Abstract
We present a generalization of the induced matching theorem of as reported by Bauer and Lesnick (in: Proceedings of the thirtieth annual symposium computational geometry 2014) and use it to prove a generalization of the algebraic stability theorem for \({\mathbb {R}}\)-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauer, U., Lesnick, M.: Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the Thirtieth Annual Symposium Computational Geometry p. 355 (2014)
Bauer, U., Lesnick, M.: Persistence diagrams as diagrams: A categorification of the stability theorem. (2016) arXiv:1610.10085
Botnan, M., Spreemann, G.: Approximating persistent homology in euclidean space through collapses. Appl. Algebra Eng. Commun. Comput. 26(1–2), 73–101 (2015). https://doi.org/10.1007/s00200-014-0247-y
Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discrete Comput. Geom. 51(3), 600–627 (2014). https://doi.org/10.1007/s00454-014-9573-x
Bubenik, P., de Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15(6), 1501–1531 (2015). https://doi.org/10.1007/s10208-014-9229-5
Bubenik, P., de Silva, V., Scott, J.: Categorification of gromov-hausdorff distance and interleaving of functors (2017). arXiv:1707.06288
Buchet, M., Chazal, F., Oudot, S.Y., Sheehy, D.R.: Efficient and robust persistent homology for measures. Comput. Geom. 58, 70–96 (2016). https://doi.org/10.1016/j.comgeo.2016.07.001
Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, LJ., Oudot, SY.: Proximity of persistence modules and their diagrams. In: Proceedings of the twenty-fifth annual symposium on computational geometry, ACM, New York, NY, USA, SCG ’09, pp. 237–246, (2009). https://doi.org/10.1145/1542362.1542407
Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules (SpringerBriefs in Mathematics). Springer, Berlin (2016)
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
Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(05), 1550066 (2015). https://doi.org/10.1142/S0219498815500668
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, p. xii+298. Cambridge University Press, Cambridge (2002)
Dey, TK., Fan, F., Wang, Y.: Graph induced complex on point data. In: Proceedings of the twenty-ninth annual symposium on computational geometry, ACM, New York, NY, USA, SoCG ’13, pp 107–116, https://doi.org/10.1145/2462356.2462387 (2013)
Dey, TK., Fan, F., Wang, Y.: Computing topological persistence for simplicial maps. In: Proceedings of the thirtieth annual symposium on computational geometry, ACM, New York, NY, USA, SOCG’14, pp 345:345–345:354, (2014) https://doi.org/10.1145/2582112.2582165
Edelsbrunner, H., Harer, J.L.: Computational Topology : an Introduction. American Mathematical Society, Providence (2010)
Friedman, G.: Survey article: an elementary illustrated introduction to simplicial sets. Rocky Mountain J. Math. 42(2), 353–423 (2012). https://doi.org/10.1216/RMJ-2012-42-2-353
Kramár, M., Levanger, R., Tithof, J., Suri, B., Xu, M., Paul, M., Schatz, M.F., Mischaikow, K.: Analysis of Kolmogorov flow and Rayleigh–Bénard convection using persistent homology. Phys D 334, 82–98 (2016)
Oudot, S.Y.: Persistence Theory: from Quiver Representations to Data Analysis, Mathematical Surveys and Monographs, vol. 209. American Mathematical Society, Providence (2015)
Sheehy, D.: Linear-size approximations to the vietoris? rips filtration. Discrete Comput. Geom. 49(4), 778–796 (2013). https://doi.org/10.1007/s00454-013-9513-1
Weibel, C.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995)
Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2004). https://doi.org/10.1007/s00454-004-1146-y
Acknowledgements
R. L. would like to thank Charles Weibel, Michael Lesnick, and Ulrich Bauer for the many insightful discussions that led to the results presented in this paper. The authors also thank the anonymous reviewers for their suggested corrections. On behalf of all authors, the corresponding author states that there is no conflict of interest.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
S. H. and K. M. were partially supported by Grants NSF-DMS-1125174, 1248071, 1521771, NIH 1R01GM126555-01 and DARPA contracts HR0011-16-2-0033, FA8750-17-C-0054. M. K. was supported by ERC Gudhi (ERC-2013-ADG-339025). R. L. was supported by DARPA contracts HR0011-17-1-0004 and HR0011-16-2-0033.
Rights and permissions
About this article
Cite this article
Harker, S., Kramár, M., Levanger, R. et al. A comparison framework for interleaved persistence modules. J Appl. and Comput. Topology 3, 85–118 (2019). https://doi.org/10.1007/s41468-019-00026-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41468-019-00026-x