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arXiv · 2608.30837

Approximate Gromov--Hausdorff continuity of magnitude and weighting

Masahiko Yoshinaga
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Abstract

Magnitude gives an effective size of a finite metric space and is now used in several data-analysis settings. For such applications, it is natural to ask how magnitude behaves under Gromov-Hausdorff perturbations, including collisions of points. However, magnitude is nowhere continuous on finite metric spaces with the Gromov--Hausdorff topology. We show that this failure is nongeneric in a precise measure-theoretic sense. After fixing the number of points in each collapsing cluster, magnitude is approximately continuous. More strongly, when clusters of points collapse to the points of a limit space, the total weighting of each cluster approximately converges to the weighting of the corresponding limit point. The main tool is a general result on approximate limits of inverses near singular matrices.

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Masahiko Yoshinaga. 2026-08-31. Approximate Gromov--Hausdorff continuity of magnitude and weighting. https://arxiv.org/abs/2608.30837

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