Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives

Brown, Adam; Wang, Bei

doi:10.1007/s00454-020-00206-y

Cited by 3 publications

(1 citation statement)

References 22 publications

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“…Cellular sheaves are a kind of "discrete" sheaves on cell complexes that allow for practical computations in finite cases. Cellular sheaves, or more generally, sheaves on finite posets, are used, for example, to describe information flows in networks [17,21,32], for sensor integration and data fusion [34,35] or for stratification learning [6]. Much more about applications of sheaves can be found in [11,16,33].…”

Section: Introductionmentioning

confidence: 99%

Persistent sheaf cohomology

Russold¹

2022

Preprint

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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.

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