mlsanityData to 7 Oct 2026

Riemannian Score-Based Generative Modelling

arXiv 2202.02763 ↗First posted 6 Feb 2022cs.LG math.PR stat.MLcited by 216

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Key facts. Cited by 216 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 48 of its 167 references that are in the map. First posted on arXiv 6 Feb 2022.

Builds on

48 of 167 references.

Closest ancestors · 12

Same direction, newest first

  1. 2022Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models17 Jan
  2. 2021Conditional Image Generation with Score-Based Diffusion Models26 Nov
  3. 2021Learning to Efficiently Sample from Diffusion Probabilistic Models7 Jun
  4. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matching5 Jun
  5. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  6. 2021Gotta Go Fast When Generating Data with Score-Based Models28 May
  7. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  8. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  9. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  10. 2020Denoising Diffusion Implicit Models6 Oct
  11. 2020Denoising Diffusion Probabilistic Models19 Jun
  12. 2020Improved Techniques for Training Score-Based Generative Models16 Jun

Other references · 36

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  3. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  4. 2018Computational Optimal Transportcited by 765
  5. 2021Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contractioncited by 306
  6. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 248
  7. 2021Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theorycited by 216
  8. 2021SNIPS: Solving Noisy Inverse Problems Stochasticallycited by 191
  9. 2020Riemannian Continuous Normalizing Flowscited by 143
  10. 2021Solving Schr"odinger Bridges via Maximum Likelihoodcited by 140
  11. 2020Normalizing Flows on Tori and Spherescited by 117
  12. 2020Flows for simultaneous manifold learning and density estimationcited by 117
24 more
  1. 2019Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoderscited by 111
  2. 2020Adversarial score matching and improved sampling for image generationcited by 95
  3. 2020Neural Manifold Ordinary Differential Equationscited by 81
  4. 2021Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Dockingcited by 81
  5. 2018geomstats: a Python Package for Riemannian Geometry in Machine Learningcited by 71
  6. 2016Normalizing Flows on Riemannian Manifoldscited by 68
  7. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  8. 2021Density Ratio Estimation via Infinitesimal Classificationcited by 59
  9. 2017Machine Learning for the Geosciences: Challenges and Opportunitiescited by 53
  10. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
  11. 2021Rectangular Flows for Manifold Learningcited by 49
  12. 2018Deep Directional Statistics: Pose Estimation with Uncertainty Quantificationcited by 45
  13. 2021Stochastic Image Denoising by Sampling from the Posterior Distributioncited by 45
  14. 2015Entropic and displacement interpolation: a computational approach using the Hilbert metriccited by 34
  15. 2019Escaping from saddle points on Riemannian manifoldscited by 33
  16. 2020Neural Ordinary Differential Equations on Manifoldscited by 31
  17. 2021Riemannian Convex Potential Mapscited by 16
  18. 2022Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithmcited by 13
  19. 2018Computing CNN Loss and Gradients for Pose Estimation with Riemannian Geometrycited by 11
  20. 2021Continuous normalizing flows on manifoldscited by 10
  21. 2021Implicit Riemannian Concave Potential Mapscited by 10
  22. 2018Directional grid maps: modeling multimodal angular uncertainty in dynamic environmentscited by 5
  23. 2016The Multivariate Generalised von Mises distribution: Inference and applicationscited by 3
  24. 2020An SQP method for equality constrained optimization on manifoldscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 22; 2023: 48; 2024: 34; 2025: 52; 2026: 60.

216 papers.

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  1. 2022Flow Matching for Generative Modelingcited by 5,426
  2. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,107
  3. 2022DiffDock: Diffusion Steps, Twists, and Turns for Molecular Dockingcited by 307
  4. 2022Torsional Diffusion for Molecular Conformer Generationcited by 262
  5. 2023Flow Matching on General Geometriescited by 260
  6. 2022A Survey on Generative Diffusion Modelcited by 238
  7. 2022Convergence of denoising diffusion models under the manifold hypothesiscited by 237
  8. 2023Practical and Asymptotically Exact Conditional Sampling in Diffusion Modelscited by 218
  9. 2023SE(3) diffusion model with application to protein backbone generationcited by 214
  10. 2023Image Restoration with Mean-Reverting Stochastic Differential Equationscited by 182
  11. 2022Generative Modelling With Inverse Heat Dissipationcited by 164
  12. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 147
38 more
  1. 2023Crystal Structure Prediction by Joint Equivariant Diffusioncited by 142
  2. 2022GENIE: Higher-Order Denoising Diffusion Solverscited by 127
  3. 2022SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusioncited by 125
  4. 2022Score-Based Generative Models Detect Manifoldscited by 121
  5. 2022Riemannian Diffusion Modelscited by 120
  6. 2022Protein Structure and Sequence Generation with Equivariant Denoising Diffusion Probabilistic Modelscited by 97
  7. 2024Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Reviewcited by 92
  8. 2023Reflected Diffusion Modelscited by 81
  9. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 78
  10. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  11. 2024Fisher Flow Matching for Generative Modeling over Discrete Datacited by 68
  12. 2023SALAD: Part-Level Latent Diffusion for 3D Shape Generation and Manipulationcited by 66
  13. 2023Particle Guidance: non-I.I.D. Diverse Sampling with Diffusion Modelscited by 65
  14. 2025Proteina: Scaling Flow-based Protein Structure Generative Modelscited by 63
  15. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 62
  16. 2023Diffusion Models for Constrained Domainscited by 60
  17. 2023On the Design Fundamentals of Diffusion Models: A Surveycited by 59
  18. 2022From Denoising Diffusions to Denoising Markov Modelscited by 58
  19. 2023Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulationcited by 51
  20. 2022Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Modelscited by 50
  21. 2023Navigating the Design Space of Equivariant Diffusion-Based Generative Models for De Novo 3D Molecule Generationcited by 50
  22. 2022Diffusion Generative Models in Infinite Dimensionscited by 49
  23. 2023GenPose: Generative Category-level Object Pose Estimation via Diffusion Modelscited by 47
  24. 2024Target Score Matchingcited by 46
  25. 2024Protein Conformation Generation via Force-Guided SE(3) Diffusion Modelscited by 44
  26. 2023Mirror Diffusion Models for Constrained and Watermarked Generationcited by 38
  27. 2023EDGI: Equivariant Diffusion for Planning with Embodied Agentscited by 35
  28. 2023AbDiffuser: Full-Atom Generation of in vitro Functioning Antibodiescited by 35
  29. 2023Str2Str: A Score-based Framework for Zero-shot Protein Conformation Samplingcited by 29
  30. 2025Continuous Diffusion Model for Language Modelingcited by 29
  31. 2023Diffusion Probabilistic Fieldscited by 28
  32. 2023Metropolis Sampling for Constrained Diffusion Modelscited by 28
  33. 2023A Unified Framework for U-Net Design and Analysiscited by 27
  34. 2024Deep Confident Steps to New Pockets: Strategies for Docking Generalizationcited by 27
  35. 2023DiffDock-PP: Rigid Protein-Protein Docking with Diffusion Modelscited by 23
  36. 2023GraspGF: Learning Score-based Grasping Primitive for Human-assisting Dexterous Graspingcited by 23
  37. 2024DimeRec: A Unified Framework for Enhanced Sequential Recommendation via Generative Diffusion Modelscited by 22
  38. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21

50 most cited of 216 All 216 (JSON)

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  1. 2023Unified framework for diffusion generative models in SO(3): applications in computer vision and astrophysics
  2. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  3. 2025What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold
  4. 2022Wavelet Score-Based Generative Modeling
  5. 2024A Good Score Does not Lead to A Good Generative Model
  6. 2022Convergence of score-based generative modeling for general data distributions
  7. 2023Geometry of Score Based Generative Models
  8. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models