# A Survey on Optimal Transport for Machine Learning: Theory and Applications

arXiv: https://arxiv.org/abs/2106.01963 · HTML: https://mlsanity.com/paper/2106.01963/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 45 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 21 of 21)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Neural Architecture Search with Bayesian Optimisation and Optimal Transport](https://mlsanity.com/paper/1802.07191/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [Theoretical Analysis of Domain Adaptation with Optimal Transport](https://mlsanity.com/paper/1610.04420/) (2016)
- [Wasserstein Embedding for Graph Learning](https://mlsanity.com/paper/2006.09430/) (2020)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [Hierarchical Optimal Transport for Document Representation](https://mlsanity.com/paper/1906.10827/) (2019)
- [Improving Sequence-to-Sequence Learning via Optimal Transport](https://mlsanity.com/paper/1901.06283/) (2019)
- [A Geometric View of Optimal Transportation and Generative Model](https://mlsanity.com/paper/1710.05488/) (2017)
- [Wasserstein Variational Inference](https://mlsanity.com/paper/1805.11284/) (2018)
- [Optimal transport: discretization and algorithms](https://mlsanity.com/paper/2003.00855/) (2020)
- [Equitable and Optimal Transport with Multiple Agents](https://mlsanity.com/paper/2006.07260/) (2020)

## Built on by (most cited first, 30 of 35)

- [InsCL: A Data-efficient Continual Learning Paradigm for Fine-tuning Large Language Models with Instructions](https://mlsanity.com/paper/2403.11435/) (2024)
- [Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey](https://mlsanity.com/paper/2305.05080/) (2023)
- [Towards Explaining Distribution Shifts](https://mlsanity.com/paper/2210.10275/) (2022)
- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [vONTSS: vMF based semi-supervised neural topic modeling with optimal transport](https://mlsanity.com/paper/2307.01226/) (2023)
- [μNCA: Texture Generation with Ultra-Compact Neural Cellular Automata](https://mlsanity.com/paper/2111.13545/) (2021)
- [Modeling Multimodal Aleatoric Uncertainty in Segmentation with Mixture of Stochastic Experts](https://mlsanity.com/paper/2212.07328/) (2022)
- [Robot Policy Learning with Temporal Optimal Transport Reward](https://mlsanity.com/paper/2410.21795/) (2024)
- [Enhancing Unsupervised Graph Few-shot Learning via Set Functions and Optimal Transport](https://mlsanity.com/paper/2501.05635/) (2025)
- [A scalable deep learning approach for solving high-dimensional dynamic optimal transport](https://mlsanity.com/paper/2205.07521/) (2022)
- [Conflict-Aware Pseudo Labeling via Optimal Transport for Entity Alignment](https://mlsanity.com/paper/2209.01847/) (2022)
- [PINS: Proximal Iterations with Sparse Newton and Sinkhorn for Optimal Transport](https://mlsanity.com/paper/2502.03749/) (2025)
- [Fair Text Classification with Wasserstein Independence](https://mlsanity.com/paper/2311.12689/) (2023)
- [Differential Motion Evolution for Fine-Grained Motion Deformation in Unsupervised Image Animation](https://mlsanity.com/paper/2110.04658/) (2021)
- [KDSTM: Neural Semi-supervised Topic Modeling with Knowledge Distillation](https://mlsanity.com/paper/2307.01878/) (2023)
- [Estimation of entropy-regularized optimal transport maps between non-compactly supported measures](https://mlsanity.com/paper/2311.11934/) (2023)
- [A Wasserstein perspective of Vanilla GANs](https://mlsanity.com/paper/2403.15312/) (2024)
- [PLOT-TAL: Prompt Learning with Optimal Transport for Few-Shot Temporal Action Localization](https://mlsanity.com/paper/2403.18915/) (2024)
- [Fair Text Classification via Transferable Representations](https://mlsanity.com/paper/2503.07691/) (2025)
- [Combating Confirmation Bias: A Unified Pseudo-Labeling Framework for Entity Alignment](https://mlsanity.com/paper/2307.02075/) (2023)
- [Optimal Transport Regularization for Speech Text Alignment in Spoken Language Models](https://mlsanity.com/paper/2508.08131/) (2025)
- [A Survey on Evaluation Metrics for Synthetic Material Micro-Structure Images from Generative Models](https://mlsanity.com/paper/2211.09727/) (2022)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [Constructing Cell-type Taxonomy by Optimal Transport with Relaxed Marginal Constraints](https://mlsanity.com/paper/2501.18650/) (2025)
- [HOTA: Hamiltonian framework for Optimal Transport Advection](https://mlsanity.com/paper/2507.17513/) (2025)
- [Cross-Fusion Distance: A Novel Metric for Measuring Fusion and Separability Between Data Groups in Representation Space](https://mlsanity.com/paper/2601.22036/) (2026)
- [Prototype Learning for Explainable Brain Age Prediction](https://mlsanity.com/paper/2306.09858/) (2023)
- [A Novel Skip Orthogonal List for Dynamic Optimal Transport Problem](https://mlsanity.com/paper/2310.18446/) (2023)
- [Trading Devil Final: Backdoor attack via Stock market and Bayesian Optimization](https://mlsanity.com/paper/2407.14573/) (2024)
- [Few-shot Scooping Under Domain Shift via Simulated Maximal Deployment Gaps](https://mlsanity.com/paper/2408.02949/) (2024)

## Abstract (arXiv)

Optimal Transport (OT) theory has seen an increasing amount of attention from the computer science community due to its potency and relevance in modeling and machine learning. It introduces means that serve as powerful ways to compare probability distributions with each other, as well as producing optimal mappings to minimize cost functions. In this survey, we present a brief introduction and history, a survey of previous work and propose directions of future study. We will begin by looking at the history of optimal transport and introducing the founders of this field. We then give a brief glance into the algorithms related to OT. Then, we will follow up with a mathematical formulation and the prerequisites to understand OT. These include Kantorovich duality, entropic regularization, KL Divergence, and Wassertein barycenters. Since OT is a computationally expensive problem, we then introduce the entropy-regularized version of computing optimal mappings, which allowed OT problems to become applicable in a wide range of machine learning problems. In fact, the methods generated from OT theory are competitive with the current state-of-the-art methods. We follow this up by breaking down research papers that focus on image processing, graph learning, neural architecture search, document representation, and domain adaptation. We close the paper with a small section on future research. Of the recommendations presented, three main problems are fundamental to allow OT to become widely applicable but rely strongly on its mathematical formulation and thus are hardest to answer. Since OT is a novel method, there is plenty of space for new research, and with more and more competitive methods (either on an accuracy level or computational speed level) being created, the future of applied optimal transport is bright as it has become pervasive in machine learning.