Barycenter in Wasserstein Spaces: Existence and Consistency

  • Thibaut Le GouicEmail author
  • Jean-Michel Loubes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9389)


We study barycenters in the Wasserstein space \(\mathcal {P}_p(E)\) of a locally compact geodesic space (Ed). In this framework, we define the barycenter of a measure \(\mathbb {P}\) on \(\mathcal {P}_p(E)\) as its Fréchet mean. The paper establishes its existence and states consistency with respect to \(\mathbb {P}\). We thus extends previous results on \(\mathbb {R}^d\), with conditions on \(\mathbb {P}\) or on the sequence converging to \(\mathbb {P}\) for consistency.


Barycenter Wasserstein space Geodesic spaces 



The author would like to thank Anonymous Referee #2 for the detailed review.


  1. [AC]
    Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. IAM J. Math. Anal. 43, 904–924 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [KP]
    Kim, Y.H., Pass, B.: Wasserstein Barycenters over Riemannian manifolds (2014)Google Scholar
  3. [BK]
    Bigot, J., Klein, T.: Consistent estimation of a population barycenter in the Wasserstein space (2012)Google Scholar
  4. [BLGL]
    Boissard, E., Le Gouic, T., Loubes, J.-M.: Distribution’s template estimate with Wasserstein metrics. Bernoulli J. 21(2), 740–759 (2015)Google Scholar
  5. [LGL]
    Le Gouic, T., Loubes, J.-M: Barycenter in Wasserstein spaces: existence and consistency (2015)Google Scholar
  6. [LG]
    Le Gouic, T.: Localisation de masse et espaces de Wasserstein, thesis manuscript (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.École Centrale de MarseilleMarseilleFrance

Personalised recommendations