MuRiT：通过Vietoris-Rips变换在多过滤标志复合体中有效计算路径持久性条形码
MuRiT: Efficient Computation of Pathwise Persistence Barcodes in Multi-Filtered Flag Complexes via Vietoris-Rips Transformations
摘要：
多参数持久同源性自然出现在持久拓扑对数据的应用中，所述数据具有取决于附加参数的额外信息，例如时间序列数据。我们引入了Vietoris-Rips变换的概念，这种方法将多过滤标志复形中的路径子复形的单参数持久同调的计算减少到某些半度量空间的Vietoris-Rips持久同调的计算。相应的路径持久性条形码跟踪环境多重过滤复合物的持久性特征，并且特别可用于恢复多参数持久性同源性中的秩不变量。我们提出了MuRiT，一种可扩展的算法，通过Vietoris-Rips变换计算多过滤标志复合物的路径持久性条形码。此外，我们提供了一个有效的软件实现的MuRiT算法诉诸Ripser的Vietoris-Rips持久性条形码的实际计算。为了证明MuRiT对真实世界数据集的适用性，我们将MuRiT建立为CoVtRec管道的一部分，用于监测当前COVID-19大流行中冠状病毒SARS-CoV-2的收敛进化。
英文摘要：
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.
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