The Universal ℓ^p-Metric on Merge Trees
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Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an ℓ^p-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.
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