A Lattice-Theoretic Perspective on the Persistence Map

TL;DR

This paper offers a lattice-theoretic framework for understanding the persistence map in topological data analysis, enabling more efficient inverse computations and connecting TDA with combinatorics.

Contribution

It introduces a local, lattice-theoretic description of the persistence map, bridging TDA and combinatorics for improved computational approaches.

Findings

Provides an isomorphic description of the persistence map

Enables potential speed-ups in inverse computations

Bridges TDA with classical combinatorics tools

Abstract

We provide a naturally isomorphic description of the persistence map from merge trees to barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our description is local, which offers the potential to speed up inverse computations, and brings classical tools in combinatorics to bear on an active area of research in topological data analysis (TDA).