发表机构
Chennai Mathematical Institute（金奈数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出用局部障碍模的同调证书替代链环锥形条件，实现HomTrim预处理，额外移除13%-41%边并加速1.55-4.61倍，同时提供正则性图与多尺度稳定性。
AI 中文摘要
现有的基于链环的组合预处理方法仅在顶点或边的链环保持为锥形时移除该顶点或边，从而加速持续同调的计算。我们将此条件替换为定量的同调证书。生成元（顶点或边）的滤流链环的约化同调定义了一个局部障碍模，其未来部分描述了删除该生成元的效果。其条形码证明要么精确删除，要么给出瓶颈误差的显式界限，而冲突着色将此保证扩展到生成元族。我们的实现HomTrim在仅支配预处理之外额外移除13%至41%的输入边，并在加权旗滤流上将后端持久化时间减少1.55至4.61倍。同一模还产生正则性图，度量生成元在变得对同调可见之前可以移动多远，以及多尺度稳定性结果。
英文摘要
Existing link-based combinatorial preprocessing methods speed up the computation of persistent homology by removing a vertex or edge only when its link remains a cone. We replace this condition with a quantitative homological certificate. The reduced homology of the filtered link of a generator (a vertex or edge) defines a local obstruction module whose future part describes the effect of deleting that generator. Its barcode certifies either exact deletion or an explicit bound on the bottleneck error, and a conflict colouring extends this guarantee to families of generators. Our implementation, HomTrim, removes an additional 13% to 41% of the input edges beyond domination-only preprocessing and reduces backend persistence time by factors ranging from 1.55 to 4.61 on weighted flag filtrations. The same module also yields regularity diagrams that measure how far a generator can move before becoming visible to homology, together with a multiscale stability result.
Comments29 pages, 2 figures, 6 tables, 1 algorithm
