Abstract
In topological data analysis (TDA), persistence diagrams (PDs) have been a successful tool. To compare them, Wasserstein and bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for PDs as a generalization of the bottleneck distance. These metrics satisfy a stability result and can be used to bound Wasserstein metrics from above and from below. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.
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