Approximate Gromov--Hausdorff continuity of magnitude and weighting
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.
Related
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- Doubling, Poincaré and Gromov-Hausdorff precompactness for tamed spaces
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BibTeX
@misc{yoshinaga2026approximate,
title = {{Approximate Gromov--Hausdorff continuity of magnitude and weighting}},
author = {Masahiko Yoshinaga},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30837},
eprint = {2608.30837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30837},
}
RIS
TY - UNPB
TI - Approximate Gromov--Hausdorff continuity of magnitude and weighting
AU - Masahiko Yoshinaga
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30837
UR - https://arxiv.org/abs/2608.30837
AB - 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.
ER -