Is magnitude 'generically continuous' for finite metric spaces?
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.
Keywords
Cite
@article{arxiv.2501.08745,
title = {Is magnitude 'generically continuous' for finite metric spaces?},
author = {Hirokazu Katsumasa and Emily Roff and Masahiko Yoshinaga},
journal= {arXiv preprint arXiv:2501.08745},
year = {2025}
}
Comments
18 pages, 3 figures. v2: Corrected a minor error in the Introduction; strengthened Conjecture 1.3 and added references to the discussion before it; added Remark 3.7