The Universal $\ell^p$-Metric on Merge Trees
This work provides a theoretical foundation for comparing topological structures in data analysis, with incremental extensions to existing distances.
The authors introduced an $\ell^p$-type extension of the interleaving distance on merge trees, proving it is a metric that upper-bounds the $p$-Wasserstein distance and is stable and universal for cellular sublevel filtrations.
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.