Given a filtration of simplicial complexes, one usually applies persistent homology and summarizes the results in barcodes. Then, in order to extract statistical information from these barcodes, one needs to compute statistical indicators over the bars of the barcode. An issue with this approach is that usually infinite bars must be deleted or cut to finite ones; however, so far there is no consensus on how to perform this procedure. In this work we propose for the first time a systematic way to analyze barcodes through the use of statistical indicators. Our approach is based on the minimization of a divergence measure that generalizes the standard Wasserstein or bottleneck distance to a new asymmetric distance-like function that we introduce and which is interesting on its own. In particular, we analyze the topology induced by this divergence and the stability of known vectorizations with respect to this topology.

翻译：给定一个单纯复形滤链，通常应用持久同调并将结果总结为条形码。为了从这些条形码中提取统计信息，需要计算条形码中各条带的统计指标。该方法存在一个问题：通常必须删除无限条带或将其截断为有限条带；然而，目前对于如何执行此操作尚未达成共识。在本工作中，我们首次提出通过统计指标分析条形码的系统方法。我们的方法基于最小化一种散度度量，该度量将标准Wasserstein距离或瓶颈距离推广至一种新的非对称类距离函数——我们引入的这一函数本身即具有研究价值。特别地，我们分析了该散度诱导的拓扑结构，以及已知向量化方法相对于该拓扑的稳定性。