Dualities in persistent (co)homology

Silva, Vin de; Morozov, Dmitriy; Vejdemo-Johansson, Mikael

doi:10.1088/0266-5611/27/12/124003

Cited by 125 publications

(182 citation statements)

References 13 publications

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“…Standard Persistence and Matrix Reduction. We review the presentation of standard persistence as in [15], where explicit chains representing a compatible homology basis are maintained. For a standard filtra-…”

Section: If a And C Injective Of Corankmentioning

confidence: 99%

“…The special case where all the arrows in the filtration have the same orientation -commonly known as standard persistence -has been extensively studied. In this case, zigzag modules are just modules over the ring of polynomials F[t], so computing their interval decomposition as in theorem 1.1 comes down to reducing a matrix to column-echelon or rowechelon form over F [t], with the additional twist that the ordering of the simplices by time of insertion in the filtration must be preserved [15,23]. Most methods use Gaussian elimination for this reduction and therefore incur a cubic worst-case time complexity in the number n of simplex insertions [20].…”

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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%

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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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“…By collecting windows beginning at every trace record, a high-dimensional point cloud is formed. We next use topological analysis [11,28] to detect circular features within the point cloud. These circular features represent cyclical behavior in the trace, as the points within such circles represent roughly the same pattern of memory access.…”

Section: Technical Overviewmentioning

confidence: 99%

“…The algorithm that computes the persistent cohomology of a sequence of simplicial complexes [11] is a modified version of the persistent homology algorithm [3,14], which in turn is a variation of the classic Smith normal form algorithm [21]. It involves a specific ordering in conducting matrix reduction on the coboundary matrices of the nested simplicial complexes.…”

Section: Detecting Circular Features In a Point Cloudmentioning

confidence: 99%

Topological analysis and visualization of cyclical behavior in memory reference traces

Choudhury

Wang

Rosen

et al. 2012

2012 IEEE Pacific Visualization Symposium

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(a) (b) Figure 1: One circular parameterization of the memory behavior of bubble sort (box (a)) morphing into another (box (b)), highlighting both their similarities and differences, and giving two views of the recurrent nature of the program. ABSTRACTWe demonstrate the application of topological analysis techniques to the rather unexpected domain of software visualization. We collect a memory reference trace from a running program, recasting the linear flow of trace records as a high-dimensional point cloud in a metric space. We use topological persistence to automatically detect significant circular structures in the point cloud, which represent recurrent or cyclical runtime program behaviors. We visualize such recurrences using radial plots to display their time evolution, offering multi-scale visual insights, and detecting potential candidates for memory performance optimization. We then present several case studies to demonstrate some key insights obtained using our techniques.

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“…Both variants lack a proof of usefulness in application scenarios. The currently fastest approaches in practice are based on the cohomological persistence algorithm by De Silva et al [dSMVJ11] and on heuristics of the standard reduction algorithm that exploit the structure of the boundary matrices [CK11].…”

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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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Dualities in persistent (co)homology

Cited by 125 publications

References 13 publications

Zigzag Persistence via Reflections and Transpositions

Zigzag Persistence via Reflections and Transpositions

Topological analysis and visualization of cyclical behavior in memory reference traces

Persistent Homology and Nested Dissection

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