Computing Persistent Homology with Various Coefficient Fields in a Single Pass

Boissonnat, Jean‐Daniel; Maria, Clément

doi:10.1007/978-3-662-44777-2_16

Cited by 12 publications

(8 citation statements)

References 8 publications

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“…Since our algorithm works similarly to the standard persistence algorithm, it is potentially amenable to the same kind optimizations. We are currently working on an adaptation for cohomology, which down the road would permit to use the optimized data structures developed in the recent years [3,4,16].…”

Section: Methodsmentioning

confidence: 99%

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Zigzag Persistence via Reflections and Transpositions

Maria¹,

Oudot²

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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We introduce a new algorithm for computing zigzag persistence, designed in the same spirit as the standard persistence algorithm. Our algorithm reduces a single matrix, maintains an explicit set of chains encoding the persistent homology of the current zigzag, and updates it under simplex insertions and removals. The total worst-case running time matches the usual cubic bound.A noticeable difference with the standard persistence algorithm is that we do not insert or remove new simplices "at the end" of the zigzag, but rather "in the middle". To do so, we use arrow reflections and transpositions, in the same spirit as reflection functors in quiver theory. Our analysis introduces new kinds of reflections in quiver representation theory: the "injective and surjective diamonds". It also introduces the "transposition diamond" which models arrow transpositions. For each type of diamond we are able to predict the changes in the interval decomposition and associated compatible bases. Arrow transpositions have been studied previously in the context of standard persistent homology, and we extend the study to the context of zigzag persistence. For both types of transformations, we provide simple procedures to update the interval decomposition and associated compatible homology basis.

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Section: Methodsmentioning

confidence: 99%

“…For instance, in this example, m is the birth time and f is the death time. 4 This matrix can be viewed as a compact encoding of the matrices R and V in the R = DV decomposition of the boundary matrix D (see [15]). strictly reduces low(col).…”

Section: If a And C Injective Of Corankmentioning

confidence: 99%

Zigzag Persistence via Reflections and Transpositions

Maria¹,

Oudot²

2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The Bockstein spectral sequence is a classical tool for recovering the integral homology from mod p homology. In the situation of weighted persistent homology, it is convenient to consider the chains with coefficients in Z/2Z (see [7,19,35]). It is important to explore the Bockstein spectral sequence on weighted persistent homology as a tool to recover the integral weighted persistent homology so that one can obtain more topological information on weighted data.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 6, we develop the Bockstein spectral sequence for weighted homology. The motivation behind using the Bockstein spectral sequence is that in persistent homology algorithms [7,35], the homology is usually computed with field coefficients. However, the integral homology groups contain more information than the homology groups with field coefficients.…”

Section: Introductionmentioning

confidence: 99%

Weighted persistent homology

Ren¹,

Wu²,

Wu³

2018

Rocky Mountain J. Math.

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In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. We also prove a theorem that allows us to calculate the mod p 2 weighted persistent homology given some information on the mod p weighted persistent homology.2010 Mathematics Subject Classification. Primary 55N35, 55T99; Secondary 55U20, 55U10.

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“…More persistent features are detected over a wide range of length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space [BDM15,BM14a], the space is represented as a sequence of simplicial complexes called a filtration. The most popular filtrations are nested sequences of increasing simplicial complexes but more advanced types of filtrations have been studied where consecutive complexes are mapped using more general simplicial maps [DFW14].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Representation for Filtrations of Simplicial Complexes

Boissonnat¹,

Karthik²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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A filtration over a simplicial complex K is an ordering of the simplices of K such that all prefixes in the ordering are subcomplexes of K. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. In order to represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica '14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest.This direction has been recently pursued for the case of maintaining simplicial complexes. For instance, Boissonnat et al. [SoCG '15] considered storing the simplices that are maximal for the inclusion and Attali et al. [IJCGA '12] considered storing the simplices that block the expansion of the complex. Nevertheless, so far there has been no data structure that compactly stores the filtration of a simplicial complex, while also allowing the efficient implementation of basic operations on the complex.In this paper, we propose a new data structure called the Critical Simplex Diagram (CSD) which is a variant of the Simplex Array List (SAL) [SoCG '15]. Our data structure allows to store in a compact way the filtration of a simplicial complex, and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Next, we show that the CSD representation admits the following construction algorithms.• A new edge-deletion algorithm for the fast construction of Flag complexes, which only depends on the number of critical simplices and the number of vertices.• A new matrix-parsing algorithm to quickly construct relaxed Delaunay complexes, depending only on the number of witnesses and the dimension of the complex.

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Computing Persistent Homology with Various Coefficient Fields in a Single Pass

Cited by 12 publications

References 8 publications

Zigzag Persistence via Reflections and Transpositions

Zigzag Persistence via Reflections and Transpositions

Weighted persistent homology

An Efficient Representation for Filtrations of Simplicial Complexes

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