Skip to main content
Log in

Combinatorial analysis of the solvability properties of the problems of recognition and completeness of algorithmic models. Part 2: Metric approach within the framework of the theory of classification of feature values

  • Mathematical Method in Pattern Recognition
  • Published:
Pattern Recognition and Image Analysis Aims and scope Submit manuscript

Abstract

The properties of solvability/regularity of problems and correctness/completeness of algorithmic models are fundamental components of the algebraic approach to pattern recognition. In this paper, we formulate the principles of the metric approach to the data analysis of poorly formalized problems and hence with obtain metric forms of the criteria of solvability, regularity, correctness, and completeness. In particular, the analysis of the compactness properties of metric configurations allowed us to obtain a set of sufficient conditions for the existence of correct algorithms. These conditions can be used for assessment of the quality of the methods of formalization of the problems for arbitrary algorithms and algorithmic models. The general schema proposed for the data analysis of poorly formalized problems includes the criteria in the cross-validation form and can assess not only the quality of formalization, but also the extent of overtraining pertaining to the procedures of generation and selection of feature descriptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Yu. I. Zhuravlev, “Correct algebras for sets of incorrect (heuristic) algorithms. Part 1,” Kibernetika, No. 4, 5–17 (1977).

    Google Scholar 

  2. Yu. I. Zhuravlev, “Correct algebras for sets of incorrect (heuristic) algorithms. Part 2,” Kibernetika, No. 6, 21–27 (1977).

    MATH  Google Scholar 

  3. Yu. I. Zhuravlev, “Correct algebras for sets of incorrect (heuristic) algorithms. Part 3,” Kibernetika, No. 2, 35–43 (1978).

    MATH  Google Scholar 

  4. Yu. I. Zhuravlev, “On algebraic approach for solving recognition and classification problems,” in Problems of Cybernetics (Nauka, Moscow, 1978), issue 33, pp. 5–68 [in Russian].

    Google Scholar 

  5. K. V. Rudakov, “On some universal limitations for classification algorithms,” Zh. Vychisl. Mat. Mat. Fiz. 26 (11), 1719–1729 (1986).

    MathSciNet  MATH  Google Scholar 

  6. K. V. Rudakov, “Universal and local limitations in heuristic algorithms correction,” Kibernetika, No. 2, 30–35 (1987).

    Google Scholar 

  7. K. V. Rudakov, “Completeness and universal limitations in the problem on correcting the heuristic classification algorithms,” Kibernetika, No. 3, 106–109 (1987).

    Google Scholar 

  8. K. V. Rudakov, “The way to apply universal limitations for researching classification algorithms,” Kibernetika, No. 1, 1–5 (1988).

    Google Scholar 

  9. I. Yu. Torshin and K. V. Rudakov, “On the theoretical basis of metric analysis of poorly formalized problems of recognition and classification,” Pattern Recogn. Image Anal. 25 (4), 577–579 (2015).

    Article  Google Scholar 

  10. I. Y. Torshin and K. V. Rudakov, “On metric spaces arising during formalization of recognition and classification problems. Part 1: Properties of compactness,” Pattern Recogn. Image Anal. 26 (2), 274–284 (2016).

    Article  Google Scholar 

  11. I. Yu. Torshin and K. V. Rudakov, “On metric spaces arising during formalization of problems of recognition and classification. Part 2: density properties,” Pattern Recogn. Image Anal. 26 (3), 483–496 (2016).

    Article  Google Scholar 

  12. I. Yu. Torshin and K. V. Rudakov, “Combinatorial analysis of the solvability properties of the problems of recognition and completeness of algorithmic models. Part 1: Factorization approach,” Pattern Recogn. Image Anal. 27 (1), 16–28 (2017).

    Article  Google Scholar 

  13. K. V. Rudakov and I. Yu. Torshin, “The way to analyze motifs informativeness on the base of solvability criteria in the problem on recognizing secondary protein structure,” Inf. Ee Primen. 6 (1), 79–90 (2012).

    Google Scholar 

  14. Yu. I. Zhuravlev, K. V. Rudakov, and I. Yu. Torshin, “Algebraic criteria of local solvability and regularity as an instrument for researching amino acid sequences,” Trudy Mosk. Fiz.-Tekhn. Inst. 3 (4), 45–54 (2011).

    Google Scholar 

  15. I. Yu. Torshin, “The study of the solvability of the genome annotation problem on sets of elementary motifs,” Pattern Recogn. Image Anal. 21 (4), 652–662 (2011).

    Article  Google Scholar 

  16. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer-Verlag, 2009).

    Book  MATH  Google Scholar 

  17. A. N. Kolmogorov, Selected Works. Probability Theory and Mathematical Statistics (Moscow, 1986) [in Russian].

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

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 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 225 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, Full member of the Russian Academy of Sciences, Head of the Department of Computational Methods of Forecasting at the Dorodnicyn 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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Torshin, I.Y., Rudakov, K.V. Combinatorial analysis of the solvability properties of the problems of recognition and completeness of algorithmic models. Part 2: Metric approach within the framework of the theory of classification of feature values. Pattern Recognit. Image Anal. 27, 184–199 (2017). https://doi.org/10.1134/S1054661817020110

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1054661817020110

Keywords

Navigation