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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 528281, 8 pages
On the Low-Rank Approximation Arising in the Generalized Karhunen-Loeve Transform
1College of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China
2Department of Mathematics, Shanghai University, Shanghai 200444, China
Received 11 March 2013; Accepted 25 April 2013
Academic Editor: Masoud Hajarian
Copyright © 2013 Xue-Feng Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Y. Hua and W. Q. Liu, “Generalized Karhunen-Loeve transform,” IEEE Signal Processing Letters, vol. 5, pp. 141–142, 1998.
- S. Kraut, R. H. Anderson, and J. L. Krolik, “A generalized Karhunen-Loeve basis for efficient estimation of tropospheric refractivity using radar clutter,” IEEE Transactions on Signal Processing, vol. 52, no. 1, pp. 48–60, 2004.
- H. Ogawa and E. Oja, “Projection filter, Wiener filter, and Karhunen-Loève subspaces in digital image restoration,” Journal of Mathematical Analysis and Applications, vol. 114, no. 1, pp. 37–51, 1986.
- Y. Yamashita and H. Ogawa, “Relative Karhumen-Loeve transform,” IEEE Transactions on Signal Process, vol. 44, pp. 371–378, 1996.
- G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996.
- P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Numerical Mathematics, vol. 27, no. 4, pp. 534–553, 1987.
- H. D. Simon and H. Zha, “Low-rank matrix approximation using the Lanczos bidiagonalization process with applications,” SIAM Journal on Scientific Computing, vol. 21, no. 6, pp. 2257–2274, 2000.
- P. Drineas, R. Kannan, and M. W. Mahoney, “Fast Monte Carlo algorithms for matrices—II. Computing a low-rank approximation to a matrix,” SIAM Journal on Computing, vol. 36, no. 1, pp. 158–183, 2006.
- A. Frieze, R. Kannan, and S. Vempala, “Fast Monte-Carlo algorithms for finding low-rank approximations,” Journal of the ACM, vol. 51, no. 6, pp. 1025–1041, 2004.
- J. P. Ye, “Generalized low rank approximations of matrices,” Machine Learning, vol. 61, pp. 167–191, 2005.
- J. Liu, S. C. Chen, Z. H. Zhou, and X. Y. Tan, “Generalized low rank approximations of matrices revisited,” IEEE Transactions on Neural Networks, vol. 21, pp. 621–632, 2010.
- Z. Z. Liang and P. F. Shi, “An analytical algorithm for generalized low rank approxiamtions of matrices,” Pattern Recognition, vol. 38, pp. 2213–2216, 2005.
- J. H. Manton, R. Mahony, and Y. Hua, “The geometry of weighted low-rank approximations,” IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 500–514, 2003.
- I. Markovsky and S. Van Huffel, “Left versus right representations for solving weighted low-rank approximation problems,” Linear Algebra and its Applications, vol. 422, no. 2-3, pp. 540–552, 2007.
- M. Schuermans, P. Lemmerling, and S. Van Huffel, “Block-row Hankel weighted low rank approximation,” Numerical Linear Algebra with Applications, vol. 13, no. 4, pp. 293–302, 2006.
- F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006.
- F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005.
- F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41–50, 2008.
- J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form AiXBi = Fi,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3500–3507, 2010.
- L. Xie, Y. Liu, and H. Yang, “Gradient based and least squares based iterative algorithms for matrix equations AXB + CXTD = F,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2191–2199, 2010.
- L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1441–1448, 2009.
- F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95–107, 2005.
- W. Xiong, W. Fan, and R. Ding, “Least-squares parameter estimation algorithm for a class of input nonlinear systems,” Journal of Applied Mathematics, vol. 2012, Article ID 684074, 14 pages, 2012.
- C. C. Paige and M. A. Saunders, “Towards a generalized singular value decomposition,” SIAM Journal on Numerical Analysis, vol. 18, no. 3, pp. 398–405, 1981.