# Optimal Transport for Machine Learners

arXiv: https://arxiv.org/abs/2505.06589 · HTML: https://mlsanity.com/paper/2505.06589/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2025.

## Built on by (most cited first, 12 of 12)

- [Multivariate Uncertainty Quantification with Tomographic Quantile Forests](https://mlsanity.com/paper/2512.16383/) (2025)
- [Learning to Emulate Chaos: Adversarial Optimal Transport Regularization](https://mlsanity.com/paper/2604.21097/) (2026)
- [Transformers for Tabular Data: A Training Perspective of Self-Attention via Optimal Transport](https://mlsanity.com/paper/2512.09530/) (2025)
- [Parametric and Generative Forecasts of EPEX Day-Ahead Energy Market Curves](https://mlsanity.com/paper/2601.20226/) (2026)
- [On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans](https://mlsanity.com/paper/2602.16265/) (2026)
- [Generative Modeling by Value-Driven Transport](https://mlsanity.com/paper/2605.22507/) (2026)
- [SinkSLOT: Sinkhorn via Sparse Lifted Optimal Transport](https://mlsanity.com/paper/2608.28262/) (2026)
- [Dual-guided Hierarchical Edge Localization for Large-scale Optimal Transport Across Dimensions](https://mlsanity.com/paper/2609.13010/) (2026)
- [Unlocking Geodesic Gromov-Wasserstein Distances for 3D Modeling](https://mlsanity.com/paper/2609.32824/) (2026)
- [Sliced Orlicz-Wasserstein](https://mlsanity.com/paper/2609.32847/) (2026)
- [Robust blind unmixing: A geometric approach to overcoming basis variation](https://mlsanity.com/paper/2610.04091/) (2026)
- [Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs](https://mlsanity.com/paper/2610.05071/) (2026)

## Abstract (arXiv)

Modern machine learning repeatedly manipulates probability measures: empirical datasets, generated samples, latent distributions, class-conditional laws, particle systems, weights of wide networks and attention patterns. Optimal transport is useful in this setting because it compares such objects by asking how mass should move. It therefore combines a statistically meaningful notion of discrepancy with a geometry of interpolation, dual certificates and variational dynamics. This makes OT a common language for losses, generative modeling, domain adaptation, robust learning, barycenters, gradient flows and mean-field descriptions of learning algorithms. This book presents the main OT techniques with these machine-learning uses in mind. It starts from finite assignment and the Monge map viewpoint, passes to Kantorovich couplings and dual potentials, and then explains the algorithmic ideas that make transport usable: linear programming, semi-discrete cells, Sinkhorn scaling and low-dimensional projections. The same objects are then reused as a geometry of measures, giving Wasserstein distances, barycenters, gradient flows, dynamic formulations and Gaussian/Bures formulas. The final chapters emphasize the variants most relevant to modern ML: divergences and adversarial losses, entropic and unbalanced relaxations, robust or spectral ground geometries, Gromov and quantum extensions, and transport-based views of generative models, mean-field networks and attention dynamics. The goal is to keep the mathematics explicit while exposing the computational and geometric intuitions needed to turn OT into a working toolbox for machine learners.