ReSFlow: Rectifying Categorical Flows on Statistical Manifolds for One-Step Generation

Manifold-valued data, and consequently the distributions they induce, are prevalent across many domains, ranging from the locations of geospatial events, such as earthquakes, to biomolecular torsion angles that encode information about three-dimensional structure. While diffusion and flow-based generative models have been successfully extended to compact manifolds, sampling typically requires tens or hundreds of sequential network evaluations. We introduce RW-Flow, a theoretically grounded framework for learning one-step generative models on compact manifolds via Wasserstein gradient flows. The main challenge is identifiability: driving the velocity field to zero should guarantee that the model distribution matches the target distribution. We establish a necessary and sufficient condition for identifiability on compact, connected Riemannian manifolds. We specifically show that, for a symmetric, Lipschitz-continuous cost function, the velocity field induced by the Sinkhorn divergence is identifiable if and only if the associated Gibbs kernel is nondegenerate. This characterization provides a general principle for designing identifiable costs on compact manifolds. It also reveals that the squared geodesic distance, the natural manifold analogue of the squared Euclidean distance, does not always guarantee identifiability. Across benchmarks involving geospatial events, protein side chain torsion angles, RNA backbone torsion angles, and general manifolds discretized as triangular meshes, RW-Flow outperforms existing one-step methods in nearly all settings under fair comparison conditions.

We introduce a new framework for one-step generative modelling on finite state spaces. To extend drifting beyond continuous domains, we use discrete Wasserstein geometry to define a target-relative KL gradient flow over the transitions of a reversible Markov kernel. We realize this probability flow at the particle level through Markov jumps and amortize the resulting transport updates into a latent-conditioned generator, so that the iterative dynamics are required only during training while inference remains one-step. In a controlled setting where the underlying distributions and transport dynamics can be computed exactly, we verify KL dissipation, consistency between the particle dynamics and the probability flow, and the predicted numerical scaling. We further show that a finite-capacity neural generator can track these exact transport targets while retaining one-step generation. These results validate the basic construction and provide a foundation for scaling Discrete Drifting to structured discrete data.

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.