# Multidimensional Scaling on Metric Measure Spaces

## TL;DR

This paper extends classical multidimensional scaling (MDS) theory to infinite metric measure spaces, exploring optimality, embeddings of spheres, and convergence properties, thereby broadening the understanding of MDS in more general geometric contexts.

## Contribution

It introduces a generalized notion of MDS for infinite metric measure spaces, analyzing embeddings of spheres and convergence behavior, which advances the theoretical framework of MDS.

## Key findings

- Generalization of MDS to infinite metric measure spaces
- Analysis of MDS embeddings of spheres like $S^1$ and $S^n$
- Results on convergence of MDS embeddings under space convergence

## Abstract

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle $S^1$ into $\mathbb{R}^m$ for all $m$, and ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $\mathbb{R}^m$. Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$?