In this paper, we establish explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance and the Wasserstein distance between two zigzag persistence modules restricted to an interval is always bounded above by the distance between the unrestricted versions. While this result is not surprising, it could have potential practical implications. We give two related applications for metric graph distances, as well as an extension for the matching distance between multi-parameter persistence modules.
Published: September 21, 2022
Citation
Gasparovic E., M. Gommel, E. Purvine, R. Sazdanovic, B. Wang, Y. Wang, and L. Ziegelmeier. 2022.Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules. In Research in Computational Topology 2. Association for Women in Mathematics Series, edited by E. Gasparovic, V. Robins and K Turner. 63–76. Cham:Springer.PNNL-SA-159311.doi:10.1007/978-3-030-95519-9_3