Toward a Unified Geometry Understanding: Riemannian Diffusion Framework for Graph Generation and Prediction
This work addresses a domain-specific problem in graph machine learning by improving geometric modeling for generation and prediction, representing an incremental advance over existing diffusion approaches.
The paper tackled the problem of graph diffusion models entangling features of different curvatures in a unified latent space, which limits geometric potential, by proposing GeoMancer, a Riemannian diffusion framework that decouples features onto task-specific manifolds and uses a manifold-constrained diffusion method, resulting in superior performance across various tasks.
Graph diffusion models have made significant progress in learning structured graph data and have demonstrated strong potential for predictive tasks. Existing approaches typically embed node, edge, and graph-level features into a unified latent space, modeling prediction tasks including classification and regression as a form of conditional generation. However, due to the non-Euclidean nature of graph data, features of different curvatures are entangled in the same latent space without releasing their geometric potential. To address this issue, we aim to construt an ideal Riemannian diffusion model to capture distinct manifold signatures of complex graph data and learn their distribution. This goal faces two challenges: numerical instability caused by exponential mapping during the encoding proces and manifold deviation during diffusion generation. To address these challenges, we propose GeoMancer: a novel Riemannian graph diffusion framework for both generation and prediction tasks. To mitigate numerical instability, we replace exponential mapping with an isometric-invariant Riemannian gyrokernel approach and decouple multi-level features onto their respective task-specific manifolds to learn optimal representations. To address manifold deviation, we introduce a manifold-constrained diffusion method and a self-guided strategy for unconditional generation, ensuring that the generated data remains aligned with the manifold signature. Extensive experiments validate the effectiveness of our approach, demonstrating superior performance across a variety of tasks.