Multidimensional scaling on metric measure spaces

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 ℝ m for all m , and ask questions about the MDS embeddings of the geodesic n -spheres S n into 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 ?