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**DOI: **10.18287/2412-6179-2019-43-4-611-617

**Pages: **611-617.

**Full text of article:** Russian language.

Abstract**:**

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

Keywords:

machine learning, topological invariants, degree of a map, image processing

Citation: Kurochkin SV. Detection of the homotopy type of an object using differential invariants of an approximating map. Computer Optics 2019; 43(4): 611-617. DOI: 10.18287/2412-6179-2019-43-4-611-617.

References:

- Carlsson G. Topology and data. Bulletin of the American Mathematical Society 2009; 46(2): 255-308. DOI: 10.1090/S0273-0979-09-01249-X.

- Zomorodian A. Topological data analysis. In Book: Zomorodian A, ed. Advances in applied and computational topology. American Mathematical Society; 2012: 1-40. ISBN: 978-0-8218-5327-6.

- Mervis J. What makes DARPA tick? Science 2016; 351(6273): 549-553.

- DARPA – Frontiers of engineering. Source: <https://www.naefrontiers.org/File.aspx?id=22017>.

- Rogozin DO, Sheremet IA, Garbuk SV, Gubinskii AM. High technologies in the USA: The experience of the Defence Ministry and other Institutions [In Russian]. Moscow: "MSU" Publisher; 2013.

- Edelsbrunner H, Letscher D, Zomorodian A. Topological persistence and simplification. Discrete and Computational Geometry 2002; 28(4): 511-533. DOI: 10.1007/s00454-002-2885-2.

- Adamaszek M, Frick F, Vakili A. On homotopy types of euclidean rips complexes. Source: <https://arxiv.org/pdf/1602.04131.pdf>.

- Postnikov M.M. Introduction to Morse theory [In Russian]. Moscow: "Nauka" Publisher; 1971.

- Fomenko A, Fuchs D. Homotopical topology. 2th ed. Switzerland: Springer International Publishing; 2016.

- Demidov EE, et al. Nonlinear correlation analysis [In Russian]. Obozrenie Prikladnoy i Promyshlennoy Matemetiki 1999; 6(1): 4-57.

- Chazal F, Cohen-Steiner D, Lieutier A. A sampling theory for compact sets in Euclidean space. Discete and Computational Geometry 2009; 41(3): 461-479. DOI: 10.1007/s00454-009-9144-8.

- Chazal F. High-dimensional topological data analysis. Handbook of discrete and computational geometry. Boca Raton, FL: CRC Press; 2017. ISBN: 978-1-4987-1139-5.

- Ribeiro M, Singh S, Guestrin C. "Why should i trust you?" Explaining the predictions of any classifier. Source: <https://arxiv.org/abs/1602.04938>.

- Katok A, Sossinsky A. Introduction to modern topology and geometry. Source: <http://www.personal.psu.edu/axk29/TOPOLOGY/>.

- Erickson, J. CS 598: Computational topology. Spring 2013. Source: <http://jeffe.cs.illinois.edu/teaching/comptop/index.html>.

- The MNIST database of handwritten digits. Source: <http://yann.lecun.com/exdb/mnist/>.

- Hennig C. Dissolution point and isolation robustness: Robustness criteria for general cluster analysis methods. Journal of Multivariate Analysis 2008; 99(6): 1154-1176. DOI: 10.1016/j.jmva.2007.07.002

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