发表机构
Robert Bosch Center for Cyber Physical Systems, IISc Bengaluru; Centre for Infrastructure, Sustainable Transportation and Urban Planning, IISc Bengaluru; Department of Computer Science and Automation, IISc Bengaluru（印度科学学院班加罗尔分校罗伯特·博世网络物理系统中心; 印度科学学院班加罗尔分校基础设施、可持续交通与城市规划中心; 印度科学学院班加罗尔分校计算机科学与自动化系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对次主导超度量建立ℓ₀型稳定性理论，揭示稀疏编辑仅通过最小生成树传播，证明相关边界的尖锐性及多编辑近加性，实验显示其得分可用于层次表示的漏洞诊断。
AI 中文摘要
次主导（极小极大）超度量是相异度矩阵的典型树状摘要，等价于单链路聚类诱导的超度量。其经典稳定性理论通常以ℓ_∞或Gromov–Hausdorff形式表述，但这类边界不适用于仅改变少数成对距离的稀疏扰动。本文针对该算子建立了ℓ_0型稳定性理论，分析表明稀疏编辑仅通过最小生成树（MST）传播：若成对超度量值的树路径穿过已编辑边，或穿过由已编辑树外边新暴露的割，则该值会发生变化。这得到了尖锐的单编辑暴露割得分及仅基于树的全局包络，进而得出汉明-利普希茨边界，限定可改变的超度量条目数量。本文还证明了尖锐性结果，表明这种对树几何的依赖不可避免：在严格割分离下，树边边界可精确达到；对于树外编辑，存在显式族使得一个已编辑距离改变Θ(n²)个超度量条目。此外，本文证明了多编辑的条件近加性原理，适用于经认证的大单编辑改变区域及可忽略的聚合重叠情况。在深度嵌入图上的实验表明，所得结构得分可为层次表示提供有用的漏洞诊断。
英文摘要
The subdominant (minmax) ultrametric is a canonical tree-structured summary of a dissimilarity matrix, arising equivalently as the ultrametric induced by single-linkage clustering. While its classical stability theory is usually formulated in $\ell_\infty$ or Gromov--Hausdorff terms, such bounds are poorly suited to sparse perturbations that alter only a few pairwise distances. We develop an $\ell_0$-type stability theory for this operator. Our analysis shows that sparse edits propagate only through the minimum spanning tree (MST): a pairwise ultrametric value can change only if its tree path crosses an edited edge or a cut newly exposed by an edited off-tree edge. This yields a sharp per-edit exposed-cut score and a tree-only global envelope, leading to Hamming--Lipschitz bounds on the number of ultrametric entries that can change. We also prove sharpness results showing that this dependence on tree geometry is unavoidable: under strict cut separation the tree-edge bound is attained exactly, and for off-tree edits there are explicit families in which one edited distance changes $Θ(n^2)$ ultrametric entries. In addition, we prove a conditional near-additivity principle for multiple edits under certified large per-edit changed regions and negligible aggregate overlap. Experiments on deep-embedding graphs show that the resulting structural scores provide useful vulnerability diagnostics for hierarchical representations.
