${\mathrm{\ell }}^p$-Distances on Multiparameter Persistence Modules

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 $p\in [1,\mathrm{\infty }]$, we in fact introduce two such generalizations $d_{\mathcal{I}}^p$ and $d_{\mathcal{M}}^p$, such that $d_{\mathcal{I}}^{\mathrm{\infty }}$ equals the interleaving distance and $d_{\mathcal{M}}^{\mathrm{\infty }}$ equals the matching distance. We show that on 1- or 2-parameter persistence modules over prime fields, $d_{\mathcal{I}}^p$ is the universal (i.e., largest) metric satisfying a natural stability property; this extends a stability theorem 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. We also show that $d_{\mathcal{M}}^p\le d_{\mathcal{I}}^p$ for all $p\in [1,\infty]$,extendinganobservationofLandiinthe$p=\infty$ case. We observe that on 2-parameter persistence modules, $d_{\mathcal{M}}^p$ can be efficiently approximated. In a forthcoming companion paper, we apply some of these results to study the stability of ($2$-parameter) multicover persistent homology.