# A new method for determining Wasserstein 1 optimal transport maps from Kantorovich potentials, with deep learning applications

arXiv: https://arxiv.org/abs/2211.00820 · HTML: https://mlsanity.com/paper/2211.00820/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 2 Nov 2022.

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

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks](https://mlsanity.com/paper/1703.10593/) (2017)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [On the regularization of Wasserstein GANs](https://mlsanity.com/paper/1709.08894/) (2017)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [A Geometric View of Optimal Transportation and Generative Model](https://mlsanity.com/paper/1710.05488/) (2017)
- [Discriminator optimal transport](https://mlsanity.com/paper/1910.06832/) (2019)
- [A fast approach to optimal transport: The back-and-forth method](https://mlsanity.com/paper/1905.12154/) (2019)
- [Wasserstein GANs with Gradient Penalty Compute Congested Transport](https://mlsanity.com/paper/2109.00528/) (2021)
- [Gradient Layer: Enhancing the Convergence of Adversarial Training for Generative Models](https://mlsanity.com/paper/1801.02227/) (2018)
- [Regularization by Adversarial Learning for Ultrasound Elasticity Imaging](https://mlsanity.com/paper/2106.00167/) (2021)
- [Trust the Critics: Generatorless and Multipurpose WGANs with Initial Convergence Guarantees](https://mlsanity.com/paper/2111.15099/) (2021)

## Built on by (most cited first, 2 of 2)

- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Fast and scalable Wasserstein-1 neural optimal transport solver for single-cell perturbation prediction](https://mlsanity.com/paper/2411.00614/) (2024)

## Abstract (arXiv)

Wasserstein 1 optimal transport maps provide a natural correspondence between points from two probability distributions, μ and nu, which is useful in many applications. Available algorithms for computing these maps do not appear to scale well to high dimensions. In deep learning applications, efficient algorithms have been developed for approximating solutions of the dual problem, known as Kantorovich potentials, using neural networks (e.g. [Gulrajani et al., 2017]). Importantly, such algorithms work well in high dimensions. In this paper we present an approach towards computing Wasserstein 1 optimal transport maps that relies only on Kantorovich potentials. In general, a Wasserstein 1 optimal transport map is not unique and is not computable from a potential alone. Our main result is to prove that if μ has a density and nu is supported on a submanifold of codimension at least 2, an optimal transport map is unique and can be written explicitly in terms of a potential. These assumptions are natural in many image processing contexts and other applications. When the Kantorovich potential is only known approximately, our result motivates an iterative procedure wherein data is moved in optimal directions and with the correct average displacement. Since this provides an approach for transforming one distribution to another, it can be used as a multipurpose algorithm for various transport problems; we demonstrate through several proof of concept experiments that this algorithm successfully performs various imaging tasks, such as denoising, generation, translation and deblurring, which normally require specialized techniques.