The ultrametric Gromov-Wasserstein distance

TL;DR

This paper introduces ultrametric variants of the Gromov-Wasserstein and Sturm distances for compact ultrametric measure spaces, analyzing their properties, relations, and providing efficient algorithms for their computation.

Contribution

It defines and studies ultrametric versions of key distances on ultrametric measure spaces, including algorithms and bounds, extending previous work on ultrametric spaces.

Findings

Ultrametric Gromov-Wasserstein distance properties analyzed

Polynomial time algorithm for p=∞ case developed

Lower bounds for the distances derived

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 $p$ 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 $p=\infty$ 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.