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Normalizing Flows on Tori and Spheres

arXiv 2002.02428 ↗First posted 6 Feb 2020stat.ML cs.LGcited by 117

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Abstract (arXiv)

Normalizing flows are a powerful tool for building expressive distributions in high dimensions. So far, most of the literature has concentrated on learning flows on Euclidean spaces. Some problems however, such as those involving angles, are defined on spaces with more complex geometries, such as tori or spheres. In this paper, we propose and compare expressive and numerically stable flows on such spaces. Our flows are built recursively on the dimension of the space, starting from flows on circles, closed intervals or spheres.

Key facts. Cited by 117 papers in the map (data through 14 Oct 2026). First posted on arXiv 6 Feb 2020.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 19; 2022: 24; 2023: 22; 2024: 19; 2025: 21; 2026: 12.

117 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. 2022Flow Annealed Importance Sampling Bootstrapcited by 127
  5. 2022Riemannian Diffusion Modelscited by 122
  6. 2023Spontaneous Symmetry Breaking in Generative Diffusion Modelscited by 80
  7. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  8. 2021Implicit-PDF: Non-Parametric Representation of Probability Distributions on the Rotation Manifoldcited by 65
  9. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 64
  10. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  11. 2023normflows: A PyTorch Package for Normalizing Flowscited by 56
  12. 2021Smooth Normalizing Flowscited by 53
38 more
  1. 2021Rectangular Flows for Manifold Learningcited by 50
  2. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  3. 2021Normalizing flows for atomic solidscited by 39
  4. 2022Learning Lattice Quantum Field Theories with Equivariant Continuous Flowscited by 37
  5. 2023SE(3) Equivariant Augmented Coupling Flowscited by 37
  6. 2022Flow-matching -- efficient coarse-graining of molecular dynamics without forcescited by 36
  7. 2021Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flowscited by 31
  8. 2022Spherical Sliced-Wassersteincited by 31
  9. 2022Diagnosing and Fixing Manifold Overfitting in Deep Generative Modelscited by 28
  10. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  11. 2022Symmetry-Based Representations for Artificial and Biological General Intelligencecited by 26
  12. 2022Aspects of scaling and scalability for flow-based sampling of lattice QCDcited by 26
  13. 2024PPFlow: Target-aware Peptide Design with Torsional Flow Matchingcited by 26
  14. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  15. 2023Riemannian Residual Neural Networkscited by 21
  16. 2024Training Neural Samplers with Reverse Diffusive KL Divergencecited by 21
  17. 2021Equivariant Manifold Flowscited by 20
  18. 2024Efficient mapping of phase diagrams with conditional Boltzmann Generatorscited by 18
  19. 2021Riemannian Convex Potential Mapscited by 17
  20. 2022Learning Riemannian Stable Dynamical Systems via Diffeomorphismscited by 17
  21. 2023Confronting Ambiguity in 6D Object Pose Estimation via Score-Based Diffusion on SE(3)cited by 17
  22. 2022Statistical Deep Learning for Spatial and Spatio-Temporal Datacited by 15
  23. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  24. 2023Canonical normalizing flows for manifold learningcited by 15
  25. 2025Temperature-Annealed Boltzmann Generatorscited by 15
  26. 2023Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modelingcited by 14
  27. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 13
  28. 2021Learning Stable Vector Fields on Lie Groupscited by 12
  29. 2025Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifoldscited by 12
  30. 2021Density estimation on low-dimensional manifolds: an inflation-deflation approachcited by 11
  31. 2021Density estimation on smooth manifolds with normalizing flowscited by 11
  32. 2021Continuous normalizing flows on manifoldscited by 10
  33. 2021Implicit Riemannian Concave Potential Mapscited by 10
  34. 2024Scalable Normalizing Flows Enable Boltzmann Generators for Macromoleculescited by 10
  35. 2025Learning Boltzmann Generators via Constrained Mass Transportcited by 9
  36. 2022TO-FLOW: Efficient Continuous Normalizing Flows with Temporal Optimization adjoint with Moving Speedcited by 8
  37. 2022On the Robustness of Normalizing Flows for Inverse Problems in Imagingcited by 8
  38. 2023Transition Path Sampling with Boltzmann Generator-based MCMC Movescited by 8

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