通过扩展 Pareto 网格计算匹配距离
Matching distance via the extended Pareto grid
AI总结：
本文利用扩展 Pareto 网格分析双参数持久化匹配距离计算中真正起作用的过滤直线，并刻画其有限特殊类型。
摘要：
多维持久化中最活跃的主题之一是不变量之间的比较。持续 Betti 数函数（或秩不变量）之间的匹配距离是该背景下研究最多的度量之一，尤其是在双参数持久化中。这一兴趣的主要原因在于，在双参数情形下，叶状化方法能够基于沿正斜率直线过滤空间，实现匹配距离的有效计算。我们的工作基于一种称为扩展 Pareto 网格的构造，对实际参与匹配距离计算的过滤直线进行了定性分析。在某些通用性假设下，我们证明这些直线必须是水平直线、垂直直线、斜率为 1 的直线，或者属于与间断现象相关的有限特殊直线集合。
英文摘要：
One of the most animated themes of multidimensional persistence is the comparison between invariants. The matching distance between persistent Betti numbers functions (or rank invariants), is among the most studied metrics in this context, particularly in 2-parameter persistence. The main reason for this interest is that, in the 2-parameter case, the foliation method allows for an effective computation of the matching distance, based on filtering the space along lines of positive slope. Our work provides a qualitative analysis, based on a construction called extended Pareto grid, of the filtering lines that actually contribute to the computation of the matching distance. Under certain genericity assumptions, we show that these lines must either be horizontal, vertical, of slope 1 or belong to a finite collection of special lines associated with discontinuity phenomena.
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