# On a Neural Implementation of Brenier's Polar Factorization

arXiv: https://arxiv.org/abs/2403.03071 · HTML: https://mlsanity.com/paper/2403.03071/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 43 references that are in the map.
- First posted on arXiv 5 Mar 2024.

## Builds on (most cited first, 20 of 20)

- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Direct Diffusion Bridge using Data Consistency for Inverse Problems](https://mlsanity.com/paper/2305.19809/) (2023)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [A Framework for Adaptive MCMC Targeting Multimodal Distributions](https://mlsanity.com/paper/1812.02609/) (2018)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Riemannian Convex Potential Maps](https://mlsanity.com/paper/2106.10272/) (2021)
- [Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows](https://mlsanity.com/paper/2209.10873/) (2022)

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

- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures](https://mlsanity.com/paper/2503.10576/) (2025)
- [Brenier Meets Adversarial Training: Optimal Transport Geometry for Robust Learning](https://mlsanity.com/paper/2609.31363/) (2026)

## Abstract (arXiv)

In 1991, Brenier proved a theorem that generalizes the polar decomposition for square matrices -- factored as PSD × unitary -- to any vector field F:ℝdrightarrow ℝd. The theorem, known as the polar factorization theorem, states that any field F can be recovered as the composition of the gradient of a convex function u with a measure-preserving map M, namely F=nabla u circ M. We propose a practical implementation of this far-reaching theoretical result, and explore possible uses within machine learning. The theorem is closely related to optimal transport (OT) theory, and we borrow from recent advances in the field of neural optimal transport to parameterize the potential u as an input convex neural network. The map M can be either evaluated pointwise using u^*, the convex conjugate of u, through the identity M=nabla u^* circ F, or learned as an auxiliary network. Because M is, in general, not injective, we consider the additional task of estimating the ill-posed inverse map that can approximate the pre-image measure M-1 using a stochastic generator. We illustrate possible applications of Brenier's polar factorization to non-convex optimization problems, as well as sampling of densities that are not log-concave.