Multidimensional Scaling: Infinite 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 study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle S^1 into R^m for all m, and to ask questions about the MDS embeddings of the geodesic n-spheres S^n into R^m. Furthermore, 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? Convergence is understood when each metric space in the sequence has the same finite number of points, or when each metric space has a finite number of points tending to infinity. We are also interested in notions of convergence when each metric space in the sequence has an arbitrary (possibly infinite) number of points.
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