# Density estimation on smooth manifolds with normalizing flows

arXiv: https://arxiv.org/abs/2106.03500 · HTML: https://mlsanity.com/paper/2106.03500/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 39 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [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)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [Machine Learning for the Geosciences: Challenges and Opportunities](https://mlsanity.com/paper/1711.04708/) (2017)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [A RAD approach to deep mixture models](https://mlsanity.com/paper/1903.07714/) (2019)

## Built on by (most cited first, 11 of 11)

- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Gradual Domain Adaptation via Normalizing Flows](https://mlsanity.com/paper/2206.11492/) (2022)
- [Learning Distributions on Manifolds with Free-Form Flows](https://mlsanity.com/paper/2312.09852/) (2023)
- [Neural Implicit Manifold Learning for Topology-Aware Density Estimation](https://mlsanity.com/paper/2206.11267/) (2022)
- [Learning Geometry and Topology via Multi-Chart Flows](https://mlsanity.com/paper/2505.24665/) (2025)
- [Manifold Aware Denoising Score Matching (MAD)](https://mlsanity.com/paper/2603.02452/) (2026)
- [Out-of-distribution detection using normalizing flows on the data manifold](https://mlsanity.com/paper/2308.13792/) (2023)
- [Normalizing Flows on Quotient Manifolds via Boundary Quotients](https://mlsanity.com/paper/2511.22882/) (2025)
- [Reparameterization through Coverings and Topological Weight Priors](https://mlsanity.com/paper/2604.23804/) (2026)
- [Elimination Geometry](https://mlsanity.com/paper/2608.17646/) (2026)

## Abstract (arXiv)

We present a framework for learning probability distributions on topologically non-trivial manifolds, utilizing normalizing flows. Current methods focus on manifolds that are homeomorphic to Euclidean space, enforce strong structural priors on the learned models or use operations that do not easily scale to high dimensions. In contrast, our method learns distributions on a data manifold by "gluing" together multiple local models, thus defining an open cover of the data manifold. We demonstrate the efficiency of our approach on synthetic data of known manifolds, as well as higher dimensional manifolds of unknown topology, where our method exhibits better sample efficiency and competitive or superior performance against baselines in a number of tasks.