Matrix Invariants of Finite Metric Spaces

TL;DR

This paper introduces a matrix-based invariant for finite metric spaces using Gromov product structures, enabling classification of spaces with more points than previously possible.

Contribution

It defines a new matrix representation for Gromov product structures and demonstrates how matrix invariants can classify finite metric spaces beyond six points.

Findings

Matrix invariants distinguish classes of 5- and 6-point metric spaces.

Matrix similarity corresponds to permutation of Gromov product structures.

New classification method simplifies understanding of finite metric space structures.

Abstract

Finite metric spaces are characterized by a polyhedral cone defined in terms of the positivity of the distance functions and the triangle inequalities. Their classification is based on the decomposition of an associated polyhedral cone, called the "metric fan". The complete classification of $n$-point metric spaces is available only for $n\le 6$. As the number of classes increases rapidly with the number of elements, it is desirable to have coarser equivalence class decompositions based on certain invariants of finite metric spaces. If $(X,d)$ is a finite metric space with elements $P_i$ and with distance functions $d_{ij}$, the Gromov product at $P_i$ is defined as $\Delta_{ijk}=1/2(d_{ij}+d_{ik}-d_{jk})$. Assuming that the set of Gromov product at $P_i$ has a unique smallest element $\Delta_{ijk}$, the association of the edge $P_jP_k$ to $P_i$ defines the "Gromov product structure". The "pendant-free" reduction of the finite metric space is the graph obtained by removing the edges $P_jP_k$ corresponding to the minimal Gromov products $\Delta_{ijk}$ at $P_i$. In the present work, we define a matrix representation for a Gromov product structure $S$ on an $n$-point metric space, by $n\times n$ matrix $G_S$. We prove that if two metric spaces have Gromov product structures that can be mapped to each other by a permutation of the indices, then their matrices are similar via the corresponding permutation matrix. Matrix invariants of $G_S$ are used to define subclasses of Gromov product structures and their application to $n=5$ and $n=6$-point spaces are given.