AI 中文总结
本文针对Conley-Morse持续同调条形码计算成本高的问题，提出用更简单的块取代指数对，通过更新矩阵分解的方法实现高效计算，优化了现有算法。
AI 中文摘要
组合动力系统的最新进展对经典离散莫尔斯理论进行了推广，推动了组合向量场的算法研究。在这方面，文献[7]的作者近期提出了Conley-Morse持续同调条形码的概念，该条形码通过同调持续同调来总结演化向量场中不变集的连续性。他们提出了一种算法，利用名为“转移图”的偏序集上所谓“指数对”的滤波来计算该条形码。由于该算法需对“笨重”的指数对结构的滤波多次执行锯齿状持续同调，计算成本很高。我们通过用结构简单得多的“块”取代指数对来克服这一困难。这些取代需要反转转移图中的某些关系，从而得到一个简单得多的算法。该算法通过更新矩阵分解来工作，类似于标准持续同调中“ vineyard（葡萄园）”的计算。
英文摘要
Recent advances in combinatorial dynamical systems that generalize the classic discrete Morse theory have prompted algorithmic studies of combinatorial vector fields. In this regard, authors in [7] recently proposed the concept of Conley-Morse persistence barcode that summarizes the continuation of invariant sets in an evolving vector field through homological persistence. They proposed an algorithm to compute this barcode using a filtration of the so called \emph{index pairs} on a poset called \emph{transition diagram}. The algorithm becomes costly due to multiple runs of zigzag persistence it executes on filtrations of `unwieldy' structures of index pairs. We overcome this difficulty by replacing the index pairs with \emph{blocks}, which are structurally much simpler. These replacements need reversal of certain relations in the transition diagram resulting in a much simpler algorithm. The algorithm works by updating matrix decompositions akin to computing `vineyard' in standard persistence.
