# Normalizing Flows on Riemannian Manifolds

arXiv: https://arxiv.org/abs/1611.02304 · HTML: https://mlsanity.com/paper/1611.02304/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Nov 2016.

## Built on by (most cited first, 30 of 68)

- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows](https://mlsanity.com/paper/2106.05275/) (2021)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Diagnosing and Fixing Manifold Overfitting in Deep Generative Models](https://mlsanity.com/paper/2204.07172/) (2022)
- [CaloMan: Fast generation of calorimeter showers with density estimation on learned manifolds](https://mlsanity.com/paper/2211.15380/) (2022)
- [Diffusion Probabilistic Fields](https://mlsanity.com/paper/2303.00165/) (2023)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [First Hitting Diffusion Models for Generating Manifold, Graph and Categorical Data](https://mlsanity.com/paper/2209.01170/) (2022)
- [Scaling Riemannian Diffusion Models](https://mlsanity.com/paper/2310.20030/) (2023)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modeling](https://mlsanity.com/paper/2304.03937/) (2023)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Stereographic Spherical Sliced Wasserstein Distances](https://mlsanity.com/paper/2402.02345/) (2024)
- [Density estimation on low-dimensional manifolds: an inflation-deflation approach](https://mlsanity.com/paper/2105.12152/) (2021)
- [Density estimation on smooth manifolds with normalizing flows](https://mlsanity.com/paper/2106.03500/) (2021)
- [Manifold Diffusion Fields](https://mlsanity.com/paper/2305.15586/) (2023)
- [Principal Component Density Estimation for Scenario Generation Using Normalizing Flows](https://mlsanity.com/paper/2104.10410/) (2021)
- [Continuous normalizing flows on manifolds](https://mlsanity.com/paper/2104.14959/) (2021)

## Abstract (arXiv)

We consider the problem of density estimation on Riemannian manifolds. Density estimation on manifolds has many applications in fluid-mechanics, optics and plasma physics and it appears often when dealing with angular variables (such as used in protein folding, robot limbs, gene-expression) and in general directional statistics. In spite of the multitude of algorithms available for density estimation in the Euclidean spaces Rn that scale to large n (e.g. normalizing flows, kernel methods and variational approximations), most of these methods are not immediately suitable for density estimation in more general Riemannian manifolds. We revisit techniques related to homeomorphisms from differential geometry for projecting densities to sub-manifolds and use it to generalize the idea of normalizing flows to more general Riemannian manifolds. The resulting algorithm is scalable, simple to implement and suitable for use with automatic differentiation. We demonstrate concrete examples of this method on the n-sphere Sn.