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 1-parameter persistence module. It has been recognized that multi-parameter version of persistence modules given by simplicial filtrations over d-dimensional integer grids Z^d is equally or perhaps more important in data science applications. However, in the multi-parameter 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 1-parameter case because there is no known generalization of the persistence algorithm for computing the decomposition of multi-parameter persistence modules. The Meataxe algorithm , a popular one known for computing such a decomposition runs in (n^{6(d+1)  on a generalized matrix reduction technique that runs in O( n ^{2\omega+1}) time for d=2 and in O(n^{d(2\omega +1)) time for d>2 where \omega<2.373 is the exponent for matrix multiplication. Various structural and computational results connecting the graded modules from commutative algebra to matrix reductions are established through the course.} (N^{6(d+1)}) time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of n simplices, our algorithm runs in O(n^{2\omega+1}) time for d=2 and in O(n^{d(2\omega + 1)}) time for d>2.}
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 the integers Z giving rise to a 1-parameter persistence module. It has been recognized that multi-parameter version of persistence modules given by simplicial filtrations over d-dimensional integer grids Z^d is equally or perhaps more important in data science applications. However, in the multi-parameter setting, one of the main challenges is that topological summaries based on algebraic structure such as decompositions and bottleneck distances cannot be as efficiently computed as in the 1-parameter case because there is no known extension of the persistence algorithm to multi-parameter persistence modules. We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module M defined over the multiparameter \mathbb{Z distinctly graded. Based on a generalized matrix reduction technique it runs in O( N ^{2\omega+1}) time ) time for d>2 where \omega<2.373 is the exponent for matrix multiplication. Various structural and computational results connecting the graded modules from commutative algebra to matrix reductions are established through the course.} where \omega<2.373 is the exponent for matrix multiplication. This is much better than the well known algorithm called Meataxe which runs in \tilde{O(N^{6(d+1)}) time on such an input. In practice, persistence modules are usually induced by simplicial filtrations. With such an input consisting of n simplices, our algorithm runs in O(n^{2\omega+1}) time for d=2 and in O(n^{d(2\omega + 1)}) time for d>2.}
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such as barcodes and distances among them
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generalization
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The Meataxe algorithm , a popular one known for computing such a decomposition runs in
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(n^{6(d+1)
We present an efficient algorithm to compute the unique decomposition of a finitely presented persistence module M defined over the multiparameter \mathbb{Z
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distinctly graded
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for d=2 and in O(n^{d(2\omega +1)
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where \omega<2.373 is the exponent for matrix multiplication. This is much better than the well known algorithm called Meataxe which runs in \tilde{O