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Isometric embedding of the n-point spaces into the space of spaces for $n \leq 4$

Benjamin Capdeville

Abstract

In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up to isomorphism which he called The space of spaces. He also introduced for a natural number n the space of all n-points metric spaces. The aim of this article is to study if the embedding of this space in the space of spaces is isometric. Using results from [Haggai Maron and Yaron Lipman. (probably) concave graph matching. Advances in Neural Information Processing Systems, 31, 2018] and [Hiroshi Maehara. Euclidean embeddings of finite metric spaces. Discrete Mathematics, 2013], we prove that it is the case for $n \leq 4$ and for Euclidean metric spaces.

Isometric embedding of the n-point spaces into the space of spaces for $n \leq 4$

Abstract

In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up to isomorphism which he called The space of spaces. He also introduced for a natural number n the space of all n-points metric spaces. The aim of this article is to study if the embedding of this space in the space of spaces is isometric. Using results from [Haggai Maron and Yaron Lipman. (probably) concave graph matching. Advances in Neural Information Processing Systems, 31, 2018] and [Hiroshi Maehara. Euclidean embeddings of finite metric spaces. Discrete Mathematics, 2013], we prove that it is the case for and for Euclidean metric spaces.
Paper Structure (20 sections, 25 theorems, 32 equations, 1 figure)

This paper contains 20 sections, 25 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The space of spaces, the n-point spaces
  4. The space of spaces
  5. Gauged spaces
  6. The n-point spaces as an injection into the space of spaces
  7. Statement of the problem
  8. Counterexample for gauged spaces
  9. Proof for spaces of negative type
  10. A convexity argument
  11. Kronecker product and vectorization
  12. Conditionally negative semi-definite matrices
  13. Negative type metric spaces
  14. Properties of metric spaces of negative type
  15. Embedding of metric spaces into an Euclidean space
  16. ...and 5 more sections

Key Result

Proposition 3.4

$\Delta\!\!\!\!\Delta_p(\mathbb{X},\mathbb{Y}) = 0$ if and only if $\mathbb{X}$ and $\mathbb{Y}$ are isomorphic.

Figures (1)

  • Figure 1: Complete bipartite graph $K_{3,2}$

Theorems & Definitions (61)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Definition 3.8
  • Definition 3.9: $L^2$-distortion distance
  • Definition 3.10
  • ...and 51 more