Discrete Level Set Persistence for Finite Discrete Functions

2025-01-29

AI Summary

This work investigates discrete persistent homology of finite discrete functions, focusing on duality between sublevel and superlevel persistence, transformation mechanisms under multi-order directions, and barcode construction on ordered sets. Methodologically, it introduces a fully discrete filtration duality framework that dispenses with continuity assumptions and Morse-theoretic prerequisites, naturally accommodating flat extrema and unifying boundary treatment. Leveraging finite poset topology and discrete persistent homology, the paper proposes the box-snake structure and an order-reversal computational paradigm, establishing a rigorous theorem for mutual conversion between sublevel and superlevel persistence. It provides computationally tractable rules for barcode generation and extends the “surgery” theory for ordered sets. The contributions yield the first topological analysis tool for discrete data that is both free of continuous approximations and algebraically complete.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Sustainability of Web economics

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.

Problem

• Discrete Function Persistence
• High-order and Low-order Persistence Relationship
• Ordered Set Barcode Generation

Innovation

• Discrete Level Set Persistence
• Morse Theory Behavior
• Sorting Direction Transformation