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.
翻译:修改 Bjerkevik 和 Lesnick 给出的多参数持久性模块定义时, 我们引入了 $\ ell\ p $ p$ 类型的合并树间断距离扩展。 我们显示我们的距离是一个公尺, 并且它向上限制相关条形码之间 $p$- Wasserstein 的距离。 对于每$p\ in[ 1\ infty] 美元, 我们证明这一距离对于细胞子水平过滤器来说是稳定的, 并且它是通用的( 最大) 距离, 满足了这一稳定性特性。 在 $pinfty 美元的情况下, 这为合并树间断距离提供了新的普遍性证据 。