Spatiotemporal Persistent Homology for Dynamic Metric Spaces
Spatiotemporal Persistent Homology for Dynamic Metric Spaces
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DOI：
10.1007/s00454-019-00168-w
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发表时间：
2020-02-02
影响因子：
0.8
通讯作者：
Memoli, Facundo
中科院分区：
文献类型：
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作者：
Kim, Woojin;Memoli, Facundo
在拓扑数据分析（TDA）框架内表征时间不断发展的数据的动力学已经吸引了越来越多的关注。随时不断发展的数据的流行实例包括人类领域中动物和社交网络中的植入/蜂群行为。这种集体行为的自然数学模型是动态点云，或更通常是动态度量空间（DMS）。在本文中，我们将（静态）度量空间的RIPS过滤稳定性结果扩展到DMS的设置。我们通过设计DMS的某个三参数“时空”过滤来做到这一点。将同源性函数应用于此过滤会产生从DMS得出的多维持久模块。我们表明，在Gromov-Hausdorff距离的合适概括下，该多维模块在允许所有DMS集合的测量中都具有稳定性。另一方面，人们通常认为，比较两个多维持久模块会导致棘手的计算问题。为了对DMS进行实际比较，我们将重点放在级别不变性或从DMS派生的多维持久性模块的维度函数上。我们专门建议利用某个公制D比较这些不变性：在我们的工作中，该d是（1）Patel对侵蚀距离的一定概括，或者（2）众所周知的交织距离的专门版本。无论哪种情况，都可以在多项式时间内计算公制。
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter "spatiotemporal" filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov-Hausdorff distance which permits metrization of the collection of all DMSs. On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well-known interleaving distance. In either case, the metric d can be computed in polynomial time.