April 2020 Multidimensional scaling on metric measure spaces
Henry Adams, Mark Blumstein, Lara Kassab
Rocky Mountain J. Math. 50(2): 397-413 (April 2020). DOI: 10.1216/rmj.2020.50.397


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


Download Citation

Henry Adams. Mark Blumstein. Lara Kassab. "Multidimensional scaling on metric measure spaces." Rocky Mountain J. Math. 50 (2) 397 - 413, April 2020. https://doi.org/10.1216/rmj.2020.50.397


Received: 19 July 2019; Accepted: 17 September 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210967
MathSciNet: MR4104382
Digital Object Identifier: 10.1216/rmj.2020.50.397

Primary: 15A18‎ , 51F99 , 62H25
Secondary: 47A05

Keywords: dimensionality reduction , Integral‎ ‎Operators , metric measure spaces , multidimensional scaling

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


This article is only available to subscribers.
It is not available for individual sale.

Vol.50 • No. 2 • April 2020
Back to Top