
Notes on pivot pairings
We present a row reduction algorithm to compute the barcode decompositio...
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Fast Minimal Presentations of Bigraded Persistence Modules
Multiparameter persistent homology is a recent branch of topological da...
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Computing Height Persistence and Homology Generators in R^3 Efficiently
Recently it has been shown that computing the dimension of the first hom...
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Computing Zigzag Persistence on Graphs in NearLinear Time
Graphs model realworld circumstances in many applications where they ma...
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Computing Minimal Presentations and Betti Numbers of 2Parameter Persistent Homology
Motivated by applications to topological data analysis, we give an effic...
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On rectangledecomposable 2parameter persistence modules
This paper addresses two questions: (1) can we identify a sensible class...
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Computing Minimal Presentations and Bigraded Betti Numbers of 2Parameter Persistent Homology
Motivated by applications to topological data analysis, we give an effic...
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Generalized Persistence Algorithm for Decomposing Multiparameter Persistence Modules
The classical persistence algorithm virtually computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over integers Z giving rise to a 1parameter persistence module. It has been recognized that multiparameter version of persistence modules given by simplicial filtrations over ddimensional integer grids Z^d is equally or perhaps more important in data science applications. However, in the multiparameter setting, one of the main bottlenecks is that topological summaries such as barcodes and distances among them cannot be as efficiently computed as in the 1parameter case because there is no known generalization of the persistence algorithm for computing the decomposition of multiparameter persistence modules. The Meataxe algorithm, the only known one for computing such a decomposition runs in Õ(n^6(d+1)) time where n is the size of input module. We present for the first time a generalization of the persistence algorithm based on a generalized matrix reduction technique that runs in O(n^2ω) time where ω<2.373 is the exponent for matrix multiplication. We also present an O(n^d+1) algorithm to convert the input filtration to a suitable matrix called presentation matrix which is ultimately decomposed. Various structural and computational results connecting the graded modules from commutative algebra to matrix reductions are established through the course.
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