Classical MDS on metric-measure spaces

We study a generalization of the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We identify spectral properties of the generalized cMDS operator thus providing a natural and rigorous mathematical formulation of cMDS. Furthermore, we characterize the cMDS output of several continuous exemplar metric measures spaces. In particular, we characterize the cMDS output for spheres ${\mathrm{\S }}^{d-1}$ (with geodesic distance) and subsets of Euclidean space. In particular, the case of spheres requires that we establish the its cMDS operator is trace class, a condition which is natural in context when the cMDS has infinite rank (such as in the case of spheres with geodesic distance). Finally, we establish the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance.