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.

基于黎曼流形的图神经网络模型中，我们提出了两个关键的图神经网络层。第一个是扩散层，其基于流形值图扩散方程，适用于任意数量节点和图的连通模式。第二个是切线多层感知机层，借鉴了向量神经元框架的思想，并在一般的情境中应用。这两个层在节点排列和特征流形的等变性方面表现出非常好的性能。在合成数据和海马右侧三角网格对阿尔茨海默病分类的数值实例中，我们的模型均取得了非常好的性能。

流形 GCN: 基于扩散的流形值图卷积神经网络

Manifold GCN: Diffusion-based Convolutional Neural Network for  Manifold-valued Graphs

We introduce a new general-purpose approach to deep learning on 3D surfaces,
based on the insight that a simple diffusion layer is highly effective for
spatial communication. The resulting networks are automatically robust to
changes in resolution and sampling of a surface -- a basic property which is
crucial for practical applications. Our networks can be discretized on various
geometric representations such as triangle meshes or point clouds, and can even
be trained on one representation then applied to another. We optimize the
spatial support of diffusion as a continuous network parameter ranging from
purely local to totally global, removing the burden of manually choosing
neighborhood sizes. The only other ingredients in the method are a multi-layer
perceptron applied independently at each point, and spatial gradient features
to support directional filters. The resulting networks are simple, robust, and
efficient. Here, we focus primarily on triangle mesh surfaces, and demonstrate
state-of-the-art results for a variety of tasks including surface
classification, segmentation, and non-rigid correspondence.

本文提出了一种新的通用的深度学习方法，基于对简单扩散层高度有效的空间通信的理解。这种方法使得网络可以自动适应于 3D 表面的分辨率和采样变化，能够在不同的几何表示上进行离散化并且具有强鲁棒性和高效率，在三角网格表面分类、分割和非刚性对应等任务上表现出了最先进的结果。