Lipschitz (non-)equivalence of the Gromov--Hausdorff distances, including on ultrametric spaces

TL;DR

This paper explores the Lipschitz (non-)equivalence of the Gromov--Hausdorff distances and their relaxations, revealing conditions under which they are equivalent or differ, especially on ultrametric spaces.

Contribution

It establishes Lipschitz equivalence conditions for the standard and modified Gromov--Hausdorff distances on bounded families and analyzes their relationship on ultrametric spaces.

Findings

Distances are Lipschitz-equivalent on bounded families of metric spaces.

Equivalence does not hold in general, especially on ultrametric spaces.

Distances are either equal or within a factor of 2 on regular simplexes.

Abstract

The Gromov--Hausdorff distance measures the difference in shape between compact metric spaces. While even approximating the distance up to any practical factor poses an NP-hard problem, its relaxations have proven useful for the problems in geometric data analysis, including on point clouds, manifolds, and graphs. We investigate the modified Gromov--Hausdorff distance, a relaxation of the standard distance that retains many of its theoretical properties, which includes their topological equivalence on a rich set of families of metric spaces. We show that the two distances are Lipschitz-equivalent on any family of metric spaces of uniformly bounded size, but that the equivalence does not hold in general, not even when the distances are restricted to ultrametric spaces. We additionally prove that the standard and the modified Gromov--Hausdorff distances are either equal or within a factor of 2 from each other when taken to a regular simplex, which connects the relaxation to some well-known problems in discrete geometry.