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Persistent Intersection Homology for the Analysis of Discrete Data

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Part of the Mathematics and Visualization book series (MATHVISUAL)

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.

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Notes

  1. 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. 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    See the unpublished notes by MacPherson on Intersection Homology and Perverse Sheaves, available under http://faculty.tcu.edu/gfriedman/notes/ih.pdf, for the origin of this name.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. 3.

    https://github.com/Submanifold/Aleph.

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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

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