$\ell^p$-Distances on Multiparameter Persistence Modules

Abstract

Motivated both by theoretical and practical considerations in topological data analysis, we generalize the p-Wasserstein distance on barcodes to multiparameter persistence modules. For each p ∈ [1, ∞], we in fact introduce two such generalizations d p I and d p M , such that d ∞ I equals the interleaving distance and d ∞ M equals the matching distance. We show that d p M ≤ d p I for all p ∈ [1, ∞], extending an observation of Landi in the p = ∞ case. We observe that the distances d p M can be efficiently approximated. Finally, we show that on 1-or 2-parameter persistence modules over prime fields, d p I is the universal (i.e., largest) metric satisfying a natural stability property; our result extends a stability result of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. In a companion paper, we apply some of these results to study the stability of (2-parameter) multicover persistent homology.