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Normalizing Flows on Riemannian Manifolds

arXiv 1611.02304 ↗First posted 7 Nov 2016stat.ML cs.AI math.STcited by 68

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Abstract (arXiv)

We consider the problem of density estimation on Riemannian manifolds. Density estimation on manifolds has many applications in fluid-mechanics, optics and plasma physics and it appears often when dealing with angular variables (such as used in protein folding, robot limbs, gene-expression) and in general directional statistics. In spite of the multitude of algorithms available for density estimation in the Euclidean spaces Rn that scale to large n (e.g. normalizing flows, kernel methods and variational approximations), most of these methods are not immediately suitable for density estimation in more general Riemannian manifolds. We revisit techniques related to homeomorphisms from differential geometry for projecting densities to sub-manifolds and use it to generalize the idea of normalizing flows to more general Riemannian manifolds. The resulting algorithm is scalable, simple to implement and suitable for use with automatic differentiation. We demonstrate concrete examples of this method on the n-sphere Sn.

Key facts. Cited by 68 papers in the map (data through 14 Oct 2026). First posted on arXiv 7 Nov 2016.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 12; 2022: 18; 2023: 13; 2024: 9; 2025: 9; 2026: 7.

68 papers.

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  1. 2023Flow Matching on General Geometriescited by 265
  2. 2022Riemannian Score-Based Generative Modellingcited by 219
  3. 2021Hyperbolic Deep Neural Networks: A Surveycited by 139
  4. 2024FlowMM: Generating Materials with Riemannian Flow Matchingcited by 109
  5. 2024Causal Representation Learning from Multiple Distributions: A General Settingcited by 68
  6. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 64
  7. 2021Smooth Normalizing Flowscited by 53
  8. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 52
  9. 2021Rectangular Flows for Manifold Learningcited by 50
  10. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  11. 2021Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flowscited by 31
  12. 2022Spherical Sliced-Wassersteincited by 31
38 more
  1. 2022Diagnosing and Fixing Manifold Overfitting in Deep Generative Modelscited by 28
  2. 2022CaloMan: Fast generation of calorimeter showers with density estimation on learned manifoldscited by 28
  3. 2023Diffusion Probabilistic Fieldscited by 28
  4. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  5. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  6. 2022First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Datacited by 18
  7. 2023Scaling Riemannian Diffusion Modelscited by 18
  8. 2022Learning Riemannian Stable Dynamical Systems via Diffeomorphismscited by 17
  9. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  10. 2023Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modelingcited by 14
  11. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 13
  12. 2021Learning Stable Vector Fields on Lie Groupscited by 12
  13. 2024Stereographic Spherical Sliced Wasserstein Distancescited by 12
  14. 2021Density estimation on low-dimensional manifolds: an inflation-deflation approachcited by 11
  15. 2021Density estimation on smooth manifolds with normalizing flowscited by 11
  16. 2023Manifold Diffusion Fieldscited by 11
  17. 2021Principal Component Density Estimation for Scenario Generation Using Normalizing Flowscited by 10
  18. 2021Continuous normalizing flows on manifoldscited by 10
  19. 2022VQ-Flows: Vector Quantized Local Normalizing Flowscited by 10
  20. 2021Interpreting diffusion score matching using normalizing flowcited by 9
  21. 2025Riemannian Consistency Modelcited by 8
  22. 2023Learning Distributions on Manifolds with Free-Form Flowscited by 7
  23. 2022Nonlinear Isometric Manifold Learning for Injective Normalizing Flowscited by 6
  24. 2022Denoising Deep Generative Modelscited by 6
  25. 2022Neural Implicit Manifold Learning for Topology-Aware Density Estimationcited by 5
  26. 2023Unified framework for diffusion generative models in SO(3): applications in computer vision and astrophysicscited by 5
  27. 2026Causal Representation Learning from General Environments under Nonparametric Mixingcited by 5
  28. 2021Spread Flows for Manifold Modellingcited by 4
  29. 2022Principal Manifold Flowscited by 4
  30. 2022Deterministic training of generative autoencoders using invertible layerscited by 4
  31. 2025Discovering Symmetry Groups with Flow Matchingcited by 3
  32. 2021Reparameterized Sampling for Generative Adversarial Networkscited by 2
  33. 2022Risk-Neutral Market Simulationcited by 2
  34. 2022ManiFlow: Implicitly Representing Manifolds with Normalizing Flowscited by 2
  35. 2025Ergodic Generative Flowscited by 2
  36. 2022Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transportcited by 1
  37. 2022Atlas flow : compatible local structures on the manifoldcited by 1
  38. 2023Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applicationscited by 1

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