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An Introduction to Multiparameter Persistence
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In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped with a multiparameter filtration. Multiparameter persistence has become one of the most active areas of research within TDA, with exciting progress on several fronts. In this article, we introduce multiparameter persistence and survey some of this recent progress, with a focus on ideas likely to lead to practical applications in the near future.
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Cited by 4 Pith papers
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The fiber of multiparameter persistent homology for simplicial complexes
For fixed simplicial complexes, the fibers of multiparameter persistent homology are trivial polyhedral bundles over each stratum, with dimension bounded by multigraded Betti numbers.
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Betti curves and kNN distributions give similar cosmological constraints in Quijote simulations, with beta0/beta1 dominating Betti information and the two statistics only partially redundant.
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Topology of Shape and Data in Material Microstructures
A dual-parameter persistence summary I, integrating Betti-1 counts over shape-distance and spatial-scale, rises monotonically with strain and jumps sharply between 8% and 12% in four EBSD ice microstructures.
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Path representations in multiparameter persistent homology
A multiparameter persistence distance is defined by taking persistence along monotone piecewise-linear paths instead of straight slices, generalizing the matching distance.
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