Bridging geometry and topology, curvature is a powerful and expressive invariant. While the utility of curvature has been theoretically and empirically confirmed in the context of manifolds and graphs, its generalization to the emerging domain of hypergraphs has remained largely unexplored. On graphs, Ollivier-Ricci curvature measures differences between random walks via Wasserstein distances, thus grounding a geometric concept in ideas from probability and optimal transport. We develop ORCHID, a flexible framework generalizing Ollivier-Ricci curvature to hypergraphs, and prove that the resulting curvatures have favorable theoretical properties. Through extensive experiments on synthetic and real-world hypergraphs from different domains, we demonstrate that ORCHID curvatures are both scalable and useful to perform a variety of hypergraph tasks in practice.
翻译:架桥几何学和地形学,曲线是一个强力和直观的变异体。虽然曲解的效用在理论上和经验上已经得到在数学和图表方面的确认,但其对新兴高地学领域的概括性基本上尚未探索。在图表上,奥利维埃-里希曲解测量了通过瓦塞斯坦距离随机行走的差异,从而将几何概念从概率和最佳运输的概念中打下基础。我们开发了ORCHID,这是一个将奥利维埃-里科曲解法通向高地学的灵活框架,并证明由此产生的曲解具有有利的理论特性。通过对不同领域的合成和现实世界高地高地进行广泛的实验,我们证明ORCHID曲解对于实际执行各种高地学任务既可扩展又有用。