Abstract
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample—not necessarily a uniform one—of some high-dimensional manifold. In such cases, persistent homology can be successfully employed to extract features, remove noise, and compare data sets. The underlying problems in some application domains, however, turn out to represent multiple manifolds with different dimensions. Algebraic topology typically analyzes such problems using intersection homology, an extension of homology that is capable of handling configurations with singularities. In this paper, we describe how the persistent variant of intersection homology can be used to assist data analysis in visualization. We point out potential pitfalls in approximating data sets with singularities and give strategies for resolving them.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
We remark that topologically, this case can often be reduced to the computation of ordinary homology, because a theorem of Goresky and MacPherson [14] ensures that for pseudomanifolds, the intersection homology groups remain the same under normalization, and if they are nonsingular, the intersection homology groups are ordinary homology groups. As it is not clear how to obtain normalizations for real-world data, the calculation of persistent intersection homology is necessary.
- 2.
See the unpublished notes by MacPherson on Intersection Homology and Perverse Sheaves, available under http://faculty.tcu.edu/gfriedman/notes/ih.pdf, for the origin of this name.
- 3.
References
Bendich, P.: Analyzing stratified spaces using persistent versions of intersection and local homology. Ph.D. thesis, Duke University (2009)
Bendich, P., Harer, J.: Persistent intersection homology. FoCM 11(3), 305–336 (2011)
Bendich, P., Wang, B., Mukherjee, S.: Local homology transfer and stratification learning. In: Rabani, Y. (ed.) Symposium on Discrete Algorithms, pp. 1355–1370. SIAM, Philadelphia (2012)
Carlsson, G.: Topological pattern recognition for point cloud data. Acta Numer. 23, 289–368 (2014)
Cheng, Y.: Mean shift, mode seeking, and clustering. IEEE TPAMI 17(8), 790–799 (1995)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discret. Comput. Geom. 37(1), 103–120 (2007)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincaré and Lefschetz duality. FoCM 9(1), 79–103 (2009)
Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Mileyko, Y.: Lipschitz functions have Lp-stable persistence. Found. Comput. Math. 10(2), 127–139 (2010)
de Silva, V., Morozov, D., Vejdemo-Johansson, M.: Dualities in persistent (co)homology. Inverse Probl. 27(12), 124003 (2011)
Donoho, D.L., Grimes, C.: Image manifolds which are isometric to Euclidean space. J. Math. Imaging Vision 23(1), 5–24 (2005)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. AMS, Providence (2010)
Edelsbrunner, H., Morozov, D.: Persistent homology: theory and practice. In: European Congress of Mathematics. EMS Publishing House, Zürich (2014)
Fefferman, C., Mitter, S., Narayanan, H.: Testing the manifold hypothesis. J. Am. Math. Soc. 29(4), 983–1049 (2016)
Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19(2), 135–162 (1980)
Hinton, G.E., Dayan, P., Revow, M.: Modeling the manifolds of images of handwritten digits. IEEE Trans. Neural Netw. 8(1), 65–74 (1997)
Kirwan, F., Woolf, J.: An Introduction to Intersection Homology Theory, 2nd edn. Chapman and Hall/CRC, Boca Raton (2006)
MacPherson, R., Vilonen, K.: Elementary construction of perverse sheaves. Invent. Math. 84(2), 403–435 (1986)
Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Heidelberg (2003)
Narayanan, H., Mitter, S.: Sample complexity of testing the manifold hypothesis. In: NIPS 23, pp. 1786–1794. Curran Associates, Inc., Red Hook, NY (2010)
Pratt, V.: Direct least-squares fitting of algebraic surfaces. ACM SIGGRAPH Comput. Graph. 21(4), 145–152 (1987)
Rieck, B., Leitte, H.: Persistent homology for the evaluation of dimensionality reduction schemes. Comput. Graph. Forum 34(3), 431–440 (2015)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)
Saul, L.K., Roweis, S.T.: Think globally, fit locally: unsupervised learning of low dimensional manifolds. J. Mach. Learn. Res. 4, 119–155 (2003)
Singh, G., Mémoli, F., Carlsson, G.: Topological methods for the analysis of high dimensional data sets and 3D object recognition. In: Eurographics Symposium on Point-Based Graphics. Eurographics Association, Prague (2007)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Rieck, B., Banagl, M., Sadlo, F., Leitte, H. (2020). Persistent Intersection Homology for the Analysis of Discrete Data. In: Carr, H., Fujishiro, I., Sadlo, F., Takahashi, S. (eds) Topological Methods in Data Analysis and Visualization V. TopoInVis 2017. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-43036-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-43036-8_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-43035-1
Online ISBN: 978-3-030-43036-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)