mlsanityData to 14 Oct 2026

Riemannian Diffusion Models

arXiv 2208.07949 ↗First posted 16 Aug 2022cs.LGcited by 122

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models are recent state-of-the-art methods for image generation and likelihood estimation. In this work, we generalize continuous-time diffusion models to arbitrary Riemannian manifolds and derive a variational framework for likelihood estimation. Computationally, we propose new methods for computing the Riemannian divergence which is needed in the likelihood estimation. Moreover, in generalizing the Euclidean case, we prove that maximizing this variational lower-bound is equivalent to Riemannian score matching. Empirically, we demonstrate the expressive power of Riemannian diffusion models on a wide spectrum of smooth manifolds, such as spheres, tori, hyperboloids, and orthogonal groups. Our proposed method achieves new state-of-the-art likelihoods on all benchmarks.

Key facts. Cited by 122 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 32 of its 68 references that are in the map. First posted on arXiv 16 Aug 2022.

Builds on

32 of 68 references.

Closest ancestors · 7

Same direction, newest first

  1. 2022Riemannian Score-Based Generative Modelling6 Feb
  2. 2021Variational Diffusion Models1 Jul
  3. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  4. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  5. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  6. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  7. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 25

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2013Auto-Encoding Variational Bayescited by 4,053
  6. 2018Neural Ordinary Differential Equationscited by 3,779
  7. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,545
  8. 2015Importance Weighted Autoencoderscited by 456
  9. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  10. 2018Neural Autoregressive Flowscited by 208
  11. 2021Neural SDEs as Infinite-Dimensional GANscited by 161
  12. 2018Neural Importance Samplingcited by 155
13 more
  1. 2020Riemannian Continuous Normalizing Flowscited by 145
  2. 2020Flows for simultaneous manifold learning and density estimationcited by 118
  3. 2020Normalizing Flows on Tori and Spherescited by 117
  4. 2019Mixed-curvature Variational Autoencoderscited by 86
  5. 2020Neural Manifold Ordinary Differential Equationscited by 81
  6. 2017REBAR: Low-variance, unbiased gradient estimates for discrete latent variable modelscited by 79
  7. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  8. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
  9. 2020Neural Ordinary Differential Equations on Manifoldscited by 31
  10. 2020The Power Spherical distributioncited by 27
  11. 2020Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolevcited by 16
  12. 2020SUMO: Unbiased Estimation of Log Marginal Probability for Latent Variable Modelscited by 14
  13. 2021Manifold Density Estimation via Generalized Dequantizationcited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 5; 2023: 23; 2024: 23; 2025: 34; 2026: 37.

122 papers.

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  1. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,114
  2. 2023ModelScope Text-to-Video Technical Reportcited by 658
  3. 2023I2VGen-XL: High-Quality Image-to-Video Synthesis via Cascaded Diffusion Modelscited by 350
  4. 2023Flow Matching on General Geometriescited by 265
  5. 2022A Survey on Generative Diffusion Modelcited by 240
  6. 2023SE(3) diffusion model with application to protein backbone generationcited by 219
  7. 2023UFOGen: You Forward Once Large Scale Text-to-Image Generation via Diffusion GANscited by 166
  8. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 150
  9. 2023Crystal Structure Prediction by Joint Equivariant Diffusioncited by 148
  10. 2022SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusioncited by 127
  11. 2024FlowMM: Generating Materials with Riemannian Flow Matchingcited by 109
  12. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 85
38 more
  1. 2024NVS-Solver: Video Diffusion Model as Zero-Shot Novel View Synthesizercited by 79
  2. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 77
  3. 2024Fisher Flow Matching for Generative Modeling over Discrete Datacited by 72
  4. 2025Proteina: Scaling Flow-based Protein Structure Generative Modelscited by 65
  5. 2023Diffusion Models for Constrained Domainscited by 63
  6. 2024Riemannian Flow Matching Policy for Robot Motion Learningcited by 63
  7. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 60
  8. 2023Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulationcited by 52
  9. 2024Target Score Matchingcited by 50
  10. 2022Diffusion Generative Models in Infinite Dimensionscited by 49
  11. 2023Mirror Diffusion Models for Constrained and Watermarked Generationcited by 42
  12. 2023EDGI: Equivariant Diffusion for Planning with Embodied Agentscited by 36
  13. 2024Full-Atom Peptide Design based on Multi-modal Flow Matchingcited by 34
  14. 2023Metropolis Sampling for Constrained Diffusion Modelscited by 28
  15. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  16. 2025Diffuse Everything: Multimodal Diffusion Models on Arbitrary State Spacescited by 21
  17. 2024Constrained Diffusion Models via Dual Trainingcited by 20
  18. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  19. 2023Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimizationcited by 18
  20. 2023Scaling Riemannian Diffusion Modelscited by 18
  21. 2024Fast and Robust Visuomotor Riemannian Flow Matching Policycited by 18
  22. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularitycited by 17
  23. 2025A Periodic Bayesian Flow for Material Generationcited by 16
  24. 2024Neural Thermodynamic Integration: Free Energies from Energy-based Diffusion Modelscited by 14
  25. 2025Carr'e du champ flow matching: better quality-generalisation tradeoff in generative modelscited by 14
  26. 2023Star-Shaped Denoising Diffusion Probabilistic Modelscited by 13
  27. 2023Generative Fractional Diffusion Modelscited by 12
  28. 2024Reflected Schr"odinger Bridge for Constrained Generative Modelingcited by 12
  29. 2025Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifoldscited by 12
  30. 2024Context-Guided Diffusion for Out-of-Distribution Molecular and Protein Designcited by 11
  31. 2023Geometric Neural Diffusion Processescited by 10
  32. 2024Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifoldscited by 9
  33. 2024Neural Approximate Mirror Maps for Constrained Diffusion Modelscited by 9
  34. 2024Counterfactual Explanations via Riemannian Latent Space Traversalcited by 9
  35. 2024Hotspot-Driven Peptide Design via Multi-Fragment Autoregressive Extensioncited by 9
  36. 2025Diffusion Generative Modeling on Lie Group Representationscited by 8
  37. 2025Riemannian Consistency Modelcited by 8
  38. 2026Riemannian MeanFlowcited by 8

50 most cited of 122 All 122 (JSON)

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Close in wording, no citation either way

  1. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models
  2. 2022Riemannian Diffusion Schr"odinger Bridge
  3. 2025What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold
  4. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  5. 2025Image Interpolation with Score-based Riemannian Metrics of Diffusion Models
  6. 2023Infinite-Dimensional Diffusion Models
  7. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions
  8. 2026DiffRGD: An Inference-Time Diffusion Guidance Through Riemannian Gradient Descent