Skip to Main content Skip to Navigation
Conference papers

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

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.
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-03160475
Contributor : Elsa Cazelles Connect in order to contact the contributor
Submitted on : Wednesday, October 27, 2021 - 12:34:02 PM
Last modification on : Tuesday, January 4, 2022 - 6:29:19 AM

File

weakBar_hal.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-03160475, version 1
  • ARXIV : 2102.13380

Citation

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-03160475v1⟩

Share

Metrics

Les métriques sont temporairement indisponibles