Sheaf-theoretic stratification learning from geometric and topological perspectives

Brown A, Wang B. 2020. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry.

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Abstract
We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.
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Date Published
2020-05-29
Journal Title
Discrete and Computational Geometry
Acknowledgement
Open access funding provided by Institute of Science and Technology (IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375. The authors would like to thank the anonymous referees for their insightful comments.
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Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 2020. doi:10.1007/s00454-020-00206-y
Brown, A., & Wang, B. (2020). Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00206-y
Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” Discrete and Computational Geometry. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00206-y.
A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric and topological perspectives,” Discrete and Computational Geometry. Springer Nature, 2020.
Brown A, Wang B. 2020. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry.
Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” Discrete and Computational Geometry, Springer Nature, 2020, doi:10.1007/s00454-020-00206-y.
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