Pruning distance of upset-decomposable persistence modules

Abstract

The pruning distance recently introduced by Bjerkevik compares persistence modules using approximate decompositions called prunings. Bjerkevik conjectures that this distance is Lipschitz equivalent to the classical interleaving distance on modules of a bounded pointwise dimension. In this article, we establish a Lipschitz equivalence with respect to the bottleneck distance for upset-decomposable persistence modules. In particular, this proves half of Bjerkevik's conjecture for these modules. More precisely, we bound the bottleneck distance by a multiple of the pruning distance, improving the conjectured bound from~$2r$ to~$(2r-1)$ where~$r$ is the maximal pointwise dimension, and show that this improved bound is sharp. We also prove the converse inequality, bounding the pruning distance by the bottleneck distance. Our approach relies on explicitly computing the pruning of upset-decomposable modules, which we carry out using a directed graph formalism.