# Riemannian Consistency Model arXiv: https://arxiv.org/abs/2510.00983 ยท HTML: https://mlsanity.com/paper/2510.00983/ ## Key facts - Cited by 8 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 46 of its 58 references that are in the map. - First posted on arXiv 1 Oct 2025. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 30 of 46) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022) - [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023) - [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021) - [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020) - [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Adversarial Diffusion Distillation](https://mlsanity.com/paper/2311.17042/) (2023) - [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022) - [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018) - [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022) - [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023) - [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023) - [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024) - [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024) - [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023) - [Diff-Instruct: A Universal Approach for Transferring Knowledge From Pre-trained Diffusion Models](https://mlsanity.com/paper/2305.18455/) (2023) - [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022) - [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023) - [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023) ## Built on by (most cited first, 8 of 8) - [Riemannian MeanFlow](https://mlsanity.com/paper/2602.07744/) (2026) - [Riemannian MeanFlow for One-Step Generation on Manifolds](https://mlsanity.com/paper/2603.10718/) (2026) - [EvoEGF-Mol: Evolving Exponential Geodesic Flow for Structure-based Drug Design](https://mlsanity.com/paper/2601.22466/) (2026) - [FLARE: Flow Matching with Local Axis-Angle Representations for Stochastic Micromagnetic Evolution](https://mlsanity.com/paper/2609.34070/) (2026) - [Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature](https://mlsanity.com/paper/2609.36568/) (2026) - [RW-Flow: One-Step Generation on Compact Manifolds via Wasserstein Gradient Flows](https://mlsanity.com/paper/2609.39271/) (2026) - [One-Step Generation via Riemannian Wasserstein Gradient Flows](https://mlsanity.com/paper/2610.04454/) (2026) - [ReSFlow: Rectifying Categorical Flows on Statistical Manifolds for One-Step Generation](https://mlsanity.com/paper/2610.13225/) (2026) ## Abstract (arXiv) Consistency models are a class of generative models that enable few-step generation for diffusion and flow matching models. While consistency models have achieved promising results on Euclidean domains like images, their applications to Riemannian manifolds remain challenging due to the curved geometry. In this work, we propose the Riemannian Consistency Model (RCM), which, for the first time, enables few-step consistency modeling while respecting the intrinsic manifold constraint imposed by the Riemannian geometry. Leveraging the covariant derivative and exponential-map-based parameterization, we derive the closed-form solutions for both discrete- and continuous-time training objectives for RCM. We then demonstrate theoretical equivalence between the two variants of RCM: Riemannian consistency distillation (RCD) that relies on a teacher model to approximate the marginal vector field, and Riemannian consistency training (RCT) that utilizes the conditional vector field for training. We further propose a simplified training objective that eliminates the need for the complicated differential calculation. Finally, we provide a unique kinematics perspective for interpreting the RCM objective, offering new theoretical angles. Through extensive experiments, we manifest the superior generative quality of RCM in few-step generation on various non-Euclidean manifolds, including flat-tori, spheres, and the 3D rotation group SO(3).