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Lanczos method for large-scale quaternion singular value decomposition

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Abstract

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.

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References

  1. Baglama, J., Reichel, L.: Augmented implicitly restarted Lanczos bidiagonalization methods. SIAM J. Sci. Comput. 27(1), 19–42 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Le Bihan, N., Sangwine, S.J.: Jacobi method for quaternion matrix singular value decomposition. Appl. Math Comput. 187(2), 1265–1271 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Björck, A.: Numerical methods for least squares problems. SIAM, Philadelphia (1996)

    Book  MATH  Google Scholar 

  4. Bunse-Gerstner, A., Byers, R., Mehrmann, V.: A quaternion QR algorithm. Numer. Math. 55, 83–95 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, J., Candés, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956–1982 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demmel, J.W.: Applied numerical linear algebra. SIAM, Philadelphia (1997)

    Book  MATH  Google Scholar 

  7. Ell, T.A., Sangwine, S.J.: Hypercomplex fourier transforms of color images. IEEE Trans. Image Processing 16(1), 22–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fletcher, P., Sangwine, S.J.: The development of the quaternion wavelet transform. Signal Process. 136, 2–15 (2017)

    Article  Google Scholar 

  9. Golub, G.H., 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Golub, G.H., Van Loan, C.F.: Matrix computation, 4th edn. The Johns Hopkins University Press, Baltimore (2013)

    MATH  Google Scholar 

  11. Hamilton, W. R.: Elements of quaternions. Chelsea, New York (1969)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rodman, L.: Topics in quaternion linear algebra. University Press Princeton, Princeton (2014)

    Book  MATH  Google Scholar 

  22. Sangwine, S.J.: Fourier transforms of colour images using quaternion or hypercomplex numbers. Electron. Lett. 32(1), 1979–1980 (1996)

    Article  Google Scholar 

  23. Sangwine, S.J., Le 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)

    MathSciNet  MATH  Google Scholar 

  24. Simon, H.D., Zha, H.: Low-rank matrix approximation using the Lanczos bidiagonalization process with applications. SIAM J. Sci Comput. 21(6), 2257–2274 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sorensen, D.C.: Implicit application of polynomial filters in a k-Step arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Subakan, O.N., Vemuri, B.C.: A quaternion framework for color image smoothing and segmentation. Int. J. Comput. Vision 91(3), 233–250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  Google Scholar 

  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(5), 416–428 (2016)

    Article  Google Scholar 

  30. Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. The Georgia Tech face database. http://www.anefian.com/research/facereco.htm

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Acknowledgements

We are grateful to the editor and the anonymous referees for their excellent comments and suggestions, which helped us to improve the original presentation.

Funding

This research is supported in part by the HKRGC GRF 1202715, 12306616, 12200317, 12300218, and HKBU RC-ICRS/16-17/03, and by the National Natural Science Foundation of China under grant 11771188.

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Authors and Affiliations

  1. School of Mathematics and Statistics & Jiangsu Key Laboratory of Education Big Data Science and Engineering, Jiangsu Normal University, Xuzhou, 221116, People’s Republic of China

    Zhigang Jia

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Michael K. Ng

  3. School of Mathematics and Information Sciences, Weifang University, Weifang, 261061, People’s Republic of China

    Guang-Jing Song

Authors
  1. Zhigang Jia
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  2. Michael K. Ng
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  3. Guang-Jing Song
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Corresponding author

Correspondence to Michael K. Ng.

Additional information

Research supported in part by the HKRGC GRF 1202715, 12306616, 12200317, 12300218, and HKBU RC-ICRS/16-17/03.

Research supported in part by National Natural Science Foundation of China under grant 11771188.

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Jia, Z., Ng, M.K. & Song, GJ. Lanczos method for large-scale quaternion singular value decomposition. Numer Algor 82, 699–717 (2019). https://doi.org/10.1007/s11075-018-0621-0

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