Applications of Mathematics, Vol. 64, No. 2, pp. 103-128, 2019

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

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Affiliations:   Iveta Hnětynková, Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic, e-mail:; 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:; Jana Žáková, Department of Mathematics, Technical University of Liberec, Studentská 1402/2, 461 17 Liberec 1, Czech Republic, e-mail:

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