Edit Distance and Persistence Diagrams Over Lattices

TL;DR

This paper introduces a functorial pipeline for persistent homology over finite metric lattices, establishing stability and equivalence of edit and bottleneck distances, thus generalizing classical persistence diagrams.

Contribution

It develops a novel functorial framework for persistent homology over lattices, proving stability and distance equivalence, extending classical concepts.

Findings

Pipeline is stable under the adapted Reeb graph edit distance.

Persistence diagrams are defined via M"obius inversion of birth-death functions.

Bottleneck and edit distances are strongly equivalent in this setting.

Abstract

We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice and the output is a persistence diagram defined as the M\"obius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are $1$-Lipschitz making our pipeline stable. Our constructions generalize the classical persistence diagram and, in this setting, the bottleneck distance is strongly equivalent to the edit distance.