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Zigzag Persistence [article]

Gunnar Carlsson, Vin de Silva
2008 arXiv   pre-print
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence.  ...  Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by  ...  Zigzag persistence addresses this limitation.  ... 
arXiv:0812.0197v1 fatcat:6iuywk2zwbab5el3bwakef25ia

Zigzag Persistence

Gunnar Carlsson, Vin de Silva
2010 Foundations of Computational Mathematics  
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence.  ...  Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by  ...  Zigzag persistence addresses this limitation.  ... 
doi:10.1007/s10208-010-9066-0 fatcat:icqprugshfd7rgwzmrohmnqn5y

Computing Zigzag Persistent Cohomology [article]

Clément Maria, Steve Oudot
2016 arXiv   pre-print
First, we compare its performance with zigzag persistent homology algorithm and show the interest of cohomology in zigzag persistence.  ...  Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence  ...  [28] for zigzag persistence.  ... 
arXiv:1608.06039v1 fatcat:ikdzsivobfhw7nivdzbejs4duu

Algebraic Stability of Zigzag Persistence Modules [article]

Magnus Bakke Botnan, Michael Lesnick
2016 arXiv   pre-print
In this paper, we establish an analogue of this algebraic stability theorem for zigzag persistence modules.  ...  To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions.  ...  Algebraic Stability of Zigzag Persistence Modules In this section, we define the fully faithful functor E sending each zigzag module to a block decomposable module, first mentioned in Section 1.  ... 
arXiv:1604.00655v2 fatcat:i2ycbw25qfci5aeyan5mao5inu

The reflection distance between zigzag persistence modules [article]

Alexander Elchesen, Facundo Mémoli
2019 arXiv   pre-print
We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the ℓ^1-bottleneck distance between their respective persistence diagrams.  ...  By invoking the reflection functors introduced by Bernstein, Gelfand, and Ponomarev in 1973, in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection  ...  This generalized setting is called zigzag persistence.  ... 
arXiv:1805.11190v3 fatcat:upezpojhsne5tpp5tljxqltwnq

Parametrized homology via zigzag persistence

Gunnar Carlsson, Vin de Silva, Sara Kališnik, Dmitriy Morozov
2019 Algebraic and Geometric Topology  
By defining persistence in terms of finite rectangle measures, we classify barcode intervals into four classes.  ...  We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or, equivalently, as a persistence diagram.  ...  Levelset Zigzag Persistence. In some situations finite zigzag diagrams carry all the needed information. Let X = (X, f ) be an R-space constructed as follows.  ... 
doi:10.2140/agt.2019.19.657 fatcat:7msdmumyprh25h5ty47r5thlay

Fast Computation of Zigzag Persistence [article]

Tamal K. Dey, Tao Hou
2022 arXiv   pre-print
to computing zigzag persistence.  ...  Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions.  ...  Zigzag persistence possesses some key differences from standard persistence.  ... 
arXiv:2204.11080v2 fatcat:5xuh7je7pzakpjd5pzf6k6f6bu

Discrete Morse Theory for Computing Zigzag Persistence [article]

Clément Maria, Hannah Schreiber
2019 arXiv   pre-print
This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda, defined for standard persistence.  ...  From a zigzag filtration of complexes (K_i), we introduce a zigzag Morse filtration whose complexes (A_i) are Morse reductions of the original complexes (K_i), and we prove that they both have same persistent  ...  in Diagrams (13) , (14) , and (15), the Morse algorithm for computing the zigzag persistence of F is depicted in Algorithm 1, where zigzag persistence algorithm(M B , M j , σ) is the zigzag persistence  ... 
arXiv:1807.05172v2 fatcat:7gw3wqfzkjcs7dwvfr46ctgviy

Algebraic stability theorem for derived categories of zigzag persistence modules [article]

Yasuaki Hiraoka, Yuichi Ike, Michio Yoshiwaki
2021 arXiv   pre-print
The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module.  ...  We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander--Reiten quivers.  ...  We call each M ∈ rep k A n , each N ∈ rep k A n (z), and each L ∈ rep k A n (a) a (ordinary) persistence module, a purely zigzag persistence module, and a zigzag persistence module, respectively.  ... 
arXiv:2006.06924v4 fatcat:pu4igkisijbk3jujab6c36qdvy

Temporal Network Analysis Using Zigzag Persistence [article]

Audun Myers and Firas Khasawneh and Elizabeth Munch
2022 arXiv   pre-print
We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence.  ...  This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA).  ...  (b) Zero-dimensional zigzag persistence.(c) One-dimensional zigzag persistence. Fig. 11 11 Fig. 11 Zigzag persistence diagrams of the coach transportation network of Great Britain.  ... 
arXiv:2205.11338v1 fatcat:cagpx6fcqrh7dloqyjahrbuuwm

Zigzag Persistence via Reflections and Transpositions [chapter]

Clément Maria, Steve Y. Oudot
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
We introduce a new algorithm for computing zigzag persistence, designed in the same spirit as the standard persistence algorithm.  ...  Arrow transpositions have been studied previously in the context of standard persistent homology, and we extend the study to the context of zigzag persistence.  ...  [13] in the context of standard persistence. The transposition diamond principle generalizes the study to zigzag persistence.  ... 
doi:10.1137/1.9781611973730.14 dblp:conf/soda/MariaO15 fatcat:ppgfcnc6szhirbl54dlehomagu

Updating Barcodes and Representatives for Zigzag Persistence [article]

Tamal K. Dey, Tao Hou
2022 arXiv   pre-print
We consider computing these vines over changing filtrations for zigzag persistence.  ...  Computing persistence over changing filtrations give rise to a stack of 2D persistence diagrams where the birth-death points are connected by the so-called 'vines'.  ...  (a) (b) Figure 2 : (a) Computing dimensions or rank function on a persistence module with support over a 2D zigzag grid (poset) can be more efficiently computed by considering zigzag persistence on an  ... 
arXiv:2112.02352v2 fatcat:esmvaunosvgn3iyhez5r2zvlbe

Persistent and Zigzag Homology: A Matrix Factorization Viewpoint [article]

Gunnar Carlsson, Anjan Dwaraknath, Bradley J. Nelson
2021 arXiv   pre-print
Two fundamental computations in this field are persistent homology and zigzag homology.  ...  Persistent homology [42] is the first example of this idea, but much work has been done in extending it, notably to zigzag persistence [7] .  ...  Zigzag homology has received less attention than persistence, but similar efforts can be found in [32, 33] .  ... 
arXiv:1911.10693v2 fatcat:b4lv6kt6xnfkzoiohwpxjbq2uy

Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence [article]

Woojin Kim, Facundo Memoli
2018 arXiv   pre-print
Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module.  ...  Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG.  ...  We also review the notions of zigzag modules (Section 3.2) and levelset zigzag persistence (Section 3.3).  ... 
arXiv:1712.04064v4 fatcat:ffjv2myg5ndwblvqt3quqzmwgq

Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and its Applications [article]

Tamal K. Dey, Woojin Kim, Facundo Mémoli
2022 arXiv   pre-print
We show that the generalized rank over a finite interval I of a ℤ^2-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly  ...  The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence  ...  It follows that the total cost due to zigzag persistence computation is bounded by O(t ω+2 ).  ... 
arXiv:2111.15058v3 fatcat:koevn56ug5czvpsc7qgcbcf5zm
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