Reducing complexes in multidimensional persistent homology theory

Állili, Madjid; Kaczyński, Tomasz; Landi, Claudia

doi:10.1016/j.jsc.2015.11.020

Cited by 22 publications

(38 citation statements)

References 27 publications

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“…Two papers, by Allili et al and Scaramuccia et al, have introduced algorithms which use discrete Morse theory to simplify a multi-filtration without altering its topological structure [2,41]. Fugacci and Kerber have recently developed a purely algebraic analogue of these: They give an algorithm that replaces a chain complex of persistence modules with a smaller one having the same quasi-isomorphism type [29].…”

Section: Related Workmentioning

confidence: 99%

“…Specifically, we run Algorithm 8 to obtain a minimal set S of generators of im ∂ 2 , and we assemble the vector representations of these generators into a bigraded matrix H. We then use H in place of [∂ 2 ] in the computation of rankf α. We also use H to construct a partially reduced surrogate J for [∂ 2 ← ] in the computation of rankf γ; see Appendix A. Since S generates im ∂ 2 , this optimization yields the correct results for rankf α and rankf γ.…”

Section: Optimizationsmentioning

confidence: 99%

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Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Lesnick¹,

Wright²

2019

Preprint

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Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K[x, y]-module M , where K is a field. The algorithm takes as input a short chain complex of free modulesmemory, where |F i | denotes the size of a basis of F i . We observe that, given the presentation computed by our algorithm, the bigraded Betti numbers of M are readily computed. We also introduce a different but related algorithm, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds. These algorithms have been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our approach outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

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Section: Related Workmentioning

confidence: 99%

Section: Optimizationsmentioning

confidence: 99%

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Lesnick¹,

Wright²

2019

Preprint

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“…To date, the approaches proposed in the literature for computing multi-parameter persistent homology are at a pioneering level and are not able to deal with the complexity and the size of real datasets. Recently, an interesting connection between multi-parameter persistent homology and discrete Morse theory has been pointed out in [14]. A formal proof is given of the equivalence between the multi-parameter persistent homology of the Morse complex defined by a Forman gradient compatible with the multifiltration and that of the underlying simplicial complex is provided.…”

Section: Discussionmentioning

confidence: 99%

“…In very recent research areas, like the analysis of higher dimensional scalar fields [13] or in the analysis of shapes based on multi-parameter persistent homology [14,15], there is a need for efficient methods capable of encoding a Forman gradient on higher dimensional simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

Computing discrete Morse complexes from simplicial complexes

2019

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We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial complexes, and we define an efficient encoding for the discrete Morse gradient on the most compact of such representations. We theoretically compare methods based on reductions and coreductions for computing a discrete Morse gradient, proving that the combination of reductions and coreductions produces new mutually equivalent approaches. We design and implement a new algorithm for computing a discrete Morse complex on simplicial complexes. We show that our approach scales very well with the size and the dimension of the simplicial complex also through comparisons with the only existing public-domain algorithm for discrete Morse complex computation. We discuss applications to the computation of multi-parameter persistent homology and of extrema graphs for visualization of time-varying 3D scalar fields.

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“…This paradigm has also a combinatorial counterpart developed for accelerating computations and based on discrete Morse theory [MN13,AKL17].…”

Section: Introductionmentioning

confidence: 99%

Morse inequalities for the Koszul complex of multi-persistence

Guidolin¹,

Landi²

2021

Preprint

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In this paper, we derive inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells.Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables.

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Reducing complexes in multidimensional persistent homology theory

Cited by 22 publications

References 27 publications

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Computing discrete Morse complexes from simplicial complexes

Morse inequalities for the Koszul complex of multi-persistence

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