From the work of Bauer and Lesnick, it is known that there is no functor from the category of pointwise finite-dimensional persistence modules to the category of barcodes and overlap matchings. In this work, we introduce sub-barcodes and show that there is a functor from the category of factorizations of persistence module homomorphisms to a poset of barcodes ordered by the sub-barcode relation. Sub-barcodes and factorizations provide a looser alternative to bottleneck matchings and interleavings that can give strong guarantees in a number of settings that arise naturally in topological data analysis. The main use of sub-barcodes is to make strong claims about an unknown barcode in the absence of an interleaving. For example, given only upper and lower bounds $g\geq f\geq \ell$ of an unknown real-valued function $f$, a sub-barcode associated with $f$ can be constructed from $\ell$ and $g$ alone. We propose a theory of sub-barcodes and observe that the subobjects in the category of functors from intervals to matchings naturally correspond to sub-barcodes.
翻译:从 Bauer 和 Lesnick 的作品中,已知在条形码和重叠匹配的类别中,没有从点数的有限维持久性模块类别到条形码和重叠匹配类别中的杀菌剂。在这项工作中,我们引入了次条形码,并显示从持久性模块同质性系数类别到由子条形码关系订购的条形码的杀菌剂类别有一个杀菌剂。子条形码和保理法提供了较松的替代物,可以用来替代瓶颈匹配和中间置物,从而在一些自然出现在地形数据分析中的环境里提供强有力的保障。次条形码的主要用途是在没有内分解的情况下对未知的条形码提出有力的索赔。例如,仅考虑到未知真实价值函数的上下界限$g\ge f\\geq\ ell$,与美元相联的子条形码可以用美元和美元单独美元来构建。我们提出了一个子条形码的理论,并观察子条形码的子条形码在自然和次形码之间对应的分级。