Solvability classes for core problems in matrix total least squares minimization. (English) Zbl 07088734

Hnětynková, Iveta; Plešinger, Martin; Žáková, Jana

Appl. Math., Praha 64, No. 2, 103-128 (2019).

Summary: 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 I. Hnětynková, M. Plešinger, and D. M. Sima [SIAM J. Matrix Anal. Appl. 37, No. 3, 861–876 (2016; Zbl 1343.15002)]. 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.

MSC:

15A06	Linear equations (linear algebraic aspects)
15A09	Theory of matrix inversion and generalized inverses
15A18	Eigenvalues, singular values, and eigenvectors
15A23	Factorization of matrices
65F20	Numerical solutions to overdetermined systems, pseudoinverses

Keywords:

linear approximation problem; core problem theory; total least squares; classification; irreducible problem; reducible problem

Citations:

Zbl 1343.15002

Software:

VanHuffel

References:

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