双重 Degree Cech 双过滤
The Dual Degree Cech Bifiltration
AI总结：
本文提出双重 Degree Cech 双过滤，一种度量空间点云的 Prohorov 稳定双复形，证明其与 Measure Dowker 双过滤同伦等价，并利用环境与内蕴构造的交织关系获得稳定性结果。
摘要：
在拓扑数据分析（TDA）中，一个长期存在的挑战是识别噪声数据中潜在的几何结构。一个具有启发性的例子是由图像给出的欧几里得空间中点云的形状。Carlsson等人提出了一种检测点云拓扑特征的方法，即先按密度进行过滤，然后应用持久同调。后来人们发展了更精细的方法，例如 Lesnick 和 Wright 的 degree Rips 复形以及多重覆盖双过滤。在本文中，我们引入了双重 Degree Cech 双过滤，它是度量空间中点云的一个 Prohorov 稳定的双复形，以点云本身作为顶点集。它与 Hellmer 和 Spaliński 的 Measure Dowker 双过滤具有相同的同伦类型，但具有不同的顶点集。双重 Degree Cech 双过滤既可以通过环境方式构造，也可以通过内蕴方式构造。内蕴双重 Degree Cech 双过滤在距离参数上与环境双重 Degree Cech 双过滤呈 $(1,2)$-交织。这种交织可用于利用内蕴定义的双重 Degree Cech 双过滤的稳定性结果。这一稳定性结果最近出现在 Hellmer 和 Spaliński 的工作中。
英文摘要：
In topological data analysis (TDA), a longstanding challenge is to recognize underlying geometric structures in noisy data. One motivating examples is the shape of a point cloud in Euclidean space given by image. Carlsson et al. proposed a method to detect topological features in point clouds by first filtering by density and then applying persistent homology. Later more refined methods have been developed, such as the degree Rips complex of Lesnick and Wright and the multicover bifiltration. In this paper we introduce the dual Degree Cech bifiltration, a Prohorov stable bicomplex of a point cloud in a metric space with the point cloud itself as vertex set. It is of the same homotopy type as the Measure Dowker bifiltration of Hellmer and Spaliński but it has a different vertex set. The dual Degree Cech bifiltration can be constructed both in an ambient and an intrinsic way. The intrinsic dual Degree Cech bifiltration is a $(1,2)$-intereaved with the ambent dual Degree Cech bifiltration in the distance parameter. This interleaving can be used to leverage a stability result for the intrinsically defined dual Degree Cech bifiltration. This stability result recently occured in work by Hellmer and Spaliński.
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