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Randomized Quaternion QLP Decomposition for Low-Rank Approximation

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Abstract

The low-rank approximation of a quaternion matrix has attracted growing attention in many applications including color image processing and signal processing. In this paper, based on quaternion normal distribution random sampling, we propose a randomized quaternion QLP decomposition algorithm for computing a low-rank approximation to a quaternion data matrix. For the theoretical analysis, we first present convergence results of the quaternion QLP decomposition, which provides slightly tighter upper bounds than the existing ones for the real QLP decomposition. Then, for the randomized quaternion QLP decomposition, the matrix approximation error and the singular value approximation error analyses are also established to show the proposed randomized algorithm can track the singular values of the quaternion data matrix with high probability. Finally, we present some numerical examples to illustrate the effectiveness and reliablity of the proposed algorithm.

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Data Availability Statement

All data generated or analysed during this study are included in this manuscript.

References

  1. Chen, Y.Y., Guo, Y.W., Wang, Y.L., Wang, D., Peng, C., He, G.P.: Denoising of hyperspectral images using nonconvex low rank matrix approximation. IEEE Trans. Geosci. Remote Sens. 55, 5366–5380 (2017)

    Article  Google Scholar 

  2. Ren, W.Q., Cao, X.C., Pan, J.S., Guo, X.J., Zuo, W.M., Yang, M.-H.: Image deblurring via enhanced low-rank prior. IEEE Trans. Image Process. 25, 3426–3437 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53, 217–288 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Rokhlin, V., Szlam, A., Tygert, M.: A randomized algorithm for principal component analysis. SIAM J. Matrix Anal. Appl. 31, 1100–1124 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Muller, N., Magaia, L., Herbst, B.M.: Singular value decomposition, eigenfaces, and 3D reconstructions. SIAM Rev. 46, 518–545 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Martinsson, P.G.: A fast randomized algorithm for computing a hierarchically semi-separable representation of a matrix. SIAM J. Matrix Anal. Appl. 32, 1251–1274 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xia, J.L., Gu, M.: Robust approximate Cholesky factorization of rank-structured symmetric positive definite matrices. SIAM J. Matrix Anal. Appl. 31, 2899–2920 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika. 1, 211–218 (1936)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Gu, M.: Subspace iteration randomization and singular value problems. SIAM J. Sci. Comput. 37, A1139–A1173 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mahoney, M.: Randomized algorithms for matrices and data. Found. Trends Mach. Learn. 3, 123–224 (2011)

  12. Tropp, J.A., Yurtsever, A., Udell, M., Cevher, V.: Practical sketching algorithms for low-rank matrix approximation. SIAM J. Matrix Anal. Appl. 38, 1454–1485 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Woodruff, D.P.: Sketching as a tool for numerical linear algebra. Found. Trends Theor. Comput. Sci. 10, 1–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shabat, G., Shmueli, Y., Aizenbud, Y., Averbuch, A.: Randomized LU decomposition. Appl. Comput. Harmon. Anal. 44, 246–272 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duersch, J.A., Gu, M.: Randomized QR with column pivoting. SIAM J. Sci. Comput. 39, C263–C291 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xiao, J.W., Gu, M., Langou, J.: Fast parallel randomized QR with column pivoting algorithms for reliable low-rank matrix approximations. 2017 IEEE 24th International Conference on High Performance Computing (HiPC), IEEE. 233–242 (2017)

  17. Duersch, J.A., Gu, M.: Randomized projection for rank-revealing matrix factorizations and low-rank approximations. SIAM Rev. 62, 661–682 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, N.C., Xiang, H.: Randomized QLP decomposition. Linear Algebra Appl. 59, 18–35 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  20. Scolarici, G., Solombrino, L.: Notes on quaternionic groups representation. Int. J. Theor. Phys. 34, 2491–2500 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bihan, N.L., Sangwine, S.J.: Quaternion principal component analysis of color images. IEEE Interniational Conference on Iamge Processing. 1, 809–812 (2003)

    Google Scholar 

  22. Jia, Z.G., Ng, M.K., Song, G.J.: Robust quaternion matrix completion with applications to image inpainting. Numer. Linear Algebra Appl. 26, e2245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jia, Z.G., Ng, M.K., Wang, W.: Color image restoration by saturation-value total variation. SIAM J. Imaging Sciences. 12, 972–1000 (2019)

    Article  MathSciNet  Google Scholar 

  24. Sangwine, S.J., Bihan, N.L.: Quaternion and octonion toolbox for Matlab. http://qtfm.sourceforge.net/

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Sangwine, S.J., Bihan, N.L.: 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 

  27. Li, Y., Wei, M.S., Zhang, F.X., Zhao, J.L.: A real structure-preserving method for the quaternion LU decomposition, revisited. Calcolo. 54, 1553–1563 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, M.H., Ma, W.H.: A structure-preserving method for the quaternion LU decomposition in quaternionic quantum theory. Comput. Phys. Comm. 184, 2182–2186 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, Y., Wei, M.S., Zhang, F.X., Zhao, J.L.: Real structure-preserving algorithms of Householder based transformations for quaternion matrices. J. Comput. Appl. Math. 305, 82–91 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jia, Z.G., Wei, M.S., Zhao, M.-X., Chen, Y.: A new real structure-preserving quaternion QR algorithm. J. Comput. Appl. Math. 343, 26–48 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Y., Wei, M.S., Zhang, F.X., Zhao, J.L.: A fast structure-preserving algorithms for computing the singular value decomposition of quaternion matrix. Appl. Math. Comput. 235, 157–167 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Wang, M.H., Ma, W.H.: A structure-preserving algorithm for the quaternion Cholesky decomposition. Appl. Math. Comput. 223, 354–361 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Chen, Y.Y., Xiao, X.L., Zhou, Y.C.: Low-rank quaternion approximation for color image processing. IEEE Trans. Image Process. 29, 1426–1439 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Bahia, B., Sacchi, M.D.: Quaternionic rank-reduction methods for vector-field seismic data processing. Digit. Signal Process. 87, 178–189 (2019)

    Article  Google Scholar 

  35. Liu, Q.H., Jia, Z.G., Ling, S.T.: Randomized quaternion singular value decomposition for low-rank approximation. SIAM J. Sci. Comput. 44, A870–A900 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  36. Stewart, G.W.: The QLP approximation to the singular value decomposition. SIAM J. Sci. Comput. 20, 1336–1348 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Huckaby, D.A., Chan, T.F.: On the convergence of Stewart’s QLP algorithm for approximating the SVD. Numer. Algorithms 32, 287–316 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jia, Z.G., Ng, M.K.: Structure preserving quaternion generalized minimal residual method. SIAM J. Matrix Anal. Appl. 42, 616–634 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ma, R.-R., Jia, Z.G., Bai, Z.-J.: A structure-preserving Jacobi algorithm for quaternion Hermitian eigenvalue problems. Comput. Math. Appl. 75, 809–820 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jia, Z.G., Wei, M.S., Ling, S.T.: A new structure-preserving method for quaternion Hermitian eigenvalue problems. J. Comput. Appl. Math. 239, 12–24 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang, F.Z.: Quaternions and matrices of quaternion. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mahoney, M.W., Duchi, J.C., Gilbert, A.C.: The Mathematics of Data. AMS, IAS/Park City Mathematics Institute, and SIAM (2018)

  43. Hron, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  44. Bernstein, D.S.: Matrix Mathematics: Theory, Facts, and Formulas, 2nd edn. Princeton University Press, Princeton and Oxford (2009)

    Book  MATH  Google Scholar 

  45. Higham, N.J.: Analysis of the Cholesky decomposition of a semi-definite matrix. In: Cox, M.G., Hammarling, S.J. (eds.) Reliable Numerical Computation, pp. 161–185. Oxford University Press, Oxford (1990)

    Google Scholar 

  46. Meier, M., Nakatsukasa, Y.: Fast randomized numerical rank estimation. preprint, arXiv: 2105.07388

  47. Mathias, R., Stewart, G.W.: A block QR algorithm and singular value decomposition. Linear Algebra Appl. 182, 91–100 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. Strogatz, L.S.: Nolinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Westview Press, Boulder, Co (2001)

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Acknowledgements

The authors are grateful to the handling editor and three anonymous referees for their useful comments and suggestions, which greatly improved the original presentation.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 360015, People’s Republic of China

    Huan Ren

  2. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, People’s Republic of China

    Ru-Ru Ma

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Qiaohua Liu

  4. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing, Xiamen University, Xiamen, 361005, People’s Republic of China

    Zheng-Jian Bai

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  1. Huan Ren
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Correspondence to Qiaohua Liu.

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Ren, H., Ma, RR., Liu, Q. et al. Randomized Quaternion QLP Decomposition for Low-Rank Approximation. J Sci Comput 92, 80 (2022). https://doi.org/10.1007/s10915-022-01917-5

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