Preprint

Inserted: 22 jan 2024
Last Updated: 15 mar 2024

Year: 2024

Abstract:

We study asymptotics of the eigenvalues and eigenfunctions of the operators used for constructing multidimensional scaling (MDS) on compact connected Riemannian manifolds, in particular on closed connected symmetric spaces. They are the limits of eigenvalues and eigenvectors of squared distance matrices of an increasing sequence of finite subsets covering the space densely in the limit. We show that for products of spheres and real projective spaces, the numbers of positive and negative eigenvalues of these operators are both infinite. We also find a class of spaces (namely $\mathbb{RP}^n$ with odd $n>1$) whose MDS defining operators are not trace class, and original distances cannot be reconstructed from the eigenvalues and eigenfunctions of these operators.

Keywords: manifold learning, mulidimensiona scaling, spectra of distance matrices, symmetric Riemannian manifolds

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• maste-mdssymm030624.pdf