Master Optimization, Université Paris-Saclay, spring 2021
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 https://bbb2.imo.universite-paris-saclay.fr/b/luc-wqz-da9 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) |
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.
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 (TAKEN)
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 (TAKEN)
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 (TAKEN)
13] Jordan, Kinderlehrer and Otto (1998). The variational formulation of the Fokker-Planck equation. pdf (TAKEN)
14] Backhoff, Conforti, Gentil, Léonard (2019). The mean field Schr¨odinger problem: ergodic behavior, entropy estimates and functional inequalities. pdf
15] Bigot, Cazelles, Papadakis (2019). Penalization of Barycenters in the Wasserstein Space. pdf (TAKEN)
16] Baradat, Monsaingeon (2019). Small noise limit and convexity for generalized incompressible flows, Schrödinger problems, and optimal transport. pdf (TAKEN)
17] McCann (1997). A Convexity Principle for Interacting Gases. pdf (TAKEN)
18] Carlier, Galichon, Santambrogio (2009). From Knothe’s transport to Brenier’s map and a continuation method for optimal transport. pdf (TAKEN)
19] Altschuler, Weed, Rigollet (2017). Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration. pdf (TAKEN)
20] Figalli (2009). The Optimal Partial Transport Problem. pdf (TAKEN)