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On metric spaces arising during formalization of problems of recognition and classification. Part 2: Density properties

  • Methematical Method in Pattern Recognition
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Abstract

In order to obtain tractable formal descriptions of poorly formalized problems within the context of the algebraic approach to pattern recognition, we develop methods for analyzing metric configurations. In this paper, using the concepts of σ-isomorphism and σ-completion of metric configurations, a system of criteria for assessing the properties of “generalized density” is obtained. The analysis of the density properties along the axes of a metric configuration allowed us to formulate methods for calculating the topological neighborhoods of points and for finding the “grains” of metric condensations. The theoretical results point to a new plethora of algorithms for searching metric condensations − methods based on the “restoration” of the set (the condensation searched) using the data on the components of the projection of the set on the axes of the metric configuration. The only mandatory parameters of any algorithm of this family of algorithms are the metric itself and the distribution of σ, which characterizes the accuracy of the values of the metric.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700, Russia

    I. Yu. Torshin & K. V. Rudakov

  2. Dorodniсyn Computing Centre, Federal Research Center “Informatics and Control,”, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    K. V. Rudakov

Authors
  1. I. Yu. Torshin
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  2. K. V. Rudakov
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Corresponding author

Correspondence to I. Yu. Torshin.

Additional information

Ivan Yur’evich Torshin. Born 1972. Graduated from the Department of Chemistry, Moscow State University, in 1995. Received Candidate’s 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 209 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 Dorodniсyn Computing Centre, Federal Research Center “Informatics and Control,” 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 problems of recognition and classification. Part 2: Density properties. Pattern Recognit. Image Anal. 26, 483–496 (2016). https://doi.org/10.1134/S1054661816030202

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  • DOI: https://doi.org/10.1134/S1054661816030202

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