Computational optimal transport

MATH-528 / 5 crédits

Enseignant: Chizat Lénaïc

Langue: Anglais

Summary

Computational aspects of measure transportation, from classical optimal transport to modern denoising diffusion models.

Content

- basics of optimal transport theory (discrete and continuous settings, duality and Brenier theorem)

- particular cases: optimal transport in 1D and between Gaussians and matching algorithms

- entropic regularization of optimal transport,  Schrödinger bridge problem, Sinkhorn's algorithm and its convergence

- Wasserstein barycenters

- W1, Integral probability metrics and divergences on the space of probability measures

- Wassersteing gradient flows (theory and applications)

- Generative models via denoising diffusion and flow matching

Keywords

optimal transport, space of probability measures,  Wasserstein, Sinkhorn, diffusion models

Learning Prerequisites

Required courses

Analysis, Linear Algebra, Probability and Statistics, notions of Partial Differential Equations

Recommended courses

Optimal Transport (fall course): it is recommended to follow this course, but not necessary as the required theory will be recalled

Important concepts to start the course

  • A good knowledge of undergraduate mathematics is important.

  • Ability to code in a scientific computing programming language of your choice (e.g. Python, Matlab, Julia). The course will involve coding exercises.

Teaching methods

Blackboard (or tablet) lectures

Assessment methods

Written exam

Resources

Virtual desktop infrastructure (VDI)

No

Bibliography

- An idea of the course content can be found in "Optimal Transport for Machine Learners" by Gabriel Peyré (available online at https://arxiv.org/pdf/2505.06589)

Références suggérées par la bibliothèque

Moodle Link

Dans les plans d'études

  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Computational optimal transport
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Computational optimal transport
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Computational optimal transport
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Computational optimal transport
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Computational optimal transport
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Ecrit (session d'été)
  • Matière examinée: Computational optimal transport
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel

Semaine de référence

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