TY - JOUR

T1 - Infinite multidimensional scaling for metric measure spaces

AU - Kroshnin, Alexey

AU - Stepanov, Eugene

AU - Trevisan, Dario

N1 - Publisher Copyright: © The authors. Published by EDP Sciences, SMAI 2022.

PY - 2022

Y1 - 2022

N2 - 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 MDSa"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).

AB - 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 MDSa"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).

KW - Assouad embedding

KW - Isometric embedding

KW - Multidimensional scaling (MDS)

UR - https://www.esaim-cocv.org/articles/cocv/abs/2022/01/cocv220011/cocv220011.html

UR - http://www.scopus.com/inward/record.url?scp=85136943432&partnerID=8YFLogxK

U2 - 10.1051/cocv/2022053

DO - 10.1051/cocv/2022053

M3 - Article

VL - 28

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

IS - 58

M1 - 58

ER -