Abstract
Low-rank matrix approximation finds wide application in the analysis of big data, in recommendation systems on the Internet, for the approximate solution of some equations of mechanics, and in other fields. In this paper, a method for approximating positive matrices by rank-one matrices on the basis of minimization of log-Chebyshev distance is proposed. The problem of approximation reduces to an optimization problem having a compact representation in terms of an idempotent semifield in which the operation of taking the maximum plays the role of addition and which is often referred to as max-algebra. The necessary definitions and preliminary results of tropical mathematics are given, on the basis of which the solution of the original problem is constructed. Using the methods and results of tropical optimization, all positive matrices at which the minimum of approximation error is reached are found in explicit form. A numerical example illustrating the application of the rank-one approximation is considered.
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References
S. A. Solovyev, “Application of the low-rank approximation technique in the Gauss elimination method for sparse linear systems,” Vychisl. Metody Programm. 15, 441–460 (2014).
K. V. Voronin and S. A. Solovyev, “Solution of the Helmholtz problem using the preconditioned low-rank approximation technique,” Vychisl. Metody Programm. 16, 268–280 (2015).
J. Lee, S. Kim, G. Lebanon, and Y. Singer, “Local low-rank matrix approximation,” in Proc. 30th Int. Conf. on Machine Learning (ICML 2013), Atlanta, GA, 2013 (International Machine Learning Society, Stroudsburg, PA, 2013), pp. 82–90.
J. Lee, S. Bengio, S. Kim, et al., “Local collaborative ranking,” in Proc. 23rd Int. Conf. on World Wide Web (WWW’14), Seoul, Korea, April 7–11, 2014 (ACM, New York, 2014), pp. 85–96. https://doi.org/10.1145/2566486.2567970.
B. Savas, Algorithms in Data Mining Using Matrix and Tensor Methods (Linköping Univ., Inst. of Technology, Linköping, Sweden, 2008), in Ser.: Linköping Studies in Science and Technology, Dissertations, Vol. 1178.
M. Ispany, G. Michaletzky, J. Reiczigel, et al., “Approximation of non–negative integer–valued matrices with application to Hungarian mortality data,” in Proc. 19th Int. Symp. on Mathematical Theory of Networks and Systems (MTNS 2010), Budapest, Hungary, July 5–9, 2010 (MTNS, 2010), pp. 831–838.
E. E. Tyrtyshnikov, Matrices, Tensors and Computations. Textbook (Mosk. Gos. Univ., Moscow, 2013) [in Russian].
H. Wang and N. Ahuja, “A tensor approximation approach to dimensionality reduction,” Int. J. Comput. Vis. 76, 217–229 (2008). https://doi.org/10.1007/s11263-007-0053-0.
Q. Yao and J. Kwok, “Greedy learning of generalized low-rank models,” inProc. 25th Int. Joint Conf. on Artificial Intelligence (IJCAI’16), New York, July 9–10, 2016 (AAAI Press, Palo Alto, CA, 2016), pp. 2294–2300.
A. V. Shlyannikov, “Generation algorithm for 3D face model by a photograph,” Nauchno-Tekh. Vestn. Inf. Tekhnol., Mekh. Opt. 69 (5), 86–90 (2010).
A. Aissa-El-Bey and K. Seghouane, “Sparse canonical correlation analysis based on rank-1 matrix approximation and its application for FMRI signals,” in Proc. 2016 IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), Shanghai, China, Mar. 20–25, 2016 (IEEE, Piscataway, NJ, 2016), 4678–4682. https://doi.org/10.1109/ICASSP.2016.7472564.
R. Luss and M. Teboulle, “Conditional gradient algorithms for rank-one matrix approximations with a sparsity constraint,” SIAM Rev. 55, 65–98 (2013). https://doi.org/10.1137/110839072.
S. Friedland, V. Mehrmann, R. Pajarola, and S. K. Suter, “On best rank one approximation of tensors,” Numer. Linear Algebra Appl. 20, 942–955 (2013). https://doi.org/10.1002/nla.1878.
A. P. da Silva, P. Comon, and A. L. F. de Almeida, Rank-1 Tensor Approximation Methods and Application to Deflation, Research Report (GIPSA-lab, 2015).
T. Saaty, The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation (McGraw-Hill, New York, 1980; Radio i Svyaz’, Moscow, 1993).
H. Shen and J. Huang, “Sparse principal component analysis via regularized low rank matrix approximation,” J. Multivar. Anal. 99, 1015–1034 (2008). https://doi.org/10.1016/j.jmva.2007.06.007.
K. Zietak, “The Chebyshev approximation of a rectangular matrix by matrices of smaller rank as the limit of lpapproximation,” J. Comput. Appl. Math. 11, 297–305 (1984). https://doi.org/10.1016/0377-0427(84)90004-9.
S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge Univ. Press, Cambridge, 2004).
N. Gillis and Y. Shitov, “Low-rank matrix approximation in the infinity norm” (2017). arXiv 1706.00078.
M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Algebra Appl. 284, 193–228 (1998). https://doi.org/10.1016/S0024-3795(98)10032-0.
N. K. Krivulin, Methods of Idempotent Algebra for Problems in Modeling and Analysis of Complex Systems (S.-Peterb. Gos. Univ., St. Petersburg, 2009) [in Russian].
N. Krivulin, “Using tropical optimization techniques to evaluate alternatives via pairwise comparisons,” in Proc. 7th SIAM Workshop on Combinatorial Scientific Computing, Albuquerque, NM, Oct. 10–12, 2016 (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016), pp. 62–72. https://doi.org/10.1137/1.9781611974690.ch7.
N. K. Krivulin, V. A. Ageev, and I. V. Gladkikh, “Application of methods of tropical optimization for evaluating alternatives based on pairwise comparisons,” Vestn. St. Peterb. Univ., Ser. 10: Prikl. Mat. Inf. Protsessy Upr. 13, 27–41 (2017). https://doi.org/10.21638/spbu10.2017.103.
N. K. Krivulin, “Eigenvalues and eigenvectors of matrices in idempotent algebra,” Vestn. St. Petersburg Univ.: Math. 39, 72–83 (2006).
N. Krivulin, “Extremal properties of tropical eigenvalues and solutions to tropical optimization problems,” Linear Algebra Appl. 468, 211–232 (2015). https://doi.org/10.1016/j.laa.2014.06.044.
V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Applications to Optimal Control Theory (Fizmatlit, Moscow, 1994) [in Russian].
J.-E. Pin, “Tropical semirings,” in Idempotency, Ed. by J. Gunawardena (Cambridge Univ. Press, Cambridge, 1998), pp. 50–69.
G. L. Litvinov, V. P. Maslov, and A. N. Sobolevskii, “Idempotent mathematics and interval analysis,” Vychisl. Tekhnol. 6 (6), 47–70 (2001).
P. Butkovič, Max-Linear Systems: Theory and Algorithms (Springer-Verlag, London, 2010), in Ser.: Springer Monographs in Mathematics. https://doi.org/10.1007/978-1-84996-299-5.
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Original Russian Text © N.K. Krivulin, E.Yu. Romanova, 2018, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2018, Vol. 63, No. 2, pp. 216–230.
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Krivulin, N.K., Romanova, E.Y. Rank-One Approximation of Positive Matrices Based on Methods of Tropical Mathematics. Vestnik St.Petersb. Univ.Math. 51, 133–143 (2018). https://doi.org/10.3103/S106345411802005X
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DOI: https://doi.org/10.3103/S106345411802005X