Ultrametric spaces and clouds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909471344689152 |
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| author | Mikhailov, I. N. |
| author_facet | Mikhailov, I. N. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ultrametric spaces and clouds Mikhailov, I. N. Metric Geometry 51F99 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. |
| title | Ultrametric spaces and clouds |
| topic | Metric Geometry 51F99 |
| url | https://arxiv.org/abs/2501.19346 |