Optimal Transport (From the classical Wasserstein distance to multi-marginal problems in physics and machine learning)
This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.
MA5934 is a semester module in English language at Master’s level which is offered irregular.
This Module is included in the following catalogues within the study programs in physics.
- Catalogue of non-physics elective courses
|Total workload||Contact hours||Credits (ECTS)|
|270 h||90 h||9 CP|
Content, Learning Outcome and Preconditions
Questions I will address include:
- How does one get from Monge's problem of how to efficiently transport a pile of sand into a hole to the celebrated Wasserstein distance on probability measures?
- Why does this distance metrize weak convergence?
- Why is it far more effective at modern machine learning tasks like pattern recognition than traditional distances between density profiles from your undergraduate courses, like L^p?
- How and why did Kantorovich modify Monge's problem into an (infinite-dimensional) linear optimization problem, and how does this allow to bring to bear powerful methods of linear (functional) analysis like weak convergence and duality?
- Why are contemporary pattern recognition and many-particle physics applications of optimal transport impeded by the 'curse of dimension'? (Rough answer: you are dealing with N-marginal problems, where N is the number of patterns in your database respectively the number of particles; Kantorovich requires you to find a ''joint probability density'' on the product of the spaces on which the marginals live; but even if marginals are crudely discretized by their values on 10 gridpoints, joint probability densities on the N -fold product space would require 10^N gridpoint values!)
- What is the state of the art in trying to overcome the curse of dimension (e.g., via convex geometry)?
provided by courses MA2003 and MA3001) will be helpful.
Courses, Learning and Teaching Methods and Literature
Courses and Schedule
|VO||4||Optimal Transport [MA5934]||Friesecke, G.|
|UE||2||Optimal Transport (Exercise Session)[MA5934]||Friesecke, G. Vögler, D.|
Learning and Teaching Methods
 C. Villani, Topics in Optimal Transportation, AMS 2003
 F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser 2015
Description of exams and course work
The exam may be repeated at the end of the semester.