We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point $q$, we focus our attention to a ball centered at $q$ whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the $q$-ball, and a relative variant where the homology of the $q$-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not $k$-critical for any finite $k$. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for $2$-parameter persistence (which usually assume $k$-criticality for some finite $k$) carry over to the $\infty$-critical case.
翻译:我们提议延长传统球联盟对持久性同族体的过滤: 确定一个点, 我们把注意力集中在一个以美元为中心的球上, 其半径由二等参数控制。 我们讨论一个绝对变量, 即该联盟仅限于美元球, 以及一个相对变量, 即考虑美元球相对于其边界的同质性。 有趣的是, 这些自然构造导致一个对任何限定美元来说都不是美元关键值的双相形简化复合体。 然而, 我们证明这些双曲线可以精确和高效地计算, 我们利用CGAL图书馆提供一种原型执行。 我们还争论说, 最近为2美元单数的持久性( 通常假定某些限定美元为美元- 临界值 ) 的算法进步, 将转到美元- 美元- 关键值的案例中 。</s>