Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many algorithms calls for a preprocessing step to reduce the input size. In general, a minimal presentation is the smallest possible representation of a persistence module. Lesnick and Wright proposed recently an algorithm (the LW-algorithm) for computing minimal presentations based on matrix reduction. In this work, we propose, implement and benchmark several improvements over the LW-algorithm. Most notably, we propose the use of priority queues to avoid extensive scanning of the matrix columns, which constitutes the computational bottleneck in the LW-algorithm, and we combine their algorithm with ideas from the multi-parameter chunk algorithm by Fugacci and Kerber. Our extensive experiments show that our algorithm outperforms the LW-algorithm and computes the minimal presentation for data sets with millions of simplices within a few seconds. Our software is publicly available.

中文翻译：

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双级持久性模块的快速最小化演示

多参数持久同源性是拓扑数据分析的一个新分支。在这个领域，数据集是通过关于两个或更多尺度参数的同源性镜头来研究的。许多算法的高计算成本需要一个预处理步骤来减少输入大小。通常，最小表示是持久性模块的最小可能表示。Lesnick 和 Wright 最近提出了一种基于矩阵约简计算最小表示的算法（LW 算法）。在这项工作中，我们针对 LW 算法提出、实施和基准测试了一些改进。最值得注意的是，我们建议使用优先级队列来避免对矩阵列进行大量扫描，这构成了 LW 算法中的计算瓶颈，我们将他们的算法与 Fugacci 和 Kerber 的多参数块算法的想法相结合。我们的大量实验表明，我们的算法优于 LW 算法，并在几秒钟内计算出具有数百万个单纯形的数据集的最小表示。我们的软件是公开可用的。