Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:15 pm
Date: 23th of January 2023
Place: Room 3001

Lénaïc CHIZAT (EPFL) “Grid-free, Debiaised, Stable Entropic Wasserstein Barycenters”

Abstract:

The notion of “Wasserstein barycenter” is a natural way to define the average a family of probability measures but suffer from a high computational and statistical complexity, particularly in high dimension. In this talk, we introduce a formulation of entropy-regularized Wasserstein barycenters that enjoys favorable optimization, approximation, and statistical properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport costs with respect to a family of given probability measures, plus an entropy term. We show that: (i) this notion of barycenter lends itself naturally to a grid-free optimization algorithm which, in the mean-field limit, converges globally at an exponential rate, (ii) it is debiased, in the sense that it is a better approximation of the unregularized Wasserstein barycenter than the naive entropic Wasserstein barycenter and that (iii) it can be estimated at the parametric rate: given n samples from each of the probability measures, the barycenter converges to the population barycenter at a rate n^{-1/2}, as measured in relative entropy.

Organizers:
Cristina BUTUCEA (CREST), Alexandre TSYBAKOV (CREST), Karim LOUNICI (CMAP) , Jaouad MOURTADA (CREST)
Sponsors:
CREST-CMAP