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Tractable Metric Spaces and Magnitude Continuity

Sara Kališnik, Davorin Lešnik

Abstract

Magnitude is an isometric invariant of metric spaces introduced by Leinster. Since its inception, it has inspired active research into its connections with integral geometry, geometric measure theory, fractal dimensions, persistent homology, and applications in machine learning. In particular, when it comes to applications, continuity and stability of invariants play an important role. Although it has been shown that magnitude is nowhere continuous on the Gromov--Hausdorff space of finite metric spaces, positive results are possible if we restrict the ambient space. In this paper, we introduce the notion of tractable metric spaces, provide a characterization of these spaces, and establish several continuity results for magnitude in this setting. As a consequence, we offer a new proof of a known result stating that magnitude is continuous on the space of compact subsets of $\mathbb{R}$ with respect to the Hausdorff metric. Furthermore, we show that the magnitude function is Lipschitz when restricted to bounded subspaces of $\mathbb{R}$.

Tractable Metric Spaces and Magnitude Continuity

Abstract

Magnitude is an isometric invariant of metric spaces introduced by Leinster. Since its inception, it has inspired active research into its connections with integral geometry, geometric measure theory, fractal dimensions, persistent homology, and applications in machine learning. In particular, when it comes to applications, continuity and stability of invariants play an important role. Although it has been shown that magnitude is nowhere continuous on the Gromov--Hausdorff space of finite metric spaces, positive results are possible if we restrict the ambient space. In this paper, we introduce the notion of tractable metric spaces, provide a characterization of these spaces, and establish several continuity results for magnitude in this setting. As a consequence, we offer a new proof of a known result stating that magnitude is continuous on the space of compact subsets of with respect to the Hausdorff metric. Furthermore, we show that the magnitude function is Lipschitz when restricted to bounded subspaces of .

Paper Structure

This paper contains 10 sections, 17 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. The Magnitude of a Matrix
  5. The Magnitude of Finite Metric Spaces
  6. Positive Definite Metric Spaces
  7. The Magnitude of Compact Metric Spaces
  8. Tractable Spaces
  9. Continuity of Magnitude on R and subspaces of R
  10. Conclusion

Key Result

Proposition 2.5

Let $M$ be a finite metric space with metric $d$ and let $a \in M$. Then is a metric on $M \times \mathbb{R}_{> 0}$. Moreover, for this metric, for every $t \in \mathbb{R}_{> 0}$ the map $t\;\!M \to M \times \mathbb{R}_{> 0}$, $x \mapsto (x, t)$ is an isometric embedding.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 36 more