Edit Distance and Persistence Diagrams over Lattices

McCleary, Alexander; Patel, Amit

doi:10.1137/20m1373700

Cited by 16 publications

(7 citation statements)

References 8 publications

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“…Invariants obtained in this way are sometimes referred to as "generalized persistence diagrams. " See, e.g., [AENYb, BBE22, BE22, BOO KM21,MP22,Pat18].…”

Section: Signed Barcodes and Möbius Inversionmentioning

confidence: 99%

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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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“…Invariants obtained in this way are sometimes referred to as "generalized persistence diagrams. " See, e.g., [AENYb, BBE22, BE22, BOO KM21,MP22,Pat18].…”

Section: Signed Barcodes and Möbius Inversionmentioning

confidence: 99%

“…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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“…Recently, A. J. Blumberg and M. Lesnick studied the stability of 2-parameter persistent homology [3]. The stability theorems for other objects were also considered, such as the stability of persistent homology for hypergraphs [22], the stability of persistent path homology for networks [11], and the stability of persistence diagrams over lattices [18].…”

Section: Introductionmentioning

confidence: 99%

The algebraic stability for persistent Laplacians

Liu¹,

Li²,

Wu³

2023

Preprint

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The stability for topological persistence is one of the fundamental issues in topological data analysis. Many methods have been proposed to deal with the stability for persistent modules or persistence diagrams. The persistent Laplacians, as a relatively new topological persistence, have gained more attention and applications in different fields. In this paper, we start to consider the stability for persistent Laplacians. We introduce the concept of the Laplacian tree to describe the collection of persistent Laplacians that persist from a given parameter. We construct the category of Laplacian trees and prove an algebraic stability theorem for persistence Laplacian trees. Specifically, our stability theorem is applied to the real-valued functions on simplicial complexes and digraphs.

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“…The earliest work in this direction was by Carlsson and Zomorodian [9], who proposed the rank invariant. Subsequent efforts to study multiparameter persistence modules have involved a plethora of tools sourced from diverse locales -these include sheaf theory [29,22], commutative and homological algebra [31,18,26], lattice theory [30,19,5] and beyond [28].…”

Section: Introductionmentioning

confidence: 99%

Harder-Narasimhan Filtrations of Persistence Modules

Marc¹,

Jacquard²,

Nanda³

et al. 2023

Preprint

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The Harder-Narasimhan type of a quiver representation is a discrete invariant parameterised by a real-valued function (called a central charge) defined on the vertices of the quiver. In this paper, we investigate the strength and limitations of Harder-Narasimhan types for several families of quiver representations which arise in the study of persistence modules. We introduce the skyscraper invariant, which amalgamates the HN types along central charges supported at single vertices, and generalise the rank invariant from multiparameter persistence modules to arbitrary quiver representations. Our four main results are as follows:(1) we show that the skyscraper invariant is strictly finer than the rank invariant in full generality, (2) we characterise the set of complete central charges for zigzag (and hence, ordinary) persistence modules, (3) we extend the preceding characterisation to rectangle-decomposable multiparameter persistence modules of arbitrary dimension; and finally, (4) we show that although no single central charge is complete for nestfree ladder persistence modules, a finite set of central charges is complete.

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Edit Distance and Persistence Diagrams over Lattices

Cited by 16 publications

References 8 publications

Homological approximations in persistence theory

Homological approximations in persistence theory

The algebraic stability for persistent Laplacians

Harder-Narasimhan Filtrations of Persistence Modules

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