The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Boissonnat, Jean‐Daniel; Dey, Tamal K.; Maria, Clément

doi:10.1007/s00453-015-9999-4

Cited by 31 publications

(49 citation statements)

References 18 publications

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“…Otherwise, if col = 0, then c / ∈ τ i i∈I . The algorithm is valid because low : {1, · · · , m} → {1, · · · , m} is injective in M (actually, the identity) and every column addition 3 Since the persistence module is represented backwards, birth and death times are reversed compared to the standard persistence setting. For instance, in this example, m is the birth time and f is the death time.…”

Section: If a And C Injective Of Corankmentioning

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: If a And C Injective Of Corankmentioning

confidence: 99%

“…Nevertheless, in practice they are observed to behave near linearly in n on typical data 1 . The most optimized ones among them [1,3] are able to process millions of simplex insertions per second on a recent machine, which is considered fast enough for many practical purposes.…”

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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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“…We present a simple algorithm for computing persistence that we refer to as the persistence algorithm in this work (Algorithm 1). The reduction strategy resembles the annotation algorithm [DFW14,BDM13] as well as the reduction implicit in the fast matrix multiplication algorithm [MMS11]. for j = 1, .…”

Section: Fill and Work In A Persistence Algorithmmentioning

confidence: 99%

“…Some algorithmic variants are analyzed in terms of additional parameters in addition to the input size, including the total (index) persistence of the filtration [CK11], the number of critical cells in a cubical complex [BKR14], or the maximum Betti number among all the complexes in the input filtration [DFW14,BDM13]; while these results partially explain the excellent behavior of persistent homology in practice, the worst-case for all these variants remains cubic in the input size. Several open source software libraries implement variations of these algorithms [BKRW14,Dio,MBGY14].…”

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confidence: 99%

Persistent Homology and Nested Dissection

Kerber

Sheehy

Škraba

2015

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

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Nested dissection exploits the underlying topology to do matrix reductions while persistent homology exploits matrix reductions to the reveal underlying topology. It seems natural that one should be able to combine these techniques to beat the currently best bound of matrix multiplication time for computing persistent homology. However, nested dissection works by fixing a reduction order, whereas persistent homology generally constrains the ordering according to an input filtration. Despite this obstruction, we show that it is possible to combine these two theories. This shows that one can improve the computation of persistent homology if the underlying space has some additional structure. We give reasonable geometric conditions under which one can beat the matrix multiplication bound for persistent homology.1 Introduction Persistent homology [ELZ02] has become the standard tool in the growing field of topological data analysis (TDA). The underlying idea is to consider a sequence of increasing shapes and track the homological changes of the shapes in this process, that is, the appearance and disappearance of holes. This information can be read off from a (generalized) LU -decomposition of the boundary matrix which is a combinatorial representation of the sequence of shapes. All persistent homology algorithms used in practice are based on matrix reduction through row or column operations on an initially sparse boundary matrix. Due to matrix fill-in, a cubic complexity in the size of the matrix can be achieved on worst-case examples [Mor05]. On the other hand, algorithms often show near-linear asymptotic behavior on most realistic inputs. It is likely that the structure of realistic examples keeps the matrices sparse during the computation, but how can we quantify this?Generalized nested dissection [LRT79] is a technique to reduce the effect of fill-in in Gaussian elimination and related matrix reduction problems for instances which

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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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The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology

Cited by 31 publications

References 18 publications

Zigzag Persistence via Reflections and Transpositions

Zigzag Persistence via Reflections and Transpositions

Persistent Homology and Nested Dissection

An Efficient Representation for Filtrations of Simplicial Complexes

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