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Correspondence modules and persistence sheaves: a unifying perspective on one-parameter persistent homology

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Abstract

We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.

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References

  1. Berkouk, N., Ginot, G.: A derived isometry theorem for constructible sheaves on \(\mathbb{R} \). arXiv:1805.09694 (2018)

  2. Berkouk, N., Ginot, G., Oudot, S.: Level-sets persistence and sheaf theory. arXiv:1907.09759 (2019)

  3. Bjerkevik, H.B.: Stability of higher-dimensional interval decomposable persistence modules. arXiv:1609.02086v2 (2016)

  4. Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebra Geom. Topol. 18(6), 3133–3204 (2018). https://doi.org/10.2140/agt.2018.18.3133

    Article  MathSciNet  MATH  Google Scholar 

  5. Botnan, M.B., Crawley-Boevey, W.: Decomposition of persistence modules. Proc. Am. Math. Soc. (in press). https://doi.org/10.1090/proc/14790

  6. Bubenik, P.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Carlsson, G., de Silva, V.: Zigzag persistence. Found. Comput. Math. 10(4), 367–405 (2010). https://doi.org/10.1007/s10208-010-9066-0

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlsson, G., de Silva, V., Kališnik, S., Morozov, D.: Parametrized homology via zigzag persistence. Algebraic Geom. Topol. 19(2), 657–700 (2019). https://doi.org/10.2140/agt.2019.19.657

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlsson, G., de Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, SCG ’09, pp. 247–256. Association for Computing Machinery, New York, NY, USA (2009). https://doi.org/10.1145/1542362.1542408

  10. Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71–93 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11(02), 149–187 (2005). https://doi.org/10.1142/S0218654305000761

    Article  MATH  Google Scholar 

  12. 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, SCG ’09, pp. 237–246. Association for Computing Machinery, New York, NY, USA (2009). https://doi.org/10.1145/1542362.1542407

  13. 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). https://doi.org/10.1111/j.1467-8659.2009.01516.x

    Article  Google Scholar 

  14. Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules. Springer Briefs in Mathematics. Springer International Publishing (2016). https://doi.org/10.1007/978-3-319-42545-0

  15. Cochoy, J., Oudot, S.: Decomposition of exact pfd persistence bimodules. Discrete Comput. Geom. 63(2), 255–293 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007). https://doi.org/10.1007/s00454-006-1276-5

    Article  MathSciNet  MATH  Google Scholar 

  17. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math. 9, 133–134 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(05), 1550066 (2015). https://doi.org/10.1142/S0219498815500668

    Article  MathSciNet  MATH  Google Scholar 

  19. Curry, J.: Sheaves, cosheaves and applications. arXiv:1303.3255 (2013)

  20. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete. Comput. Geom. 28(4), 511–533 (2002). https://doi.org/10.1007/s00454-002-2885-2

    Article  MathSciNet  MATH  Google Scholar 

  21. Frosini, P.: A distance for similarity classes of submanifolds of a Euclidean space. Bull. Aust. Math. Soc. 42(3), 407–415 (1990). https://doi.org/10.1017/S0004972700028574

    Article  MathSciNet  MATH  Google Scholar 

  22. Frosini, P., Landi, C., Mémoli, F.: The persistent homotopy type distance. Homol. Homotopy Appl. 21(2), 231–259 (2019). https://doi.org/10.4310/hha.2019.v21.n2.a13

    Article  MathSciNet  MATH  Google Scholar 

  23. Gabriel, P.: Unzerlegbare darstellungen I. Manuscr. Math. 6(1), 71–103 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hang, H., Mémoli, F., Mio, W.: A topological study of functional data and Fréchet functions of metric measure spaces. J. Appl. Comput. Topol. 3(4), 359–380 (2019). https://doi.org/10.1007/s41468-019-00037-8

    Article  MathSciNet  MATH  Google Scholar 

  25. Kashiwara, M., Schapira, P.: Persistent homology and microlocal sheaf theory. J. Appl. Comput. Topol. 2(1–2), 83–113 (2018). https://doi.org/10.1007/s41468-018-0019-z

    Article  MathSciNet  MATH  Google Scholar 

  26. Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15(3), 613–650 (2015). https://doi.org/10.1007/s10208-015-9255-y

    Article  MathSciNet  MATH  Google Scholar 

  27. Lesnick, M., Wright, M.: Interactive visualization of 2-d persistence modules. arXiv:1512.00180 (2015)

  28. Milnor, J.: On the Steenrod homology theory (first distributed 1961). In: S.C. Ferry, A. Ranicki, J.M. Rosenberg (eds.) Novikov Conjectures, Index Theorems, and Rigidity: Oberwolfach 1993, London Mathematical Society Lecture Note Series, vol. 1, pp. 79–96. Cambridge University Press (1995). https://doi.org/10.1017/CBO9780511662676.005

  29. Oudot, S.Y.: Persistence theory: from quiver representations to data analysis, vol. 209. American Mathematical Society Providence (2015)

  30. Patel, A.: Generalized persistence diagrams. J. Appl. Comput. Topol. 1(3–4), 397–419 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Robins, V.: Towards computing homology from finite approximations. In: Topology Proceedings, vol. 24, pp. 503–532 (1999)

  32. Skryzalin, J., Carlsson, G.: Numeric invariants from multidimensional persistence. J. Appl. Comput. Topol. 1(1), 89–119 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete. Comput. Geom. 33(2), 249–274 (2005). https://doi.org/10.1007/s00454-004-1146-y

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was partially supported by NSF grant DMS-1722995.

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  1. Department of Mathematics, Florida State University, Tallahassee, FL, 32306-4510, USA

    Haibin Hang & Washington Mio

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  1. Haibin Hang
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  2. Washington Mio
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Hang, H., Mio, W. Correspondence modules and persistence sheaves: a unifying perspective on one-parameter persistent homology. Japan J. Indust. Appl. Math. 40, 41–93 (2023). https://doi.org/10.1007/s13160-022-00517-y

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  • DOI: https://doi.org/10.1007/s13160-022-00517-y

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