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The geometry of magnitude for finite metric spaces

Karel Devriendt

TL;DR

This work provides a geometric reinterpretation of magnitude for finite metric spaces by embedding X into Euclidean space via a similarity embedding, linking magnitude to the circumradius via $|X| = \frac{1}{1-2R(S)^2}$; this yields a unified view of magnitude across subspaces and scales. The authors develop a comprehensive matrix-theoretic framework around the similarity matrix $Z$, its centering $K$, and the Moore–Penrose inverse $K^\dagger$, establishing precise relations that propagate magnitude information to subspaces and enable Schur-complement based formulas. A novel class of strongly positive definite metric spaces is identified, closed under subspaces and capturing the asymptotics of large-scale rescalings $tX$; within this regime, they prove submodularity-type properties for magnitude as a set function, analogous to Euler-characteristic-type valuative behavior. Collectively, these results provide a geometric and algebraic toolkit for understanding magnitude, its asymptotics, subspace interactions, and combinatorial properties, with potential implications for data analysis and biodiversity metrics where magnitude is used as a diversity proxy.

Abstract

The main result of this article is a geometric interpretation of magnitude, a real-valued invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric space $X$ such that the magnitude of $X$ can be expressed in terms of the `circumradius' of its embedding $S$. The circumradius is the radius of the unique sphere that goes through $S$. We give three applications: First, we describe the asymptotic behaviour of the magnitude of $tX$ as $t\rightarrow \infty$, in terms of the circumradius. Second, we develop a matrix theory for magnitude that leads to explicit relations between the magnitude of $X$ and the magnitude of its subspaces. Third, we identify a new regime in the limiting behaviour of $tX$, and use this to show submodularity-type results for magnitude as a function on subspaces.

The geometry of magnitude for finite metric spaces

TL;DR

This work provides a geometric reinterpretation of magnitude for finite metric spaces by embedding X into Euclidean space via a similarity embedding, linking magnitude to the circumradius via ; this yields a unified view of magnitude across subspaces and scales. The authors develop a comprehensive matrix-theoretic framework around the similarity matrix , its centering , and the Moore–Penrose inverse , establishing precise relations that propagate magnitude information to subspaces and enable Schur-complement based formulas. A novel class of strongly positive definite metric spaces is identified, closed under subspaces and capturing the asymptotics of large-scale rescalings ; within this regime, they prove submodularity-type properties for magnitude as a set function, analogous to Euler-characteristic-type valuative behavior. Collectively, these results provide a geometric and algebraic toolkit for understanding magnitude, its asymptotics, subspace interactions, and combinatorial properties, with potential implications for data analysis and biodiversity metrics where magnitude is used as a diversity proxy.

Abstract

The main result of this article is a geometric interpretation of magnitude, a real-valued invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric space such that the magnitude of can be expressed in terms of the `circumradius' of its embedding . The circumradius is the radius of the unique sphere that goes through . We give three applications: First, we describe the asymptotic behaviour of the magnitude of as , in terms of the circumradius. Second, we develop a matrix theory for magnitude that leads to explicit relations between the magnitude of and the magnitude of its subspaces. Third, we identify a new regime in the limiting behaviour of , and use this to show submodularity-type results for magnitude as a function on subspaces.
Paper Structure (14 sections, 31 theorems, 64 equations, 2 figures)

This paper contains 14 sections, 31 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of results
  4. Magnitude and circumradius
  5. Definitions: magnitude, weighting & positive definite
  6. Similarity embedding of positive definite metric spaces
  7. Magnitude and circumradius
  8. Magnitude asymptotics
  9. Matrix theory & magnitude of subspaces
  10. Matrix theory of $Z,K$ and $K^\dagger$
  11. Magnitude and weightings for subspaces
  12. Strongly positive definite metric spaces
  13. Definition and first properties
  14. Magnitude submodularity

Key Result

Proposition 2.3

For any metric space $X$ and $t\gg 0$, $tX$ is positive definite.

Figures (2)

  • Figure 1: The magnitude $\vert tX^{(2)}\vert$ of the metric space with two points at distance $d=1$ is well-approximated by $2e^{-t}$ at large scales $t\gg 0$. This is explained by Theorem \ref{['thm: asymptotics of q']}.
  • Figure 2: Both panels: The magnitude $\vert tX\vert$ (black line) of the three-point metric space in Example \ref{['ex: 3-point']}; note the logarithmic scale for the horizontal axis. Left panel: The relative contribution $w_x/\vert tX\vert$ for points 1 and 2 (blue line) and 3 (red line). Right panel: The change contribution $2w^2_x/(\bar{c}_x\vert tX\vert)$ for points 1 and 2 (blue line) and 3 (red line).

Theorems & Definitions (49)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Definition 2.1: Weighting, magnitude
  • Definition 2.2
  • Proposition 2.3: leinster_magnitude_2013
  • Proposition 2.4
  • Example 2.5: Negative type metric
  • Definition 2.6: Similarity embedding
  • ...and 39 more