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Inverse problems in topological persistence. (English) Zbl 1447.55006
Baas, Nils (ed.) et al., Topological data analysis. Proceedings of the Abel symposium 2018, Geiranger, Norway, June 4–8, 2018. Cham: Springer. Abel Symp. 15, 405-433 (2020).
Summary: 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.
For the entire collection see [Zbl 1448.62008].
MSC:
55N31 Persistent homology and applications, topological data analysis
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
68T09 Computational aspects of data analysis and big data
62R07 Statistical aspects of big data and data science
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[1] Atiyah, M., Mcdonald, I.: Commutative algebra, Addison-Wesley. Reading Mass (1969)
[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) · Zbl 1395.68289
[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)
[4] Carlsson, G.: Topology and data. Bulletin of the American Mathematical Society 46(2), 255-308 (2009) · Zbl 1172.62002
[5] Carlsson, G.: Going deeper: Understanding how convolutional neural networks learn using TDA (2018). https://www.ayasdi.com/blog/artificial-intelligence/going-deeper-understanding-convolutional-neural-networks-learn-using-tda/
[6] Carriere, M., Oudot, S., Ovsjanikov, M.: Local signatures using persistence diagrams (2015)
[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) · Zbl 1380.68387
[8] Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The structure and stability of persistence modules. Springer (2016) · Zbl 1362.55002
[9] Chazal, F., De Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geometriae Dedicata 173(1), 193-214 (2014) · Zbl 1320.55003
[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) · Zbl 1281.68176
[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) · Zbl 1387.68252
[12] Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and its Applications 14(05), 1550066 (2015) · Zbl 1345.16015
[13] Curry, J.: The fiber of the persistence map. arXiv preprint arXiv:1706.06059 (2017)
[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)
[15] Dey, T.K., Shi, D., Wang, Y.: Comparing graphs via persistence distortion. arXiv preprint arXiv:1503.07414 (2015) · Zbl 1378.05037
[16] Edelsbrunner, H., Harer, J.: Computational topology: an introduction. American Mathematical Soc. (2010) · Zbl 1193.55001
[17] Gameiro, M., Hiraoka, Y., Obayashi, I.: Continuation of point clouds via persistence diagrams. Physica D: Nonlinear Phenomena 334, 118-132 (2016) · Zbl 1415.55006
[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) · Zbl 1422.55037
[19] Ghrist, R.: Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society 45(1), 61-75 (2008) · Zbl 1391.55005
[20] Ghrist, R., Levanger, R., Mai, H.: Persistent homology and euler integral transforms. arXiv preprint arXiv:1804.04740 (2018) · Zbl 07089249
[21] Ghrist, R.W.: Elementary applied topology, vol. 1. Createspace Seattle (2014) · Zbl 1427.55001
[22] Giesen, J., Cazals, F., Pauly, M., Zomorodian, A.: The conformal alpha shape filtration. The Visual Computer 22(8), 531-540 (2006)
[23] Hatcher, A.: Algebraic topology (2005) · Zbl 1044.55001
[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)
[25] Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics 15(3), 613-650 (2015) · Zbl 1335.55006
[26] Munkres, J.R.: Elements of algebraic topology. CRC Press (2018) · Zbl 0673.55001
[27] Oudot, S., Solomon, E.: Barcode embeddings for metric graphs. arXiv:1712.03630 (2017)
[28] Oudot, S.Y.: Persistence theory: from quiver representations to data analysis, vol. 209. American Mathematical Society Providence, RI (2015) · Zbl 1335.55001
[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)
[30] Schapira, P.: Tomography of constructible functions. In: International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 427-435. Springer (1995) · Zbl 0878.44002
[31] Solomon, Y.E.: Euler curves. https://github.com/IsaacSolomon/EulerCurves (2018)
[32] Turner, K. · Zbl 06840289
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