The rank invariant (RI), one of the best known invariants of persistence modules $M$ over a given poset $P$, is defined as the map sending each comparable pair $p\leq q$ in $P$ to the rank of the linear map $M(p\leq q)$.The recently introduced notion of generalized rank invariant (GRI) acquires more discriminating power than the RI at the expense of enlarging the domain of RI to the set Int$(P)$ of intervals of $P$ or to the even larger set Con$(P)$ of connected subposets of $P$. Given that the size of Int$(P)$ and Con$(P)$ can be much larger than the domain of the RI, restricting the domain of the GRI to smaller, more manageable subcollections $\mathcal{I}$ would be desirable to reduce the total cost of computing the GRI. This work studies the tension between computational efficiency and strength when restricting the domain of the GRI to different choices of $\mathcal{I}$. In particular, we prove that in terms of discriminating power, the GRI over restricted collections $\mathcal{I}$ faithfully interpolates between the RI and the GRI over Int$(P)$. We also establish that for suitable collections $\mathcal{I}$, the GRI over $\mathcal{I}$ is stable. Finally, we introduce the notion of Zigzag-path-Indexed Barcode (ZIB) for persistence modules $M$ over a finite 2d-grid, which is a function that sends each zigzag path $\Gamma$ in the 2d-grid to the barcode of the restriction of $M$ to $\Gamma$. Since the RI is equivalent to the fibered barcode (i.e. the ZIB induced by monotone paths), the ZIB is a natural refinement of the RI. Motivated by a recent finding that zigzag persistence can be used to compute the GRI of $M$, we compare the discriminating power of the ZIB with that of the GRI. Clarifying the connection between the GRI and the ZIB is necessary to understand to what extent zigzag persistence algorithms can be exploited for computing the GRI.

翻译：秩不变量（RI）是给定偏序集 $P$ 上持久模 $M$ 最著名的不变量之一，定义为将 $P$ 中每个可比较对 $p\leq q$ 映射为线性映射 $M(p\leq q)$ 秩的函数。最近引入的广义秩不变量（GRI）通过将RI的定义域扩展至 $P$ 的区间集 Int$(P)$ 甚至更大的连通子偏序集集 Con$(P)$，获得了比RI更强的判别能力。鉴于 Int$(P)$ 和 Con$(P)$ 的规模可能远大于RI的定义域，将GRI的定义域限制为更小、更易处理的子集合 $\mathcal{I}$ 将有助于降低计算GRI的总成本。本文研究限制GRI定义域至不同 $\mathcal{I}$ 时计算效率与判别能力之间的权衡。特别地，我们证明在判别能力方面，限制集合 $\mathcal{I}$ 上的GRI能在RI与Int$(P)$上的GRI之间实现忠实插值。同时证明对于合适的集合 $\mathcal{I}$，$\mathcal{I}$ 上的GRI具有稳定性。最后，我们引入有限二维网格上持久模 $M$ 的之字形路径索引条形码（ZIB）概念，该函数将二维网格中每条之字形路径 $\Gamma$ 映射为 $M$ 限制在 $\Gamma$ 上的条形码。由于RI等价于纤维化条形码（即单调路径诱导的ZIB），ZIB是RI的自然细化。受近期发现的之字形持久可用于计算 $M$ 的GRI这一成果启发，我们比较了ZIB与GRI的判别能力。阐明GRI与ZIB之间的联系，对于理解之字形持久算法在何种程度上可用于计算GRI至关重要。