Zigzag Persistence

Carlsson, Gunnar; Silva, Vin de

doi:10.1007/s10208-010-9066-0

Cited by 271 publications

(407 citation statements)

References 12 publications

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“…Functorial clustering schemes produce diagrams of sets of varying shapes which can be used to construct simplicial complexes which in turn reflect the topology of connected components [Car09]. Functorial schemes also can be used to construct "zig-zag diagrams" [CdS10] which reflect the stability of clusters produced over a family of independent samples.…”

Section: Discussionmentioning

confidence: 99%

Classifying Clustering Schemes

Carlsson

Mémoli

2013

Found Comput Math

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Abstract. Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality.Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg [Kle02], in which for example one obtains an existence and uniqueness theorem instead of a non-existence result.We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

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Section: Discussionmentioning

confidence: 99%

Classifying Clustering Schemes

Carlsson

Mémoli

2013

Found Comput Math

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“…In general, see [3] for a comprehensive treatment of this topic, a zigzag module U is a sequence of vector spaces over the field K, …”

Section: A ∞ -Coalgebrasmentioning

confidence: 99%

“…After observing that A ∞ -persistence contains classical persistence as the first invariant of the family, we see that, unlike in this classical setting, homology classes may be born and die several times along the filtration with respect to any high diagonal ∆ n , see Theorem 3.1. Nevertheless, using the classification of finitely generated zigzag modules [3], we are able to prove that A ∞ -persistence of the given filtration is characterized in term of barcodes, see Theorem 2.7.…”

Section: Introductionmentioning

confidence: 99%

$$A_\infty $$ A ∞ -persistence

Belchí

Murillo

2014

AAECC

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We introduce and study A ∞ -persistence of a given homology filtration of topological spaces. This is a family, one for each n ≥ 1, of homological invariants which provide information not readily available by the (persistent) Betti numbers of the given filtration. This may help to detect noise, not just in the simplicial structure of the filtration but in further geometrical properties in which the higher codiagonals of the A ∞ -structure are translated. Based in the classification of zigzag modules, a characterization of the A ∞ -persistence in terms of its associated barcode is given.

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“…A more recent development is zigzag persistence [5], which is a generalization of ordinary persistence built on algebraic insights. Together with an efficient algorithm in the homological setting, zigzag persistence has already resolved open Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.…”

Section: Introductionmentioning

confidence: 99%

Zigzag persistent homology in matrix multiplication time

Milosavljević¹,

Morozov

Škraba

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

118

107

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We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M (n) time, our algorithm runs in O(M (n) + n 2 log 2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n 2.376 ), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n 3 ) time in the worst case.

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Zigzag Persistence

Cited by 271 publications

References 12 publications

Classifying Clustering Schemes

Classifying Clustering Schemes

$$A_\infty $$ A ∞ -persistence

Zigzag persistent homology in matrix multiplication time

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