The ultrametric Gromov-Wasserstein distance

In this paper, we investigate compact ultrametric measure spaces which form a subset U of the collection of all metric measure spaces M. Similar as for the ultrametric Gromov-Hausdorff distance on the collection of ultrametric spaces U , we define ultrametric versions of two metrics on U, namely of Sturm’s distance of order p and of the GromovWasserstein distance of order p. We study the basic topological and geometric properties of these distances as well as their relation and derive for p “ 8 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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