Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology

TL;DR

This paper introduces a unifying algebraic framework for one-parameter persistent homology using correspondence modules, enabling richer geometric insights and a categorical understanding of various persistence structures.

Contribution

It develops a categorical and sheaf-theoretic approach to persistent homology, unifying different architectures and providing new decomposition and isometry theorems.

Findings

Unified treatment of persistence and zigzag modules

Interval decomposition theorems for persistence sheaves

Construction of richer barcodes and slices for 2-parameter modules

Abstract

We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.