Master Optimization, Université Paris-Saclay, spring 2020
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 | |
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.
1] Korman and McCann (2012). Optimal transportation with capacity constraints. pdf (TAKEN)
2] Agueh and Carlier (2010). Barycenters in the Wasserstein space. pdf (TAKEN)
3] Cotar, Friesecke and Klüppelberg (2011). Density Functional Theory and Optimal Transportation with Coulomb Cost. pdf (TAKEN)
4] Carlier, Duval, Peyré and Schmitzer (2017). Convergence of entropic schemes for optimal transport and gradient flows. pdf (TAKEN)
5] Buttazzo and Santambrogio (2009). A Mass Transportation Model for the Optimal Planning of an Urban Region. pdf (TAKEN)
6] Blanchet and Carlier (2012). Optimal Transport and Cournot-Nash equilibria. pdf
7] Solomon, De Goes, Peyré, Cuturi, Butscher, Nguyen and Guibas (2015). Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains. pdf (TAKEN)
8] Niles-Weed and Bach (2017). Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. pdf
9] Di marino and Louet (2017). The entropic regularization of the Monge problem on the real line. pdf (TAKEN)
10] Gangbo and Święch (1998). Optimal maps for the multidimensional Monge‐Kantorovich problem. pdf (TAKEN)
11] Benamou, Carlier and Santambrogio (2016). Variational Mean Field Games. pdf (TAKEN)
12] Brenier (1991). Polar factorization and monotone rearrangement of vector‐valued functions. pdf
13] Jordan, Kinderlehrer and Otto (1998). The variational formulation of the Fokker-Planck equation. pdf (TAKEN)