Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules
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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, 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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