Abstract
Topology is the branch of mathematics that studies shapes and maps among them. From the algebraic definition of topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of \(k-\)dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of \(k-\)dimensional holes along such sequence is called the persistence barcode. \(k-\)dimensional holes with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological feature associated to the given data (i.e. the filtration). In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy.
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Acknowledgments
Authors are partially supported by IMUS, University of Seville under grant VPPI-US and Spanish Government under grant MTM2015-67072-P (MINECO/FEDER, UE). We also thank the reviewers for their valuable and constructive comments.
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Atienza, N., Gonzalez-Diaz, R., Rucco, M. (2016). Separating Topological Noise from Features Using Persistent Entropy. In: Milazzo, P., Varró, D., Wimmer, M. (eds) Software Technologies: Applications and Foundations. STAF 2016. Lecture Notes in Computer Science(), vol 9946. Springer, Cham. https://doi.org/10.1007/978-3-319-50230-4_1
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DOI: https://doi.org/10.1007/978-3-319-50230-4_1
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