mlsanityData to 7 Oct 2026

Matching Normalizing Flows and Probability Paths on Manifolds

arXiv 2207.04711 ↗First posted 11 Jul 2022stat.ML cs.LGcited by 57

Not on the map. Why?

Cited by 33 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF's defining vector field. PPD has several key benefits over existing methods: it sidesteps the need to solve an ODE per iteration, readily applies to manifold data, scales to high dimensions, and is compatible with a large family of target paths interpolating pure noise and data in finite time. Theoretically, PPD is shown to bound classical probability divergences. Empirically, we show that CNFs learned by minimizing PPD achieve state-of-the-art results in likelihoods and sample quality on existing low-dimensional manifold benchmarks, and is the first example of a generative model to scale to moderately high dimensional manifolds.

Key facts. Cited by 57 papers in the map (data through 7 Oct 2026). Builds on 14 of its 25 references that are in the map. First posted on arXiv 11 Jul 2022.

Builds on

14 of 25 references.

References · 14

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  5. 2018Neural Ordinary Differential Equationscited by 3,733
  6. 2015Variational Inference with Normalizing Flowscited by 2,161
  7. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  8. 2020Riemannian Continuous Normalizing Flowscited by 143
  9. 2020Normalizing Flows on Tori and Spherescited by 117
  10. 2020Neural Manifold Ordinary Differential Equationscited by 81
  11. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  12. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
2 more
  1. 2020Neural Ordinary Differential Equations on Manifoldscited by 31
  2. 2019Radial and Directional Posteriors for Bayesian Neural Networkscited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 3; 2023: 15; 2024: 14; 2025: 17; 2026: 8.

57 papers.

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  1. 2022Flow Matching for Generative Modelingcited by 5,426
  2. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  3. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  4. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  5. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  6. 2023Flow Matching on General Geometriescited by 260
  7. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 147
  8. 2022Action Matching: Learning Stochastic Dynamics from Samplescited by 94
  9. 2023Error Bounds for Flow Matching Methodscited by 88
  10. 2024ZigMa: A DiT-style Zigzag Mamba Diffusion Modelcited by 84
  11. 2023Simulation-free Schr"odinger bridges via score and flow matchingcited by 81
  12. 2024Variational Flow Matching for Graph Generationcited by 78
38 more
  1. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  2. 2024Fisher Flow Matching for Generative Modeling over Discrete Datacited by 68
  3. 2023Functional Flow Matchingcited by 63
  4. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  5. 2023Parallelizing non-linear sequential models over the sequence lengthcited by 45
  6. 2024Categorical Flow Matching on Statistical Manifoldscited by 43
  7. 2023Generative Sliced MMD Flows with Riesz Kernelscited by 36
  8. 2024Full-Atom Peptide Design based on Multi-modal Flow Matchingcited by 34
  9. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  10. 2024Around the World in 80 Timesteps: A Generative Approach to Global Visual Geolocationcited by 23
  11. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processescited by 21
  12. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  13. 2025Towards Hierarchical Rectified Flowcited by 19
  14. 2023Scaling Riemannian Diffusion Modelscited by 18
  15. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  16. 2025Exponential Family Variational Flow Matching for Tabular Data Generationcited by 15
  17. 2025Video Latent Flow Matching: Optimal Polynomial Projections for Video Interpolation and Extrapolationcited by 11
  18. 2024Generative Modeling of Discrete Joint Distributions by E-Geodesic Flow Matching on Assignment Manifoldscited by 9
  19. 2024Hotspot-Driven Peptide Design via Multi-Fragment Autoregressive Extensioncited by 9
  20. 2025Riemannian Variational Flow Matching for Material and Protein Designcited by 9
  21. 2025Controlled Generation with Equivariant Variational Flow Matchingcited by 9
  22. 2023Learning Distributions on Manifolds with Free-Form Flowscited by 7
  23. 2026Flow Actor-Critic for Offline Reinforcement Learningcited by 6
  24. 2024Adaptive deep density approximation for stochastic dynamical systemscited by 4
  25. 2025TFG-Flow: Training-free Guidance in Multimodal Generative Flowcited by 4
  26. 2025Riemannian Denoising Diffusion Probabilistic Modelscited by 4
  27. 2025Kuramoto Orientation Diffusion Modelscited by 4
  28. 2026Beyond Steering Vector: Flow-based Activation Steering for Inference-Time Interventioncited by 4
  29. 2025Discovering Symmetry Groups with Flow Matchingcited by 3
  30. 2026Riemannian MeanFlow for One-Step Generation on Manifoldscited by 3
  31. 2025Efficient Diffusion Models for Symmetric Manifoldscited by 2
  32. 2025Graph Generation with Spectral Geodesic Flow Matchingcited by 2
  33. 2023A survey of manifold learning and its applications for multimediacited by 1
  34. 2025Riemannian Neural Geodesic Interpolantcited by 1
  35. 2026Manifold Aware Denoising Score Matching (MAD)cited by 1
  36. 2026When Can Conditional Flow Matching Replace Pointwise Negative Log-Likelihood?cited by 1
  37. 2024ReMatching Dynamic Reconstruction Flowcited by 0
  38. 2025Time-reversed Flow Matching with Worst Transport in High-dimensional Latent Space for Image Anomaly Detectioncited by 0

50 most cited of 57 All 57 (JSON)