Barycenters for the hellinger-Kantorovich distance over

Gero Friesecke, Daniel Matthes, Bernhard Schmitzer

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5 Scopus citations


We study the barycenter of the Hellinger-Kantorovich metric over nonnegative measures on compact, convex subsets of \BbbR d. The article establishes existence, uniqueness (under suitable assumptions), and equivalence between a coupled-two-marginal and a multimarginal formulation. We analyze the HK barycenter between Dirac measures in detail, and find that it differs substantially from the Wasserstein barycenter by exhibiting a local ``clustering"" behavior, depending on the length scale of the input measures. In applications it makes sense to simultaneously consider all choices of this scale, leading to a 1-parameter family of barycenters. We demonstrate the usefulness of this family by analyzing point clouds sampled from a mixture of Gaussians and inferring the number and location of the underlying Gaussians.

Original languageEnglish
Pages (from-to)62-110
Number of pages49
JournalSIAM Journal on Mathematical Analysis
Issue number1
StatePublished - 2021


  • Barycenter
  • Data analysis
  • Multimarginal
  • Optimal transport
  • Unbalanced


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