Inverse Problems in Topological Persistence

  • Steve OudotEmail author
  • Elchanan Solomon
Conference paper
Part of the Abel Symposia book series (ABEL, volume 15)


In this survey, we review the literature on inverse problems in topological persistence theory. The first half of the survey is concerned with the question of surjectivity, i.e. the existence of right inverses, and the second half focuses on injectivity, i.e. left inverses. Throughout, we highlight the tools and theorems that underlie these advances, and direct the reader’s attention to open problems, both theoretical and applied.


  1. 1.
    Atiyah, M., Mcdonald, I.: Commutative algebra, Addison-Wesley. Reading Mass (1969)Google Scholar
  2. 2.
    Bauer, U., Lesnick, M.: Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the thirtieth annual symposium on Computational geometry, p. 355. ACM (2014)Google Scholar
  3. 3.
    Belton, R.L., Fasy, B.T., Mertz, R., Micka, S., Millman, D.L., Salinas, D., Schenfisch, A., Schupbach, J., Williams, L.: Learning simplicial complexes from persistence diagrams. arXiv preprint arXiv:1805.10716 (2018)Google Scholar
  4. 4.
    Carlsson, G.: Topology and data. Bulletin of the American Mathematical Society 46(2), 255–308 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carlsson, G.: Going deeper: Understanding how convolutional neural networks learn using TDA (2018).
  6. 6.
    Carriere, M., Oudot, S., Ovsjanikov, M.: Local signatures using persistence diagrams (2015)Google Scholar
  7. 7.
    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. ACM (2009)Google Scholar
  8. 8.
    Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules. Springer (2016)Google Scholar
  9. 9.
    Chazal, F., De Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geometriae Dedicata 173(1), 193–214 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chazal, F., Guibas, L.J., Oudot, S.Y., Skraba, P.: Persistence-based clustering in riemannian manifolds. Journal of the ACM (JACM) 60(6), 41 (2013)Google Scholar
  11. 11.
    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proceedings of the twenty-first annual symposium on Computational geometry, pp. 263–271. ACM (2005)Google Scholar
  12. 12.
    Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and its Applications 14(05), 1550066 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Curry, J.: The fiber of the persistence map. arXiv preprint arXiv:1706.06059 (2017)Google Scholar
  14. 14.
    Curry, J., Mukherjee, S., Turner, K.: How many directions determine a shape and other sufficiency results for two topological transforms. arXiv preprint arXiv:1805.09782 (2018)Google Scholar
  15. 15.
    Dey, T.K., Shi, D., Wang, Y.: Comparing graphs via persistence distortion. arXiv preprint arXiv:1503.07414 (2015)Google Scholar
  16. 16.
    Edelsbrunner, H., Harer, J.: Computational topology: an introduction. American Mathematical Soc. (2010)Google Scholar
  17. 17.
    Gameiro, M., Hiraoka, Y., Obayashi, I.: Continuation of point clouds via persistence diagrams. Physica D: Nonlinear Phenomena 334, 118–132 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gasparovic, E., Gommel, M., Purvine, E., Sazdanovic, R., Wang, B., Wang, Y., Ziegelmeier, L.: A complete characterization of the one-dimensional intrinsic čech persistence diagrams for metric graphs. In: Research in Computational Topology, pp. 33–56. Springer (2018)Google Scholar
  19. 19.
    Ghrist, R.: Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society 45(1), 61–75 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ghrist, R., Levanger, R., Mai, H.: Persistent homology and euler integral transforms. arXiv preprint arXiv:1804.04740 (2018)Google Scholar
  21. 21.
    Ghrist, R.W.: Elementary applied topology, vol. 1. Createspace Seattle (2014)Google Scholar
  22. 22.
    Giesen, J., Cazals, F., Pauly, M., Zomorodian, A.: The conformal alpha shape filtration. The Visual Computer 22(8), 531–540 (2006)CrossRefGoogle Scholar
  23. 23.
    Hatcher, A.: Algebraic topology (2005)Google Scholar
  24. 24.
    Lee, Y., Barthel, S.D., Dłotko, P., Moosavi, S.M., Hess, K., Smit, B.: Quantifying similarity of pore-geometry in nanoporous materials. Nature communications 8, 15396 (2017)CrossRefGoogle Scholar
  25. 25.
    Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics 15(3), 613–650 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Munkres, J.R.: Elements of algebraic topology. CRC Press (2018)Google Scholar
  27. 27.
    Oudot, S., Solomon, E.: Barcode embeddings for metric graphs. arXiv:1712.03630 (2017)Google Scholar
  28. 28.
    Oudot, S.Y.: Persistence theory: from quiver representations to data analysis, vol. 209. American Mathematical Society Providence, RI (2015)Google Scholar
  29. 29.
    Poulenard, A., Skraba, P., Ovsjanikov, M.: Topological function optimization for continuous shape matching. In: Computer Graphics Forum, vol. 37, pp. 13–25. Wiley Online Library (2018)Google Scholar
  30. 30.
    Schapira, P.: Tomography of constructible functions. In: International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 427–435. Springer (1995)Google Scholar
  31. 31.
    Solomon, Y.E.: Euler curves. (2018)
  32. 32.
    Turner, K., Mukherjee, S., Boyer, D.M.: Persistent homology transform for modeling shapes and surfaces. Information and Inference: A Journal of the IMA 3(4), 310–344 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Inria SaclayIle-de-FrancePalaiseauFrance
  2. 2.DukeDurhamUSA

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