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Geometric interpretation of magnitude

Yasuhiko Asao, Kiyonori Gomi

TL;DR

The paper provides a comprehensive geometric interpretation of magnitude by linking Mag$\,Z$ to the circumsphere radius in a (pseudo-)Euclidean representation of $Z$, establishing Mag$\,Z = \frac{1}{1-R^2}$ for positive definite $Z$ with $Z_{ii}=1$ and $Z=V^tV$. It extends the interpretation to general symmetric $Z$ with $Z_{ii}=1$ via quasi-spheres, derives an upper bound for magnitude in negative-type spaces, and develops multiple equivalent geometric descriptions and weighting criteria that illuminate magnitudes, spreads, and related quantities. The results connect magnitude to curvature of quasi-spheres, provide a practical criterion for the existence of magnitude and positive weightings, and offer alternate formulations such as Mag$\,Z = 4R^2$, enhancing understanding of magnitude as a geometric invariant with broad implications for embeddings and clustering in finite metric spaces.

Abstract

For an $n\times n$ positive definite symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$, we show that there exists a set of vectors $V_Z\subset \mathbb{R}^n$ such that the radius $R$ of the circumsphere of $V_Z$ satisfies ${\rm Mag}\ Z = (1-R^2)^{-1}$. This leads us to interpret geometrically several known and new facts on magnitude. In particular, we show that ${\rm Mag}\ Z_{X}< n$ for an $n$-point metric space $X$ of negative type with $n>1$. This result gives a negative answer to a problem given by Gomi--Meckes \cite{GM}. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix $Z$ with $Z_{ii} = 1$ for all $i$. In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product.

Geometric interpretation of magnitude

TL;DR

The paper provides a comprehensive geometric interpretation of magnitude by linking Mag to the circumsphere radius in a (pseudo-)Euclidean representation of , establishing Mag for positive definite with and . It extends the interpretation to general symmetric with via quasi-spheres, derives an upper bound for magnitude in negative-type spaces, and develops multiple equivalent geometric descriptions and weighting criteria that illuminate magnitudes, spreads, and related quantities. The results connect magnitude to curvature of quasi-spheres, provide a practical criterion for the existence of magnitude and positive weightings, and offer alternate formulations such as Mag, enhancing understanding of magnitude as a geometric invariant with broad implications for embeddings and clustering in finite metric spaces.

Abstract

For an positive definite symmetric matrix with for all , we show that there exists a set of vectors such that the radius of the circumsphere of satisfies . This leads us to interpret geometrically several known and new facts on magnitude. In particular, we show that for an -point metric space of negative type with . This result gives a negative answer to a problem given by Gomi--Meckes \cite{GM}. Furthermore, we also have a similar geometric description of magnitude for general real symmetric matrix with for all . In this case, the radius corresponds to that of a circum-quasi-sphere, namely the set of points having a prescribed norm in a vector space endowed with an indefinite inner product.

Paper Structure

This paper contains 13 sections, 31 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pseudo-Euclidean space
  4. Affine space
  5. Quasi-sphere
  6. Magnitude
  7. Main result
  8. Upper bound for magnitude
  9. Geometric interpretations of the other facts on magnitude
  10. Criterion for the existence of magnitude weighting and positive weighting
  11. Rayleigh quotient-like expression of magnitude
  12. Spread and magnitude
  13. Another geometric description

Key Result

Theorem 1.1

Let $Z \in M_n(\mathbb{R})$ be a positive definite symmetric matrix with $Z_{ii} = 1$ for all $i$. We can choose a non-degenerate square matrix $V = (v_1\ \dots\ v_n)$ satisfying $Z=V^t\cdot V$. Let $R$ be the radius of the circumsphere of points $\{v_1, \dots, v_n\} \subset \mathbb{R}^n$. Namely, $ Then we have Furthermore, for $Va \in \mathbb{R}^n$ being the center of the circumsphere, we have

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 56 more