计算流形类复形上水平集之字形的最优持续环
Computing Optimal Persistent Cycles for Levelset Zigzag on Manifold-like Complexes
AI总结：
针对PL函数诱导的持续同调，提出水平集持续环序列以刻画拓扑特征演化，并给出弱伪流形上计算最优环序列的多项式时间算法，实验证明其质量良好。
摘要：
在标准持续同调中，一个与持续区间（条）同生共死的持续环将该条与一个具体的拓扑代表相关联，这提供了从条形码有效地导航回拓扑空间的方法。在可能存在的众多环中，最优持续环由于具有有保证的质量而带来进一步的信息。然而，拓扑特征通常在条的整个生命周期中会经历变化，而单个持续环可能无法捕捉到这些变化。因此，对于由PL函数诱导的持续同调，我们提出了水平集持续环，它由一系列环组成，描绘了同调特征从出生到死亡的演化。我们的定义基于水平集之字形持续，它涉及四种类型的持续区间，而标准持续只有两种类型。对于这四种类型中的每一种，我们提出了一种多项式时间算法，用于计算所谓的弱$(p+1)$-伪流形上的最优水平集持续$p$-环序列。鉴于同调的最优环问题在一般情况下是NP难的，我们的结果在实践中很有用，因为弱伪流形确实出现在应用中。我们的算法借鉴了一种将最优环与图中的最小割联系起来的思想，该思想先前已被用于标准持续环。请注意，水平集之字形对该方法提出了非平凡的挑战，因为在这种情况下需要计算一个最优环序列而不是单个环。我们展示了一些经验证据，表明我们实现的软件产生的最优环具有良好的质量。
英文摘要：
In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the topological space. Among the possibly many, optimal persistent cycles bring forth further information due to having guaranteed quality. However, topological features usually go through variations in the lifecycle of a bar which a single persistent cycle may not capture. Hence, for persistent homology induced from PL functions, we propose levelset persistent cycles consisting of a sequence of cycles that depict the evolution of homological features from birth to death. Our definition is based on levelset zigzag persistence which involves four types of persistence intervals as opposed to the two types in standard persistence. For each of the four types, we present a polynomial-time algorithm computing an optimal sequence of levelset persistent $p$-cycles for the so-called weak $(p+1)$-pseudomanifolds. Given that optimal cycle problems for homology are NP-hard in general, our results are useful in practice because weak pseudomanifolds do appear in applications. Our algorithms draw upon an idea of relating optimal cycles to min-cuts in a graph that was exploited earlier for standard persistent cycles. Notice that levelset zigzag poses non-trivial challenges for the approach because a sequence of optimal cycles instead of a single one needs to be computed in this case. We show some empirical evidence that optimal cycles produced by our implemented software have nice quality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
