Degenerating families of dendrograms

TL;DR

This paper explores the mathematical structure of dendrograms using nonarchimedean geometry and $p$-adic representations, revealing their connection to algebraic moduli spaces and stochastic classification.

Contribution

It introduces a $p$-adic geometric framework for dendrograms, linking them to moduli spaces and providing new insights into their topology and classification.

Findings

Dendrograms can be modeled as subtrees of Bruhat-Tits trees.

Moduli spaces in algebraic geometry serve as $p$-adic parameter spaces for dendrograms.

The topology of the hidden part of a dendrogram is characterized.

Abstract

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are $p$-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.