Meta-Diagrams for 2-Parameter Persistence
Meta-Diagrams for 2-Parameter Persistence
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DOI：
10.48550/arxiv.2303.08270
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发表时间：
2023-03
期刊：
影响因子：
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通讯作者：
Nathaniel Clause;T. Dey;Facundo M'emoli;Bei Wang
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文献类型：
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作者：
Nathaniel Clause;T. Dey;Facundo M'emoli;Bei Wang
我们首先为2参数持久模块介绍了元级的概念，该模块的不变式捕获了模块1D切片之间的形态图像背后的信息。然后，我们将2参数持久模块的元数据定义为m \“ {o} bius bius倒置，从而获得了从签名的1参数持久模块中获取值的函数。我们显示了该函数。元级和元数据包含等同于等级不变的信息，并且该对等相等。用于计算2参数模块的元级和元数据，该算法是由$ o o（n^3）$ o（n^3）$ o（n^3）$ o（n^3）$ o（$ n $ n $ n $ n $ n $ n $ n $ n $ n）索引的算法。签名的条形码，具有$ O（n^4）$运行时。 $ m $的分解为$ o（n^4）$到$ o（n^3）$。相对于交织距离，元级和元数据是稳定的。最后，可以以直观的方式将元数据视为图表的持久图，该图在1参数设置中概括了良好的持久图。
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.