## optimal transport and Wasserstein barycentres

**F**ollowing my musing about using medians versus means, a few days ago, Arnaud Doucet sent me a paper he recently wrote with Marco Cuturi for the incoming ICML meeting in Beijing. (The program is full of potentially interesting papers on computational methods.) The starting point is the *Wasserstein distance* between two probability measures, which amounts to finding the most correlated copula associated with these measures. Correlation being measured by a certain Euclidean distance. A second notion is a *Wasserstein barycentre*, which is the measure minimising a sum or average of Wasserstein distances to several measures. The connection with the random measures of the previous post is to find an estimator as an empirical measure that minimises the average of Wasserstein distances to several empirical measures. When the support of the empirical distribution is fixed, the weights are derived by Cuturi and Doucet by subgradient methods. When the support is free (but with a maximal size), they propose an alternating optimisation extension to derive the Wasserstein barycentre. Those algorithms being extremely costly, the authors move to a smoothed and much less intensive version. As I am completely novice in this topic, I cannot say much about the method, but the illustration on a digit restoration dataset is certainly impressive!

May 28, 2014 at 4:20 pm

Thanks, Márcio. Guillaume Carlier happens to be one of my colleagues, so I hope we’ll have the opportunity to discuss that in the coffee room one of these days..!

May 28, 2014 at 12:26 am

Another use of optimal transport in a statistical problem can be viewed in a recent working paper by Carlier, Chernozhukov and Galichion, “Vector Quantile Regression”, that propose a formulation of vector quantile regression and conditional quantiles as a solution of an optimal transport problem.

http://www.mit.edu/~vchern/papers/PaperVQR11Apr2014.pdf