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Lanczos method for large-scale quaternion singular value decomposition. (English) Zbl 07107363
Summary: In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

MSC:
65 Numerical analysis
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[1] Baglama, J.; Reichel, L., Augmented implicitly restarted Lanczos bidiagonalization methods, SIAM J. Sci. Comput., 27, 19-42, (2005) · Zbl 1087.65039
[2] Bihan, N.; Sangwine, SJ, Jacobi method for quaternion matrix singular value decomposition, Appl. Math Comput., 187, 1265-1271, (2007) · Zbl 1114.65321
[3] Björck, A.: Numerical methods for least squares problems. SIAM, Philadelphia (1996) · Zbl 0847.65023
[4] Bunse-Gerstner, A.; Byers, R.; Mehrmann, V., A quaternion QR algorithm, Numer. Math., 55, 83-95, (1989) · Zbl 0681.65024
[5] Cai, J.; Candés, EJ; Shen, Z., A singular value thresholding algorithm for matrix completion, SIAM J. Optim., 20, 1956-1982, (2010) · Zbl 1201.90155
[6] Demmel, J.W.: Applied numerical linear algebra. SIAM, Philadelphia (1997) · Zbl 0879.65017
[7] Ell, TA; Sangwine, SJ, Hypercomplex fourier transforms of color images, IEEE Trans. Image Processing, 16, 22-35, (2007) · Zbl 1279.94014
[8] Fletcher, P.; Sangwine, SJ, The development of the quaternion wavelet transform, Signal Process., 136, 2-15, (2017)
[9] Golub, GH; Kahan, W., Calculating the singular values and Pseudo-Inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2, 205-224, (1965) · Zbl 0194.18201
[10] Golub, G.H., Van Loan, C.F.: Matrix computation, 4th edn. The Johns Hopkins University Press, Baltimore (2013) · Zbl 1268.65037
[11] Hamilton, W. R.: Elements of quaternions. Chelsea, New York (1969)
[12] Hernández, V., Román, J.E., Tomás, A.: Restarted Lanczos Bidiagonalization for the SVD in SLEPc SLEPc Technical Report STR-8. Available at http://slepc.upv.es (2007)
[13] Jia, Z.; Niu, D., An implicitly restarted refined bidiagonalization Lanczos method for computing a partial singular value decomposition, SIAM J. Matrix Anal. Appl., 25, 246-265, (2003) · Zbl 1063.65030
[14] Jia, Z.; Wei, M.; Ling, S., A new structure-preserving method for quaternion hermitian eigenvalue problems, J. Comput. Appl Math., 239, 12-24, (2013) · Zbl 1255.65079
[15] Jia, Z., Ling, S., Zhao, M.: Color two-dimensional principal component analysis for face recognition based on quaternion model. LNCS 10361, ICIC (1):177-189 (2017)
[16] Jia, Z., Ng, M., Song, G.: A quaternion framework for robust color images completion. Preprint, May 2018. http://www.math.hkbu.edu.hk/ mng/QMC.pdf
[17] Jia, Z.; Wei, M.; Zhao, M.; Chen, Y., A new real structure-preserving quaternion QR algorithm, J. Comput. Appl. Math., 343, 26-48, (2018) · Zbl 06892252
[18] Larsen, R.M.: Lanczos bidiagonalization with partial reorthogonalization. Technical Report PB-537. Department of Computer Science, University of Aarhus, Aarhus (1998). Available at http://www.daimi.au.dk/PB/537
[19] Li, Y.; Wei, M.; Zhang, F.; Zhao, J., A fast structure-preserving method for computing the singular value decomposition of quaternion matrix, Appl. Math Comput., 235, 157-167, (2014) · Zbl 1336.65057
[20] Li, Y.; Wei, M.; Zhang, F.; Zhao, J., Real structure-preserving algorithms of householder based transformations for quaternion matrices, J. Comput. Appl. Math., 305, 82-91, (2016) · Zbl 1386.65120
[21] Rodman, L.: Topics in quaternion linear algebra. University Press Princeton, Princeton (2014) · Zbl 1304.15004
[22] Sangwine, SJ, Fourier transforms of colour images using quaternion or hypercomplex numbers, Electron. Lett., 32, 1979-1980, (1996)
[23] Sangwine, SJ; Bihan, N., Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations, Appl. Math. Comput., 182, 727-738, (2006) · Zbl 1109.65037
[24] Simon, HD; Zha, H., Low-rank matrix approximation using the Lanczos bidiagonalization process with applications, SIAM J. Sci Comput., 21, 2257-2274, (2000) · Zbl 0962.65038
[25] Sorensen, DC, Implicit application of polynomial filters in a k-Step arnoldi method, SIAM J. Matrix Anal. Appl., 13, 357-385, (1992) · Zbl 0763.65025
[26] Subakan, ON; Vemuri, BC, A quaternion framework for color image smoothing and segmentation, Int. J. Comput. Vision, 91, 233-250, (2011) · Zbl 1235.68315
[27] Via, J., Vielva, L., Santamaria, I., Palomar, D.P.: Independent component analysis of quaternion gaussian vectors. In: IEEE Sensor Array and Multichannel Signal Processing Workshop, pp. 145-148 (2010)
[28] Wang, X., Zha, H.: An implicitly restarted bidiagonal Lanczos method for large-scale singular value problems technical report 42472. Scientific Computing Division, Lawrence Berkeley National Laboratory, Berkeley (1998)
[29] Zeng, R.; Wu, J.; Shao, Z.; Chen, Y.; Chen, B.; Senhadji, L.; Shu, H., Color image classification via quaternion principal component analysis network, Neurocomputing, 216, 416-428, (2016)
[30] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57, (1997) · Zbl 0873.15008
[31] The Georgia Tech face database. http://www.anefian.com/research/facereco.htm
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