English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Gromov--Hausdorff distance measures the difference in shape betweencompact metric spaces. While even approximating the distance up to anypractical factor poses an NP-hard problem, its relaxations have proven usefulfor 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 theoreticalproperties, which includes their topological equivalence on a rich set offamilies of metric spaces. We show that the two distances areLipschitz-equivalent on any family of metric spaces of uniformly bounded size,but that the equivalence does not hold in general, not even when the distancesare restricted to ultrametric spaces. We additionally prove that the standardand the modified Gromov--Hausdorff distances are either equal or within afactor of 2 from each other when taken to a regular simplex, which connects therelaxation to some well-known problems in discrete geometry.

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