Riemannian MeanFlow for One-Step Generation on Manifolds

TL;DR

Riemannian MeanFlow (RMF) introduces a novel manifold-valued generative modeling approach that avoids trajectory simulation, enabling efficient one-step sampling on complex geometric spaces.

Contribution

RMF extends MeanFlow to Riemannian manifolds by defining an intrinsic average-velocity field, simplifying geometric computations, and enhancing sampling efficiency.

Findings

RMF achieves competitive one-step sampling on various manifolds.

RMF reduces sampling costs significantly compared to traditional methods.

RMF supports conditional generation with classifier-free guidance.

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, SO(3), and SE(3) demonstrate competitive one-step sampling with improved quality-efficiency trade-offs and substantially reduced sampling cost.