Pruning distance of upset-decomposable persistence modules

TL;DR

A Lipschitz equivalence with respect to the bottleneck distance for upset-decomposable persistence modules is established, which proves half of Bjerkevik's conjecture for these modules and bound the bottleneck distance by a multiple of the pruning distance.

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.

Figures (1)

• Figure 1: A CI problem on the left and the graph associated to it in the middle. On the right is an example of two upset-decomposable modules $M = M_1 \oplus M_2~$and $N = N_1 \oplus N_2$ associated to the CI problem, constructed using the proof of bjerkevik_stabilizing (with $C = 4$). The solid black, dashed black, solid red and dashed red curves are the boundaries of the upsets $M_1$, $M_2$, $N_1$ and $N_2$, respectively. For every $i$ in $\{0, 1, 2, 3\}$, $p_i = (8i,-8i) \in \mathbb{R}^2$ and each diagonal step is adding $(1,1)$ to the point before it. This construction gives us that $d_I(M,N)=3$ using the same lemma.

Theorems & Definitions (50)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2: botnan_crawley-boevey, azumaya1950corrections
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 2.7
• Definition 2.8
• Definition 2.9