论文标题
一个子键的理论
A Theory of Sub-Barcodes
论文作者
论文摘要
从鲍尔（Bauer）和莱斯尼克（Lesnick）的工作来看，众所周知，没有函数来自点式有限维持续模块的类别，即条形码和重叠匹配的类别。在这项工作中，我们介绍了子键，并表明从持久性模块同构的因素化类别中有一个函子，到由子键相关关系订购的条形码的poset。亚键盘和因素化为瓶颈匹配和交织提供了一种更宽松的替代方法，可以在拓扑数据分析中自然出现的许多环境中提供强大的保证。子键的主要用途是在没有交织的情况下对未知条形码提出强烈的主张。例如，只有一个未知的实价函数$ f $的上限和下限$ g \ geq f \ geq \ ell $，可以单独从$ \ ell $和$ g $构建与$ f $相关的子标准。我们提出了一个亚键模的理论，并观察到函子类别中的子对象从间隔到匹配的类别自然对应于亚键模。
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.
