Generalized Persistence Diagrams for Persistence Modules over Posets

Kim, Woojin; Memoli, Facundo

doi:10.48550/arxiv.1810.11517

Cited by 6 publications

(8 citation statements)

References 30 publications

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“…Extending these notions to a general 'persistence' setting in TDA, leads to the study of persistent invariants, which are designed to extract and quantify important information about the TDA structures. As an important example, we refer the readers to the rank invariant for persistent vector spaces [KM18]. The main focus of this paper is the persistent cup-length invariant of persistent graded rings.…”

Section: Persistent Invariantsmentioning

confidence: 99%

Persistent Cup-Length

Contessoto¹,

Mémoli²,

Stefanou³

et al. 2021

Preprint

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Cohomological ideas have recently been injected into persistent homology and have been utilized for both enriching and accelerating the calculation of persistence diagrams. For instance, the software Ripser fundamentally exploits the computational advantages offered by cohomological ideas. The cup product operation which is available at cohomology level gives rise to a graded ring structure which extends the natural vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the Cup-Length, which is efficient at discriminating spaces.In this paper, we lift the cup-length into the Persistent Cup-Length invariant for the purpose of extracting non-trivial information about the evolution of the cohomology ring structure across a filtration. We show that the Persistent Cup-Length can be computed from a family of representative cocycles and devise a polynomial time algorithm for the computation of the Persistent Cup-Length invariant. We furthermore show that this invariant is stable under suitable interleaving-type distances. Along the way, we identify an invariant which we call the Cup-Length Diagram, which is stronger than persistent cup-length but can still be computed efficiently.

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Section: Persistent Invariantsmentioning

confidence: 99%

Persistent Cup-Length

Contessoto¹,

Mémoli²,

Stefanou³

et al. 2021

Preprint

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“…For such t, there does not always exist ǫ > 0 such that f −1 (t ± ǫ) is the same nesting poset as for f −1 (t), and so computing the nesting poset of an interlevel set does not make sense in our context. Nonetheless, there are modifications [29] of this approach that hold promise for applications.…”

Section: The Zigzag Of Posetsmentioning

confidence: 99%

Moduli Spaces of Morse Functions for Persistence

Catanzaro,

Curry,

Fasy

et al. 2019

Preprint

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We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse-Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in R 3 . Their level-set invariant, a poset generated by nested disks and annuli from levels sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode.

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“…It is possible to define and compute persistence in other categories than simplicial complexes or topological spaces [3,1] and, in a different sense, [20,15]. The present paper introduces a further class of generalized persistence functions (gp-functions), defined on (R, ≤)-indexed diagrams in a given category, that can be described via persistence diagrams.…”

Section: Introductionmentioning

confidence: 99%

Steady and ranging sets in graph persistence

Bergomi¹,

Ferri²,

Tavaglione³

2020

Preprint

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Generalised persistence functions (gp-functions) are defined on (R, ≤)-indexed diagrams in a given category. A sufficient condition for stability is also introduced. In the category of graphs, a standard way of producing gp-functions is proposed: steady and ranging sets for a given feature. The example of steady and ranging hubs is studied in depth; their meaning is investigated in three concrete networks.

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Generalized Persistence Diagrams for Persistence Modules over Posets

Cited by 6 publications

References 30 publications

Persistent Cup-Length

Persistent Cup-Length

Moduli Spaces of Morse Functions for Persistence

Steady and ranging sets in graph persistence

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