Graded Persistence Diagrams and Persistence Landscapes

Betthauser, Leo; Bubenik, Peter; Edwards, Parker B.

doi:10.1007/s00454-021-00316-1

Cited by 6 publications

(2 citation statements)

References 12 publications

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“…We denote by the result of applying the resulting Möbius inversion formula to the invariant . Invariants obtained in this way are sometimes referred to as “generalized persistence diagrams.” See, e.g., [AENYb, BBE22, BE22, BOO KM21, MP22, Pat18].…”

Section: Motivation and Related Invariantsmentioning

confidence: 99%

Homological approximations in persistence theory

BLANCHETTE¹,

Brüstle²,

Hanson³

2022

Can. J. Math.-J. Can. Math.

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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 homological data 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 the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement. Contents 1. Introduction 1 2. Notation and Terminology 4 3. Motivation and related invariants 6 4. Relative homological algebra 9 5. Classes and morphisms of spread modules. 14 6. Homological invariants in persistence theory 17 7. Examples and comparison to other invariants 19 References 23

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Section: Motivation and Related Invariantsmentioning

confidence: 99%

Homological approximations in persistence theory

BLANCHETTE¹,

Brüstle²,

Hanson³

2022

Can. J. Math.-J. Can. Math.

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“…The persistence landscape [3] is a second invariant that is better suited for machine learning and statistical methods [4,16]. Betthauser, Bubenik, and Edwards use the Möbius inversion formula to describe an elegant connection between persistence diagrams and persistence landscapes [2].…”

Section: Introductionmentioning

confidence: 99%

Edit Distance and Persistence Diagrams Over Lattices

McCleary,

Patel

2020

Preprint

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We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite lattice, and the output is a persistence diagram defined as the Möbius inversion of a certain monotone integral function. We adapt the Reeb graph edit distance of Landi et. al. to each of our categories and prove that both functors in our pipeline are 1-Lipschitz making our pipeline stable.

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