Generalized Wasserstein Flow Matching: Nested Couplings for Point Cloud Generation

Abstract

We formulate flow matching on probability measures over probability measures. Directly from the Wasserstein-on-Wasserstein (WoW) continuity equation, we show that admissible nested outer and inner couplings induce a valid metameasure flow and non-local velocity field. Validity requires neither optimal plans nor a smooth manifold. Practically, this opens up the design space of multisample flow matching: exact Wasserstein plans are distinguished by energy and straightness, while sliced and lazy linear plans yield scalable alternatives. Point-cloud experiments, including controlled Bures-Wasserstein and Euclidean minibatch baselines, demonstrate the resulting geometry-cost trade-off.

arXiv commentKeywords: flow matching, Wasserstein over Wasserstein, hierarchical optimal transport, multisample flow matching, minibatch optimal transport, sliced Wasserstein, Bures-Wasserstein, point cloud generation, generative models for distributions, measure-valued data, Wasserstein gradient flow, rectified flow, straight flows