Papers › The ultrametric Gromov-Wasserstein distance
The ultrametric Gromov-Wasserstein distance
Facundo Mémoli, Axel Munk, Zhengchao Wan, Christoph Weitkamp
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In this paper, we investigate compact ultrametric measure spaces which form a subset 𝒰ʷ of the collection of all metric measure spaces ℳʷ. Similar as for the ultrametric Gromov-Hausdorff distance on the collection of ultrametric spaces 𝒰, we define ultrametric versions of two metrics on 𝒰ʷ, namely of Sturm's distance of order p and of the Gromov-Wasserstein distance of order p. We study the basic topological and geometric properties of these distances as well as their relation and derive for p=∞ a polynomial time algorithm for their calculation. Further, several lower bounds for both distances are derived and some of our results are generalized to the case of finite ultra-dissimilarity spaces.
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