更新离散 Morse 向量场以计算下星过滤葡萄园
Updating a Discrete Morse Vector Field for a Lower Star Filtration Vineyard
- Michigan State University（密歇根州立大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出 colex 向量场以计算下星过滤的持续同调，支持顶点函数修改时的快速更新，并验证其运行时间与标准矩阵化简相当。
中文摘要 AI 辅助
本文构造了一个与单纯复形上下星过滤相关的全序相容的无环离散向量场，称为 colex 向量场。我们证明 colex 向量场诱导一个过滤的无环向量场，其产生的 Morse 复形计算了底层单纯复形上下星过滤的持续同调。我们证明，当通过相邻顶点交换修改诱导下星过滤的顶点函数时，colex 向量场可以快速重新计算。我们提供了一个框架，用于存储和计算单纯复形中胞腔之间的路径数量，并给出一种在 colex 向量场变化时快速更新这些值的方法。最后，我们为所展示的思想提供了公开可用的概念验证代码。将其应用于持续同调变换时，我们表明其运行时间与更标准的矩阵化简方法相当。
英文摘要
In this paper, we provide a construction of an acyclic discrete vector field that is compatible with a total order associated with a lower star filtration on a simplicial complex, called the colex vector field. We show that the colex vector field induces a filtered acyclic vector field, whose resulting Morse complex computes the persistent homology of the lower star filtration on the underlying simplicial complex. We show that the colex vector field can be recomputed quickly when the vertex function that induces the lower star filtration is modified via order-adjacent vertex swaps. We provide a framework for storing and computing the number of paths between cells in the simplicial complex, as well as a method to update these values quickly when the colex vector field changes. Finally, we provide publicly available proof-of-concept code for the ideas shown. When applying it to the Persistent Homology Transform, we show that its runtime is comparable to a more standard matrix reduction approach.
