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

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## Abstract

Linear matrix approximation problems *AX* ≈ *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## MSC 2010

15A06 15A09 15A18 15A23 65F20## Preview

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## Notes

### Acknowledgements

We wish to thank the anonymous referee for her or his careful reading the paper and useful comments which led to improvements of our manuscript.

## 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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [3]
*I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, S. Van Huffel*: The total least squares problem in*AX*≈*B*: a new classification with the relationship to the classical works. SIAM J. Matrix Anal. Appl.*32*(2011), 748–770.MathSciNetCrossRefzbMATHGoogle Scholar - [4]
*I. Hnětynková, M. Plešinger, Z. Strakoš*: The core problem within a linear approximation problem*AX*≈*B*with multiple right-hand sides. SIAM J. Matrix Anal. Appl.*34*(2013), 917–931.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [6]
*I. Hnětynková, M. Plešinger, J. Žáková*: Modification of TLS algorithm for solving*ℱ*_{2}linear data fitting problems. PAMM, Proc. Appl. Math. Mech.*17*(2017), 749–750.CrossRefzbMATHGoogle Scholar - [7]
*I. Markovsky, S. Van Huffel*: Overview of total least-squares methods. Signal Process.*87*(2007), 2283–2302.CrossRefzbMATHGoogle Scholar - [8]
*C. C. Paige, Z. Strakoš*: Core problems in linear algebraic systems. SIAM J. Matrix Anal. Appl.*27*(2005), 861–875.MathSciNetCrossRefzbMATHGoogle Scholar - [9]
*D. A. Turkington*: Generalized Vectorization, Cross-Products, and Matrix Calculus. Cambridge University Press, Cambridge, 2013.Google Scholar - [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.Google Scholar - [11]
*X.-F. Wang*: Total least squares problem with the arbitrary unitarily invariant norms. Linear Multilinear Algebra*65*(2017), 438–456.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [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.MathSciNetCrossRefzbMATHGoogle Scholar - [14]
*S. Yan, K. Huang*: The original TLS solution sets of the multidimensional TLS problem. Int. J. Comput. Math.*73*(2000), 349–359.MathSciNetCrossRefzbMATHGoogle Scholar