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Robinson spaces and their representation in low-dimensional metric spaces

Francisco Arrepol, Mauricio Soto-Gomez, Christopher Thraves Caro

TL;DR

It is proved that some subclasses of Robinson spaces always admit embeddings in a caterpillar, and the existence of Robinson spaces that cannot be embedded in any real tree is established.

Abstract

Robinson spaces are structures equipped with a total order that encodes comparative dissimilarity relationships. We study the problem of representing Robinson dissimilarity spaces into low-dimensional metric spaces. These representations aim to preserve the relative dissimilarity relationships between elements rather than their exact values. While low dimensional Euclidean spaces such as $\mathbb{R}^1$ and $\mathbb{R}^2$ are natural candidates for such embeddings, previous work has shown that not all Robinson spaces admit a valid embedding in the real line that respects their structural constraints. Motivated by this limitation, we explore the broader class of real trees, which retain low-dimensional interpretability while allowing greater flexibility. To address the embedding problem, we develop two key tools: a combinatorial representation of Robinson spaces and a topological characterization of caterpillars, a restricted class of real trees. These tools enable a formulation of the embedding problem as a linear program, providing both computational and theoretical insights. We prove that some subclasses of Robinson spaces always admit embeddings in a caterpillar, and we establish the existence of Robinson spaces that cannot be embedded in any real tree. These results clarify the geometric limitations of representing ordered dissimilarity structures and open new directions for studying the interaction between dissimilarity, order, and metric geometry.

Robinson spaces and their representation in low-dimensional metric spaces

TL;DR

It is proved that some subclasses of Robinson spaces always admit embeddings in a caterpillar, and the existence of Robinson spaces that cannot be embedded in any real tree is established.

Abstract

Robinson spaces are structures equipped with a total order that encodes comparative dissimilarity relationships. We study the problem of representing Robinson dissimilarity spaces into low-dimensional metric spaces. These representations aim to preserve the relative dissimilarity relationships between elements rather than their exact values. While low dimensional Euclidean spaces such as and are natural candidates for such embeddings, previous work has shown that not all Robinson spaces admit a valid embedding in the real line that respects their structural constraints. Motivated by this limitation, we explore the broader class of real trees, which retain low-dimensional interpretability while allowing greater flexibility. To address the embedding problem, we develop two key tools: a combinatorial representation of Robinson spaces and a topological characterization of caterpillars, a restricted class of real trees. These tools enable a formulation of the embedding problem as a linear program, providing both computational and theoretical insights. We prove that some subclasses of Robinson spaces always admit embeddings in a caterpillar, and we establish the existence of Robinson spaces that cannot be embedded in any real tree. These results clarify the geometric limitations of representing ordered dissimilarity structures and open new directions for studying the interaction between dissimilarity, order, and metric geometry.
Paper Structure (20 sections, 9 theorems, 64 equations, 12 figures, 1 table)

This paper contains 20 sections, 9 theorems, 64 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions, notation, and our results
  3. Caterpillars.
  4. Left-Right Center.
  5. Related work
  6. Tree valid drawing problem
  7. MILP formulation.
  8. Reduction to strict Robinson spaces
  9. Strong-four-point-condition
  10. Feasibility problem formulation
  11. Matrix formulation.
  12. Kernel characterization.
  13. Applications of Lemma \ref{['thm:kernel_characterization']}
  14. Existence of Valid Drawings in Trees.
  15. Case 1: No such columns exist.
  16. ...and 5 more sections

Key Result

Theorem 6

Given an $(n \times n)$ distance matrix $\mathbf{D}$. There is an unrooted tree whose path metric is compatible with $\mathbf{D}$ if and only if the metric defined by $\mathbf{D}$ satisfies Four-point-condition.

Figures (12)

  • Figure 1: Example of a caterpillar graph
  • Figure 2: (a) Example of a Robinson space represented by its dissimilarity matrix S. (b) Matrix of centers C(S) of the space S.
  • Figure 3: (a) Example of a Robinson space S represented by its dissimilarity matrix. (b) The strict mapping for the space S represented by its dissimilarity matrix.
  • Figure 4: Caterpillar construction for the metric space $(\{1,2,3,4\},d)$ satisfying the Strong-four-point-condition. The distances $d_{ijk}$ and $d_{ijkl}$ in the caterpillar are defined using the distance $d$ in the metric space as follows: $d_{ijk}=\frac{1}{2}(d(i,j)+d(j,k)-d(i,k))$, and $d_{ijkl}=d(i,l)+d(j,k)-(d(i,j)+d(k,l))$.
  • Figure 5: Construction of a caterpillar for a metric satisfying the Strong-four-point-condition.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Definition 1: Robinson Space
  • Definition 2: Valid Drawing
  • Definition 3: Real Tree
  • Definition 4: Matrix of Centers
  • Definition 5: Four-point-condition
  • Theorem 6: Buneman Buneman1974
  • Lemma 7
  • proof
  • Definition 8: Strong-four-point-condition
  • Theorem 9
  • ...and 10 more