The ultrametric Gromov-Wasserstein distance
2021-01-14 | preprint
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- Authors
- Mémoli, Facundo; Munk, Axel ; Wan, Zhengchao; Weitkamp, Christoph
- Abstract
- In this paper, we investigate compact ultrametric measure spaces which form a subset $\mathcal{U}^w$ of the collection of all metric measure spaces $\mathcal{M}^w$. Similar as for the ultrametric Gromov-Hausdorff distance on the collection of ultrametric spaces $\mathcal{U}$, we define ultrametric versions of two metrics on $\mathcal{U}^w$, namely of Sturm's distance of order $ and of the Gromov-Wasserstein distance of order $. We study the basic topological and geometric properties of these distances as well as their relation and derive for =\infty$ 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.
- Issue Date
- 14-January-2021
- Project
- EXC 2067: Multiscale Bioimaging
- Working Group
- RG Munk