Lecturer

Dr. Mikhail Chebunin

---

Time and Place

Lectures

TBA

Exercises

TBA

Format and Language

2+1 (2 hours lecture + 1 hour exercise session per week), 42 hours (28 lecture hours + 14 exercise hours). The course will be taught in English.

---

Requirements

Students should be familiar with basic courses in Analysis and Probability. Also helpful, but not required, is basic knowledge in Functional and Convex analysis.

---

Target groups

Master Math, DaSci, MaBi, Fin, WiMa, CSE.

---

Description

Optimal transport is a mathematical theory that connects probability, optimization, functional analysis, and convex analysis. Originally, it was introduced by Monge in 1781 as a problem of optimally transporting soil, and then became a fundamental tool in modern probability, machine learning, and data science. This course is an extension of the first part (Introduction to Optimal Transport) to statistical optimal transport, which studies what can be inferred about transport costs, maps, and dynamics when probability measures are unknown and only finite samples are observed. Building on the deterministic theory developed in the first part of the course, this sequel focuses on sample complexity, statistical rates of convergence, minimax lower bounds, and regularised discrepancies between probability distributions.

We start with a selective bridge from Part I, recalling the Monge and Kantorovich formulations, duality and Wasserstein metrics, Brenier maps, entropic computation, and the dynamic formulation. The remaining lectures develop genuinely new material: estimation of Wasserstein distances and transport maps, statistical properties of entropic transport, Wasserstein gradient flows, mean-field models, metric geometry, and Wasserstein barycentres.

The course primarily follows the monograph by Sinho Chewi, Jonathan Niles-Weed, and Philippe Rigollet (Springer, 2025). Students will develop proof-based and computational skills relevant to probability, statistics, machine learning, and data science.

Topics:

• Review - The Monge-Kantorovich Framework, Wasserstein Metrics, Optimal Maps, Computation, and Dynamics.
• Empirical Measures in Wasserstein Distance
• The Primal Multiscale Method 
• Dual Chaining Bounds
• Statistical Applications and Optimality
• Faster Rates for Smooth Measures
• Regularised Distances Between Distributions
• Estimation of Transport Maps: Formulation and Stability
• Slow and Fast Rates for Transport Maps
• Statistical Theory of Entropic Optimal Transport
• Riemannian Structure and Otto Calculus
• Gradient Flows for Variational Inference and Sampling
• Mean-Field Models and Machine Learning
• Metric Geometry Beyond Geodesics
• Wasserstein Barycentres and Their Statistics
   

---

Exam

Final written or oral examination, depending on the number of participants. The examination form will be announced in advance, at least 4 weeks before the examination date. The prerequisite for taking the exam is to achieve at least 50 % of the practice points.

Exercise sessions

Problem-solving sessions emphasizing computational examples and analysis-probabilistic proofs. Exercise sheets will be distributed every one or two weeks in the Moodle course.

Literature

• Chewi, Sinho; Niles-Weed, Jonathan; and Rigollet, Philippe. Statistical Optimal Transport, Lecture Notes in Mathematics 2364, Springer, 2025.
• Friesecke, Gero. Optimal Transport: A Comprehensive Introduction to Modeling, Analysis, Simulation, Applications, SIAM, 2024.
• Santambrogio, Filippo. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Birkhäuser, 2015.
• Peyré, Gabriel and Cuturi, Marco. Computational Optimal Transport, Foundations and Trends in Machine Learning, 2019.
• Ambrosio, Luigi; Gigli, Nicola; and Savaré, Giuseppe. Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed., Birkhäuser, 2008.