Stratifying Multiparameter Persistent Homology
Stratifying Multiparameter Persistent Homology
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DOI：
10.1137/18m1224350
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发表时间：
2019-01-01
影响因子：
1.2
通讯作者：
Tillmann, Ulrike
中科院分区：
文献类型：
--
作者：
Harrington, Heather A.;Otter, Nina;Tillmann, Ulrike
拓扑数据分析中的一个基本工具是持久同源性，它允许以鲁棒的方式从复杂数据集中提取信息。持久同调将主理想整环上的模分配给从数据中获得的单参数空间族。在应用中，数据往往依赖于几个参数，在这种情况下，人们感兴趣的是研究与数据相关联的多参数空间族的持久同调。虽然单参数族的持久同调理论已经很好理解，但多参数族的情况更加微妙。在Carlsson和Zomorodian的基础上，我们在多阶代数的背景下，提出了多阶Hilbert级数、多阶相伴素数和局部上同调作为研究多参数持久同调的不变量.多阶相伴素数提供了多阶模不为零的区域的分层，而多阶希尔伯特级数和局部上同调给出了不同层上支持的模的分量的大小的度量。这些不变量在适当的意义上推广了单参数情况下的不变量。
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications, data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian, we recast the problem in the setting of multigraded algebra, and we propose rnultigraded Hilbert series, rnultigraded associated primes, and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.