On TLS formulation and core reduction for data fitting with generalized models
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.
Total least squares problem (TLS), Error-in-variables modeling, Orthogonal regression, Multiple observations, Core problem, Tensor approximation problem