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Applications of Mathematics

, Volume 64, Issue 2, pp 103–128 | Cite as

Solvability classes for core problems in matrix total least squares minimization

  • Iveta Hnětynková
  • Martin PlešingerEmail author
  • Jana Žáková
Article
  • 14 Downloads

Abstract

Linear matrix approximation problems AXB 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 

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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.

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Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2019

Authors and Affiliations

  • Iveta Hnětynková
    • 1
  • Martin Plešinger
    • 2
    • 3
    Email author
  • Jana Žáková
    • 2
  1. 1.Department of Numerical Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  2. 2.Department of MathematicsTechnical University of LiberecLiberec 1Czech Republic
  3. 3.Institute of Computer ScienceCzech Academy of SciencesPraha 8Czech Republic

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