Robinson spaces and their representation in low-dimensional metric spaces

TL;DR

This paper investigates how Robinson dissimilarity spaces can be embedded into low-dimensional metric spaces, especially real trees, to preserve their order-based dissimilarity relationships, revealing both possibilities and limitations.

Contribution

It introduces combinatorial and topological tools for representing Robinson spaces and formulates the embedding problem as a linear program, advancing understanding of their geometric representations.

Findings

Some Robinson spaces can be embedded in caterpillars (a class of real trees).

Not all Robinson spaces admit embeddings into any real tree.

The paper provides a linear programming approach to the embedding problem.

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.