Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics
Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics
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DOI：
10.1007/s11083-023-09643-9
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发表时间：
2019-12
期刊：
影响因子：
--
通讯作者：
Woojin Kim;Facundo M'emoli;Anastasios Stefanou
中科院分区：
文献类型：
--
作者：
Woojin Kim;Facundo M'emoli;Anastasios Stefanou
Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distancebetween poset maps (i.e. order-preserving maps), e.g. the Gromov-Hausdorff distance between metric spaces can be reformulated in this way. We propose a representation of a poset mapas a join (i.e. supremum)of simpler poset maps(for a join dense subset) which in turn yields a decomposition ofinto a product metric. The decomposition ofis simple, but its ramifications are manifold: (1) We can construct a geodesic path between any poset mapsandwith \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_{\textrm{I}}}(\varvec{\textbf{F}},\varvec{\textbf{G}})<\varvec{\infty }$$\end{document} by assembling geodesics between alls ands via the join operation. This construction generalizes at least three constructions of geodesic paths that have appeared in the literature. (2) We can extend the Gromov-Hausdorff distance to a distance between simplicial filtrations over an arbitrary poset with a flow, preserving its universality and geodesicity. (3) We can clarify equivalence between several known metrics on multiparameter hierarchical clusterings. (4) We can illuminate the relationship between theerosion distanceby Patel and thegraded rank functionby Betthauser, Bubenik, and Edwards, which in turn takes us to an interpretation on the representationas a generalization of persistence landscapes and graded rank functions.
Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distancebetween poset maps (i.e. order-preserving maps), e.g. the Gromov-Hausdorff distance between metric spaces can be reformulated in this way. We propose a representation of a poset mapas a join (i.e. supremum)of simpler poset maps(for a join dense subset) which in turn yields a decomposition ofinto a product metric. The decomposition ofis simple, but its ramifications are manifold: (1) We can construct a geodesic path between any poset mapsandwith \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d_{\textrm{I}}}(\varvec{\textbf{F}},\varvec{\textbf{G}})<\varvec{\infty }$$\end{document} by assembling geodesics between alls ands via the join operation. This construction generalizes at least three constructions of geodesic paths that have appeared in the literature. (2) We can extend the Gromov-Hausdorff distance to a distance between simplicial filtrations over an arbitrary poset with a flow, preserving its universality and geodesicity. (3) We can clarify equivalence between several known metrics on multiparameter hierarchical clusterings. (4) We can illuminate the relationship between theerosion distanceby Patel and thegraded rank functionby Betthauser, Bubenik, and Edwards, which in turn takes us to an interpretation on the representationas a generalization of persistence landscapes and graded rank functions.