Computing persistent homology with various coefficient fields in a single pass

Boissonnat, Jean‐Daniel; Maria, Clément

doi:10.1007/s41468-019-00025-y

Cited by 10 publications

(8 citation statements)

References 15 publications

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“…We have not discussed in this paper any algorithmic complexity issues. The fact that we need rational coefficients certainly introduces new challenges, since most persistent homology related algorithms work with coefficients in the field Z/2Z [31] (but see also [5]). It will be interesting to develop efficient algorithms for computing harmonic representatives and weight vectors of bars as defined in this paper.…”

Section: Discussionmentioning

confidence: 99%

Harmonic Persistent Homology

Basu¹,

Cox²

2021

Preprint

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We introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result we can associate concrete subspaces of cycles to each bar of the barcode of the filtration. We prove stability of the harmonic persistent homology subspaces under small perturbations of functions defining them. We relate the notion of "essential simplices" introduced in an earlier work to identify simplices which play a significant role in the birth of a bar, with that of harmonic persistent homology. We prove that the harmonic representatives of simple bars maximizes the "relative essential content" amongst all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices.

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Section: Discussionmentioning

confidence: 99%

Harmonic Persistent Homology

Basu¹,

Cox²

2021

Preprint

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“…The paper by Edelsbrunner et al 1) showed an algorithm to compute a PD, and the algorithm has been refined theoretically and practically by subsequent researches. 2,[48][49][50][51][52] Parallel and distributed algorithms has been also studied. 53,54) Many researchers on PH have developed various data analysis software using PH, including Javaplex (http://ap pliedtopology.github.io/javaplex/), Perseus 55) (http://people.maths.ox.ac.uk/nanda/perseus/), PHAT 56) (https://bitbucket.org/phatcode/ph at/), Dipha 54) (https://github.com/DIPHA/dipha), Ripser 57) (https://github.com/Ripser/ripser), Gudhi (https://gudhi.inria.fr/), Dyonisys (https: //mrzv.org/sof tware/dionysus/, https://mr zv.org/sof tware/dionysus2/), R-TDA (https: //cran.r-project.org/package=TDA), EIRENE 51) (http://gregoryhenselman.org/eirene/), Cubical-Ripser (https://github.com/CubicalRipser), RIVET (https://github.com/rivetTDA/rivet), giotto-tda 58) (https://github.com/giottoai/giotto-tda), jHoles, 59) and HomCloud (https://homcloud.dev).…”

Section: Softwarementioning

confidence: 99%

Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud

Obayashi,

Nakamura,

Hiraoka

2021

Preprint

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This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We also review some applications of persistent homology to materials researches and software for persistent homology data analysis. HomCloud, one of persistent homology software, is especially featured in this paper.

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“…The successes of persistent homology stem in part from a strong theoretical foundation [52,23,35] and a focus on efficient algorithmic implementations [5,6,9,20,48]. That said, the inherent algorithmic problems are far from being solved-see [34] for a recent survey-and current theoretical approaches to abstract computations are limited to very specific cases [1,2,3].…”

Section: Introductionmentioning

confidence: 99%

Künneth Formulae in Persistent Homology

Gakhar,

Perea

2019

Preprint

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The classical Künneth formula in algebraic topology describes the homology of a product space in terms of that of its factors. In this paper, we prove Künneth-type theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. In addition to comparing the two products, we present two applications in the setting of Vietoris-Rips complexes: one towards more efficient algorithms for product metric spaces with the maximum metric, and the other recovering persistent homology calculations on the n-torus. Contents 1. Introduction 1.1. Sketch of proof 1.2. Organization of the paper 1.3. Related work 2. Preliminaries 2.1. Persistent homology 2.2. The Classical Künneth Theorems 2.3. Homotopy Colimits 3. Products of Diagrams of Spaces 3.1. A comparison theorem 4. A Persistent Künneth formula for the categorical product 5. A Persistent Künneth formula for the generalized tensor product 5.1. The tensor product and Tor of graded modules 5.

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Computing persistent homology with various coefficient fields in a single pass

Cited by 10 publications

References 15 publications

Harmonic Persistent Homology

Harmonic Persistent Homology

Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud

Künneth Formulae in Persistent Homology

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