On the signature of squared distance matrices of metric measure spaces

TL;DR

This paper investigates the eigenvalue signatures of squared distance matrices in metric measure spaces, revealing their limits, relation to embeddings in Hilbert and pseudo-Euclidean spaces, and implications for various sample spaces.

Contribution

It establishes the almost sure existence and nonrandom nature of eigenvalue limits, connecting them to multidimensional scaling and space embeddings, with explicit examples and asymptotic behaviors.

Findings

Limits of eigenvalue counts exist almost surely and are nonrandom.

These limits determine the isometric embeddability into Hilbert space.

For certain spaces, the number of negative eigenvalues diverges as sample size increases.

Abstract

We consider the numbers of positive and negative eigenvalues of matrices of squared distances between randomly sampled i.i.d. points in a given metric measure space. These numbers and their limits, as the number of points grows, in fact contain some important information about the whole space. In particular, by knowing them, we can determine whether this space can be isometrically embedded in the Hilbert space. We show that the limits of these numbers exist almost surely, are nonrandom and the same for all Borel probability measures of full support, and, moreover, are naturally related to the operators defining the multidimensional scaling (MDS) method. We also relate them to the signature of the pseudo-Euclidean space in which the given metric space can be isometrically embedded. In addition, we provide several examples of explicit calculations or just estimates of those limits for sample metric spaces. In particular, for a large class of countable spaces (for instance, containing all graphs with bounded intrinsic metrics), we get that the number of negative eigenvalues increases to infinity as the size of samples grows. However, we are able to provide examples when the number of samples grows to infinity and the numbers of both negative and positive eigenvalues increases to infinity, or the number of positive eigenvalues is bounded (but as large as desired), and the number of positive ones is fixed. Finally, we consider the example of the universal countable Rado graph.