ℓ^p-Distances on Multiparameter Persistence Modules
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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_ℐ^p and d_ℳ^p, such that d_ℐ^∞ equals the interleaving distance and d_ℳ^∞ equals the matching distance. We show that on 1- or 2-parameter persistence modules over prime fields, d_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_ℳ^p≤ d_ℐ^p for all p∈ [1,∞], extending an observation of Landi in the p=∞ case. We observe that on 2-parameter persistence modules, d_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.
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