Ultrametric spaces and clouds
I. N. Mikhailov
TL;DR
The paper addresses how to extend the ultrametric construction $\mathbf{U}$ to the full Gromov–Hausdorff landscape, including unbounded spaces, and analyzes how clouds behave under this map. It proves $\mathbf{U}$ is $1$-Lipschitz with respect to $d_{GH}$ in general and that the product mapping $\Psi: X\mapsto X\times A$ with a dotted-connected $A$ preserves $d_{GH}$ and serves as a vehicle for inverting $\mathbf{U}$ on appropriate classes. It further establishes that the class Ult is closed within $[ riangle_1]$ and derives structural consequences: clouds cannot simultaneously host unbounded and dotted-connected spaces, and ultrametric-containing clouds interact with $\mathbf{U}$ in ways that keep $\mathbf{U}([X])$ within $[X]$ in specific cases. These results deepen understanding of unbounded Gromov–Hausdorff geometry and provide tools for stable hierarchical clustering in unbounded settings.
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 $Ψ\colon X\mapsto X\times A$ 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 $Ψ$. 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.