On the Structure of Isometrically Embeddable Metric Spaces

TL;DR

This paper provides structural characterizations for finite metric spaces that can be embedded into a Hilbert space, extending geometric understanding relevant to graph clustering and vector generalizations.

Contribution

It introduces new structural criteria for when finite metric spaces are isometrically embeddable into Hilbert spaces, building on recent geometric generalizations of the Fiedler vector.

Findings

Structural characterizations for isometric embedding into Hilbert spaces

Connections to geometric generalizations of the Fiedler vector

Insights relevant to graph clustering methods

Abstract

Since its popularization in the 1970s the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced geometric generalizations of the Fiedler vector. Motivated by questions stemming from their work we provide structural characterizations for when a finite metric space can be isometrically embedded in a Hilbert space.