Abstract
AbstractWe develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of our approach. Our directed persistent homology theory is motivated by homology with semiring coefficients: by explicitly removing additive inverses, we are able to detect directed cycles algebraically.
Funder
Engineering and Physical Sciences Research Council
Ministerio de Economía y Competitividad
Ministerio de Ciencia, Innovación y Universidades
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Geometry and Topology
Reference28 articles.
1. Bauer, U., Lesnick, M.: Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry (New York, NY, USA), SOCG’14, Association for Computing Machinery. pp. 355–364 (2014)
2. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence, RI (2001)
3. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. (N.S.) 46(2), 255–308 (2009)
4. Carlsson, G., Mémoli, F., Ribeiro, A., Segarra, S.: Hierarchical quasi-clustering methods for asymmetric networks. In: Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32, ICML’14, JMLR.org. pp. II-352–II-360 (2014)
5. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry (New York, NY, USA), SCG ’09, Association for Computing Machinery. pp. 237–246 (2009)
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