Is magnitude 'generically continuous' for finite metric spaces?

2025-01-15

AI Summary

This paper investigates the continuity of magnitude—a metric invariant defined on finite metric spaces—under the Gromov–Hausdorff (GH) topology. Contrary to its frequent, implicit assumption as continuous in applied data analysis, the authors prove that magnitude is **nowhere continuous** on the space of compact metric spaces equipped with the GH metric. However, they establish a novel notion of *generic continuity*: magnitude is continuous on a dense Gδ subset, i.e., it preserves limits on a Baire-typical set. This result is rigorously derived by integrating tools from GH geometry, metric invariant theory, and Baire category methods. The paper introduces and systematically develops generic continuity as a new theoretical framework, arguing that typicality—rather than pointwise continuity—is the more meaningful and practically relevant stability property in topological data analysis (TDA) and related applications. These findings provide essential theoretical foundations for the robust use of magnitude as a stable topological summary statistic.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Amplitude Concept
• Gromov-Hausdorff Topology
• Data Analysis

Innovation

• Gromov-Hausdorff topology
• amplitude stability
• data analysis significance