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Increasing the Accuracy of Solving Discrete Ill-Posed Problems by the Random Projection Method

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Abstract

To solve discrete ill-posed problems by the random projection method, the error bias and variance that arise from averaging over the random matrix realizations are studied. An estimate for the input vector is obtained that makes possible to significantly improve the accuracy of solving such problems using the random projection method.

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References

  1. Yu. L. Zabulonov, Yu. M. Korostil, and E. G. Revunova, “Optimization of inverse problem solution to obtain the distribution density function for surface contaminations,” Modeling and Information Technologies, Iss. 39, 77–83 (2006).

    Google Scholar 

  2. D. A. Rachkovskij and E. G. Revunova, “Intelligent gamma-ray data processing for environmental monitoring,” in: Intelligent Data Analysis in Global Monitoring for Environmental and Security, ITHEA, Kiev–Sofia (2011), pp. 136–157.

  3. V. N. Starkov, Constructive Methods of Computational Physics in Interpretation Problems [in Russian], Naukova Dumka, Kyiv (2002).

  4. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia (1998).

    Book  Google Scholar 

  5. A. N. Tikhonov and V. Y. Arsenin, “Solution of Ill-Posed Problems,” V. H. Winston, Washington (1977).

    MATH  Google Scholar 

  6. E. G. Revunova and D. A. Rachkovskij, “Using randomized algorithms for solving discrete ill-posed problems,” International Journal “Information Theories and Applications”, Vol. 2, No. 16, 176–192 (2009).

    Google Scholar 

  7. E. G. Revunova, “Study of error components for solution of the inverse problem using random projections,” Mathematical Machines and Systems, No. 4, 33–42 (2010).

  8. D. A. Rachkovskij and E. G. Revunova, “Randomized method for solving discrete ill-posed problems,” Cybernetics and Systems Analysis, Vol. 48, No. 4, 621–635 (2012).

    Article  MathSciNet  Google Scholar 

  9. E. G. Revunova, “Analytical study of error components for the solution of discrete ill-posed problems using random projections,” Cybernetics and Systems Analysis, Vol. 51, No. 6, 978–991 (2015).

    Article  MathSciNet  Google Scholar 

  10. E. G. Revunova, “Model selection criteria for a linear model to solve discrete ill-posed problems on the basis of singular decomposition and random projection,” Cybernetics and Systems Analysis, Vol. 52, No. 4, 647–664 (2016).

    Article  MathSciNet  Google Scholar 

  11. E. G. Revunova, “Averaging over matrices in solving discrete ill-posed problems on the basis of random projection,” in: Proc. CSIT’17, Vol. 1 (2017), pp. 473–478.

  12. E. G. Revunova, “Solution of the discrete ill-posed problem on the basis of singular value decomposition and random projection,” in: Advances in Intelligent Systems and Computing II, Springer, Cham (2017), pp. 434–449.

    Google Scholar 

  13. E. M. Kussul, T. N. Baidyk, V. V. Lukovich, and D. A. Rachkovskij, “Adaptive neural network classifier with multifloat input coding,” in: Proc. 6th Intern. Conf. “Neural Networks and Their Industrial and Cognitive Applications (Neuro-Nimes’93)” (1993), pp. 209–216.

  14. V. V. Lukovich, A. D. Goltsev, and D. A. Rachkovskij, “Neural network classifiers for micromechanical equipment diagnostics and micromechanical product quality inspection,” in: Proc. EUFIT’97, Vol. 1 (1997), pp. 534–536.

  15. E. M. Kussul, L. M. Kasatkina, D. A. Rachkovskij, and D. C. Wunsch, “Application of random threshold neural networks for diagnostics of micro machine tool condition,” in: Proc. IEEE World Congress on Computational Intelligence; Neural Networks Proceedings, Vol. 1, 241–244 (1998).

  16. D. A. Rachkovskij, S. V. Slipchenko, E. M. Kussul, and T. N. Baidyk, “Properties of numeric codes for the scheme of random subspaces RSC,” Cybernetics and Systems Analysis, Vol. 41, No. 4, 509–520 (2005).

    Article  MathSciNet  Google Scholar 

  17. D. A. Rachkovskij, I. S. Misuno, and S. V. Slipchenko, “Randomized projective methods for construction of binary sparse vector representations,” Cybernetics and Systems Analysis, Vol. 48, No. 1, 146–156 (2012).

    Article  Google Scholar 

  18. V. I. Gritsenko, D. A. Rachkovskij, A. D. Goltsev, V. V. Lukovych, I. S. Misuno, E. G. Revunova, S. V. Slipchenko, A. M. Sokolov, and S. A. Talayev, “Neural network distributed representations for intelligent information technology and modelling of thinking,” Cybernetics and Computer Engineering, Iss. 3 (173), 7–24 (2013).

    Google Scholar 

  19. D. Kleyko, E. Osipov, and D. A. Rachkovskij, “Modification of holographic graph neuron using sparse distributed representations”, Procedia Computer Science, 88, 39–45 (2016).

    Article  Google Scholar 

  20. V. I. Gritsenko, D. A. Rachkovskij, A. A. Frolov, R. Gayler, D. Kleyko, and E. Osipov, “Neural distributed autoassociative memories: A survey,” Cybernetics and Computer Engineering, Iss. 2 (188), 5–35 (2017).

    Google Scholar 

  21. D. Nowicki, P. Verga, and H. Siegelmann, “Modeling reconsolidation in kernel associative memory,” PLoS ONE, Vol. 8, Iss. 8: e68189 (2013). DOI: https://doi.org/10.1371/journal.pone.0068189.

    Article  Google Scholar 

  22. D. Nowicki and H. Siegelmann, “Flexible kernel memory,” PLoS ONE, Vol. 5, Iss. 6: e10955. DOI: https://doi.org/10.1371/journal.pone.0010955 (2010).

    Article  Google Scholar 

  23. N. F. Kirichenko, A. M. Reznik, and S. P. Shchetinyuk, “Matrix pseudoinversion in the problem of design of associative memory,” Cybernetics and Systems Analysis, Vol. 37, No. 3, 308–316 (2001).

    Article  MathSciNet  Google Scholar 

  24. S. Geman, E. Bienenstock, and R. Dourstat, “Neural networks and the bias/variance dilemma”, Neural Computation, Vol. 4, No. 1, 1–58 (1992).

    Article  Google Scholar 

  25. S. S. Haykin, Nerual Networks: A Comprehensive Foundation, Prentic Hall, Upper Saddle River (1999).

    MATH  Google Scholar 

  26. P. Niyogi and F. Girosi, “Generalization bounds for bunction approximation from scattered noisy data,” Advances in Computational Mathematics, Vol. 10, No. 1, 51–80 (1999).

    Article  MathSciNet  Google Scholar 

  27. T. L. Marzetta, G. H. Tucci, and S. H. Simon, “A random matrix-theoretic approach to handling singular covariance estimates,” IEEE Transactions on Information Theory, Vol. 57, No. 9, 6256–6271 (2011).

    Article  MathSciNet  Google Scholar 

  28. R. J. Durrant and A. Kaban, “A tight bound on the performance of Fishers linear discriminant in randomly projected data spaces,” Pattern Recognition Letters, Vol. 33, No. 7, 911–919 (2012).

    Article  Google Scholar 

  29. R. Durrant and A. Kaban, “Random projections as regularizers: Learning a linear discriminant from fewer observations than dimensions,” Machine Learning, Vol. 99, No. 2, 257–286 (2015).

    Article  MathSciNet  Google Scholar 

  30. D. Woodruff, “Sketching as a tool for numerical linear algebra,” Found. Trends Theor. Comput. Sci., Vol. 10, Nos. 1, 2, 1–157 (2014).

    Article  MathSciNet  Google Scholar 

  31. G. H. Tucci and M. V. Vega “A note on averages over Gaussian random matrix ensembles,” Journal of Probability and Statistics, Vol. 2013, Article ID 941058, 1–6 (2013).

    Article  MathSciNet  Google Scholar 

  32. P. C. Hansen, “Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms, Vol. 6, No. 1, 1–35 (1994).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. International Scientific and Training Center of Information Technologies and Systems, National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, Kyiv, Ukraine

    E. G. Revunova

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  1. E. G. Revunova
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Correspondence to E. G. Revunova.

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Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2018, pp. 181–192.

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Revunova, E.G. Increasing the Accuracy of Solving Discrete Ill-Posed Problems by the Random Projection Method. Cybern Syst Anal 54, 842–852 (2018). https://doi.org/10.1007/s10559-018-0086-0

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  • DOI: https://doi.org/10.1007/s10559-018-0086-0

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