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Multidimensional Total Least Squares Problem with Linear Equality Constraints
Many recent data analysis models are mathematically characterized by a multidimensional total least squares problem with linear equality constraints (TLSE). In this paper, an explicit solution is firstly derived for the multidimensional TLSE problem, as well as the solvability conditions. With applying the perturbation theory of invariant subspace, the multidimensional TLSE problem is proved equivalent to a multidimensional unconstrained weighed total least squares problem in the limit sense. The Kronecker product-based formulae are also given for the normwise, mixed, and componentwise condition numbers of the multidimensional TLSE solution of minimum Frobenius norm, and their computable upper bounds are also provided to reduce the storage and computational cost. All these results are appropriate for the single right-hand-side case and the multidimensional total least squares problem, which are two especial cases of the multidimensional TLSE problem. In numerical experiments, the multidimensional TLSE model is successfully applied to color image deblurring and denoising for the first time, and the numerical results also indicate the effectiveness of the condition numbers.
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