Prerequisites

• Linear algebra (any course of linear algebra)

• Real Analysis (any course with multi-variable analysis)

• Functional analysis (Banach and Hilbert spaces, dual spaces, and convergence in these spaces) 

• Probability theory or measure theory (rigorous definition of measures, Lebesgue integration, and $L^p$ spaces)

 

Aim of the course

Optimal Transport (OT) is a classical mathematical theory that was introduced in the 18th century to study the optimal allocations of resources and has since had an impact on areas within and beyond Mathematics, including the theory of Partial Differential Equations, Geometric Analysis, Dynamical Systems, Fluid Mechanics, Physical and Engineering Sciences. In recent years, OT has found extensive applications in a wider range of subjects such as Data Science, Statistics, and Theoretical Chemistry, bringing with it a need to further develop both theoretical and computational aspects of Optimal Transport.

 

The course aims to (1) rigorously introduce the theory of Optimal Transport leaning on classical tools from functional analysis and probability theory and (2) demonstrate its utility by presenting the most relevant applications to machine learning, urban planning, and image processing.

 

By the end of the course, the student should be able to

 

• state the Monge and Kantorovich problems and discuss the well-posedness of these problems.

• derive the dual Kantorovich problem and discuss the well-posedness of the dual problem.

• construct optimal solutions to an OT problem in 1-dimension.

• state the dynamical formulation of OT problems, discuss the well-posedness of the dynamical problem, and relate the problem to other formulations.

• state branched optimal transport and irrigation problems and to derive their well-posedness theory.

• derive and implement algorithmic approaches to compute solutions to OT problems. 

• state the Entropic formulation of OT and the Sinkhorn algorithm.

• apply OT approaches to machine learning methods, in particular generative models.

Rules about Homework/Exam

The course counts for 8 EC. The learning goals will be assessed as follows:

• There will be two graded hand-in assignments during the course (30% of the final grade). Practice exercises will be assigned every week.

• There will be a final written (or oral) exam including exercises covering the material presented at the lectures (70% of the final grade)

A minimum grade of 5.0 for the exam is required to pass the course.

The hand-in assignments count as part of the grade for the resit examination.

Lecture Notes/Literature

Suggested books:

• Optimal Transport for Applied Mathematicians by Filippo Santambrogio. 

• Computational Optimal Transport by Gabriel Peyré and Marco Cuturi (freely available at https://arxiv.org/pdf/1803.00567.pdf)

Lecture notes will be provided.