论文标题
杯产品持久性及其有效的计算
Cup Product Persistence and Its Efficient Computation
论文作者
论文摘要
众所周知，同源圈的结构比同源群具有更丰富的结构。但是，直到最近，在持久性设置中的使用限于加速条形码计算。最近引入的一些不变式，即持续的杯状，持久的杯子模块和持久的steenrod模块，在某种程度上填补了这一空白。当添加到标准持久性条形码中时，它们会导致不变性比标准持续条形码更具歧视性。在这项工作中，我们设计了一个$ O（d n^4）$算法用于计算所有$ k \ in \ {2，\ dots，d \} $的持续$ k $ cup模块，其中$ d $表示过滤式复合体的维度，$ n $表示其大小。此外，我们注意到，由于可以作为计算的副产品获得持续的杯状长度，因此这会导致更快的算法以$ d> 3 $计算。最后，我们引入了一种新的稳定不变性杯产品的分区模块，该模块比持续的$ k $ cup模块更具歧视性，并设计了$ o（c（d）n^4）$计算它的算法，其中$ c（d）$在$ d $中是subppectient的。
It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an $O(d n^4)$ algorithm for computing the persistent $k$-cup modules for all $k \in \{2, \dots, d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for $d>3$. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent $k$-cup modules and devise an $O(c(d)n^4)$ algorithm for computing it, where $c(d)$ is subexponential in $d$.
