论文标题
跟踪各种交织
Tracking the variety of interleavings
论文作者
论文摘要
在拓扑数据分析中，使用持久模块将有限数据集的合法拓扑特征与噪声区分开。持久模块之间的交织在分析中突出。可以表明，对于$ε$阳性，两个持久模块$ m $和$ n $之间的$ε$ - 互插图的收集具有仿射品种的结构，因此，与非空品种相对应的$ε$的最小值是相互交织的距离。考虑到这一点，很自然地想知道这种品种如何随$ε$的价值而变化，以及从它们的品种的知识中可以看出有关$ m $和$ n $的信息。 在本文中，我们专注于特殊情况，其中$ m $和$ n $是间隔模块。在这种情况下，我们对品种的所有可能进度进行了分类，并确定进度中存在有关$ m $和$ n $的信息。
In topological data analysis persistence modules are used to distinguish the legitimate topological features of a finite data set from noise. Interleavings between persistence modules feature prominantly in the analysis. One can show that for $ε$ positive, the collection of $ε$-interleavings between two persistence modules $M$ and $N$ has the structure of an affine variety, Thus, the smallest value of $ε$ corresponding to a nonempty variety is the interleaving distance. With this in mind, it is natural to wonder how this variety changes with the value of $ε$, and what information about $M$ and $N$ can be seen from just the knowledge of their varieties. In this paper, we focus on the special case where $M$ and $N$ are interval modules. In this situation we classify all possible progressions of varieties, and determine what information about $M$ and $N$ is present in the progression.
