# Normalizing Flows on Quotient Manifolds via Boundary Quotients

arXiv: https://arxiv.org/abs/2511.22882 · HTML: https://mlsanity.com/paper/2511.22882/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 28 Nov 2025.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [Equivariant Manifold Flows](https://mlsanity.com/paper/2107.08596/) (2021)
- [Density estimation on smooth manifolds with normalizing flows](https://mlsanity.com/paper/2106.03500/) (2021)
- [VQ-Flows: Vector Quantized Local Normalizing Flows](https://mlsanity.com/paper/2203.11556/) (2022)
- [Learning Distributions on Manifolds with Free-Form Flows](https://mlsanity.com/paper/2312.09852/) (2023)

## Abstract (arXiv)

We introduce boundary quotients and present a framework for learning densities on manifolds that arise as boundary quotients of simpler domains. We show that this framework can be used to construct normalizing flows on quotient manifolds N/G, where a discrete group G acts on N. We instantiate this construction for genus-g surfaces Sigmag. When G is finite, we show applicability to symmetry aware learning; we demonstrate this on cyclic quotients of the 3-sphere. Experiments on lens spaces show that simple pre-quotient RealNVP models can achieve strong results while being substantially cheaper to evaluate.