# Scaling Riemannian Diffusion Models

arXiv: https://arxiv.org/abs/2310.20030 · HTML: https://mlsanity.com/paper/2310.20030/

## 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.

## Placed in

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

## Builds on (most cited first, 30 of 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Out-of-Distribution Detection with Deep Nearest Neighbors](https://mlsanity.com/paper/2204.06507/) (2022)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [SSD: A Unified Framework for Self-Supervised Outlier Detection](https://mlsanity.com/paper/2103.12051/) (2021)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [How to Exploit Hyperspherical Embeddings for Out-of-Distribution Detection?](https://mlsanity.com/paper/2203.04450/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)

## Built on by (most cited first, 18 of 18)

- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)
- [Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity](https://mlsanity.com/paper/2505.09922/) (2025)
- [Riemannian Consistency Model](https://mlsanity.com/paper/2510.00983/) (2025)
- [Stiefel Flow Matching for Moment-Constrained Structure Elucidation](https://mlsanity.com/paper/2412.12540/) (2024)
- [Riemannian Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2505.04338/) (2025)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Efficient Diffusion Models for Symmetric Manifolds](https://mlsanity.com/paper/2505.21640/) (2025)
- [Riemannian Neural Geodesic Interpolant](https://mlsanity.com/paper/2504.15736/) (2025)
- [Local-Time Riemannian Score Matching on the Quantum Pure-State Manifold](https://mlsanity.com/paper/2605.03573/) (2026)
- [Determinism of Randomness: Prompt-Residual Seed Shaping for Diffusion Generation](https://mlsanity.com/paper/2511.07756/) (2025)
- [DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions](https://mlsanity.com/paper/2602.10221/) (2026)
- [Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport](https://mlsanity.com/paper/2606.21256/) (2026)
- [Gaussian Linear Functional Manifold Method for Massive Point Cloud Data](https://mlsanity.com/paper/2609.05744/) (2026)
- [FLARE: Flow Matching with Local Axis-Angle Representations for Stochastic Micromagnetic Evolution](https://mlsanity.com/paper/2609.34070/) (2026)
- [Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature](https://mlsanity.com/paper/2609.36568/) (2026)

## 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.