# Unbalanced Optimal Transport, from Theory to Numerics

arXiv: https://arxiv.org/abs/2211.08775 · HTML: https://mlsanity.com/paper/2211.08775/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 178 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Builds on (most cited first, 30 of 58)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [FlowNet3D: Learning Scene Flow in 3D Point Clouds](https://mlsanity.com/paper/1806.01411/) (2018)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Unbalanced minibatch Optimal Transport; applications to Domain Adaptation](https://mlsanity.com/paper/2103.03606/) (2021)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Fused Gromov-Wasserstein distance for structured objects: theoretical foundations and mathematical properties](https://mlsanity.com/paper/1811.02834/) (2018)
- [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)

## Built on by (most cited first, 30 of 46)

- [Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport](https://mlsanity.com/paper/2305.14777/) (2023)
- [Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation](https://mlsanity.com/paper/2311.15100/) (2023)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction](https://mlsanity.com/paper/2312.16611/) (2023)
- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Unsupervised Cross-Domain Image Retrieval via Prototypical Optimal Transport](https://mlsanity.com/paper/2402.18411/) (2024)
- [Scalable Simulation-free Entropic Unbalanced Optimal Transport](https://mlsanity.com/paper/2410.02656/) (2024)
- [Uncertainty-Aware Knowledge Distillation for Compact and Efficient 6DoF Pose Estimation](https://mlsanity.com/paper/2503.13053/) (2025)
- [On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it](https://mlsanity.com/paper/2301.03662/) (2023)
- [Dude: Dual Distribution-Aware Context Prompt Learning For Large Vision-Language Model](https://mlsanity.com/paper/2407.04489/) (2024)
- [Inductive Domain Transfer In Misspecified Simulation-Based Inference](https://mlsanity.com/paper/2508.15593/) (2025)
- [MGPATH: Vision-Language Model with Multi-Granular Prompt Learning for Few-Shot WSI Classification](https://mlsanity.com/paper/2502.07409/) (2025)
- [Learning Partial Graph Matching via Optimal Partial Transport](https://mlsanity.com/paper/2410.16718/) (2024)
- [A primer on optimal transport for causal inference with observational data](https://mlsanity.com/paper/2503.07811/) (2025)
- [Fast and Large-Scale Unbalanced Optimal Transport via its Semi-Dual and Adaptive Gradient Methods](https://mlsanity.com/paper/2602.10697/) (2026)
- [Generative Molecular Morphing for Flexible-Size Design via Unbalanced Optimal Transport](https://mlsanity.com/paper/2606.07239/) (2026)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [Collective Counterfactual Explanations: Balancing Individual Goals and Collective Dynamics](https://mlsanity.com/paper/2402.04579/) (2024)
- [Deep Privacy Funnel Model: From a Discriminative to a Generative Approach with an Application to Face Recognition](https://mlsanity.com/paper/2404.02696/) (2024)
- [Unsupervised Point Cloud Completion through Unbalanced Optimal Transport](https://mlsanity.com/paper/2410.02671/) (2024)
- [Model alignment using inter-modal bridges](https://mlsanity.com/paper/2505.12322/) (2025)
- [Reweighted Flow Matching via Unbalanced OT for Label-free Long-tailed Generation](https://mlsanity.com/paper/2509.25713/) (2025)
- [Globally Solving Unbalanced Optimal Transport and Density Control for Gaussian Distributions](https://mlsanity.com/paper/2605.04246/) (2026)
- [Geometry-Aware Dataset Condensation for Diffusion Model Training](https://mlsanity.com/paper/2606.05883/) (2026)
- [Bridging Vision and Language Concepts through Optimal Transport Semantic Flow](https://mlsanity.com/paper/2606.26891/) (2026)
- [Randomized Transport Plans via Hierarchical Fully Probabilistic Design](https://mlsanity.com/paper/2408.02701/) (2024)
- [An Eye for an Ear: Zero-shot Audio Description Leveraging an Image Captioner using Audiovisual Distribution Alignment](https://mlsanity.com/paper/2410.05997/) (2024)
- [On Barycenter Computation: Semi-Unbalanced Optimal Transport-based Method on Gaussians](https://mlsanity.com/paper/2410.08117/) (2024)

## Abstract (arXiv)

Optimal Transport (OT) has recently emerged as a central tool in data sciences to compare in a geometrically faithful way point clouds and more generally probability distributions. The wide adoption of OT into existing data analysis and machine learning pipelines is however plagued by several shortcomings. This includes its lack of robustness to outliers, its high computational costs, the need for a large number of samples in high dimension and the difficulty to handle data in distinct spaces. In this review, we detail several recently proposed approaches to mitigate these issues. We insist in particular on unbalanced OT, which compares arbitrary positive measures, not restricted to probability distributions (i.e. their total mass can vary). This generalization of OT makes it robust to outliers and missing data. The second workhorse of modern computational OT is entropic regularization, which leads to scalable algorithms while lowering the sample complexity in high dimension. The last point presented in this review is the Gromov-Wasserstein (GW) distance, which extends OT to cope with distributions belonging to different metric spaces. The main motivation for this review is to explain how unbalanced OT, entropic regularization and GW can work hand-in-hand to turn OT into efficient geometric loss functions for data sciences.