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arXiv · 2201.09385

Classical Multidimensional Scaling on Metric Measure Spaces

Sunhyuk Lim·Facundo Memoli
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Abstract

We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure space. Furthermore, the cMDS output of several non-finite exemplar metric measures spaces, in particular the cMDS for spheres S^{d-1} and subsets of Euclidean space, are studied. Finally, we prove the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance.

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Sunhyuk Lim, Facundo Memoli. 2024-05-13. Classical Multidimensional Scaling on Metric Measure Spaces. https://arxiv.org/abs/2201.09385

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