$\ell^p$-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 $p\in [1,\infty]$, we in fact introduce two such generalizations $d_{\mathcal I}^p$ and $d_{\mathcal M}^p$, such that $d_{\mathcal I}^\infty$ equals the interleaving distance and $d_{\mathcal M}^\infty$ equals the matching distance. We show that $d_{\mathcal M}^p\leq d_{\mathcal I}^p$ for all $p\in [1,\infty]$, extending an observation of Landi in the $p=\infty$ case. We observe that the distances $d_{\mathcal M}^p$ can be efficiently approximated. Finally, 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; 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.