Multi-parameter persistent homology naturally arises in applications of persistent topology to data that come with extra information depending on additional parameters, like for example time series data. We introduce the concept of a Vietoris-Rips transformation, a method that reduces the computation of the one-parameter persistent homology of pathwise subcomplexes in multi-filtered flag complexes to the computation of the Vietoris-Rips persistent homology of certain semimetric spaces. The corresponding pathwise persistence barcodes track persistence features of the ambient multi-filtered complex and can in particular be used to recover the rank invariant in multi-parameter persistent homology. We present MuRiT, a scalable algorithm that computes the pathwise persistence barcodes of multi-filtered flag complexes by means of Vietoris-Rips transformations. Moreover, we provide an efficient software implementation of the MuRiT algorithm which resorts to Ripser for the actual computation of Vietoris-Rips persistence barcodes. To demonstrate the applicability of MuRiT to real-world datasets, we establish MuRiT as part of our CoVtRec pipeline for the surveillance of the convergent evolution of the coronavirus SARS-CoV-2 in the current COVID-19 pandemic.
翻译：多参数持久性同质学自然产生于对数据应用的持久性地形学,这些数据中包含取决于额外参数的额外信息,例如时间序列数据。我们引入了越远-里普斯变异的概念,这个方法可以降低多过滤国旗复合体中路径性亚复合体的单参数持久性同质体的计算,以计算某些半径空间的越远-里普斯持久性同质体。相应的路径性持久性条码跟踪环境多过滤复合体的持久性特征,特别是可用于恢复多参数持久性同质体中的异差等级。我们介绍了可缩放的算法MuriT,这是一种可缩放的算法,通过越远-里普斯变异体变异体变异体变异体变异体的计算法。此外,我们提供了高效的 Murit算法软件应用软件,该算法是使用提取器实际计算越远端-里普斯过滤复合复合复合复合复合体结构的条码,用以证明 MuRIT在现实-D-D变异变异体的变异性管中,我们为C-CSAR-C-C-C-C-CRO-C-CHR的变异变变变变变变动中我们建立的CS-CO-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-CS-CS-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-
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