# Is magnitude 'generically continuous' for finite metric spaces?

arXiv: https://arxiv.org/abs/2501.08745 · HTML: https://mlsanity.com/paper/2501.08745/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 29 references that are in the map.
- First posted on arXiv 15 Jan 2025.

## Builds on (most cited first, 4 of 4)

- [Metric Space Magnitude for Evaluating the Diversity of Latent Representations](https://mlsanity.com/paper/2311.16054/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Approximating Metric Magnitude of Point Sets](https://mlsanity.com/paper/2409.04411/) (2024)
- [Stability for Inference with Persistent Homology Rank Functions](https://mlsanity.com/paper/2307.02904/) (2023)

## Built on by (most cited first, 1 of 1)

- [Measuring What Persists: Conditioning Mechanisms and a Geometric Framework for AI Agent Identity](https://mlsanity.com/paper/2606.21843/) (2026)

## Abstract (arXiv)

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.