论文标题
持续扩展和类似条：持久性条形码之间的数据诱导的关系
Persistent Extension and Analogous Bars: Data-Induced Relations Between Persistence Barcodes
论文作者
论文摘要
拓扑数据分析中的一个核心挑战是对条形码的解释。解释同源性类别的古典代数论方法是为同源性带有我们理解的语义的空间建造图，然后吸引功能。但是，我们通常在真实数据中缺乏这样的地图。取而代之的是，我们必须依靠我们对系统的观察和参考之间的交叉差异度量。在本文中，我们开发了一对计算同源代数方法，用于关联持续的同源性类别和条形码：持续的延伸，该方法枚举了在同一顶点集合上建立的两个复合物的循环之间的潜在关系，以及类似条的方法，该方法利用了持续的延伸和与交叉式关系的建立关系，从而提供了持续的延伸和见证人。我们提供了这些方法的实现，并证明了它们在比较来自同一度量空间的两个样本之间的循环中的用途，并确定在聚类和降低尺寸降低下是否维持或破坏了拓扑。
A central challenge in topological data analysis is the interpretation of barcodes. The classical algebraic-topological approach to interpreting homology classes is to build maps to spaces whose homology carries semantics we understand and then to appeal to functoriality. However, we often lack such maps in real data; instead, we must rely on a cross-dissimilarity measure between our observations of a system and a reference. In this paper, we develop a pair of computational homological algebra approaches for relating persistent homology classes and barcodes: persistent extension, which enumerates potential relations between cycles from two complexes built on the same vertex set, and the method of analogous bars, which utilizes persistent extension and the witness complex built from a cross-dissimilarity measure to provide relations across systems. We provide an implementation of these methods and demonstrate their use in comparing cycles between two samples from the same metric space and determining whether topology is maintained or destroyed under clustering and dimensionality reduction.
