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Riemannian Consistency Model
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Proposes Riemannian Consistency Models (RCM), extending consistency distillation and training to Riemannian manifolds via exponential-map parameterization and covariant derivatives, with a simplified loss, evaluated on spheres, tori, and SO(3) for few-step generation.
Problem
Few-step generation on Riemannian manifolds is hard because consistency models assume Euclidean geometry; consistency parameterizations must stay on the manifold and time derivatives must account for curvature.
Approach
Parameterize the consistency function as f(x,t)=exp_x(kappa_t v(x,t)), derive a continuous-time consistency loss using the covariant derivative and exponential-map differentials, show RCD and RCT are equivalent via marginalization, and propose a simplified loss that drops the exponential-map differentials.
Key results
- On the 2-sphere Flood dataset, sRCD achieves KLD 2.27 vs. 8.17 for RFM-2 and 2.81 for naive CD, with 2-step generation.
- On SO(3) Line dataset, RCD achieves MMD 2.39e-2 vs. 15.50e-2 for RFM-2 in 2 steps.
- On 128-dimensional tori, RCT Fréchet distance 0.62 vs. 1.83 for RFM and 35.16 for naive CT under 2 steps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use sRCD instead of RFM-2 | Earthquake generation on the 2-sphere · Earthquake | KLD: 10.99 -> 2.13 | Helped |
| Use sRCD instead of RFM-2 | Volcano generation on the 2-sphere · Volcano | KLD: 35.40 -> 3.36 | Helped |
| Use sRCD instead of RFM-2 | Fire generation on the 2-sphere · Fire | KLD: 9.79 -> 1.65 | Helped |
| Use sRCT instead of RFM-2 | Flood generation on the 2-sphere · Flood | KLD: 8.17 -> 2.23 | Helped |
| Simplify RCD to dRCD (ablation) | 2D flat-torus generation · Board | MMD / 10^-2: 0.61 -> 0.37 | Helped |
| Use dRCD instead of RCD (ablation) | Protein generation on the 2D flat torus · Protein | MMD / 10^-2: 3.84 -> 3.79 | Helped |
| Use dRCD instead of RCD (ablation) | RNA generation on the 2D flat torus · RNA | MMD / 10^-2: 7.82 -> 3.39 | Helped |
| Use RCD instead of RFM-2 | Swiss Roll generation on SO(3) · Swiss Roll | MMD / 10^-2: 19.64 -> 1.47 | Helped |
| Use RCT instead of RFM-2 | Cone generation on SO(3) · Cone | MMD / 10^-2: 19.96 -> 3.78 | Helped |
| Use RCD instead of RFM-2 | Fisher generation on SO(3) · Fisher | MMD / 10^-2: 17.41 -> 5.71 | Helped |
| Use RCD instead of RFM-2 | Line generation on SO(3) · Line | MMD / 10^-2: 15.50 -> 2.39 | Helped |
| Use the simplified RCD objective instead of RCD (ablation) | RCM generation across reported datasets · 2-sphere datasets | KLD | Helped |
Surprising findings
- The discrete-time RCD (dRCD) performs well on the flat torus but worse on SO(3), attributed to discretization error on curved geometry.
- Naive Euclidean consistency training degrades sharply in higher-dimensional tori, which the authors attribute to the difficulty of satisfying manifold constraints.
Limitations
- Evaluations are on low-dimensional synthetic and small real datasets, not large-scale protein or image generation.
- The simplified loss relies on an approximation that the two exponential-map differentials are equal, exact only for flat tori or parallel vectors.
- Sample quality depends on the pre-trained RFM teacher for distillation variants.
Experimental setup
Datasets: checkerboard flat torus (synthetic) protein torsion angles RNA torsion angles SO(3) synthetic Cone/Fisher/Line/Swiss Roll earthquake, volcano, flood, wildfire location datasets on sphere high-dimensional wrapped Gaussian on tori
Benchmarks: earth location datasets on 2-sphere (KLD) checkerboard, protein, RNA on 2D/7D tori (MMD) SO(3) synthetic datasets (MMD) high-dimensional tori Fréchet distance
Metrics: KL divergence of kernel density maximum mean discrepancy Fréchet distance
Baselines: RFM-100 RFM-2 CD naive CT naive dRCD
Builds on: consistency models Riemannian flow matching Euclidean consistency training EDM2
Closest papers by meaning
- Flow Matching on General Geometries (ICLR 2024 · similarity 0.90)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.88)
- Categorical Flow Matching on Statistical Manifolds (NeurIPS 2024 · similarity 0.87)
- Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models (NeurIPS 2025 · similarity 0.86)
- How to build a consistency model: Learning flow maps via self-distillation (NeurIPS 2025 · similarity 0.86)
- Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction (NeurIPS 2025 · similarity 0.86)
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