Riemannian Flow Matching

Riemannian flow matching learns transformations between distributions while keeping trajectories on curved spaces, which can help ensure generated states remain valid—for example, rotations, motions, or structured representations. Common approaches define geodesic or optimal-transport probability paths and train vector fields in tangent spaces using tools such as log maps and parallel transport; flow-map methods aim to reduce sampling to one or a few steps. Applications include generation on spheres and Lie groups, human motion and robot actions, image and 3D-scene generation, and scientific simulation. A central challenge is balancing faithful geometry with computational cost, especially when manifolds lack simple geodesics, and establishing when geometric modeling improves results over Euclidean methods.