Infinite multidimensional scaling for metric measure spaces^*

¹ HSE University and Institute for Information Transmission Problems, Moscow, Russia
² St. Petersburg Branch of the Steklov Mathematical Institute of the Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
³ HSE University, Moscow, Russia
⁴ Department of Mathematical Physics, Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
⁵ Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

^** Corresponding author: stepanov.eugene.sns@gmail.com

Received: 24 January 2022
Accepted: 24 July 2022

Abstract

For a given metric measure space (X, d, μ) 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 μ. 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).