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On metric spaces arising during formalization of recognition and classification problems. Part 1: Properties of compactness

Abstract

In the context of the algebraic approach to recognition of Yu.I. Zhuravlev’s scientific school, metric analysis of feature descriptions is necessary to obtain adequate formulations for poorly formalized recognition/classification problems. Formalization of recognition problems is a cross-disciplinary issue between supervised machine learning and unsupervised machine learning. This work presents the results of the analysis of compact metric spaces arising during the formalization of recognition problems. Necessary and sufficient conditions of compactness of metric spaces over lattices of the sets of feature descriptions are analyzed, and approaches to the completion of the discrete metric spaces (completion by lattice expansion or completion by variation of estimate) are formulated. It is shown that the analysis of compactness of metric spaces may lead to some heuristic cluster criteria commonly used in cluster analysis. During the analysis of the properties of compactness, a key concept of a ρ-network arises as a subset of points that allows one to estimate an arbitrary distance in an arbitrary metric configuration. The analysis of compactness properties and the conceptual apparatus introduced (ρ-networks, their quality functionals, the metric range condition, i- and ρ-spectra, ε-neighborhood in a metric cone, ε-isomorphism of complete weighted graphs, etc.) allow one to apply the methods of functional analysis, probability theory, metric geometry, and graph theory to the analysis of poorly formalized problems of recognition and classification.

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Correspondence to I. Yu. Torshin.

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Ivan Yur’evich Torshin. Born 1972. Graduated from the Department of Chemistry, Moscow State University, in 1995. Received candidates degrees in chemistry in 1997 and in physics and mathematics in 2011. Currently is an associate professor at Moscow Institute of Physics and Technology, lecturer at the Faculty of Computational Mathematics and Cybernetics, Moscow State University, leading scientist at the Russian Branch of the Trace Elements Institute for UNESCO, and a member of the Center of Forecasting and Recognition. Author of 205 publications in peer-reviewed journals in biology, chemistry, medicine, and informatics and of 3 monographs in the series “Bioinformatics in Post-genomic Era» (Nova Biomedical Publishers, NY, 2006-2009).

Konstantin Vladimirovich Rudakov. Born 1954. Russian mathematician, corresponding member of the Russian Academy of Sciences, Head of the Department of Computational Methods of Forecasting at the Dortodnicyn Computing Center, Russian Academy of Sciences, and Head of the Chair “Intelligent Systems” at the Moscow Institute of Physics and Technology.

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Torshin, I.Y., Rudakov, K.V. On metric spaces arising during formalization of recognition and classification problems. Part 1: Properties of compactness. Pattern Recognit. Image Anal. 26, 274–284 (2016). https://doi.org/10.1134/S1054661816020255

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Keywords

  • algebraic approach
  • metric analysis of data
  • theory of classification of feature values
  • compact metric spaces
  • clustering
  • combinatorial theory of solvability