Stable Signatures for Dynamic Metric Spaces via Zigzag Persistent Homology
Stable Signatures for Dynamic Metric Spaces via Zigzag Persistent Homology
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通过 Zigzag 持久同调实现动态度量空间的稳定签名
DOI：
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发表时间：
2017
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通讯作者：
F. Mémoli
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作者：
Woojin Kim;F. Mémoli
在研究动物的群集行为时，人们感兴趣的是量化和比较动物在不同群体中聚集和分离所引起的群集动态。基于此，我们研究了基于时间相关度量数据的持久同调摘要的获取问题。给定一个有限动态度量空间（DMS），我们将Rips简单复结构（具有固定的尺度参数）应用于该有限动态度量空间，构造了由之字形简单过滤引起的之字形简单过滤。在传递与域系数的$0$-th同源性之后，我们获得了一个之字形持久性模块，并且根据标准结果，我们又从这个之字形持久性模块获得了一个持久性图或条形码。我们证明了这些条形码在输入DMS的扰动下是稳定的。为了形式化扰动的概念，我们在dms之间引入了一个合适的距离，然后我们证明了任意两个dms之间的这个距离的值承认与两个输入dms相关联的Rips条形码之间的瓶颈距离的下界。这个下界可以在多项式时间内从DMS输入计算出来。
When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent metric data. Given a finite dynamic metric space (DMS), we construct the zigzag simplicial filtration arising from applying the Rips simplicial complex construction (with a fixed scale parameter) to this finite DMS. Upon passing to $0$-th homology with field coefficients, we obtain a zigzag persistence module and, based on standard results, we in turn obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs. This lower bound can be computed in polynomial time from the DMS inputs.
Along the way, we propose a summarization of dynamic metric spaces that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering.
In order to elucidate the relationship between our distance between two dynamic metric spaces and the bottleneck distance between their Rips zigzag barcodes, we exploit recent advances in the stability of zigzag persistence (due to Botnan and Lesnick). By providing explicit constructions, we prove that for each integer $kgeq 1$ there exist pairs of DMSs at finite interleaving distance whose $k$-th persistent homology barcodes are at infinite barcode distance.