Computing multiparameter persistent homology through a discrete Morse-based approach

Scaramuccia, Sara; Iuricich, Federico; Floriani, Leila De; Landi, Claudia

doi:10.1016/j.comgeo.2020.101623

Cited by 16 publications

(10 citation statements)

References 54 publications

(99 reference statements)

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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

“…As we mentioned in Section 2, a number of invariants for multipersistence have been proposed, and a few implementations are available. Let us name some of them, addressing the interested reader to recent works like [29] for a more complete list of references. In [2] the authors propose an efficient algorithm to compute invariants associated with resolutions of modules constructed from Z m -filtrations, although some restrictive assumptions are made on the type of filtrations; a more general framework is studied in [5].…”

Section: Generalizing the Rank Invariant In The Finite Casementioning

confidence: 99%

Computing Multipersistence by Means of Spectral Systems

Guidolin

Divasón

Romero

et al. 2019

Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation

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In their original setting, both spectral sequences and persistent homology are algebraic topology tools defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. Recently, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a new invariant providing information on multifiltrations and applications of our algorithms to spaces of infinite type.

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“…Motivated by this fact, the algorithms Iuricich et al 2016) retrieve a discrete gradient vector field consistent with a multi-filtration. The Morse-based reduction preprocessing for computing MPH invariants is shown to be effective in Scaramuccia et al (2020).…”

Section: Introductionmentioning

confidence: 99%

“…The construction of such gradient vector field is efficiently achieved, actually in linear time in Iuricich et al (2016) and whenever the worst-case size of a cell star is negligible with respect to the whole complex size. The algorithm in Iuricich et al (2016) improves that in in terms of speed but it is equivalent to it in terms of retrieved critical cells Scaramuccia et al (2020). However, even for these algorithms, the question about whether they retrieve the minimum, also known as optimal, number of critical cells necessary to get the same persistence modules was left as an open problem.…”

Section: Introductionmentioning

confidence: 99%

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

Landi

Scaramuccia

2021

J Comb Optim

Self Cite

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The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

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Computing multiparameter persistent homology through a discrete Morse-based approach

Cited by 16 publications

References 54 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

Computing Multipersistence by Means of Spectral Systems

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

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