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Title: Classical multidimensional scaling on metric measure spaces
Abstract

We study a generalization of the classical multidimensional scaling procedure (cMDS) which is applicable in the setting of metric measure spaces. Metric measure spaces can be seen as natural ‘continuous limits’ of finite data sets. Given a metric measure space ${\mathcal{X}} = (X,d_{X},\mu _{X})$, the generalized cMDS procedure involves studying an operator which may have infinite rank, a possibility which leads to studying its traceability. We establish that several continuous exemplar metric measure spaces such as spheres and tori (both with their respective geodesic metrics) induce traceable cMDS operators, a fact which allows us to obtain the complete characterization of the metrics induced by their resulting cMDS embeddings. To complement this, we also exhibit a metric measure space whose associated cMDS operator is not traceable. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov–Wasserstein distance.

 
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NSF-PAR ID:
10505784
Author(s) / Creator(s):
;
Publisher / Repository:
Oxford University Press
Date Published:
Journal Name:
Information and Inference: A Journal of the IMA
Volume:
13
Issue:
2
ISSN:
2049-8772
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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