# Flows on convex polytopes

arXiv: https://arxiv.org/abs/2503.10232 · HTML: https://mlsanity.com/paper/2503.10232/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 13 Mar 2025.

## Builds on (most cited first, 14 of 14)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Aitchison's Compositional Data Analysis 40 Years On: A Reappraisal](https://mlsanity.com/paper/2201.05197/) (2022)
- [Bayesian Metabolic Flux Analysis reveals intracellular flux couplings](https://mlsanity.com/paper/1804.06673/) (2018)
- [PolytopeWalk: Sparse MCMC Sampling over Polytopes](https://mlsanity.com/paper/2412.06629/) (2024)

## Built on by (most cited first, 3 of 3)

- [Simplex-to-Euclidean Bijections for Categorical Flow Matching](https://mlsanity.com/paper/2510.27480/) (2025)
- [Simplex-to-Euclidean Bijection for Conjugate and Calibrated Multiclass Gaussian Process Classification](https://mlsanity.com/paper/2603.16621/) (2026)
- [EP-Flow: Disordered Crystal Structure Prediction without Site-Level Annotations](https://mlsanity.com/paper/2610.01315/) (2026)

## Abstract (arXiv)

We present a framework for modeling complex, high-dimensional distributions on convex polytopes by leveraging recent advances in discrete and continuous normalizing flows on Riemannian manifolds. We show that any full-dimensional polytope is homeomorphic to a unit ball, and our approach harnesses flows defined on the ball, mapping them back to the original polytope. Furthermore, we introduce a strategy to construct flows when only the vertex representation of a polytope is available, employing maximum entropy barycentric coordinates and Aitchison geometry. Our experiments take inspiration from applications in metabolic flux analysis and demonstrate that our methods achieve competitive density estimation, sampling accuracy, as well as fast training and inference times.