# Scalable Unbalanced Optimal Transport using Generative Adversarial Networks

arXiv: https://arxiv.org/abs/1810.11447 · HTML: https://mlsanity.com/paper/1810.11447/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2018.

## Built on by (most cited first, 30 of 42)

- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Neural Monge Map estimation and its applications](https://mlsanity.com/paper/2106.03812/) (2021)
- [Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation](https://mlsanity.com/paper/2311.15100/) (2023)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Multi-marginal Schr"odinger Bridges with Iterative Reference Refinement](https://mlsanity.com/paper/2408.06277/) (2024)
- [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022)
- [Diffusion Earth Mover's Distance and Distribution Embeddings](https://mlsanity.com/paper/2102.12833/) (2021)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Neural Unbalanced Optimal Transport via Cycle-Consistent Semi-Couplings](https://mlsanity.com/paper/2209.15621/) (2022)
- [On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error](https://mlsanity.com/paper/2202.03618/) (2022)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [Extremal Domain Translation with Neural Optimal Transport](https://mlsanity.com/paper/2301.12874/) (2023)
- [Unbalanced Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2306.09099/) (2023)
- [Inferring biological processes with intrinsic noise from cross-sectional data](https://mlsanity.com/paper/2410.07501/) (2024)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Analyzing and Improving Optimal-Transport-based Adversarial Networks](https://mlsanity.com/paper/2310.02611/) (2023)
- [Neural Lagrangian Schr"odinger Bridge: Diffusion Modeling for Population Dynamics](https://mlsanity.com/paper/2204.04853/) (2022)
- [On Partial Optimal Transport: Revising the Infeasibility of Sinkhorn and Efficient Gradient Methods](https://mlsanity.com/paper/2312.13970/) (2023)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [A High-Quality Robust Diffusion Framework for Corrupted Dataset](https://mlsanity.com/paper/2311.17101/) (2023)
- [Stochastic force inference via density estimation](https://mlsanity.com/paper/2310.02366/) (2023)
- [Oh SnapMMD! Forecasting Stochastic Dynamics Beyond the Schr"odinger Bridge's End](https://mlsanity.com/paper/2505.16082/) (2025)
- [Correlational Lagrangian Schr"odinger Bridge: Learning Dynamics with Population-Level Regularization](https://mlsanity.com/paper/2402.10227/) (2024)

## Abstract (arXiv)

Generative adversarial networks (GANs) are an expressive class of neural generative models with tremendous success in modeling high-dimensional continuous measures. In this paper, we present a scalable method for unbalanced optimal transport (OT) based on the generative-adversarial framework. We formulate unbalanced OT as a problem of simultaneously learning a transport map and a scaling factor that push a source measure to a target measure in a cost-optimal manner. In addition, we propose an algorithm for solving this problem based on stochastic alternating gradient updates, similar in practice to GANs. We also provide theoretical justification for this formulation, showing that it is closely related to an existing static formulation by Liero et al. (2018), and perform numerical experiments demonstrating how this methodology can be applied to population modeling.