Rocky Mountain Journal of Mathematics
- Rocky Mountain J. Math.
- Volume 50, Number 2 (2020), 397-413.
Multidimensional scaling on metric measure spaces
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 into for all , and ask questions about the MDS embeddings of the geodesic -spheres into . Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space , then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of ?
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 397-413.
Received: 19 July 2019
Accepted: 17 September 2019
First available in Project Euclid: 29 May 2020
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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.)
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