Introduction to Optimal Transport Theory

Lénaïc Chizat and Luca Nenna

Master Optimization, Université Paris-Saclay, spring 2020

Mandatory registration


Optimal transport is a powerful mathematical theory at the interface between optimization and probability theory with far reaching applications. It defines a natural tool to study probability distributions in the many situations where they appear: data science, partial differential equations, statistics or shape processing. In this course we will present the classical theory of optimal transport, efficient algorithms to compute it and applications.

Pre-requisites: Notions on measure theory, weak convergence, and convex analysis. Some basic knowledge of Python.

Language: The class will be taught in French or English, depending on attendance (all slides and class notes are in English).

Evaluation: Practical sessions to finish at home + written in-class exam report on a research article (change due to lockdown!)


Classes will be held in the room 1A13 (1st floor, Laboratoire de Mathématiques d'Orsay), Wednesday afternoon from 13:30 to 17:00. Class notes will be made available. Practical sessions will be held on laptops with Python 3 and Jupyter notebooks (please make sure to install it before March 4, and run this script). If you are not familiar with Jupyter notebooks, check out this introduction exercise.

Lecturer Date Topic Class notes/code
LN 5 February Monge and Kantorovich problems (existence of minimizers, duality) lecture1.pdf
LC 26 February Optimality conditions and consequences (c-concavity, univariate and square cost cases) lecture2.pdf
LC 4 March Wasserstein space (topology, metric, geodesics) + Practical session lecture3.pdf, TP1.pdf
LN 11 March Numerical methods (entropic regularization) lecture4.pdf
LN 18 March Wasserstein barycenters + Practical session (Sections 1,2,3)
LC 25 March Functionals on Wasserstein space (displacement convexity, gradient flows) (Canceled)
LC+LN 1 April Active research topics lenaic.pdf luca.pdf
8 April Exam (replaced by report)

List of research papers for the report

Send us an email to reserve a paper and get the published version (papers which are already taken are indicated). Report due on April 22nd.