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Multidimensional scaling on metric measure spaces. (English) Zbl 1441.62934

Summary: 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 \(\mathbb{R}^m\) for all \(m\), and ask questions about the MDS embeddings of the geodesic \(n\)-spheres \(S^n\) into \(\mathbb{R}^m\). 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 \(X\)?

MSC:

62R20 Statistics on metric spaces
62H25 Factor analysis and principal components; correspondence analysis
51F99 Metric geometry
15A18 Eigenvalues, singular values, and eigenvectors
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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References:

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