上的离散化产生了一类新的广义等变图神经网络，将ACE和MACE形式推广到了黎曼流形上的数据。