Introduction to Optimal Transport Theory

Lénaïc Chizat and Luca Nenna

Master Optimization, Université Paris-Saclay, spring 2021

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).

Organisation: lectures will take place on this BigBlueButton server on Wednesdays afternoon between 2pm and 5pm. This format is subject to modifications in the coming days, register to receive the updates.

Evaluation: Practical session (short) reports & short report (10 pages max) on a research article


Lecturer Date Topic Class notes/code
LN 17 February Monge and Kantorovich problems (existence of minimizers, duality) lecture1, blackboard
LC 24 February Optimality conditions and consequences (c-concavity, univariate and square cost cases) lecture2
LN 3 March Numerical methods (entropic regularization) lecture3, blackboard
LC 10 March Wasserstein space (topology, metric, geodesics, barycenters) + Practical session TP1, lecture4, blackboard
LN 17 March Functionals on Wasserstein space (displacement convexity, gradient flows) + Practical session TP2, lecture5
LC 24 March Divergence between probability measures (MMD, W2, Sinkhorn) + Practical session TP3, TP3 (.ipynb), lecture6, blackboard
31 March (no exam)

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 marked as such in the list). You also have the possibility to choose a published article of your choice from the optimal transport literature, send us an email with your suggestion if you wish to do so. Report and practical sessions due on April 7th via an email addressed to both of us.