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

04/21/2022
by   Vladyslav Oles, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/21/2019

Efficient estimation of a Gromov--Hausdorff distance between unweighted graphs

Gromov--Hausdorff distances measure shape difference between the objects...
research
08/02/2023

Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs

The Frechet distance is often used to measure distances between paths, w...
research
10/07/2021

The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation

The Gromov-Hausdorff distance (d_GH) provides a natural way of quantifyi...
research
02/05/2020

Fast and Robust Comparison of Probability Measures in Heterogeneous Spaces

The problem of comparing distributions endowed with their own geometry a...
research
04/30/2018

A Data-Dependent Distance for Regression

We develop a new data-dependent distance for regression problems to comp...
research
07/25/2023

Computing the Gromov–Hausdorff distance using first-order methods

The Gromov–Hausdorff distance measures the difference in shape between c...
research
08/05/2020

Persistent Homology in ℓ_∞ Metric

Proximity complexes and filtrations are a central construction in topolo...

Please sign up or login with your details

Forgot password? Click here to reset