Qualitative Euclidean embedding of Disjoint Sets of Points

TL;DR

This paper establishes conditions under which two disjoint point sets can be jointly embedded into Euclidean space, preserving certain distance relations and enabling combined geometric analysis of the sets.

Contribution

It introduces sufficient conditions for joint Euclidean embedding of two disjoint sets when at least one set can be embedded, along with a method to generate inter-set distances.

Findings

Conditions for joint embedding are derived.

A specific relation-preserving function for distances is proposed.

Inter-set distances can be constructed from arbitrary proximity functions.

Abstract

We consider two disjoint sets of points. If at least one of the sets can be embedded into an Euclidean space, then we provide sufficient conditions for the two sets to be jointly embedded in one Euclidean space. In this joint Euclidean embedding, the distances between the points are generated by a specific relation-preserving function. Consequently, the mutual distances between two points of the same set are specific qualitative transformations of their mutual distances in their original space; the pairwise distances between the points of different sets can be constructed from an arbitrary proximity function.