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On the theoretical basis of metric analysis of poorly formalized problems of recognition and classification

Abstract

In many fields of modern science, there are problems adequate formalization of which is indispensable for obtaining practically and theoretically important results. In the terminology of the scientific school of academician Yu.I. Zhuravlev, a formalized problem is uniquely defined by the matrix of information and the information matrix. In the present paper, a whole class of issues related to the formalization of recognition/classification problems is considered, and a universal formalism is proposed for carrying out a metric analysis of poorly formalized problems. Thus, the formalization of a problem can be represented as a successive transition from the set of original descriptions to a particular topology, then to a lattice, and then to a certain metric space. It is shown that the property of Zhuravlev’s regularity is sufficient for the existence of bijective mappings between these mathematical constructs. The possibilities of application of the apparatus developed are illustrated by several issues important for the formalization of the problems: introduction of metrics on the sets of the features and metrics on the sets of objects and analysis of “interactions” between dissimilar feature descriptions.

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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 per-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 Dortodnitsyn Computing Centre, 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 the theoretical basis of metric analysis of poorly formalized problems of recognition and classification. Pattern Recognit. Image Anal. 25, 577–587 (2015). https://doi.org/10.1134/S1054661815040252

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Keywords

  • algebraic approach
  • metric analysis
  • the theory of classification of feature values
  • metric condensations
  • combinatorial theory of solvability