[Paper Review] Persistent sheaf cohomology

Florian Russold|arXiv (Cornell University)|Apr 28, 2022

Topological and Geometric Data Analysis4 citations

TL;DR

This paper introduces two novel constructions of persistent sheaf cohomology by extending persistence theory to sheaf cohomology, enabling tracking of cohomological features across algebraic (sheaf variation on a fixed space) and topological (space variation with fixed sheaf) dimensions. The key contribution is the formalization of sheaf persistence modules of type A and copersistence modules of type T, with conditions under which they can be reduced to one another, and the construction of 2D persistence modules combining both dimensions.

ABSTRACT

We expand the toolbox of (co)homological methods in computational topology by applying the concept of persistence to sheaf cohomology. Since sheaves (of modules) combine topological information with algebraic information, they allow for variation along an algebraic dimension and along a topological dimension. Consequently, we introduce two different constructions of sheaf cohomology (co)persistence modules. One of them can be viewed as a natural generalization of the construction of simplicial or singular cohomology copersistence modules. We discuss how both constructions relate to each other and show that, in some cases, we can reduce one of them to the other. Moreover, we show that we can combine both constructions to obtain two-dimensional (co)persistence modules with a topological and an algebraic dimension. We also show that some classical results and methods from persistence theory can be generalized to sheaves. Our results open up a new perspective on persistent cohomology of filtrations of simplicial complexes.

Motivation & Objective

• To extend persistence theory to sheaf cohomology by introducing two distinct constructions: one for sheaf variation on a fixed topological space (type A), and one for space variation with a fixed sheaf (type T).
• To establish a theoretical foundation for persistent sheaf cohomology by generalizing key results such as the ZC-representation theorem to sheaves.
• To demonstrate that in certain cases, the topological-type copersistence module can be reduced to the algebraic-type persistence module.
• To show how both constructions can be combined into two-dimensional (co)persistence modules with separate algebraic and topological dimensions.
• To explore the potential of persistent sheaf cohomology as a tool for analyzing labeled data sets and higher-dimensional structures beyond simplicial homology.

Proposed method

• Constructs a persistence module of type A by considering a linear diagram of sheaves and sheaf morphisms on a fixed topological space X, tracking sheaf cohomology H^k(X, F_i) along the diagram.
• Constructs a copersistence module of type T by pulling back a fixed sheaf along continuous maps between topological spaces, inducing cohomology morphisms and forming a diagram of cohomology modules.
• Applies the ZC-representation theorem to show that both persistence and copersistence modules of vector space-valued sheaves are interval decomposable, yielding a persistence barcode.
• Reduces the type T copersistence module to a type A persistence module in cases where the sheaf is constant or satisfies certain compatibility conditions.
• Combines both constructions into a 2D persistence module by parameterizing both the topological space and the sheaf structure simultaneously.
• Uses sheaves of graded modules to enable efficient computation of persistent sheaf cohomology via standard cohomological techniques.

Experimental results

Research questions

• RQ1 How can persistence theory be generalized to sheaf cohomology, given that sheaves combine topological and algebraic information?
• RQ2 What are the two distinct constructions of persistent sheaf cohomology, and how do they differ in terms of what varies (sheaf or space)?
• RQ3 Under what conditions can a copersistence module of type T (space variation) be reduced to a persistence module of type A (sheaf variation)?
• RQ4 Can both constructions be combined to form a two-dimensional persistence module with independent algebraic and topological parameters?
• RQ5 What is the role of sheaf cohomology in analyzing labeled data sets, such as colored simplicial filtrations?

Key findings

• The paper establishes that both type A persistence modules and type T copersistence modules of vector space-valued sheaves are interval decomposable, yielding a persistence barcode that tracks persistent cohomology classes.
• In the case of constant sheaves on filtered simplicial complexes, the type T copersistence module can be reduced to a type A persistence module, linking classical persistent homology to sheaf cohomology.
• The zero-dimensional persistent sheaf cohomology of a labeled filtration captures the number of unicolored components across the filtration, as demonstrated in the example with labeled points.
• The construction of 2D persistence modules by combining both type A and type T modules allows for simultaneous analysis of algebraic and topological variation in sheaf cohomology.
• The method enables efficient computation of persistent sheaf cohomology by reducing it to the cohomology of a sheaf of graded modules.
• The framework provides a new perspective on persistent cohomology of filtered simplicial complexes by viewing them as (co)sheaves of graded modules.