Abstract When quantifying the topological properties of a metric dataset using persistent homology, the first step is to produce a simplicial filtration on the data. The Cech and Vietoris-Rips filtrations are two of the workhorses of persistent homology, but over time, various other filtrations which capture different properties of data or have different computational burdens have been introduced. Towards a program of characterizing all the possible simplicial filtrations on a metric dataset, we introduce and develop the framework of valuation-induced stable filtration functors. This framework is based on the concept of curvature sets due to Gromov, and encapsulates the Vietoris-Rips and various other filtrations while simultaneously providing a model for generating families of novel filtration functors that capture diverse features present in datasets. We further extend this foundation by incorporating the notion of basepoint-dependent filtration functors and proving the associated functoriality and stability properties. This rich theoretical framework provides a unifying language for various extant simplicial filtrations, and is also a mechanism for generating arbitrarily large families of novel filtration functors with control over basepoint dependence/independence as well as the locality of the filtration. We exemplify our constructions on both toy datasets and on 3D shapes from a publicly available shape database. Our paper is accompanied by a Matlab software package incorporating an interactive platform for visualizing and testing new filtrations on datasets.

中文翻译：

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新的稳定单纯过滤函子系列

摘要 当使用持久同源性量化度量数据集的拓扑属性时，第一步是对数据进行简单过滤。Cech 和 Vietoris-Rips 过滤是持久同源性的两个主力，但随着时间的推移，已经引入了各种其他过滤，它们捕获不同的数据属性或具有不同的计算负担。为了在度量数据集上表征所有可能的简单过滤的程序，我们引入并开发了估值诱导的稳定过滤函子的框架。该框架基于 Gromov 的曲率集概念，并封装了 Vietoris-Rips 和各种其他过滤，同时提供了一个模型来生成新的过滤函子家族，这些函子可以捕获数据集中存在的各种特征。我们通过结合基点相关过滤函子的概念并证明相关的函子性和稳定性属性来进一步扩展这个基础。这个丰富的理论框架为各种现存的简单过滤提供了统一的语言，也是一种生成任意大家族的新型过滤函子的机制，可以控制基点相关性/独立性以及过滤的局部性。我们举例说明了我们在玩具数据集和来自公开可用形状数据库的 3D 形状上的构造。