The Theory of Multidimensional Persistence
The Theory of Multidimensional Persistence
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DOI：
10.1007/s00454-009-9176-0
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发表时间：
2009-07-01
影响因子：
0.8
通讯作者：
Zomorodian, Afra
中科院分区：
文献类型：
--
作者：
Carlsson, Gunnar;Zomorodian, Afra
持久同调以完全离散不变量的形式捕获过滤的拓扑——一个递增空间的单参数族。这个不变量是一个多区间集，表示过滤中拓扑实体的生存期。在拓扑学的许多应用中，我们需要研究多重滤波：一组沿多个几何维度参数化的空间。在本文中，我们证明了多维持久性不存在类似的完全离散不变量。相反，我们提出了秩不变量，一个离散不变量，用于多重滤波中Betti数的鲁棒估计，并证明了它在一维上的完备性。
Persistent homology captures the topology of a filtration-a one-parameter family of increasing spaces-in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.