论文标题
从样品到持续分层的同喻类型
From Samples to Persistent Stratified Homotopy Types
论文作者
论文摘要
在应用中，奇异空间的自然发生导致最近在分层框架中对执行拓扑数据分析（TDA）进行了研究。在许多应用中，没有关于哪些要点被视为单数或常规的先验信息。为此，我们描述了一个完全可实现的过程，该过程可证明从足够接近的非分层样品中近似大型两轴惠特尼分层空间的分层。此外，在这项工作中，我们建立了从具有两个层的样品获得的持续分层同拷贝类型的概念。类似于TDA中未分类的应用，这些应用依赖于（持续的）同质拷贝类型的一系列方便属性，我们表明，我们的持续分层同置类型的表现与非分离的对应物的表现很大，并且表现出许多属性（例如稳定性和稳定性结果，例如用于tda中的稳定性和分离结果）。总的来说，我们的结果结合了一个抽样定理，以保证（持续）分层同拷贝类型的（近似）推断的（近似）的定理类型的足够规则的二层惠特尼分层空间。
The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a stratified framework. In many applications, there is no a priori information on what points should be regarded as singular or regular. For this purpose we describe a fully implementable process that provably approximates the stratification for a large class of two-strata Whitney stratified spaces from sufficiently close non-stratified samples. Additionally, in this work, we establish a notion of persistent stratified homotopy type obtained from a sample with two strata. In analogy to the non-stratified applications in TDA which rely on a series of convenient properties of (persistent) homotopy types of sufficiently regular spaces, we show that our persistent stratified homotopy type behaves much like its non-stratified counterpart and exhibits many properties (such as stability, and inference results) necessary for an application in TDA. In total, our results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types of sufficiently regular two-strata Whitney stratified spaces.
