Course information

Instructor: Jonathan Niles-Weed (


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


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


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)


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

Office Hours

By appointment.


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.

The final project is not intended to be difficult. If you opt for the written summary approach, a mathematical write-up of ~3 pages is sufficient. Feel free to skip unimportant or tecnical preliminaries: it’s more important to capture the essential pieces of the argument.

Here are some papers that could be good sources for projects:

  • “Combinatorial optimization over two random point sets” (Barthe and Bordenave, ‘13)
  • “Central limit theorems for empirical transportation cost in general dimension” (del Barrio and Loubes, ‘19)
  • “Frechet means and Procrustes Analysis in Wasserstein Space” (Zemel and Panaretos, ‘18)
  • “Sublinear time algorithms for earth mover’s distance” (Do Ba et al., 09)
  • “Existence and consistency of Wasserstein barycenters” (Le Gouic and Loubes, ‘16)
  • “Minimax estimation of smooth densities in Wasserstein distance” (Niles-Weed and Berthet, ‘20)
  • “Minimax estimation of smooth optimal transport maps” (Hutter and Rigollet, ‘20)
  • “Interpolating between Optimal Transport and MMD using Sinkhorn Divergences” (Feydy et al., ‘18)
  • “Faster Wasserstein distance estimation with the sinkhorn divergence” (Chizat et al., ‘20)
  • “Quantifying distributional model risk via optimal transport” (Blanchet and Murthy, ‘19)
  • “Empirical regularized optimal transport: Statistical theory and applications” (Klatt et al. ‘20)