On the Rank-One Approximation of Positive Matrices Using Tropical Optimization Methods

  • N. K. KrivulinEmail author
  • E. Yu. RomanovaEmail author


An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.


tropical mathematics tropical optimization max-algebra rank-one matrix approximation log-Chebyshev distance function 


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  1. 1.
    Yao Q., Kwok J., “Greedy learning of generalized low-rank models”, Proc. 25th Intern. Joint Conf. on Artificial Intelligence (IJCAI’16), 2294–2300 (AAAI Press, 2016).Google Scholar
  2. 2.
    Elden L., “Numerical linear algebra in data mining”, Acta Numer. 15, 327–384 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ruhe A., Numerical computation of principal components when several observations are missing (Research report, Umea Univ., 1974).Google Scholar
  4. 4.
    Friedland S., Mehrmann V., Miedlar A., Nkengla M., “Fast low rank approximations of matrices and tensors”, Electron. J. Linear Algebra 22, 1031–1048 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Koyuturk M., Grama A., Ramakrishnan N., “Compression, clustering, and pattern discovery in very high-dimensional discrete-attribute data sets”, IEEE Trans. Knowledge Data Eng. 17 (4), 447–461 (2005). CrossRefGoogle Scholar
  6. 6.
    Kumar N.K., Schneider J., “Literature survey on low rank approximation of matrices”, Linear Multilinear Algebra 65 (11), 2212–2244 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gillis N., “Introduction to nonnegative matrix factorization”, SIAG/OPT Views and News 25 (1), 7–16 (2017).MathSciNetGoogle Scholar
  8. 8.
    Aissa-El-Bey A., Seghouane K., “Sparse canonical correlation analysis based on rank-1 matrix approximation and its application for FMRI signals”, 2016 IEEE Intern. Conf. on Acoustics, Speech and Signal Processing (ICASSP), 4678–4682 (2016). Google Scholar
  9. 9.
    Saaty T., The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation (McGraw-Hill, New York, 1980).zbMATHGoogle Scholar
  10. 10.
    Luss R., Teboulle M. “Conditional gradient algorithms for rank-one matrix approximations with a sparsity constraint”, SIAM Review 55 (1), 65–98 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Shi Z., Wang L., Shi L., “Approximation method to rank-one binary matrix factorization”, IEEE Intern. Conf. on Automation Science and Engineering (CASE), 800–805 (2014). Google Scholar
  12. 12.
    Gillis N., Shitov Y., “Low-rank matrix approximation in the infinity norm”, Computing Research Repository, arXiv:1706.00078 (2017).Google Scholar
  13. 13.
    Krivulin N. K., Methods of idempotent algebra for problems in modeling and analysis of complex systems (St. Petersburg University Press, St. Petersburg, 2009). (In Russian)Google Scholar
  14. 14.
    Krivulin N., “Rating alternatives from pairwise comparisons by solving tropical optimization problems”, 12th Intern. Conference on Fuzzy Systems and Knowledge Discovery (FSKD), 162–167 (2015). Google Scholar
  15. 15.
    Krivulin N., “Using tropical optimization techniques to evaluate alternatives via pairwise comparisons”, 2016 Proc. 7th SIAM Workshop on Combinatorial Scientific Computing, 62–72 (Philadelphia: SIAM, 2016). Google Scholar
  16. 16.
    Krivulin N.K., Romanova E.Yu., “Rank-one approximation of positive matrices based on methods of tropical mathematics”, Vestnik St. Petersburg Univ. Math. 51 (2), 133–143 (2018 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Maslov V.P., Kolokoltsov V.N., Idempotent Analysis and Its Applications to Optimal Control Theory. (Nauka Publ., Moscow, 1994). (In Russian)Google Scholar
  18. 18.
    Butkovič P., Max-linear systems, in Springer Monographs in Mathematics (Springer, London, 2010). zbMATHGoogle Scholar
  19. 19.
    McEneaney W.M., Max-Plus Methods for Nonlinear Control and Estimation, in Systems and Control: Foundations and Applications (Birkhäuser, Boston, 2006). zbMATHGoogle Scholar
  20. 20.
    Krivulin N., Tropical optimization problems, in Advances in Economics and Optimization (Economic Issues, Problems and Perspectives), 195–214 (Nova Sci. Publ., New York, 2014).Google Scholar
  21. 21.
    Krivulin N., “Extremal properties of tropical eigenvalues and solutions to tropical optimization problems”, Linear Algebra Appl. 468, 211–232 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Krivulin N., “Tropical optimization problems in time-constrained project scheduling”, Optimization 66 (2), 205–224 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Krivulin N. K., “An extremal property of the eigenvalue of irreducible matrices in idempotent algebra and solution of the Rawls location problem”, Vestnik St. Petersburg Univ. Math. 44 (4), 272–281 (2011). MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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