Riemannian MeanFlow for One-Step Generation on Manifolds

• URL: http://arxiv.org/abs/2603.10718v1
• Date: Wed, 11 Mar 2026 12:41:46 GMT
• Title: Riemannian MeanFlow for One-Step Generation on Manifolds
• Abstract summary: Flow Matching enables simulation-free training of generative models.<n>MeanFlow can be extended to manifold-valued generation where velocities lie in location-dependent tangent spaces.<n>Experiments on spheres, tori, and SO(3) demonstrate competitive one-step sampling with improved quality-efficiency trade-offs and substantially reduced sampling cost.
• Score: 54.09734511705173
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Flow Matching enables simulation-free training of generative models on Riemannian manifolds, yet sampling typically still relies on numerically integrating a probability-flow ODE. We propose Riemannian MeanFlow (RMF), extending MeanFlow to manifold-valued generation where velocities lie in location-dependent tangent spaces. RMF defines an average-velocity field via parallel transport and derives a Riemannian MeanFlow identity that links average and instantaneous velocities for intrinsic supervision. We make this identity practical in a log-map tangent representation, avoiding trajectory simulation and heavy geometric computations. For stable optimization, we decompose the RMF objective into two terms and apply conflict-aware multi-task learning to mitigate gradient interference. RMF also supports conditional generation via classifier-free guidance. Experiments on spheres, tori, and SO(3) demonstrate competitive one-step sampling with improved quality-efficiency trade-offs and substantially reduced sampling cost.

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