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Conference papers

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

Elsa Cazelles 1 Felipe Tobar 2 Joaquin Fontbona 2 
1 IRIT-SC - Signal et Communications
IRIT - Institut de recherche en informatique de Toulouse
Abstract : We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlan et 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.
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Submitted on : Friday, January 14, 2022 - 6:47:21 PM
Last modification on : Monday, July 4, 2022 - 8:54:23 AM


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  • HAL Id : hal-03160475, version 2
  • ARXIV : 2102.13380


Elsa Cazelles, Felipe Tobar, Joaquin Fontbona. A novel notion of barycenter for probability distributions based on optimal weak mass transport. Thirty-fifth Conference on Neural Information Processing Systems (NeurIPS 2021), NIPS: Neural Information Processing Systems Foundation, Dec 2021, Online, France. ⟨hal-03160475v2⟩



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