We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the M\"{o}bius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module $M$ indexed by a bifiltration of $n$ simplices in $O(n^3)$ time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has $O(n^4)$ runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of $M$ from $O(n^4)$ to $O(n^3)$. In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting.

翻译：我们首先引入2-参数持久性模的元秩概念，该不变量捕捉了模的1维切片之间态射像背后的信息。进而将2-参数持久性模的元图定义为元秩的Möbius反演，得到一个取值为带符号1-参数持久性模的函数。我们证明元秩和元图包含的信息等价于秩不变量和带符号条形码。这一等价性带来了计算优势，我们提出一种算法，可在$O(n^3)$时间内计算由$n$个单纯形双过滤索引的2-参数模$M$的元秩和元图。这改进了现有带符号条形码算法（时间复杂度为$O(n^4)$），同时将$M$的秩分解中矩形数量的上界从$O(n^4)$改进为$O(n^3)$。此外，我们定义了元秩之间和元图之间的侵蚀距离概念，并证明在这些距离下，元秩和元图关于交错距离具有稳定性。最后，元图可直观地可视化为图的持久性图，推广了1-参数设定中广为人知的持久性图。