Rocky Mountain Journal of Mathematics

Multidimensional scaling on metric measure spaces

Henry Adams, Mark Blumstein, and Lara Kassab

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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 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 ?

Article information

Source
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 397-413.

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

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1590739278

Digital Object Identifier
doi:10.1216/rmj.2020.50.397

Mathematical Reviews number (MathSciNet)
MR4104382

Zentralblatt MATH identifier
07210967

Subjects
Primary: 62H25: Factor analysis and principal components; correspondence analysis 51F99: None of the above, but in this section 15A18: Eigenvalues, singular values, and eigenvectors
Secondary: 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

Keywords
multidimensional scaling metric measure spaces dimensionality reduction integral operators

Citation

Adams, Henry; Blumstein, Mark; Kassab, Lara. Multidimensional scaling on metric measure spaces. Rocky Mountain J. Math. 50 (2020), no. 2, 397--413. doi:10.1216/rmj.2020.50.397. https://projecteuclid.org/euclid.rmjm/1590739278


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