# Algorithms for Weak Optimal Transport with an Application to Economics

arXiv: https://arxiv.org/abs/2205.09825 · HTML: https://mlsanity.com/paper/2205.09825/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 19 May 2022.

## Builds on (most cited first, 13 of 13)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport](https://mlsanity.com/paper/1905.10812/) (2019)
- [Unsupervised Hyperalignment for Multilingual Word Embeddings](https://mlsanity.com/paper/1811.01124/) (2018)
- [Regularized Optimal Transport is Ground Cost Adversarial](https://mlsanity.com/paper/2002.03967/) (2020)
- [Convex Color Image Segmentation with Optimal Transport Distances](https://mlsanity.com/paper/1503.01986/) (2015)
- [Unbalanced Optimal Transport: Dynamic and Kantorovich Formulation](https://mlsanity.com/paper/1508.05216/) (2015)
- [Weak optimal transport with unnormalized kernels](https://mlsanity.com/paper/2203.16227/) (2022)

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

- [Entropic Neural Optimal Transport via Diffusion Processes](https://mlsanity.com/paper/2211.01156/) (2022)
- [Barycentric Weak Inner-Product Gromov-Wasserstein](https://mlsanity.com/paper/2608.25145/) (2026)

## Abstract (arXiv)

The theory of weak optimal transport (WOT), introduced by [Gozlan et al., 2017], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear. In the so-called barycentric version of WOT, the cost for transporting a point x only depends on x and on the barycenter of the points it is matched with. This aggregation property of WOT is appealing in machine learning, economics and finance. Yet algorithms to compute WOT have only been developed for the special case of quadratic barycentric WOT, or depend on neural networks with no guarantee on the computed value and matching. The main difficulty lies in the transportation constraints which are costly to project onto. In this paper, we propose to use mirror descent algorithms to solve the primal and dual versions of the WOT problem. We also apply our algorithms to the variant of WOT introduced by [Chon'e et al., 2022] where mass is distributed from one space to another through unnormalized kernels (WOTUK). We empirically compare the solutions of WOT and WOTUK with classical OT. We illustrate our numerical methods to the economic framework of [Chon'e and Kramarz, 2021], namely the matching between workers and firms on labor markets.