论文标题
关于零维多组分持久性的复杂性
On the complexity of zero-dimensional multiparameter persistence
论文作者
论文摘要
多参数持久性是众所周知的持续同源性的自然扩展，引起了很多兴趣。但是，存在主要的理论障碍，阻止了这种有前途的理论的全面发展。 在本文中，我们考虑了零维的多参数持久性的有趣特殊情况，可以被视为多参数聚类的一种形式。特别是，我们考虑了有限生成的过滤拓扑空间的零维同源性的多参数持续模块。在某些假设下，我们表征了此类模块并研究它们的分解。特别是，我们确定了分解并可以扩展为零维多参数持续模块的自然表示形式。 我们对这组表示形式的研究得出的结论是，尽管受到限制，但在此组中仍然有许多类别的不可传剂。
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules. Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.
