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On TLS formulation and core reduction for data fitting with generalized models. (English) Zbl 1420.65053
Summary: The total least squares (TLS) framework represents a popular data fitting approach for solving matrix approximation problems of the form \(\mathcal{A}(X) \equiv A X \approx B\). A general linear mapping on spaces of matrices \(\mathcal{A} : X \longmapsto B\) can be represented by a fourth-order tensor which is in the \(A X \approx B\) 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 \(\mathcal{A}\): 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.
Reviewer: Reviewer (Berlin)
MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A09 Theory of matrix inversion and generalized inverses
15A21 Canonical forms, reductions, classification
15A69 Multilinear algebra, tensor calculus
65F25 Orthogonalization in numerical linear algebra
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