Magnitude与加权的近似Gromov–Hausdorff连续性
Approximate Gromov--Hausdorff continuity of magnitude and weighting
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中文总结 AI 辅助
本文针对有限度量空间的Magnitude在Gromov–Hausdorff拓扑下的不连续性，从测度论角度证明其不连续性非通有，固定簇点数后Magnitude近似连续，簇坍缩时加权值近似收敛，核心工具为奇异矩阵逆的近似极限结果。
中文摘要 AI 辅助
Magnitude（规模）可有效衡量有限度量空间的大小，目前已应用于多种数据分析场景。针对此类应用，自然会提出Magnitude在Gromov–Hausdorff扰动（包括点的碰撞）下的表现问题。然而，在有限度量空间的Gromov–Hausdorff拓扑中，Magnitude处处不连续。我们从精确测度论意义上证明，这种不连续性并非通有现象。在固定每个坍缩簇的点数后，Magnitude是近似连续的；更重要的是，当点簇坍缩为极限空间的点时，每个簇的总加权值会近似收敛到对应极限点的加权值。主要工具是关于奇异矩阵附近逆矩阵近似极限的一般性结果。
英文摘要
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.
发表机构
- Department of Mathematics, The University of Osaka（大阪大学数学系）
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