Graph representation learning has been widely studied and demonstrated effectiveness in various graph tasks. Most existing works embed graph data in the Euclidean space, while recent works extend the embedding models to hyperbolic or spherical spaces to achieve better performance on graphs with complex structures, such as hierarchical or ring structures. Fusing the embedding from different manifolds can further take advantage of the embedding capabilities over different graph structures. However, existing embedding fusion methods mostly focus on concatenating or summing up the output embeddings, without considering interacting and aligning the embeddings of the same vertices on different manifolds, which can lead to distortion and impression in the final fusion results. Besides, it is also challenging to fuse the embeddings of the same vertices from different coordinate systems. In face of these challenges, we propose the Fused Manifold Graph Neural Network (FMGNN), a novel GNN architecture that embeds graphs into different Riemannian manifolds with interaction and alignment among these manifolds during training and fuses the vertex embeddings through the distances on different manifolds between vertices and selected landmarks, geometric coresets. Our experiments demonstrate that FMGNN yields superior performance over strong baselines on the benchmarks of node classification and link prediction tasks.

翻译：图表示学习已被广泛研究，并在各种图任务中展现出有效性。现有大多数工作将图数据嵌入到欧几里得空间中，而近期研究将嵌入模型扩展到双曲空间或球面空间，以在具有复杂结构（如层次结构或环状结构）的图上取得更优性能。融合来自不同流形的嵌入，可以进一步利用不同图结构上的嵌入能力。然而，现有的嵌入融合方法大多侧重于对输出嵌入进行拼接或求和，而未考虑对齐和交互同一顶点在不同流形上的嵌入，这可能导致最终融合结果出现扭曲和模糊。此外，从不同坐标系融合同一顶点的嵌入也具有挑战性。针对这些挑战，我们提出了融合流形图神经网络（FMGNN），这是一种新颖的GNN架构，它将图嵌入到不同的黎曼流形中，在训练过程中实现这些流形之间的交互与对齐，并通过不同流形上顶点与选定地标（几何核心集）之间的距离来融合顶点嵌入。我们的实验表明，在节点分类和链接预测任务的基准测试中，FMGNN相较于强基线方法展现出更优的性能。