Ultrametric spaces and clouds

TL;DR

This paper explores the properties of ultrametric spaces derived from general metric spaces, showing Lipschitz continuity of the construction, and investigates the relationship between ultrametric and dotted connected spaces within the Gromov-Hausdorff framework.

Contribution

It extends the Lipschitz property of the ultrametric construction to all metric spaces and characterizes the inverse relationship between specific mappings involving ultrametric and dotted connected spaces.

Findings

The map $ extbf{U}$ is 1-Lipschitz for all metric spaces, not just bounded ones.

The mapping $ extbf{U}$ is inverse to the map $ extbf{ extit{ extPsi}}$, which preserves Gromov-Hausdorff distances.

Ultrametric spaces and dotted connected spaces cannot be close in Gromov-Hausdorff distance within the same class.

Abstract

In ``Characterization, stability and convergence of hierarchical clustering methods'' by G. E. Carlsson, F. Memoli, the natural way to construct an ultrametric space from a given metric space was presented. It was shown that the corresponding map $\textbf{U}$ is $1$-Lipschitz for every pair of bounded metric spaces, with respect to the Gromov-Hausdorff distance. We make a simple observation that $\textbf{U}$ is $1$-Lipschitz for pairs of all, not necessarily bounded, metric spaces. We then study the properties of the mapping $\textbf{U}$. We show that, for a given dotted connected metric space $A$, the mapping $\Psi\colon X\mapsto X\times A$ from the proper class of all bounded ultrametric spaces ($X\times A$ is endowed with the Manhattan metric) preserves the Gromov-Hausdorff distance. Moreover, the mapping $\textbf{U}$ is inverse to $\Psi$. By a dotted connected metric space, we mean a metric space in which for an arbitrary $\varepsilon > 0$ and every two points $p,\,q$, there exist points $x_0 = p,\,x_1,\,\ldots,\,x_n = q$ such that $\max_{0\le j \le n-1}|x_jx_{j+1}|\le \varepsilon$. At the end of the paper, we prove that each class (proper or not) consisting of unbounded metric spaces on finite Gromov-Hausdorff distances from each other cannot contain an ultrametric space and a dotted connected space simultaneously.