Structured conditioning theory for the total least squares problem with linear equality constraint and their estimation

Mahvish Samar; Xinzhong Zhu; Mahvish Samar; Xinzhong Zhu

doi:10.3934/math.2023575

AIMS Mathematics

2023, Volume 8, Issue 5: 11350-11372. doi: 10.3934/math.2023575

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Research article

Structured conditioning theory for the total least squares problem with linear equality constraint and their estimation

Mahvish Samar ^{1
,
,},
Xinzhong Zhu ^1,2

1.
College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China
2.
AI Research Institute of Beijing Geekplus Technology Co., Ltd., Beijing 100101, China

Received: 23 August 2022 Revised: 02 December 2022 Accepted: 11 December 2022 Published: 13 March 2023
MSC : 15A12, 15A60, 65F20, 65F30, 65F35

This article is devoted to the structured and unstructured condition numbers for the total least squares with linear equality constraint (TLSE) problem. By making use of the dual techniques, we investigate three distinct kinds of unstructured condition numbers for a linear function of the TLSE solution and three structured condition numbers for this problem, i.e., normwise, mixed, and componentwise ones, and present their explicit expressions under both unstructured and structured componentwise perturbations. In addition, the relations between structured and unstructured normwise, componentwise, and mixed condition numbers for the TLSE problem are investigated. Furthermore, using the small-sample statistical condition estimation method, we also consider the statistical estimation of both unstructured and structured condition numbers and propose three algorithms. Theoretical and experimental results show that structured condition numbers are always smaller than the corresponding unstructured condition numbers.
Citation: Mahvish Samar, Xinzhong Zhu. Structured conditioning theory for the total least squares problem with linear equality constraint and their estimation[J]. AIMS Mathematics, 2023, 8(5): 11350-11372. doi: 10.3934/math.2023575

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Abstract

This article is devoted to the structured and unstructured condition numbers for the total least squares with linear equality constraint (TLSE) problem. By making use of the dual techniques, we investigate three distinct kinds of unstructured condition numbers for a linear function of the TLSE solution and three structured condition numbers for this problem, i.e., normwise, mixed, and componentwise ones, and present their explicit expressions under both unstructured and structured componentwise perturbations. In addition, the relations between structured and unstructured normwise, componentwise, and mixed condition numbers for the TLSE problem are investigated. Furthermore, using the small-sample statistical condition estimation method, we also consider the statistical estimation of both unstructured and structured condition numbers and propose three algorithms. Theoretical and experimental results show that structured condition numbers are always smaller than the corresponding unstructured condition numbers.

References

[1]	Q. Liu, Z. Jia, On the condition number of the total least squares problem with linear equality constraint, Numer. Algor., 90 (2022), 363–385. https://doi.org/10.1007/s11075-021-01191-w doi: 10.1007/s11075-021-01191-w
[2]	G. H. Golub, C. F. Van Loan, An analysis of total least squares problem, SIAM J. Matrix Anal. Appl., 17 (1980), 883–893. https://doi.org/10.1137/0717073 doi: 10.1137/0717073
[3]	S. Van Huffel, J. Vandevalle, The Total Least Squares Problems: Computational Aspects and Analysis, Philadelphia: SIAM, 1991.
[4]	G. H. Golub, C. F. Van Loan, Matrix Computations, Baltimore: Johns Hopkins University Press, 2013. https://doi.org/10.2307/3621013
[5]	G. Stewart, On the weighting method for least squares problems with linear equality constraints, BIT Numer. Math., 37 (1997), 961–967. https://doi.org/10.1007/BF02510363 doi: 10.1007/BF02510363
[6]	A. J. Cox, N. J. Higham, Accuracy and stability of the null space method for solving the equality constrained least squares problem, BIT Numer. Math., 39 (1999), 34–50. https://doi.org/10.1023/A:1022365107361 doi: 10.1023/A:1022365107361
[7]	H. Diao, Condition numbers for a linear function of the solution of the linear least squares problem with equality constraints, J. Comput. Appl. Math., 344 (2018), 640–656. https://doi.org/10.1016/j.cam.2018.05.050 doi: 10.1016/j.cam.2018.05.050
[8]	H. Li, S. Wang, Partial condition number for the equality constrained linear least squares problem, Calcolo, 54 (2017), 1121–1146. https://doi.org/10.1007/s10092-017-0221-8 doi: 10.1007/s10092-017-0221-8
[9]	Q. Liu, M. Wang, On the weighting method for mixed least squares-total least squares problems, Numer. Linear Algebra Appl., 24 (2017), e2094. https://doi.org/10.1002/nla.2094 doi: 10.1002/nla.2094
[10]	E. M. Dowling, R. D. Degroat, D. A. Linebarger, Total least squares with linear constraints, IEEE Int. Conf. Acoust., 5 (1992), 341–344. https://doi.org/10.1109/ICASSP.1992.226613 doi: 10.1109/ICASSP.1992.226613
[11]	B. Schaffrin, A note on constrained total least squares estimation, Linear Algebra Appl., 417 (2006), 245–258. https://doi.org/10.1016/j.laa.2006.03.044 doi: 10.1016/j.laa.2006.03.044
[12]	B. Schaffrin, Y. A. Felus, An algorithmic approach to the total least-squares problem with linear and quadratic constraints, Stud. Geophys. Geod., 53 (2009), 1–16. https://doi.org/10.1007/s11200-009-0001-2 doi: 10.1007/s11200-009-0001-2
[13]	Q. Liu, S. Jin, L. Yao, D. Shen, The revisited total least squares problems with linear equality constraint, Appl. Numer. Math., 152 (2020), 275–284. https://doi.org/10.1016/j.apnum.2019.11.021 doi: 10.1016/j.apnum.2019.11.021
[14]	Q. Liu, C. Chen, Q. Zhang, Perturbation analysis for total least squares problems with linear equality constraint, Appl. Numer. Math., 161 (2021), 69–81. https://doi.org/10.1016/j.apnum.2020.10.025 doi: 10.1016/j.apnum.2020.10.025
[15]	J. Rice, A theory of condition, SIAM J. Numer. Anal., 3 (1966), 287–310. https://doi.org/10.1137/0703023
[16]	I. Gohberg, I. Koltracht, Mixed, componentwise, and structured condition numbers, SIAM J. Matrix Anal Appl., 14 (1993), 688–704. https://doi.org/10.1137/0614049 doi: 10.1137/0614049
[17]	P. Burgisser, F. Cucker, Condition: The Geometry of Numerical Algorithms, Heidelberg: Springer, 2013. https://doi.org/10.1007/978-3-642-38896-5
[18]	M. Baboulin, S. Gratton, A contribution to the conditioning of the total least-squares problem, SIAM J.Matrix Anal. Appl., 32 (2011), 685–699. https://doi.org/10.1137/090777608 doi: 10.1137/090777608
[19]	S. Gratton, D. Titley-Peloquin, J. T. Ilunga, Sensitivity and conditioning of the truncated total least squares solution, SIAM J. Matrix Anal. Appl., 34 (2013), 1257–1276. https://doi.org/10.1137/120895019 doi: 10.1137/120895019
[20]	Z. Jia, B. Li, On the condition number of the total least squares problem, Numer. Math., 125 (2013), 61–87. https://doi.org/10.1007/s00211-013-0533-9 doi: 10.1007/s00211-013-0533-9
[21]	B. Zheng, L. Meng, Y. Wei, Condition numbers of the multidimensional total least squares problem, SIAM J. Matrix Anal. Appl., 38 (2017), 924–948. https://doi.org/10.1137/15M1053815 doi: 10.1137/15M1053815
[22]	B. Zheng, Z. Yang, Perturbation analysis for mixed least squares-total least squares problems, Numer. Linear Algebra Appl., 26 (2019), 22–39. https://doi.org/10.1002/nla.2239 doi: 10.1002/nla.2239
[23]	L. Zhou, L. Lin, Y. Wei, S. Qiao, Perturbation analysis and condition numbers of scaled total least squares problems, Numer. Algor., 51 (2009), 381–399. https://doi.org/10.1007/s11075-009-9269-0 doi: 10.1007/s11075-009-9269-0
[24]	S. Wang, H. Li, H. Yang, A note on the condition number of the scaled total least squares problem, Calcolo, 55 (2018), 46. https://doi.org/10.1007/s10092-018-0289-9 doi: 10.1007/s10092-018-0289-9
[25]	J. Kamm, J. G. Nagy, A total least squares method for Toeplitz systems of equations, BIT, 38 (1998), 560–582. https://doi.org/10.1007/BF02510260 doi: 10.1007/BF02510260
[26]	P. Lemmerling, S. Van Huffel, Analysis of the structured total least squares problem for Hankel/Toeplitz matrices, Numer. Algorithms, 27 (2001), 89–114. https://doi.org/10.1023/A:1016775707686 doi: 10.1023/A:1016775707686
[27]	I. Markovsky, S. Van Huffel, Overview of total least-squares methods, Signal Process., 87 (2007), 2283–2302. https://doi.org/10.1016/j.sigpro.2007.04.004 doi: 10.1016/j.sigpro.2007.04.004
[28]	H. Diao, Y. Sun, Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem, Linear Algebra Appl., 544 (2018), 1–29. https://doi.org/10.1016/j.laa.2018.01.008 doi: 10.1016/j.laa.2018.01.008
[29]	H. Diao, Y. Wei, P. Xie, Small sample statistical condition estimation for the total least squares problem, Numer. Algorithms, 75 (2017), 435–455. https://doi.org/10.1007/s11075-016-0185-9 doi: 10.1007/s11075-016-0185-9
[30]	Q. Meng, H. Diao, Z. Bai, Condition numbers for the truncated total least squares problem and their estimations, Numer. Linear Algebra Appl., 28 (2021), e2369. https://doi.org/10.1002/nla.2369 doi: 10.1002/nla.2369
[31]	B. Li, Z. Jia, Some results on condition numbers of the scaled total least squares problem, Linear Algebra Appl., 435 (2011), 674–686. https://doi.org/10.1016/J.LAA.2010.07.022 doi: 10.1016/J.LAA.2010.07.022
[32]	Q. Liu, Q. Zhanga, D. Shen, Condition numbers of the mixed least squares-total least squares problem revisited, 2022. https://doi.org/10.1080/03081087.2022.2094861
[33]	M. Baboulin, S. Gratton, Using dual techniques to derive componentwise and mixed condition numbers for a linear function of a linear least squares solution, BIT Numer. Math., 49 (2009), 3–19. https://doi.org/10.1007/s10543-009-0213-4 doi: 10.1007/s10543-009-0213-4
[34]	H. Diao, L. Liang, S. Qiao, A condition analysis of the weighted linear least squares problem using dual norms, Linear Algebra Appl., 66 (2018), 1085–1103. https://doi.org/10.1080/03081087.2017.1337059 doi: 10.1080/03081087.2017.1337059
[35]	M. Samar, Condition numbers for a linear function of the solution to the constrained and weighted least squares problem and their statistical estimation, Taiwanese J. Math., 25 (2021), 717–741. https://doi.org/10.11650/tjm/201202 doi: 10.11650/tjm/201202
[36]	C. S. Kenney, A. J. Laub, Small-sample statistical condition estimates for general matrix functions, SIAM J. Sci. Comput., 15 (1994), 36–61. https://doi.org/10.1137/0915003 doi: 10.1137/0915003
[37]	N. J. Higham, Accuracy and Stability of Numerical Algorithms, Philadelphia: SIAM, 2002.
[38]	P. Xie, H. Xiang, Y. Wei, A contribution to perturbation analysis for total least squares problems, Numer. Algor., 75 (2017), 381–395. https://doi.org/10.1007/s11075-017-0285-1 doi: 10.1007/s11075-017-0285-1
[39]	S. Wang, H. Yang, H. Li, Condition numbers for the nonlinear matrix equation and their statistical estimation, Linear Algebra Appl., 482 (2015), 221–240. https://doi.org/10.1016/j.laa.2015.06.011 doi: 10.1016/j.laa.2015.06.011
[40]	M. Samar, H. Li, Y. Wei, Condition numbers for the K-weighted pseudoinverse $L^{\dagger}_{K}$ and their statistical estimation, Linear Multilinear Algebra, 69 (2021), 752–770. https://doi.org/10.1080/03081087.2019.1618235 doi: 10.1080/03081087.2019.1618235
[41]	M Samar, F. Lin, Perturbation and condition numbers for the Tikhonov regularization of total least squares problem and their statistical estimation, J. Comput. Appl. Math., 411 (2022), 114230. https://doi.org/10.1016/j.cam.2022.114230 doi: 10.1016/j.cam.2022.114230
[42]	M. Baboulin, S. Gratton, R. Lacroix, A. J. Laub, Statistical estimates for the conditioning of linear least squares problems, Lect. Notes Comput. Sci., 8384 (2014), 124–133. https://doi.org/10.1007/978-3-642-55224-3-13 doi: 10.1007/978-3-642-55224-3-13

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