Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

Carlsson, Gunnar; Dwaraknath, Anjan; Nelson, Bradley J.

doi:10.48550/arxiv.1911.10693

Cited by 5 publications

(9 citation statements)

References 29 publications

(70 reference statements)

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“…There are two algorithms that explicitly deal with computing the barcode bases associated to the interval decomposition. In Carlsson et al (2019) the authors use matrix factorisation techniques to obtain bases in which the matrices are in echelon form. This technique also applies to zizag modules.…”

Section: Related Workmentioning

confidence: 99%

The space of barcode bases for persistence modules

Jacquard

Nanda

Tillmann

2022

J Appl. and Comput. Topology

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The barcode of a persistence module serves as a complete combinatorial invariant of its isomorphism class. Barcodes are typically extracted by performing changes of basis on a persistence module until the constituent matrices have a special form. Here we describe a new algorithm for computing barcodes which also keeps track of, and outputs, such a change of basis. Our main result is an explicit characterisation of the group of transformations that sends one barcode basis to another. Armed with knowledge of the entire space of barcode bases, we are able to show that any map of persistence modules can be represented via a partial matching between bars provided that neither source nor target admits nested bars in its barcode. We also generalise the algorithm and results described above to work for zizag modules.

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Section: Related Workmentioning

confidence: 99%

The space of barcode bases for persistence modules

Jacquard

Nanda

Tillmann

2022

J Appl. and Comput. Topology

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“…• Parallel versions [9,27] of the algorithms for computing standard and zigzag exist. While the computation of standard persistence in our FZZ algorithm can directly utilize the existing parallelization techniques, we ask if the conversions done in FZZ can be efficiently parallelized.…”

Section: Discussionmentioning

confidence: 99%

Fast Computation of Zigzag Persistence

Dey¹,

Hou²

2022

Preprint

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Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. This paper sheds light on narrowing this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell -wise non-zigzag filtration of a ∆-complex with the same length, where the cells are copies of the input simplices. This conversion step incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares.

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“…There are extensions to theory of persistent homology, such as zig-zag persistent homology [52,53] or multiparameter persistent homology [55,56]. The computation of these and other homological objects are actively developed and implemented to be accessible for applications [54,126,131,152,169,190,232]. An active area of research for studying higher-order analogues of directed networks is the mathematical field of directed algebraic topology, which aims to capture periodicity in data [93].…”

Section: 11mentioning

confidence: 99%

What are higher-order networks?

Bick¹,

Gross²,

Harrington³

2021

Preprint

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Modeling complex systems and data using the language of graphs and networks has become an essential topic across a range of different disciplines. Arguably, this network-based perspective derives is success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.

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Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

Cited by 5 publications

References 29 publications

The space of barcode bases for persistence modules

The space of barcode bases for persistence modules

Fast Computation of Zigzag Persistence

What are higher-order networks?

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