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arXiv · 2112.12165

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

Robert Cardona·Justin Curry·Tung Lam·Michael Lesnick
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Abstract

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-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 $p\in[1,\infty]$, 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 $p=\infty$ case, this gives a novel proof of universality for the interleaving distance on merge trees.

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BibTeXRIS

Robert Cardona, Justin Curry, Tung Lam, Michael Lesnick. 2021-12-22. The Universal $\ell^p$-Metric on Merge Trees. https://arxiv.org/abs/2112.12165

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