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 under node permutations and the feature manifold's isometries. These properties have led to a beneficial inductive bias in many deep-learning tasks. Furthermore, they enable novel, more flexible feature designs. Numerical examples on synthetic data and an Alzheimer's classification application on triangle meshes of the right hippocampus demonstrate the usefulness of our new layers: While they apply to a much broader class of problems, they outperform task-specific state-of-the-art networks.

翻译：[translated abstract in Chinese] 我们针对特征位于黎曼流形中的图，提出了两种图神经网络层。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迁移到我们的通用设置中，我们建模了一种切空间多层感知机。这两个层在节点置换和特征流形等距变换下均具有等变性。这些特性在许多深度学习任务中产生了有益的归纳偏置。此外，它们使得设计更新颖、更灵活的特征成为可能。关于合成数据的数值实验以及针对右海马体三角网格的阿尔茨海默症分类应用，证明了我们新层的有用性：尽管它们适用于更广泛的问题类别，但在性能上超越了任务特定的最先进网络。