mlsanityData to 14 Oct 2026

Scaling Riemannian Diffusion Models

arXiv 2310.20030 ↗First posted 30 Oct 2023cs.LG math.DG stat.MLcited by 18

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans"atze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

Key facts. Cited by 18 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 33 of its 52 references that are in the map. First posted on arXiv 30 Oct 2023.

Builds on

33 of 52 references.

Closest ancestors · 7

Same direction, newest first

  1. 2023Reflected Diffusion Models10 Apr
  2. 2023Denoising Diffusion Samplers27 Feb
  3. 2022Riemannian Diffusion Models16 Aug
  4. 2022Riemannian Score-Based Generative Modelling6 Feb
  5. 2021Variational Diffusion Models1 Jul
  6. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  7. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 26

  1. 2020A Simple Framework for Contrastive Learning of Visual Representationscited by 12,049
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  3. 2022Flow Matching for Generative Modelingcited by 5,702
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  5. 2018Neural Ordinary Differential Equationscited by 3,779
  6. 2016Gaussian Error Linear Units (GELUs)cited by 3,502
  7. 2020Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hyperspherecited by 1,405
  8. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  9. 2022Out-of-Distribution Detection with Deep Nearest Neighborscited by 600
  10. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  11. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 384
  12. 2022DiffDock: Diffusion Steps, Twists, and Turns for Molecular Dockingcited by 313
14 more
  1. 2021SSD: A Unified Framework for Self-Supervised Outlier Detectioncited by 264
  2. 2022Torsional Diffusion for Molecular Conformer Generationcited by 263
  3. 2020Riemannian Continuous Normalizing Flowscited by 145
  4. 2022How to Exploit Hyperspherical Embeddings for Out-of-Distribution Detection?cited by 128
  5. 2020Flows for simultaneous manifold learning and density estimationcited by 118
  6. 2020Sampling using SU(N) gauge equivariant flowscited by 84
  7. 2020Neural Manifold Ordinary Differential Equationscited by 81
  8. 2016Normalizing Flows on Riemannian Manifoldscited by 68
  9. 2021Moser Flow: Divergence-based Generative Modeling on Manifoldscited by 67
  10. 2022Matching Normalizing Flows and Probability Paths on Manifoldscited by 57
  11. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
  12. 2020Neural Ordinary Differential Equations on Manifoldscited by 31
  13. 2021Equivariant Manifold Flowscited by 20
  14. 2023Conformal Generative Modeling on Triangulated Surfacescited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 5; 2025: 6; 2026: 7.

18 papers.

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  1. 2024Riemannian Flow Matching Policy for Robot Motion Learningcited by 63
  2. 2024Target Score Matchingcited by 50
  3. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  4. 2024Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groupscited by 19
  5. 2025Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularitycited by 17
  6. 2025Riemannian Consistency Modelcited by 8
  7. 2024Stiefel Flow Matching for Moment-Constrained Structure Elucidationcited by 5
  8. 2025Riemannian Denoising Diffusion Probabilistic Modelscited by 4
  9. 2026Polynomial Convergence of Riemannian Diffusion Modelscited by 3
  10. 2025Efficient Diffusion Models for Symmetric Manifoldscited by 2
  11. 2025Riemannian Neural Geodesic Interpolantcited by 1
  12. 2026Local-Time Riemannian Score Matching on the Quantum Pure-State Manifoldcited by 1
6 more
  1. 2025Determinism of Randomness: Prompt-Residual Seed Shaping for Diffusion Generationcited by 0
  2. 2026DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutionscited by 0
  3. 2026Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transportcited by 0
  4. 2026Gaussian Linear Functional Manifold Method for Massive Point Cloud Datacited by 0
  5. 2026FLARE: Flow Matching with Local Axis-Angle Representations for Stochastic Micromagnetic Evolutioncited by 0
  6. 2026Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvaturecited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes
  2. 2026Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks
  3. 2023Diffusion Models for Constrained Domains
  4. 2026Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants
  5. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  6. 2025Toward a Unified Geometry Understanding: Riemannian Diffusion Framework for Graph Generation and Prediction
  7. 2023Manifold Diffusion Fields
  8. 2025What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold