Discrete Morse Theory for Computing Zigzag Persistence

Maria, Clément; Schreiber, Hannah

doi:10.1007/978-3-030-24766-9_39

Cited by 9 publications

(10 citation statements)

References 41 publications

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“…We will focus primarily on sequential approaches to persistent homology computation. Other, non-sequential approaches include the chunk algorithm [3], spectral sequence procedures [46,22], Morse-theoretic batch reduction [32,33,58,6,29,34,48,59,21], distributed algorithms [4,53,44], GPU acceleration [63,38], streaming [41], and homotopy collapse [9,20,8]. There are closely related techniques in matrix factorization and zigzag persistence [50,11,10].…”

Section: Related Literaturementioning

confidence: 99%

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

Hang¹,

Giusti²,

Ziegelmeier³

et al. 2021

Preprint

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Persistent homology is a leading tool in topological data analysis (TDA). Many problems in TDA can be solved via homological -and indeed, linear -algebra. However, matrices in this domain are typically large, with rows and columns numbered in billions. Low-rank approximation of such arrays typically destroys essential information; thus, new mathematical and computational paradigms are needed for very large, sparse matrices.We present the U-match matrix factorization scheme to address this challenge. U-match has two desirable features. First, it admits a compressed storage format that reduces the number of nonzero entries held in computer memory by one or more orders of magnitude over other common factorizations. Second, it permits direct solution of diverse problems in linear and homological algebra, without decompressing matrices stored in memory. These problems include look-up and retrieval of rows and columns; evaluation of birth/death times, and extraction of generators in persistent (co)homology; and, calculation of bases for boundary and cycle subspaces of filtered chain complexes. Such bases are key to unlocking a range of other topological techniques for use in TDA, and U-match factorization is designed to make such calculations broadly accessible to practitioners.As an application, we show that individual cycle representatives in persistent homology can be retrieved at time and memory costs orders of magnitude below current state of the art, via global duality. Moreover, the algebraic machinery needed to achieve this computation already exists in many modern solvers.

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Section: Related Literaturementioning

confidence: 99%

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

Hang¹,

Giusti²,

Ziegelmeier³

et al. 2021

Preprint

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show abstract

“…Second, there have been efforts to reduce the inherent size of computations using methods that preserve the homotopy type of a space while reducing the size of its combinatorial representation [4,12,30,36]. Zigzag homology has received less attention than persistence, but similar efforts can be found in [28,29]. The use of non-inclusion maps in persistent and zigzag homology has been somewhat limited in topological data analysis, although the case of simplicial maps has been investigated in [11,23], based on a strategy that uses zigzag homology to compute a persistence barcode.…”

Section: Contributions and Related Workmentioning

confidence: 99%

“…In this paper, we address the computational questions that are necessary for computing arbitrary induced maps on homology, and an explicit algorithm for computing interval indecomposables. In contrast to [6] and existing algorithms for zigzag homology [28,29,31], our algorithm does not explicitly use right filtrations, and instead use an approach that involves producing a matrix factorization.…”

Section: Contributions and Related Workmentioning

confidence: 99%

Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

Carlsson,

Dwaraknath,

Nelson

2019

Preprint

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Over the past two decades, topological data analysis has emerged as a young field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type-A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes beyond the capabilities of existing software for topological data analysis. Furthermore, this framework offers multiple opportunities for parallelization which have not been previously explored.

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“…Specifically, the associated quivers are mutation equivalent to a relation extension of the tensor products of two type A quivers. This is echoed by one-dimensional persistence homology in Topological Data Analysis in [EH14,MS19] where two zig-zag Morse filtrations are shown to have the same persistent homology. Recently, Dyckerhoff, Jasso and Lekili [DJL19] connect Morsifications of the Lefschetz fibrations of Auroux [Aur10] to higher Auslander algebras of type A.…”

Section: Introductionmentioning

confidence: 99%

The combinatorics of tensor products of higher Auslander algebras of type $A$

McMahon¹,

Williams²

2020

Preprint

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We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two drepresentation-finite algebras has a d-precluster-tilting subcategory. Finally we relate mutations of these collections to a form of tilting for these algebras. Contents 1. Introduction 1 Acknowledgements 4 2. Background 4 2.1. Grassmannian cluster algebras 4 2.2. Triangulations of cyclic polytopes 5 3. Non-l-intertwining collections 6 4. Higher precluster-tilting subcategories 10 5. Mutations 15 5.1. Higher APR tilting 15 5.2. Uniterated mutation of non-l-intertwining collections 17 References 22 PRECLUSTER-TILTING FOR TYPE A d n ⊗ A d m

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Discrete Morse Theory for Computing Zigzag Persistence

Cited by 9 publications

References 41 publications

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology

Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

The combinatorics of tensor products of higher Auslander algebras of type $A$

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