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arXiv · 2501.08745

Is magnitude 'generically continuous' for finite metric spaces?

Hirokazu Katsumasa·Emily Roff·Masahiko Yoshinaga
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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.

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Hirokazu Katsumasa, Emily Roff, Masahiko Yoshinaga. 2025-01-15. Is magnitude 'generically continuous' for finite metric spaces?. https://arxiv.org/abs/2501.08745

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