  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, the Ollivier-Ricci curvature measures differences
between random walks via Wasserstein distances, thus grounding a geometric
concept in ideas from probability theory 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.
