Tri-partitions and Bases of an Ordered Complex

Edelsbrunner, Herbert; Ölsböck, Katharina

doi:10.1007/s00454-020-00188-x

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…In addition to facilitating this application, the results of 3 and 5 may be of independent interest for the following reasons. Theorem 3.3 is a new interpretation of a duality relationship that manifests in many contexts such as the correspondence between persistent homology and persistent relative cohomology [5], symmetries in extended persistence diagrams [3], and a discrete Helmoltz-Hodge decomposition [11]. In [13], we show that the filtered discrete Morse chain complexes also exhibit this duality.…”

Section: Discussionmentioning

confidence: 92%

“…At its most basic level, the algebraic relationship between the persistent homology of two dual filtered cell complexes is similar to that between persistent homology and persistent relative cohomology. The latter corresponds to taking the anti-transpose of the boundary matrix [5] or equivalently, leaving the boundary matrix as is and applying the row reduction algorithm instead of the column-reduction algorithm [5,11]. Lemma 3.1 shows that the same relationship applies to the boundary matrices of dual filtered cell complexes.…”

Section: Related Workmentioning

confidence: 97%

“…The same reversal of ordering holds for the dual filtered cell complexes defined here, so we obtain a similar relationship between the persistence diagrams. Our proof of the correspondence between persistence pairs in dual filtrations uses the matrix rank function and pairing uniqueness lemma in a similar way to the combinatorial Helmoltz-Hodge decomposition of [11]. Nonetheless, Theorem 3.3 interprets the underlying linear algebra in the setting of dual filtered complexes and only uses the concept of persistent homology without using the connection to persistent relative cohomology, which makes it more accessible.…”

Section: The Persistent Homology Of Dual Filtered Complexesmentioning

confidence: 99%

See 2 more Smart Citations

The Persistent Homology of Dual Digital Image Constructions

Bleile¹,

Garin²,

Heiss³

et al. 2021

Preprint

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To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation.

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Section: Discussionmentioning

confidence: 92%

Section: Related Workmentioning

confidence: 97%

Section: The Persistent Homology Of Dual Filtered Complexesmentioning

confidence: 99%

See 1 more Smart Citation

The Persistent Homology of Dual Digital Image Constructions

Bleile¹,

Garin²,

Heiss³

et al. 2021

Preprint

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“…Exhaustive reduction. We review the exhaustive reduction, discussed in [10], even if the idea was already present in [11,27]. The idea is that after the pivot of the reduced matrix has been identified, further (left-to-right) column additions are performed to eliminate nonzero entries with indices smaller than the pivot.…”

Section: Input: Boundary Matrix Dmentioning

confidence: 99%

Keeping it sparse: Computing Persistent Homology revisited

Bauer¹,

Masood²,

Giunti³

et al. 2022

Preprint

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In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes. We propose two novel variants of the standard algorithm, called swap and retrospective reductions, which improve upon state-of-the-art techniques on several examples in practice. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worstcase complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

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“…Though the concept of spanning acycles was introduced by Kalai [23] in 1983, minimal spanning acycles have only received attention in recent years [16] and especially in the context of random simplicial complexes [19,40,18,24].…”

Section: Introductionmentioning

confidence: 99%

Central Limit Theorem for Euclidean Minimal Spanning Acycles

Škraba¹,

Yogeshwaran²

2022

Preprint

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We investigate asymptotics for the minimal spanning acycles of the (Alpha)-Delaunay complex on a stationary Poisson process on R d , d ≥ 2. Minimal spanning acycles are topological (or higher-dimensional) generalization of minimal spanning trees. We establish a central limit theorem for total weight of the minimal spanning acycle on a Poisson-Delaunay complex. Our approach also allows us to establish central limit theorems for sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of minimal spanning acycles which proceeds by establishing suitable chain maps and uses matroidal properties of minimal spanning acycles. In contrast to the proof of weak-stabilization for Euclidean minimal spanning trees via percolation-theoretic estimates, our weak-stabilization proof is algebraic in nature and provides an alternative proof even in the case of minimal spanning trees.

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Tri-partitions and Bases of an Ordered Complex

Cited by 7 publications

References 19 publications

The Persistent Homology of Dual Digital Image Constructions

The Persistent Homology of Dual Digital Image Constructions

Keeping it sparse: Computing Persistent Homology revisited

Central Limit Theorem for Euclidean Minimal Spanning Acycles

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