Course information

Instructor: Jonathan Niles-Weed (jnw@cims.nyu.edu)

Description

This course will focus on recent research in statistical aspects of optimal transport, including limit theorems, concentration inequalities, and the statistical impact of regularization.

Prerequisites

Probability. Real analysis would also be helpful, but is not required.

Books

This class does not require prior knowledge of optimal transport. However, if you would like to learn more about this subject, two good texts are:

  • Optimal Transport for Applied Mathematicians (Santambrogio)
  • Topics in Optimal Transportation (Villani)

Tentative syllabus

This class will likely cover the following topics/papers.

  • Law of large numbers: ‘‘On optimal matchings’’ (Ajtai et al. ‘84), ‘‘On the mean speed of convergence of empirical and occupation measures in Wasserstein distance’’ (Boissard and Le Gouic, ‘14)
  • Central limit theorem: ‘‘Central limit theorems for the Wasserstein distance between the empirical and the true distributions’’ (del Barrio et al. ‘99), ‘‘Central limit theorems for empirical transportation cost in general dimension’’ (del Barrio and Loubes, ‘19)
  • Non-asymptotic concentration inequalities
  • Statistical aspects of regularization: ‘‘Sample complexity of Sinkhorn divergences’’ (Genevay et al. ‘18)
  • Minimax rates of estimation: ‘‘Minimax distribution estimation in Wasserstein distance’’ (Singh and Poczos, ‘18)

Logistics

Class: Mondays, 3:20 - 5:10 pm, Warren Weaver 101 (and online)

Office Hours

By appointment.

Grading

Final project, which can be:

  • a piece of original research
  • a written mathematical summary of a paper or topic not covered in class

The project is due Monday, March 22 at 5 PM Eastern Time.