Multidimensional Scaling on Metric Measure Spaces
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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 R^m for all m, and ask questions about the MDS embeddings of the geodesic n-spheres S^n into 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?
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