Zigzag persistent homology in matrix multiplication time

Milosavljević, Nikola; Morozov, Dmitriy; Škraba, Primož

doi:10.1145/1998196.1998229

Cited by 120 publications

(111 citation statements)

References 27 publications

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“…We show that converting a reduced matrix for a nested dissection order into a reduced matrix for any other filtration can be done in O(n 2+β ) time. Even though this bound is worse than the best known complexity bound of O(n ω ) for persistent homology [MMS11], it leads to a persistent homology algorithm that is purely based on elementary reductions and yields a subcubic bound for a large class of instances.…”

Section: Vineyards In Separable Complexesmentioning

confidence: 87%

“…In the deterministic case, R(n) = O(n ω ) [IMH82], resulting in a complexity of O(C (1−δ)Γ n 2.373 ). This is asymptotically inferior to the asymptotically best known persistence algorithm that runs in O(n ω ) time [MMS11]. However, using Wiedemann's algorithm [KDS91] to compute ranks yields a randomized, Monte-Carlo algorithm for the Γ-persistent homology that runs in Õ (C (1−δ)Γ n 2 ) time, whereÕ means that logarithmic factors in n are ignored.…”

Section: Vineyards In Separable Complexesmentioning

confidence: 98%

“…We present a simple algorithm for computing persistence that we refer to as the persistence algorithm in this work (Algorithm 1). The reduction strategy resembles the annotation algorithm [DFW14,BDM13] as well as the reduction implicit in the fast matrix multiplication algorithm [MMS11]. for j = 1, .…”

Section: Fill and Work In A Persistence Algorithmmentioning

confidence: 99%

“…The only unconditional subcubic complexity bound by Milosavljević et al [MMS11] employs fast matrix multiplication to run in O(n ω ) time. Edelsbrunner and Parsa show that computing Betti numbers, and thus also persistent homology, is at least as hard as computing ranks of sparse n×n matrices [EP14].…”

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confidence: 99%

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Persistent Homology and Nested Dissection

Kerber

Sheehy

Škraba

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Nested dissection exploits the underlying topology to do matrix reductions while persistent homology exploits matrix reductions to the reveal underlying topology. It seems natural that one should be able to combine these techniques to beat the currently best bound of matrix multiplication time for computing persistent homology. However, nested dissection works by fixing a reduction order, whereas persistent homology generally constrains the ordering according to an input filtration. Despite this obstruction, we show that it is possible to combine these two theories. This shows that one can improve the computation of persistent homology if the underlying space has some additional structure. We give reasonable geometric conditions under which one can beat the matrix multiplication bound for persistent homology.1 Introduction Persistent homology [ELZ02] has become the standard tool in the growing field of topological data analysis (TDA). The underlying idea is to consider a sequence of increasing shapes and track the homological changes of the shapes in this process, that is, the appearance and disappearance of holes. This information can be read off from a (generalized) LU -decomposition of the boundary matrix which is a combinatorial representation of the sequence of shapes. All persistent homology algorithms used in practice are based on matrix reduction through row or column operations on an initially sparse boundary matrix. Due to matrix fill-in, a cubic complexity in the size of the matrix can be achieved on worst-case examples [Mor05]. On the other hand, algorithms often show near-linear asymptotic behavior on most realistic inputs. It is likely that the structure of realistic examples keeps the matrices sparse during the computation, but how can we quantify this?Generalized nested dissection [LRT79] is a technique to reduce the effect of fill-in in Gaussian elimination and related matrix reduction problems for instances which

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Section: Vineyards In Separable Complexesmentioning

confidence: 87%

Section: Vineyards In Separable Complexesmentioning

confidence: 98%

Section: Fill and Work In A Persistence Algorithmmentioning

confidence: 99%

mentioning

confidence: 99%

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Persistent Homology and Nested Dissection

Kerber

Sheehy

Škraba

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…An algorithm in matrix multiplication time was introduced in [MMS11]. The algorithms present in [ZC05,MN13] have cubical worst-time complexity, but they seem to behave better in practice.…”

Section: Persistent Homologymentioning

confidence: 99%

Two Measures for the Homology Groups of Binary Volumes

Gonzalez-Lorenzo

Bac

Mari

et al. 2016

Discrete Geometry for Computer Imagery

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Abstract. Given a binary object (2D or 3D), its Betti numbers characterize the number of holes in each dimension. They are obtained algebraically, and even though they are perfectly defined, there is no unique way to display these holes. We propose two geometric measures for the holes, which are uniquely defined and try to compensate the loss of geometric information during the homology computation: the thickness and the breadth. They are obtained by filtering the information of the persistent homology computation of a filtration defined through the signed distance transform of the binary object.

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Discrete Morse Theory for Computing Zigzag Persistence

Maria

Schreiber

2019

Lecture Notes in Computer Science

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We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology.From a zigzag filtration of complexes (Xi), we introduce a zigzag Morse filtration whose complexes (Ai) are Morse reductions of the original complexes (Xi), and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda [34], defined for standard persistence.The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.

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Zigzag persistent homology in matrix multiplication time

Cited by 120 publications

References 27 publications

Persistent Homology and Nested Dissection

Persistent Homology and Nested Dissection

Two Measures for the Homology Groups of Binary Volumes

Discrete Morse Theory for Computing Zigzag Persistence

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