Browsing by Author "Hnětynková, Iveta"

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    On TLS formulation and core reduction for data fitting with generalized models
    (2019-07-24) Hnětynková, Iveta; Plešinger, Martin; Žáková, Jana
    The total least squares (TLS) framework represents a popular data fitting approach for solving matrix approximation problems of the form A(X) ≡ AX ≈ B. A general linear mapping on spaces of matrices A ∶ X → B can be represented by a fourth-order tensor which is in the AX ≈ B case highly structured. This has a direct impact on solvability of the corresponding TLS problem, which is known to be complicated. Thus this paper focuses on several generalizations of the model A: the bilinear model, the model of higher Kronecker rank, and the fully tensorized model. It is shown how the corresponding generalization of the TLS formulation induces enrichment of the search space for the data corrections. Solvability of the resulting minimization problem is studied. Furthermore, extension of the so-called core reduction to the bilinear model is presented. For the fully tensor model, its relation to a particular single right-hand side TLS problem is derived. Relationships among individual formulations are discussed.
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    Solvability classes for core problems in matrix total least squares minimization
    (ACAD SCIENCES CZECH REPUBLIC, INST MATHEMATICS, ZITNA 25, PRAHA 1, 115 67, CZECH REPUBLIC, 2019-01-01) Hnětynková, Iveta; Plešinger, Martin; Žáková, Jana
    Linear matrix approximation problems AX approximate to 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 Hnetynkova, Pleinger, 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.