Applications of Mathematics, Vol. 62, No. 2, pp. 105-120, 2017
Filter factors of truncated TLS regularization with multiple observations
Iveta Hnětynková, Martin Plešinger, Jana Žáková
Received August 18, 2016. First published February 28, 2017.
Abstract: The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems $Ax\approx b$ were analyzed by Fierro, Golub, Hansen, and O'Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of $A$ applied to $b$. This paper focuses on the situation when multiple observations $b_1,\ldots,b_d$ are available, i.e., the T-TLS method is applied to the problem $AX\approx B$, where $B=[b_1,\ldots,b_d]$ is a matrix. It is proved that the filtering representation of the T-TLS solution can be generalized to this case. The corresponding filter factors are explicitly derived.
Keywords: truncated total least squares; multiple right-hand sides; eigenvalues of rank-$d$ update; ill-posed problem; regularization; filter factors
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Affiliations: Iveta Hnětynková, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 3, 121 16 Praha 2, Czech Republic, e-mail: iveta.hnetynkova@mff.cuni.cz; Martin Plešinger (corresponding author), Technical University of Liberec, Department of Mathematics, Studentská 1402/2, 461 17 Liberec, Czech Republic and Institute of Computer Science of the Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07 Praha 8, Czech Republic, e-mail: martin.plesinger@tul.cz; Jana Žáková, Technical University of Liberec, Department of Mathematics, Studentská 1402/2, 461 17 Liberec, Czech Republic, e-mail: jana.zakova@tul.cz
