Abstract
Metrics of interest in topological data analysis (TDA) are often explicitly or implicitly in the form of an interleaving distance \({d_{\textrm{I}}}\) between 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 map \(\textbf{F}:{\mathcal {P}}\rightarrow {\mathcal {Q}}\) as a join (i.e. supremum) \(\bigvee _{b\in B} \varvec{\textbf{F}}_{b}\) of simpler poset maps \(\varvec{\textbf{F}}_{b}\) (for a join dense subset \({B}\subset {\mathcal {Q}}\)) which in turn yields a decomposition of \({d_{\textrm{I}}}\) into a product metric. The decomposition of \({d_{\textrm{I}}}\) is simple, but its ramifications are manifold: (1) We can construct a geodesic path between any poset maps \(\textbf{F}\) and \(\textbf{G}\) with \({d_{\textrm{I}}}(\varvec{\textbf{F}},\varvec{\textbf{G}})<\varvec{\infty }\) by assembling geodesics between all \(\varvec{\textbf{F}}_{b}\)s and \(\varvec{\textbf{G}}_{b}\)s 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 the erosion distance by Patel and the graded rank function by Betthauser, Bubenik, and Edwards, which in turn takes us to an interpretation on the representation \(\bigvee _{b} \varvec{\textbf{F}}_{b}\) as a generalization of persistence landscapes and graded rank functions.
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References
Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)
Edelsbrunner, H., Harer, J.: Persistent homology-a survey. Contemp. Math. 453, 257–282 (2008)
Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. 45(1), 61–75 (2008)
Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discrete Comput. Geom. 51(3), 600–627 (2014)
Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp. 237–246 (2009). ACM
Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebr. Geom. Topol. 18(6), 3133–3204 (2018)
Botnan, M., Curry, J., Munch, E.: A relative theory of interleavings. ar**v preprint ar**v:2004.14286 (2020)
Bubenik, P., De Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15(6), 1501–1531 (2015)
Curry, J.: Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania (2013)
De Silva, V., Munch, E., Patel, A.: Categorified Reeb graphs. Discrete Comp. Geom. 55(4), 854–906 (2016)
de Silva, V., Munch, E., Stefanou, A.: Theory of interleavings on categories with a flow. Theory Appl. Categ. 33(21), 583–607 (2018)
Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15(3), 613–650 (2015)
Scoccola, L.N.: Locally persistent categories and metric properties of interleaving distances. PhD thesis, The University of Western Ontario (2020)
Bjerkevik, H.B., Botnan, M.B., Kerber, M.: Computing the interleaving distance is NP-hard. Found. Comput. Math. 1–35 (2019)
Bjerkevik, H.B., Kerber, M.: Asymptotic improvements on the exact matching distance for 2-parameter persistence. ar**v preprint ar**v:2111.10303 (2021)
Cerri, A., Fabio, B.D., Ferri, M., Frosini, P., Landi, C.: Betti numbers in multidimensional persistent homology are stable functions. Math. Methods Appl. Sci. 36(12), 1543–1557 (2013)
Kerber, M., Lesnick, M., Oudot, S.: Exact Computation of the Matching Distance on 2-Parameter Persistence Modules. In: 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), vol. 129, pp. 46–14615. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2019). https://doi.org/10.4230/LIPIcs.SoCG.2019.46. http://drops.dagstuhl.de/opus/volltexte/2019/10450
Landi, C.: The rank invariant stability via interleavings. In: Research in Computational Topology, pp. 1–10. Springer, Switzerland (2018)
Patel, A.: Generalized persistence diagrams. J. Appl. Comput. Topol. 1(3–4), 397–419 (2018)
Kozlov, D.: Combinatorial Algebraic Topology vol. 21. Springer, (2008)
Mémoli, F.: A distance between filtered spaces via tripods. ar**v preprint ar**v:1704.03965 (2017)
Mémoli, F., Okutan, O.B.: Quantitative simplification of filtered simplicial complexes. Discrete Comput. Geom. 65, 554–583 (2021)
Blumberg, A.J., Lesnick, M.: Universality of the homotopy interleaving distance, Transaction of the American Mathematical Society, https://doi.org/10.1090/tran/8738
Bauer, U., Landi, C., Mémoli, F.: The Reeb graph edit distance is universal. Foundations of Computational Mathematics (2020)
Kim, W., Mémoli, F., Smith, Z.: Analysis of dynamic graphs and dynamic metric spaces via zigzag persistence. In: Topological Data Analysis, pp. 371–389. Springer, ??? (2020)
Carlsson, G., Mémoli, F.: Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res. 11(Apr), 1425–1470 (2010)
Cai, C., Kim, W., Mémoli, F., Wang, Y.: Elder-rule staircodes for augmented metric spaces. In: Proceedings of the Thirty-sixth International Symposium on Computational Geometry (SoCG 2020) (2020)
Carlsson, G.,Mémoli, F.: Multiparameter hierarchical clustering methods. In: Classification as a Tool for Research, pp. 63–70. Springer, ??? (2010)
Rolle, A., Scoccola, L.: Stable and consistent density-based clustering. ar** structure. In: Workshop on Algorithms and Data Structures, pp. 219–230 (2013). Springer
Kim, W., Mémoli, F.: Formigrams: Clustering summaries of dynamic data. In: Proceedings of the Thirtieth Canadian Conference on Computational Geometry, pp. 180–188 (2018)
Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27(4), 733–767 (2001)
Sokal, R.R., Rohlf, F.J.: The comparison of dendrograms by objective methods. Taxon, 33–40 (1962)
Woese, C.R., Kandler, O., Wheelis, M.L.: Towards a natural system of organisms: proposal for the domains archaea, bacteria, and eucarya. Proc. Nat. Acad. Sci. 87(12), 4576–4579 (1990)
Griffiths, R.C., Marjoram, P.: An ancestral recombination graph. In: Donnelly, P. and Tavaré, S. (Eds.), Progress in Population Genetics and Human Evolution, IMA Volumes in Mathematics and Its Applications vol. 87, pp. 257–270. Springer, Berlin (1997)
Huson, D.H., Rupp, R., Scornavacca, C.: Phylogenetic Networks: Concepts. Algorithms and Applications. Cambridge University Press, Cambridge, UK (2010)
Martin, S.H., Dasmahapatra, K.K., Nadeau, N.J., Salazar, C., Walters, J.R., Simpson, F., Blaxter, M., Manica, A., Mallet, J., Jiggins, C.D.: Genome-wide evidence for speciation with gene flow in heliconius butterflies. Genome Res. 23(11), 1817–1828 (2013)
Parida, L., Utro, F., Yorukoglu, D., Carrieri, A.P., Kuhn, D., Basu, S.: Topological signatures for population admixture. In: International Conference on Research in Computational Molecular Biology, pp. 261–275 (2015). Springer
Gasparovic, E., Munch, E., Oudot, S., Turner, K., Wang, B., Wang, Y.: Intrinsic interleaving distance for merge trees. ar**v preprint ar**v:1908.00063 (2019)
Morozov, D., Beketayev, K., Weber, G.: Interleaving distance between merge trees. In: Proceedings of Topology-Based Methods in Visualization (2013)
Bauer, U., Munch, E., Wang, Y.: Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. In: Arge, L., Pach, J. (eds.) 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), vol. 34, pp. 461–475. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2015). https://doi.org/10.4230/LIPIcs.SOCG.2015.461
Chambers, E.W., Munch, E., Ophelders, T.: A Family of Metrics from the Truncated Smoothing of Reeb Graphs. In: 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), vol. 189, pp. 22–12217. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2021). https://doi.org/10.4230/LIPIcs.SoCG.2021.22. https://drops.dagstuhl.de/opus/volltexte/2021/13821
Stefanou, A.: Tree decomposition of Reeb graphs, parametrized complexity, and applications to phylogenetics. J. Appl. Comput. Topol. (2020). https://doi.org/10.1007/s41468-020-00051-1
Kim, W., Mémoli, F.: Extracting persistent clusters in dynamic data via Möbius inversion. ar**v preprint ar**v:1712.04064v5 (2022), to appear in Discrete Comput. Geom.
Weibel, C.A.: The K-book: An Introduction to Algebraic K-theory, vol. 145. American Mathematical Society, Providence, RI (2013)
Kim, W., Mémoli, F.: Generalized persistence diagrams for persistence modules over posets. J. Appl. Comput. Topol. 5(4), 533–581 (2021)
Puuska, V.: Erosion distance for generalized persistence modules. Homol. Homotopy Appl. 22(1), 233–254 (2020)
Kim, W., Mémoli, F.: Spatiotemporal persistent homology for dynamic metric spaces. Discrete Comput. Geom. 66(3), 831–875 (2021)
Clause, N., Kim, W.: Spatiotemporal Persistent Homology Computation Tool. https://github.com/ndag/PHoDMSs (2020)
Bubenik, P., De Silva, V., Scott, J.: Interleaving and Gromov-Hausdorff distance. ar**v preprint ar**v:1707.06288 (2017)
Betthauser, L., Bubenik, P., Edwards, P.B.: Graded persistence diagrams and persistence landscapes. Discrete Comput. Geom. 67(1), 203–230 (2022)
Bubenik, P., et al.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015)
Chazal, F., Cohen-Steiner, D., Guibas, L.J., Mémoli, F., Oudot, S.Y.: Gromov-Hausdorff stable signatures for shapes using persistence. Comput. Graph. Forum 28(5), 1393–1403 (2009). Wiley Online Library
Chazal, F., De Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicata 173(1), 193–214 (2014)
Erné, M., Šešelja, B., Tepavčević, A.: Posets generated by irreducible elements. Order 20(1), 79–89 (2003)
Roman, S.: Lattices and Ordered Sets. Springer, New York (2008)
Erné, M.: Compact generation in partially ordered sets. J. Aust. Math. Soc. 42(1), 69–83 (1987)
Curry, J., Patel, A.: Classification of constructible cosheaves. Theory Appl. Categ. 35(27), 1012–1047 (2020)
Cardona, G., Mir, A., Rosselló, F., Rotger, L., Sánchez, D.: Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf. BMC Bioinform. 14(1), 3 (2013)
Munch, E., Stefanou, A.: The \(\ell ^{\infty }\)-cophenetic metric for phylogenetic trees as an interleaving distance. In: Research in Data Science, pp. 109–127. Springer, Switzerland (2019)
Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162–191 (2015)
Serra, J.: Hausdorff distances and interpolations. Comput. Imaging Vis. 12, 107–114 (1998)
Chowdhury, S.: Geodesics in persistence diagram space. ar**v preprint ar**v:1905.10820 (2019)
Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules, vol. 10. Springer, Switzerland (2016)
Burago, D., Burago, I.D., Burago, Y., Ivanov, S.A., Ivanov, S.: A Course in Metric Geometry, vol. 33. American Mathematical Soc, Providence, Rhode Island (2001)
Chowdhury, S., Mémoli, F.: Explicit geodesics in Gromov-Hausdorff space. Electron. Res. Announc. 25, 48–59 (2018)
Munkres, J.R.: Elements of Algebraic Topology. CRC Press, Boca Raton, FL, USA (2018)
Barmak, J.A.: On quillen’s theorem a for posets. J. Comb. Theory Ser. A 118(8), 2445–2453 (2011)
Bakke Bjerkevik, H.: On the stability of interval decomposable persistence modules. Discrete Comput. Geom. 66(1), 92–121 (2021)
Schmiedl, F.: Computational aspects of the gromov-hausdorff distance and its application in non-rigid shape matching. Discrete Comput. Geom. 57(4), 854–880 (2017)
Erickson, J.: Algorithms. Independent Publish, Urbana-Champaign, IL (2019)
McCleary, A., Patel, A.: Edit distance and persistence diagrams over lattices. SIAM J. Appl. Algebra Geom. 6(2), 134–155 (2022)
Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer, New York (2013)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Courier Corporation, Mineola, NY (2001)
Kerber, M., Morozov, D., Nigmetov, A.: Geometry helps to compare persistence diagrams. J. Exp. Algorithmics 22, 1–20 (2017)
Acknowledgements
WK thanks Parker Edwards, Alex McCleary and Justin Curry for beneficial discussions. We also thank Zane Smith for hel** with Theorem E.1. The authors thank the anonymous reviewer for their suggestions and corrections.
Funding
WK and FM were supported by NSF through grants DMS-1723003, CCF-1740761, RI-1901360, and CCF-1526513. AS was supported by NSF through grants CCF-1740761, DMS-1440386 and RI-1901360.
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Kim, W., Mémoli, F. & Stefanou, A. Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of Geodesics. Order (2023). https://doi.org/10.1007/s11083-023-09643-9
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DOI: https://doi.org/10.1007/s11083-023-09643-9