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On eigenvalues and eigenfunctions of the operators defining multidimensional scaling on some symmetric spaces

Tianyu Ma, Eugene Stepanov

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.

On eigenvalues and eigenfunctions of the operators defining multidimensional scaling on some symmetric spaces

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 with odd ) whose MDS defining operators are not trace class, and original distances cannot be reconstructed from the eigenvalues and eigenfunctions of these operators.
Paper Structure (11 sections, 10 theorems, 61 equations)

This paper contains 11 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Asymptotics of MDS embeddings
  3. Questions and results
  4. Notation and preliminaries
  5. Review on Riemannian symmetric spaces
  6. The MDS map for closed connected symmetric spaces
  7. MDS for elementary symmetric spaces
  8. Products of spheres
  9. Projective spaces
  10. Signature of $T$ for projective spaces
  11. Spectral asymptotics of $T$ odd-dimensional projective spaces

Key Result

Lemma 2.1

If the operator $T$ is Hilbert-Schmidt, then for any MDS maps $\mathcal{M}^1,\mathcal{M}^2$, we have Moreover, in this case the right-hand side of eq_aeinj11 is independent of the choice of eigenfunctions $\phi_j$ of $\mathcal{K}$.

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • Corollary 4.2
  • Proposition 4.3
  • ...and 7 more