We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant with respect to node permutations and isometries of the feature manifold. These properties have been shown to lead to a beneficial inductive bias in many deep learning tasks. Numerical examples on synthetic data as well as on triangle meshes of the right hippocampus to classify Alzheimer's disease demonstrate the very good performance of our layers.

翻译：我们提出了两种适用于黎曼流形上特征图的图神经网络层。首先，基于流形值图扩散方程，我们构建了一个可应用于任意数量节点和任意图连接模式的扩散层。其次，通过将向量神经元框架的思想推广至通用场景，我们建模了一个切空间多层感知器。这两个层均具有节点置换等变性和特征流形等距等变性。这些性质已被证明能在众多深度学习任务中带来有益的归纳偏置。针对合成数据以及用于阿尔茨海默病分类的右侧海马三角网格的数值实验，验证了我们所提层结构的优异性能。