迈向统一几何理解：图生成与预测的黎曼扩散框架
Toward a Unified Geometry Understanding: Riemannian Diffusion Framework for Graph Generation and Prediction
AI总结：
本文提出GeoMancer框架，通过黎曼扩散方法解决图数据生成与预测中的几何潜力释放问题，提升模型对复杂流形结构的学习能力。
摘要：
图扩散模型在学习结构化图数据方面取得了显著进展，并在预测任务中展现出强大的潜力。现有方法通常将节点、边和图层面的特征嵌入到统一的潜在空间中，将分类和回归等预测任务建模为条件生成的形式。然而，由于图数据的非欧几里得性质，不同曲率的特征在同一个潜在空间中纠缠在一起，未能释放其几何潜力。为了解决这个问题，我们旨在构建一个理想的黎曼扩散模型，以捕捉复杂图数据的不同流形特征，并学习其分布。这一目标面临两个挑战：编码过程中由于指数映射导致的数值不稳定性，以及扩散生成过程中的流形偏差。为了解决这些挑战，我们提出了GeoMancer：一种新的黎曼图扩散框架，用于生成和预测任务。为了缓解数值不稳定性，我们用等距不变的黎曼 gyrokernel 方法替代指数映射，并将多级特征解耦到各自的任务特定流形上以学习最优表示。为了解决流形偏差问题，我们引入了流形约束的扩散方法和无条件生成的自引导策略，确保生成的数据保持与流形特征一致。广泛的实验验证了我们方法的有效性，证明了其在各种任务上的优越性能。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
