# Riemannian Diffusion Models

arXiv: https://arxiv.org/abs/2208.07949 · HTML: https://mlsanity.com/paper/2208.07949/

## Key facts

- Cited by 122 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 32 of its 68 references that are in the map.
- First posted on arXiv 16 Aug 2022.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models](https://mlsanity.com/paper/1703.07370/) (2017)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [The Power Spherical distribution](https://mlsanity.com/paper/2006.04437/) (2020)
- [Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev](https://mlsanity.com/paper/2010.05263/) (2020)

## Built on by (most cited first, 30 of 122)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [ModelScope Text-to-Video Technical Report](https://mlsanity.com/paper/2308.06571/) (2023)
- [I2VGen-XL: High-Quality Image-to-Video Synthesis via Cascaded Diffusion Models](https://mlsanity.com/paper/2311.04145/) (2023)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [A Survey on Generative Diffusion Model](https://mlsanity.com/paper/2209.02646/) (2022)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [UFOGen: You Forward Once Large Scale Text-to-Image Generation via Diffusion GANs](https://mlsanity.com/paper/2311.09257/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [NVS-Solver: Video Diffusion Model as Zero-Shot Novel View Synthesizer](https://mlsanity.com/paper/2405.15364/) (2024)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [Proteina: Scaling Flow-based Protein Structure Generative Models](https://mlsanity.com/paper/2503.00710/) (2025)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [Diffusion Generative Models in Infinite Dimensions](https://mlsanity.com/paper/2212.00886/) (2022)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [EDGI: Equivariant Diffusion for Planning with Embodied Agents](https://mlsanity.com/paper/2303.12410/) (2023)
- [Full-Atom Peptide Design based on Multi-modal Flow Matching](https://mlsanity.com/paper/2406.00735/) (2024)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [Diffuse Everything: Multimodal Diffusion Models on Arbitrary State Spaces](https://mlsanity.com/paper/2506.07903/) (2025)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)

## Abstract (arXiv)

Diffusion models are recent state-of-the-art methods for image generation and likelihood estimation. In this work, we generalize continuous-time diffusion models to arbitrary Riemannian manifolds and derive a variational framework for likelihood estimation. Computationally, we propose new methods for computing the Riemannian divergence which is needed in the likelihood estimation. Moreover, in generalizing the Euclidean case, we prove that maximizing this variational lower-bound is equivalent to Riemannian score matching. Empirically, we demonstrate the expressive power of Riemannian diffusion models on a wide spectrum of smooth manifolds, such as spheres, tori, hyperboloids, and orthogonal groups. Our proposed method achieves new state-of-the-art likelihoods on all benchmarks.