  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.
