# Institute of Mathematics

## Solvability classes for core problems in matrix total least squares minimization

#### Iveta Hnětynková, Martin Plešinger, Jana Žáková

###### Received September 20, 2018.   Published online February 19, 2019.

Abstract:  Linear matrix approximation problems $AX\approx B$ are often solved by the total least squares minimization (TLS). Unfortunately, the TLS solution may not exist in general. The so-called core problem theory brought an insight into this effect. Moreover, it simplified the solvability analysis if $B$ is of column rank one by extracting a core problem having always a unique TLS solution. However, if the rank of $B$ is larger, the core problem may stay unsolvable in the TLS sense, as shown for the first time by Hnětynková, Plešinger, and Sima (2016). Full classification of core problems with respect to their solvability is still missing. Here we fill this gap. Then we concentrate on the so-called composed (or reducible) core problems that can be represented by a composition of several smaller core problems. We analyze how the solvability class of the components influences the solvability class of the composed problem. We also show on an example that the TLS solvability class of a core problem may be in some sense improved by its composition with a suitably chosen component. The existence of irreducible problems in various solvability classes is discussed.
Keywords:  linear approximation problem; core problem theory; total least squares; classification; (ir)reducible problem
Classification MSC:  15A06, 15A09, 15A18, 15A23, 65F20
DOI:  10.21136/AM.2019.0252-18

References:
[1] G. H. Golub, C. F. Van Loan: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17 (1980), 883-893. DOI 10.1137/0717073 | MR 0595451 | Zbl 0468.65011
[2] I. Hnětynková, M. Plešinger, D. M. Sima: Solvability of the core problem with multiple right-hand sides in the TLS sense. SIAM J. Matrix Anal. Appl. 37 (2016), 861-876. DOI 10.1137/15M1028339 | MR 3523076 | Zbl 1343.15002
[3] I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, S. Van Huffel: The total least squares problem in $AX\approx B$: a new classification with the relationship to the classical works. SIAM J. Matrix Anal. Appl. 32 (2011), 748-770. DOI 10.1137/100813348 | MR 2825323 | Zbl 1235.15016
[4] I. Hnětynková, M. Plešinger, Z. Strakoš: The core problem within a linear approximation problem $AX\approx B$ with multiple right-hand sides. SIAM J. Matrix Anal. Appl. 34 (2013), 917-931. DOI 10.1137/120884237 | MR 3073354 | Zbl 1279.65041
[5] I. Hnětynková, M. Plešinger, Z. Strakoš: Band generalization of the Golub-Kahan bidiagonalization, generalized Jacobi matrices, and the core problem. SIAM J. Matrix Anal. Appl. 36 (2015), 417-434. DOI 10.1137/140968914 | MR 3335497 | Zbl 1320.65057
[6] I. Hnětynková, M. Plešinger, J. Žáková: Modification of TLS algorithm for solving $\mathcal{F}_2$ linear data fitting problems. PAMM, Proc. Appl. Math. Mech. 17 (2017), 749-750. DOI 10.1002/pamm.201710342
[7] I. Markovsky, S. Van Huffel: Overview of total least-squares methods. Signal Process. 87 (2007), 2283-2302. DOI 10.1016/j.sigpro.2007.04.004 | Zbl 1186.94229
[8] C. C. Paige, Z. Strakoš: Core problems in linear algebraic systems. SIAM J. Matrix Anal. Appl. 27 (2005), 861-875. DOI 10.1137/040616991 | MR 2208340 | Zbl 1097.15003
[9] D. A. Turkington: Generalized Vectorization, Cross-Products, and Matrix Calculus. Cambridge University Press, Cambridge (2013). DOI 10.1017/CBO9781139424400 | MR 3154976 | Zbl 1307.15001
[10] S. Van Huffel, J. Vandewalle: The Total Least Squares Problem: Computational Aspects and Analysis. Frontiers in Applied Mathematics 9, Society for Industrial and Applied Mathematics, Philadelphia (1991). DOI 10.1137/1.9781611971002 | MR 1118607 | Zbl 0789.62054
[11] X.-F. Wang: Total least squares problem with the arbitrary unitarily invariant norms. Linear Multilinear Algebra 65 (2017), 438-456. DOI 10.1080/03081087.2016.1189493 | MR 3589611 | Zbl 1367.15017
[12] M. S. Wei: Algebraic relations between the total least squares and least squares problems with more than one solution. Numer. Math. 62 (1992), 123-148. DOI 10.1007/BF01396223 | MR 1159048 | Zbl 0761.65030
[13] M. Wei: The analysis for the total least squares problem with more than one solution. SIAM J. Matrix Anal. Appl. 13 (1992), 746-763. DOI 10.1137/0613047 | MR 1168020 | Zbl 0758.65039
[14] S. Yan, K. Huang: The original TLS solution sets of the multidimensional TLS problem. Int. J. Comput. Math. 73 (2000), 349-359. DOI 10.1080/00207160008804902 | MR 1756273 | Zbl 0949.65037

Affiliations:   Iveta Hnětynková, Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic, e-mail: iveta.hnetynkova@mff.cuni.cz; Martin Plešinger (corresponding author), Department of Mathematics, Technical University of Liberec, Studentská 1402/2, 461 17 Liberec 1, Czech Republic & Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 271/2, 182 07 Praha 8, Czech Republic, e-mail: martin.plesinger@tul.cz; Jana Žáková, Department of Mathematics, Technical University of Liberec, Studentská 1402/2, 461 17 Liberec 1, Czech Republic, e-mail: jana.zakova@tul.cz

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