A rank-based distance for interval modules and Wasserstein stability of persistence landscapes

Abstract

Barcodes and persistence diagrams have a canonical one-parameter family of distances called Wasserstein distances. These distances depend on a choice of distance for interval modules. The usual choices are all Lipschitz equivalent. We observe that they may be written in terms of the dimension function. We instead define a distance for interval modules using the rank function. The rank function contains the information of dimension function and unlike the dimension function, also contains information on persistence. Using this distance, we show that persistence landscapes are stable with respect to the 1-Wasserstein distance. This gives us a 1-Lipschitz embedding of persistence diagrams with the 1-Wasserstein distance into a Banach space given by an $L^1$ function space, and also provides a lower bound for the 1-Wasserstein distance. In addition, we give a stability theorem for the mapping from chains to barcodes. This result leads to a heuristic for truncating infinite bars.

17 pages, 4 figures, fixed typos