Euclidean Embeddings

Abstract

Distances are functions that can be defined on any set of objects. Euclidean distances, in contrast, are functions that can only be defined on sets that possess a particular structure. Given a set of proximities, one can test whether these values are distances and, moreover, whether they can even be interpreted as Euclidean distances. More generally, one can ask the same questions allowing for particular transformations of the given proximities such as adding a constant to each value. For ordinal transformations, the hypothesis that proximities are Euclidean distances is trivially true. Hence, in ordinal MDS, we learn nothing from the fact that the proximities can be represented in a Euclidean space. In interval MDS, in contrast, Euclidean embedding is nontrivial. If the data can be mapped into Euclidean distances, one can ask how many dimensions at most are necessary for a perfect representation. A further question, related to classical MDS, is how to find an interval transformation that leads to approximate Euclidean distances, while keeping the dimensionality of the MDS space as low as possible.