Optimal Transportfor Machine Learners

OT4ML is a global project for learning and using optimal transport in machine learning: a PDF book, an interactive MyST book, reproducible figure notebooks with a searchable database, teaching notebooks, course slides, and curated references/software.

The Book

The manuscript develops optimal transport from finite assignments to Monge and Kantorovich formulations, Sinkhorn algorithms, generalized distances, gradient flows, and transportation-based generative models.

arXiv snippet
External

arXiv version

Canonical public version of the manuscript.

Open arXiv Book PDF snippet
Local

Book PDF

PDF built from the LaTeX sources shipped in this repository.

Open PDF Compact handout snippet
Teaching

Compact handout

A shortened teaching version preserving the core mathematics.

Open compact PDF
How to cite

Cite the book

Gabriel Peyré. Optimal Transport for Machine Learners. arXiv:2505.06589 [stat.ML], submitted May 10, 2025; cross-listed in cs.AI and math.OC. DOI: 10.48550/arXiv.2505.06589.

arXiv details

Interactive Book

The web version follows the manuscript and places interactive panels beside the mathematical figures, so the reading flow stays centered on the book.

Interactive Kantorovich coupling panel preview Interactive image and histogram transport panel preview Interactive semidiscrete Laguerre cell panel preview Interactive graph Wasserstein transport panel preview

Figure Notebooks

The figure gallery is a searchable database: filter by concept or book section, inspect thumbnails, open notebooks on GitHub, and launch them in Colab.

Book figure preview for McCann shape interpolation Book figure preview for Sinkhorn plans at several regularization strengths Book figure preview for Wasserstein barycenters Book figure preview for Fokker--Planck gradient-flow representations

Teaching Notebooks

Self-contained notebooks for classroom use and quick experimentation. Each can be opened in GitHub or launched directly in Colab.

Course Slides

Four slide decks provide a lecture-oriented route through the computational OT material.

Further Resources

Books, long reviews, surveys, and software are kept on a dedicated page so the homepage stays compact.

Book cover preview for Optimal Transport for Applied Mathematicians Book cover preview for Optimal Transport Methods in Economics Book cover preview for Gradient Flows Book cover preview for Modelling and Optimisation of Flows on Networks Book cover preview for Computational Optimal Transport Book cover preview for Topics in Optimal Transportation

BibTeX citation

Use this entry when citing the arXiv version of the book.

@misc{peyre2025optimaltransportmachinelearners,
  title         = {Optimal Transport for Machine Learners},
  author        = {Gabriel Peyr{\'e}},
  year          = {2025},
  eprint        = {2505.06589},
  archivePrefix = {arXiv},
  primaryClass  = {stat.ML},
  doi           = {10.48550/arXiv.2505.06589},
  url           = {https://arxiv.org/abs/2505.06589}
}