Infinite multidimensional scaling for metric measure spaces

TL;DR

This paper introduces an infinite-dimensional limit of multidimensional scaling (MDS) for metric measure spaces, revealing that the embedding often results in snowflake embeddings rather than distance-preserving maps.

Contribution

It formalizes the concept of infinite MDS as a limit of finite samples and analyzes its stability and geometric properties in the context of metric measure spaces.

Findings

Infinite MDS converges to an embedding into a Hilbert space as sample size grows.

The embedding is stable under convergence of metric measure spaces.

Often produces snowflake embeddings rather than bi-Lipschitz embeddings.

Abstract

For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithm with this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure $\mu$. This limit can be viewed as "infinite MDS" embedding of the original space, now not anymore into a finite-dimensional space but rather into an infinitedimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show that in many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances).