Discrete Level Set Persistence for Finite Discrete Functions

TL;DR

This paper explores the duality and computation of persistent homology for discrete functions on finite ordered sets, providing new theoretical insights without relying on classical Morse theory.

Contribution

It introduces a duality framework for sublevel and superlevel set persistence in discrete settings, extending persistent homology theory beyond continuous functions.

Findings

Proves duality of filtrations for sublevel and superlevel sets in discrete functions.

Shows Morse-like behavior can occur without genericity assumptions.

Provides methods to compute one type of persistence from the other via duality.

Abstract

We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. Finite ordered sets also serve as the codomain of our functions making all arguments finite and discrete. We prove duality of filtrations of sublevel sets and superlevel sets that undergirths a range of duality results of sublevel set persistent homology without the need to invoke complications of continuous functions or classical Morse theory. We show that Morse-like behavior can be achieved for flat extrema without assuming genericity. Additionally, we show that with inversion of order, one can compute sublevel set persistence from superlevel set persistence, and vice versa via a duality result that does not require the boundary to be treated as a special case. Furthermore, we discuss aspects of barcode construction rules, surgery of circular and linearly ordered sets, as well as surgery on auxiliary structures such as box snakes, which segment the ordered set by extrema and monotones.