Symplectic Structure-Aware Hamiltonian (Graph) Embeddings
This work addresses the problem of adapting GNNs to diverse graph geometries for researchers and practitioners in graph machine learning, representing an incremental improvement over existing Hamiltonian GNNs.
The paper tackles the limitation of fixed embedding manifolds in Graph Neural Networks (GNNs) by proposing Symplectic Structure-Aware Hamiltonian GNN (SAH-GNN), which generalizes Hamiltonian dynamics for flexible node feature updates, achieving superior performance in node classification tasks across multiple graph datasets.
In traditional Graph Neural Networks (GNNs), the assumption of a fixed embedding manifold often limits their adaptability to diverse graph geometries. Recently, Hamiltonian system-inspired GNNs have been proposed to address the dynamic nature of such embeddings by incorporating physical laws into node feature updates. We present Symplectic Structure-Aware Hamiltonian GNN (SAH-GNN), a novel approach that generalizes Hamiltonian dynamics for more flexible node feature updates. Unlike existing Hamiltonian approaches, SAH-GNN employs Riemannian optimization on the symplectic Stiefel manifold to adaptively learn the underlying symplectic structure, circumventing the limitations of existing Hamiltonian GNNs that rely on a pre-defined form of standard symplectic structure. This innovation allows SAH-GNN to automatically adapt to various graph datasets without extensive hyperparameter tuning. Moreover, it conserves energy during training meaning the implicit Hamiltonian system is physically meaningful. Finally, we empirically validate SAH-GNN's superiority and adaptability in node classification tasks across multiple types of graph datasets.