The Universal $\ell^p$-Metric on Merge Trees

Adapting a definition given by Bjerkevik and Lesnick [9] for multiparameter persistence modules, we introduce an `-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p ∈ [1, ∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p = ∞ case, this gives a novel proof of universality for the interleaving distance on merge trees. 2012 ACM Subject Classification Mathematics of computing → Algebraic topology; Theory of computation → Unsupervised learning and clustering; Theory of computation → Computational geometry

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