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On the Local Ultrametricity of Finite Metric Data

Patrick Erik Bradley

TL;DR

The paper addresses how to quantify and exploit local ultrametric structure in finite metric data. It develops a framework that builds local ultrametrics from Vietoris-Rips graphs, encodes local pieces in p-adic spaces via embeddings into Bruhat-Tits trees, and assigns equity measures from regular differential 1-forms on Mumford curves. The main contributions are: (i) a precise construction of local ultrametrics, (ii) a p-adic encoding scheme with an equity measure, and (iii) invariants such as the local epsilon-delta-genus and connectedness-based metrics; and an experimental demonstration on the Iris dataset showing increased ultrametricity within clusters and a concrete 3-adic interpolation example. This approach suggests a new direction for data analysis with potential computational benefits through hierarchical/p-adic representations.

Abstract

New local ultrametricity measures for finite metric data are proposed through the viewpoint that their Vietoris-Rips corners are samples from p-adic Mumford curves endowed with a Radon measure coming from a regular differential 1-form. This is experimentally applied to the iris dataset.

On the Local Ultrametricity of Finite Metric Data

TL;DR

The paper addresses how to quantify and exploit local ultrametric structure in finite metric data. It develops a framework that builds local ultrametrics from Vietoris-Rips graphs, encodes local pieces in p-adic spaces via embeddings into Bruhat-Tits trees, and assigns equity measures from regular differential 1-forms on Mumford curves. The main contributions are: (i) a precise construction of local ultrametrics, (ii) a p-adic encoding scheme with an equity measure, and (iii) invariants such as the local epsilon-delta-genus and connectedness-based metrics; and an experimental demonstration on the Iris dataset showing increased ultrametricity within clusters and a concrete 3-adic interpolation example. This approach suggests a new direction for data analysis with potential computational benefits through hierarchical/p-adic representations.

Abstract

New local ultrametricity measures for finite metric data are proposed through the viewpoint that their Vietoris-Rips corners are samples from p-adic Mumford curves endowed with a Radon measure coming from a regular differential 1-form. This is experimentally applied to the iris dataset.
Paper Structure (7 sections, 1 theorem, 22 equations, 3 figures, 3 tables)

This paper contains 7 sections, 1 theorem, 22 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Finite locally ultrametric spaces
  3. Local ultrametrics
  4. Local $p$-adic encodings
  5. Invariants of a finite metric space
  6. Experiments
  7. Conclusion

Key Result

Lemma 2.3

It holds true that for $\delta\ge\epsilon>0$. $(X,d)$ is ultrametric, if and only if for all $\delta\ge\epsilon>0$, the right hand side is an equality.

Figures (3)

  • Figure 1: Vietoris-Rips graphs $\Gamma_\epsilon$ for $\epsilon=0.64$ (top left), $\epsilon=0.7$ (top right), $\epsilon=1.640$ (bottom left), $\epsilon=1.650$ (bottom right). Only the non-singleton connected components are shown.
  • Figure 2: The coarse graphs $\Gamma_\epsilon^\delta$ for $(\epsilon,\delta)=(0.64,1.64)$ (left), and $(\epsilon,\delta)=(0.7,1.64)$ (right).
  • Figure 3: Single-linkage dendrogram of the cluster $C_3$ in $\Gamma_{0.64}$.

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof