Prerequisites: Analysis in R^n, Linear Algebra and Applications.Probability I, Optimization and/or Functional Analysis are recommended.

(1) Optimal Transport Theory
Monge Problem
Relaxation and Kantorovich duality
Wp distances
Remember: Lower semicontinuity, weak convergence∗, convex functions.

(2) Computational aspects
Discrete Optimal Transport as Linear Programming
Entropic regularization of the Optimal Transport algorithm and Sinkhorn
Sinkhorn algorithm: the programmer’s point of view
Sinkhorn algorithm: a mathematical point of view

(3) Mathematical analysis
Existence of Entropy-Kantorovich potentials
Characterization of solutions to the primal problem
Interpolation between Wp and Maximum Mean Discrepancy

(4) Metric side of Optimal Transport
Existence of Kantorovich potentials and Duality
Existence of Optimal Transport maps
The p-Wasserstein distance
Characterization of Convergence

(5) Statistical aspects
Wasserstein loss training
Calculating the gradient of Wasserstein distances
Wasserstein barycenters
Few comments on sample complexity

(6) If time permits: Gradient flows in the space of probability measures (back to theory)
Gradient flows in R^d
Gradient flows in metric spaces and De Giorgi minimizing motions
Heat equations, Fokker-Planck and porous media
Possible choices: round-trip Wasserstein gradient flows, JKOnet, sampling as
a convex optimization, Kernalized Wasserstein Flows.

References:
1. An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows
by Alessio Figalli and Federico Glaudo (2021).
2. Computational Optimal Transport by Marco Cuturi and Gabriel Peyr ́e (available online).
3. Lectures on Optimal Transport by Luigi Ambrosio, Elia Bru ́e, Daniele Semola.

Note: This course is offered as a master’s course. In the doctorate, it has additional requirements.

 

* Basic syllabus. The teacher has the autonomy to make any changes.