Acyclic Partial Matchings for Multidimensional Persistence: Algorithm and Combinatorial Interpretation
Acyclic Partial Matchings for Multidimensional Persistence: Algorithm and Combinatorial Interpretation
复制标题
多维持久性的非循环部分匹配：算法和组合解释
DOI：
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发表时间：
2019
影响因子：
2
通讯作者：
Filippo Masoni
中科院分区：
文献类型：
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作者：
M. Allili;T. Kaczynski;C. Landi;Filippo Masoni
给定一个单纯复形及其顶点上的向量值函数，我们提出了与给定函数兼容的复形单元上的非循环部分匹配的算法构造。这意味着该构造可用于构建简化的过滤复合体，其具有与由函数的子级别集过滤的原始复合体相同的多维持久同源性。证明了算法的正确性，并分析了算法的复杂度。本文首次介绍了基于多维离散莫尔斯函数概念的算法的组合解释。数值实验表明，该算法可大幅减少细胞数量。
Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex compatible with the given function. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. The correctness of the algorithm is proved, and its complexity is analyzed. A combinatorial interpretation of our algorithm based on the concept of a multidimensional discrete Morse function is introduced for the first time in this paper. Numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.