Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Is magnitude 'generically continuous' for finite metric spaces?

Hirokazu Katsumasa, Emily Roff, Masahiko Yoshinaga

TL;DR

This study investigates the stability of magnitude, an invariant of metric spaces, under the Gromov--Hausdorff topology. It establishes that magnitude is nowhere continuous on the finite metric space space $\operatorname{FMet}^\circ$ (Formulation A), using wedge and join constructions to produce discontinuities and even arbitrary limiting values along certain paths. It then develops a framework of lines in GH space and proves a first generic-stability result: for finite $X$ with invertible $Z_X$, there exists a dense open subset of lines approaching $X$ for which $\lim_{t\to0}|Y_{f,t}|=|X|$, indicating directional (along lines) generic continuity. These results together suggest a nuanced notion of generic stability for magnitude and raise open questions about broader forms of generic continuity and positivity-definite settings, with significant implications for applications in data analysis.

Abstract

Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data analysis, this paper investigates the stability of magnitude: its continuity properties with respect to the Gromov-Hausdorff topology. We show that magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces. Yet, we find evidence to suggest that it may be 'generically continuous', in the sense that generic Gromov-Hausdorff limits are preserved by magnitude. We make the case that, in fact, 'generic stability' is what matters for applicability.

Is magnitude 'generically continuous' for finite metric spaces?

TL;DR

This study investigates the stability of magnitude, an invariant of metric spaces, under the Gromov--Hausdorff topology. It establishes that magnitude is nowhere continuous on the finite metric space space (Formulation A), using wedge and join constructions to produce discontinuities and even arbitrary limiting values along certain paths. It then develops a framework of lines in GH space and proves a first generic-stability result: for finite with invertible , there exists a dense open subset of lines approaching for which , indicating directional (along lines) generic continuity. These results together suggest a nuanced notion of generic stability for magnitude and raise open questions about broader forms of generic continuity and positivity-definite settings, with significant implications for applications in data analysis.

Abstract

Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data analysis, this paper investigates the stability of magnitude: its continuity properties with respect to the Gromov-Hausdorff topology. We show that magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces. Yet, we find evidence to suggest that it may be 'generically continuous', in the sense that generic Gromov-Hausdorff limits are preserved by magnitude. We make the case that, in fact, 'generic stability' is what matters for applicability.
Paper Structure (4 sections, 11 theorems, 61 equations, 3 figures)

This paper contains 4 sections, 11 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Magnitude is nowhere continuous on Gromov--Hausdorff space
  3. Lines in Gromov--Hausdorff space
  4. Magnitude behaves continuously for generic limits along lines

Key Result

Lemma 2.3

Let $A$ and $B$ be metric spaces which both have magnitude.

Figures (3)

  • Figure 1: The wedge $A\vee_{a,b}B$ and join $A+_\ell B$.
  • Figure 2: The dashed graph shows the magnitude function of $K_2$. The solid graph shows the pointwise limit, as $t \to 0$, of the magnitude functions of the spaces $X_t = \{\ast\} +_1 tY_{\alpha}$, for $\alpha = 2$.
  • Figure 3: A line in Gromov--Hausdorff space from $Y$ to $X$.

Theorems & Definitions (29)

  • Conjecture 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Example 2.6
  • Theorem 2.7
  • proof
  • ...and 19 more