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Computing Persistent Homology

Afra Zomorodian, Gunnar Carlsson
2004 Discrete & Computational Geometry  
It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field.  ...  Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domains in any dimension.  ...  Acknowledgments The first author thanks Herbert Edelsbrunner and John Harer for discussions on Z-homology, and Leo Guibas for providing support and encouragement.  ... 
doi:10.1007/s00454-004-1146-y fatcat:7mrh5xexjfhhlmjwwwxhoszfji

Computing persistent homology

Afra Zomorodian, Gunnar Carlsson
2004 Proceedings of the twentieth annual symposium on Computational geometry - SCG '04  
It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field.  ...  Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domains in any dimension.  ...  Acknowledgments The first author thanks Herbert Edelsbrunner and John Harer for discussions on Z-homology, and Leo Guibas for providing support and encouragement.  ... 
doi:10.1145/997817.997870 dblp:conf/compgeom/ZomorodianC04 fatcat:ncwwi4fcyjeizh2bcz663yqroy

Fast computation of persistent homology representatives with involuted persistent homology [article]

Matija Čufar, Žiga Virk
2021 arXiv   pre-print
In a nutshell, we first compute persistent cohomology and use the obtained information to significantly improve the running time of the direct persistent homology computations.  ...  Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives.  ...  Conclusions We have demonstrated the feasibility of the involuted persistent homology computations to obtain the persistent homology representatives.  ... 
arXiv:2105.03629v1 fatcat:ij3c3vs7h5esnmssle76y7226i

Interleaved computation for persistent homology [article]

Mikael Vejdemo-Johansson
2011 arXiv   pre-print
We describe an approach to bounded-memory computation of persistent homology and betti barcodes, in which a computational state is maintained with updates introducing new edges to the underlying neighbourhood  ...  graph and percolating the resulting changes into the simplex stream feeding the persistence algorithm.  ...  the order they appear in the simplex stream, and computes persistent homology interleaved with the generation of new simplices.  ... 
arXiv:1105.6305v1 fatcat:ewlgktg2mrdxnjhyfyph3zwowy

Distributed Computation of Persistent Homology [chapter]

Ulrich Bauer, Michael Kerber, Jan Reininghaus
2013 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
Advances in algorithms for computing persistent homology have reduced the computation time drastically -as long as the algorithm does not exhaust the available memory.  ...  Persistent homology is a popular and powerful tool for capturing topological features of data.  ...  This research is partially supported by the Toposys project FP7-ICT-318493-STREP and the Max Planck Center for Visual Computing and Communication.  ... 
doi:10.1137/1.9781611973198.4 dblp:conf/alenex/BauerKR14 fatcat:bqmaiuppefetbojr6kbun2macy

Distributed computation of persistent homology [article]

Ulrich Bauer, Michael Kerber, Jan Reininghaus
2013 arXiv   pre-print
Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not exhaust the available memory.  ...  Persistent homology is a popular and powerful tool for capturing topological features of data.  ...  Contribution We present a scalable algorithm for computing persistent homology in parallel in a distributed memory environment.  ... 
arXiv:1310.0710v1 fatcat:m2gwmsaitrfo7pg5ha27dz6rha

Singular persistent homology with geometrically parallelizable computation [article]

Boris Goldfarb
2019 arXiv   pre-print
We state and prove versions of the Mayer-Vietoris theorem for persistent homology under mild and commonplace assumptions.  ...  This is done through the use of a new theory, the singular persistent homology, better suited for handling coverings of data sets.  ...  There are surely other attempts to parallelize the computation of persistent homology.  ... 
arXiv:1607.01257v4 fatcat:fusqsfi2d5erbhyt5wyket6b24

Computational Tools in Weighted Persistent Homology [article]

Shiquan Ren, Chengyuan Wu, Jie Wu
2019 arXiv   pre-print
In this paper, we study further properties and applications of weighted homology and persistent homology.  ...  We also prove a theorem that allows us to calculate the mod p^2 weighted persistent homology given some information on the mod p weighted persistent homology.  ...  The purpose of this paper is to reformulate some classical computational tools in homology theory in the context of weighted persistent homology.  ... 
arXiv:1711.09211v3 fatcat:2ikzkcz37rgwhghuven3deugda

Computing persistent homology under random projection

Karthikeyan Natesan Ramamurthy, Kush R. Varshney, Jayaraman J. Thiagarajan
2014 2014 IEEE Workshop on Statistical Signal Processing (SSP)  
In this paper, we investigate random linear projection of point clouds followed by topological data analysis for computing persistence diagrams and Betti numbers.  ...  We further investigate how the mean of the persistence diagrams from several random projections can be used favorably in Betti number recovery.  ...  Complexes can be directly computed on R(X ), and since these depend only on the Euclidean distance between the samples, the persistent homology will be approximately preserved.  ... 
doi:10.1109/ssp.2014.6884586 dblp:conf/ssp/RamamurthyVT14 fatcat:x7ouawon7ra4fknluj6lmib7iq

Regularization of Persistent Homology Gradient Computation [article]

Padraig Corcoran, Bailin Deng
2020 arXiv   pre-print
Persistent homology is a method for computing the topological features present in a given data.  ...  Computing the gradients of persistent homology is an ill-posed inverse problem with infinitely many solutions.  ...  sequence independently, persistent homology computes the homology of the inclusions R p (X) − → R q (X) for all p < q [15] .  ... 
arXiv:2011.05804v2 fatcat:3nv2al337rfy7o5wwvih5o54ci

Matroid Filtrations and Computational Persistent Homology [article]

Gregory Henselman, Robert Ghrist
2017 arXiv   pre-print
This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell  ...  These resources are commonly accessible for computations in persistent homology, as described in the following section.  ...  This technical report is intended for a reader familiar with homology, persistent homology, computational homology, and (elementary) discrete Morse theory: as such, we pass over the usual literature review  ... 
arXiv:1606.00199v2 fatcat:shgzvlqrofde7ohkhzxoua3hoy

Computing persistent homology within Coq/SSReflect

Jónathan Heras, Thierry Coquand, Anders Mörtberg, Vincent Siles
2013 ACM Transactions on Computational Logic  
In this paper, we report on the formal development of certified programs to compute persistent Betti numbers, an instrumental tool of persistent homology, using the Coq proof assistant together with the  ...  Persistent homology is one of the most active branches of Computational Algebraic Topology with applications in several contexts such as optical character recognition or analysis of point cloud data.  ...  persistent homology. Moreover, we have formalized relevant theorems like the Fundamental Lemma of Persistent Homology.  ... 
doi:10.1145/2528929 fatcat:fpin6ia3pvgyhpzipzeoftvdaa

Computing Persistent Homology within Coq/SSReflect [article]

Jónathan Heras and Thierry Coquand and Anders Mörtberg and Vincent Siles
2012 arXiv   pre-print
In this paper, we report on the formal development of certified programs to compute persistent Betti numbers, an instrumental tool of persistent homology, using the Coq proof assistant together with the  ...  Persistent homology is one of the most active branches of Computational Algebraic Topology with applications in several contexts such as optical character recognition or analysis of point cloud data.  ...  Moreover, we have implemented certified programs to compute persistent Betti numbers, an instrumental tool in the context of persistent homology. The rest of this paper is organized as follows.  ... 
arXiv:1209.1905v1 fatcat:nkgtdzs26berfbbwfso7doph3m

Computing homology and persistent homology using iterated Morse decomposition [article]

Paweł Dłotko, Hubert Wagner
2012 arXiv   pre-print
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology.  ...  Computations of homology and persistent homology is a well established area of research with a rich history.  ...  There are many software libraries to compute homology and persistent homology. For a homology software the reader should consult [3, 6, 24] . For persistent homology [23, 8, 32] are recommended.  ... 
arXiv:1210.1429v2 fatcat:a2vansvxdfbfvcfpslov67gsna

Dory: Overcoming Barriers to Computing Persistent Homology [article]

Manu Aggarwal, Vipul Periwal
2021 arXiv   pre-print
We present Dory, an efficient and scalable algorithm that can compute the persistent homology of large data sets.  ...  Persistent homology (PH) is an approach to topological data analysis (TDA) that computes multi-scale topologically invariant properties of high-dimensional data that are robust to noise.  ...  To introduce these extant computational limitations, we briefly introduce some terminology. A general and detailed exposition on persistent homology can be found in Edelsbrunner and Harer [2008] .  ... 
arXiv:2103.05608v3 fatcat:gnvq4k7usrdhvox2cpxpzthosy
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