A novel notion of barycenter for probability distributions based on optimal weak mass transport

TL;DR

This paper introduces weak barycenters of probability distributions based on optimal weak transport, providing theoretical insights, algorithms for computation, and demonstrating their effectiveness on synthetic and real-world data.

Contribution

It proposes a new notion of barycenter based on weak transport, offering theoretical analysis, algorithms, and practical validation, distinct from traditional Wasserstein barycenters.

Findings

Weak barycenters encode shared geometric information via a latent variable.

An iterative and a stochastic algorithm for computing weak barycenters are developed.

Weak barycenters outperform Wasserstein barycenters in certain applications.

Abstract

We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlanet al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.