Classical Multidimensional Scaling on Metric Measure Spaces

2022/01/23 by Sunhyuk Lim, Lim, Sunhyuk, Facundo Mémoli +1
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2201.09385

openalex publication_date 2022/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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