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.
翻译:地形数据分析中的一项中心挑战是解释条形码。典型的代数-地形学解释同系类的方法是绘制地图,使同族体带有我们所理解的语义学,然后又能吸引交替性。然而,我们往往在真实数据中缺乏这种地图;相反,我们必须依靠一种不同测量尺度,测量一个系统和一个参考的系统。在本文件中,我们为与持久性同系类和条形码相联系而开发了一套计算式同系数代数方法:持续的扩展,其中列出了同一顶点组建立的两个复合体之间周期的潜在关系,以及类似条形法的方法,其中利用了持续延伸和从跨异度测量中构建的证人复杂体,以提供跨系统的关系。我们提供这些方法的实施,并表明它们用于比较同一计量空间的两个样本之间的周期,并确定在集群和维度减少下是否保持或销毁了表理学。