Generalized persistence diagrams for persistence modules over posets

Kim, Woojin; Mémoli, Facundo

doi:10.1007/s41468-021-00075-1

Cited by 35 publications

(30 citation statements)

References 45 publications

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“…One can view persistence modules as sheaves on the underlying poset and hence investigate persistence modules by the means of sheaf theory. This approach is discussed in [2,7,11,23,24].…”

Section: Related Workmentioning

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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“…One can view persistence modules as sheaves on the underlying poset and hence investigate persistence modules by the means of sheaf theory. This approach is discussed in [2,7,11,23,24].…”

Section: Related Workmentioning

confidence: 99%

Persistent sheaf cohomology

Russold¹

2022

Preprint

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“…Our approach to the problem of establishing a bottleneck stability result for multigraded Betti numbers is inspired by recent work on signed barcodes and their stability [7], which takes its roots in the line of work on generalized persistence diagrams [1,13,17,20]. Going back to the example from Figure 1, we see that there is a matching of cost ε if we allow for the matching of generators coming from the same module but in different degrees, e.g., matching (ε, ε) ∈ β 1 (N ) with (ε, 0) ∈ β 0 (N ).…”

Section: Contextmentioning

confidence: 99%

On the stability of multigraded Betti numbers and Hilbert functions

Oudot¹,

Scoccola²

2021

Preprint

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Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory-it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution-as well as in practice-it is sometimes easy to visualize and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no bottleneck stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes, and allows matching of bars of the same module in homological degrees of different parity, as well as for matchings of bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the 2-parameter case, a 1-Wasserstein stability result for Hilbert functions with respect to the 1-presentation distance of Bjerkevik and Lesnick.

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“…It has since been extended to categories of persistence modules over arbitrary posets, including infinite ones [KM21]. We note that the rank invariant is strictly finer than the dimension vector, since for all a ∈ P one has dim M (a) = rank M (a, a).…”

Section: Introductionmentioning

confidence: 99%

Homological approximations in persistence theory

Blanchette¹,

Brüstle²,

Hanson³

2021

Preprint

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We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some exact structure and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. Contents 1. Introduction 1 1.1. Organization and main results 2 1.2. Acknowledgements 3 2. Relative homological algebra 3 3. Spread modules 6 3.1. Morphisms between spread modules 8 4. Homological invariants in persistence theory 10 4.1. Projectivisation 10 4.2. Homological invariants 12 5. Examples and comparison to other invariants 13 5.1. The dimension vector 13 5.2. The barcode 13 5.3. The rank invariant 13 5.4. Compressed multiplicities 15 References 15 2020 Mathematics Subject Classification. 55N31, 16E20 (primary); 16Z05, 18G35 (secondary). Key words and phrases. persistence modules, invariants, Grothendieck groups, relative homological algebra, exact structures. 1 There is an even more general notion of a persistence module over a small category, see [BCB20].

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Generalized persistence diagrams for persistence modules over posets

Cited by 35 publications

References 45 publications

Persistent sheaf cohomology

Persistent sheaf cohomology

On the stability of multigraded Betti numbers and Hilbert functions

Homological approximations in persistence theory

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