论文标题
普遍的等级不变：Möbius的不可逆转，歧视力量和与其他不变的连接
The Generalized Rank Invariant: Möbius invertibility, Discriminating Power, and Connection to Other Invariants
论文作者
论文摘要
除了固有的计算挑战外，在多参数持续的同源性中缺乏量化“持久性”的规范方法仍然是其应用的障碍。 多参数持久同源性持久性持久性的最著名量化之一是等级不变式，最近通过自然扩展其域而发展成为广义级别不变（GRI）。此扩展使我们能够量化与等级不变的索引POSET中更广泛区域的持久性。但是，GRI域的大小通常是可怕的，因此希望将其域限制为用于计算目的的更易于管理的子集。有关该领域限制的最重要的问题是：（1）如何限制GRI的域而不会丢失任何信息？ （2）我们什么时候可以更紧凑地将GRI编码为“持续图”？ （3）随着域的限制数量而变化，计算效率与GRI的区分能力之间的权衡是什么？ （4）在多参数设置中可以从GRI派生的多参数设置中存在哪些代理？为了解决前三个问题，我们通过我们提出的GRI的Möbius倒置概念来概括和公理化持续同源性的经典基本引理。该扩展还将已知结果与（广义）等级不变的莫比乌斯反演理论不变。我们在Möbius的不顾性与其他与持久模块的结构简单性有关的现有概念之间进行了全面比较。 我们通过动机不变的概念解决了第四个问题。我们证明，许多文献中的许多不变性既可以源自GRI，又可以作为动机不变。
In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of persistence for multi-parameter persistent homology is the rank invariant, which has recently evolved into the generalized rank invariant (GRI) by naturally extending its domain. This extension enables us to quantify persistence across a broader range of regions in the indexing poset compared to the rank invariant. However, the size of the domain of the GRI is generally formidable, making it desirable to restrict its domain to a more manageable subset for computational purposes. The foremost questions regarding such a restriction of the domain are: (1) How to restrict, if possible, the domain of the GRI without any loss of information? (2) When can we more compactly encode the GRI as a `persistence diagram'? (3) What is the trade-off between computational efficiency and the discriminating power of the GRI as the amount of the restriction on the domain varies? (4) What proxies exist for persistence diagrams in the multi-parameter setting that can be derived from the GRI? To address the first three questions, we generalize and axiomatize the classic fundamental lemma of persistent homology via the notion of Möbius invertibility of the GRI which we propose. This extension also contextualizes known results regarding the (generalized) rank invariant within the classical theory of Möbius inversion. We conduct a comprehensive comparison between Möbius invertibility and other existing concepts related to the structural simplicity of persistence modules. We address the fourth question through the notion of motivic invariants. We demonstrate that many invariants from the literature can be both derived from the GRI and recast as motivic invariants.
