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Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes
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Proposes Riemannian Diffusion Mixture, a generative diffusion framework on manifolds built as a mixture of logarithm/spectral bridge processes, trained with a divergence-free two-way bridge matching objective.
Problem
Generative modeling of data distributions on Riemannian manifolds; existing manifold diffusion models need heat kernel approximations or divergence computation, and CNF/flow methods need many in-training simulation steps on general manifolds.
Approach
Construct a manifold diffusion process as a mixture of bridge processes (logarithm bridges or spectral bridges) whose drift is a weighted mean of tangent directions toward data points; train with two-way bridge matching, a divergence-free regression on bridge drifts with time-scaled importance sampling, avoiding heat kernel estimation.
Key results
- On earth science datasets, LogBM achieves -9.52 test NLL on Volcano and -2.47 on Fire, significantly outperforming baselines such as RFM (-7.93) and RDM (-6.61).
- On general closed manifolds, SpecBM outperforms RFM using only 5% of in-training simulation steps (15 steps), with a reported 12.8x training speedup over RFM.
- On high-dimensional tori, the method scales to dimension 10^4 and reports up to 5x training speedup over RSGM with implicit score matching.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LogBM instead of the strongest reported baseline | manifold generative modeling · Volcano and Fire | test NLL: Volcano: -7.93 -> -9.52; Fire: -1.86 -> -2.47 | Helped |
| Use LogBM instead of the strongest reported baseline | manifold generative modeling · Earthquake and Flood | test NLL: Earthquake: -0.38 -> -0.30; Flood: 0.25 -> 0.42 | Hurt |
| Use LogBM instead of the strongest reported baseline | protein distribution modeling · General (2D), Glycine (2D), Pre-Pro (2D), and RNA (7D) | test NLL: General: 1.01 -> 1.01; Glycine: 1.90 -> 1.89; Pre-Pro: 1.18 -> 1.18; RNA: -5.20 -> -5.27 | Mixed |
| Use LogBM instead of RDM | protein distribution modeling · Proline (2D) | test NLL: 0.12 -> 0.14 | Hurt |
| Use time-scaled sampling instead of uniformly distributed time (ablation) | Riemannian Diffusion Mixture · time-scaled training objective experiment | test NLL | Helped |
| Use the same bridge samples for forward and backward directions instead of different samples (ablation) | two-way bridge matching · two-way bridge matching ablation | performance | No clear effect |
| Use stochastic diffusion mixture rather than a deterministic flow | general closed manifold modeling · triangular mesh distributions | performance | Helped |
| Use two-way bridge simulation rather than one-way simulation | Riemannian Diffusion Mixture training · bridge trajectory simulation | training speedup: up to ×34.9 | Helped |
| Use stochastic mixture process rather than deterministic RFM | general manifold training · general closed manifolds | training speedup: ×12.8 | Helped |
Surprising findings
- Uniform time sampling instead of the time-scaled distribution causes a significant drop in performance, showing the time distribution matters for likelihood.
- Riemannian flow matching (deterministic) fails with few in-training simulation steps, whereas the stochastic mixture works with 15 steps, suggesting stochasticity is necessary on curved manifolds.
Limitations
- Logarithm bridge requires a closed-form or cheaply computable logarithm map; spectral bridge requires eigenfunctions of the Laplace-Beltrami operator computed in advance.
- Likelihood evaluation uses ODE solvers (Dormand-Prince) or many Euler steps, which is costly.
- Heat-kernel-based Brownian bridge is not used in the main method due to intractability.
Experimental setup
Datasets: Volcano Earthquake Flood Fire General Glycine Proline Pre-Pro RNA Stanford Bunny Spot the Cow
Benchmarks: Volcano Earthquake Flood Fire General Glycine Proline Pre-Pro RNA Stanford Bunny Spot the Cow
Metrics: test NLL log-likelihood in bits training speedup
Baselines: RCNF Moser Flow CNFM RFM StereoSGM RSGM RDM RSGM-improved MoPS
Builds on: diffusion mixture representation Riemannian Flow Matching Riemannian score-based models Brownian bridge
Closest papers by meaning
- Efficient Diffusion Models for Symmetric Manifolds (ICML 2025 · similarity 0.91)
- Learning Distributions on Manifolds with Free-Form Flows (NeurIPS 2024 · similarity 0.91)
- Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity (NeurIPS 2025 · similarity 0.91)
- Manifold Diffusion Fields (ICLR 2024 · similarity 0.90)
- Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion (ICLR 2025 · similarity 0.90)
- Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction (NeurIPS 2025 · similarity 0.89)
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