Overview of total least-squares methods
Signal Processing
The matrix-restricted total least-squares problem
Signal Processing
On a quadratic eigenproblem occurring in regularized total least squares
Computational Statistics & Data Analysis
Structured Total Maximum Likelihood: An Alternative to Structured Total Least Squares
SIAM Journal on Matrix Analysis and Applications
Efficient determination of the hyperparameter in regularized total least squares problems
Applied Numerical Mathematics
Large-scale Tikhonov regularization of total least squares
Journal of Computational and Applied Mathematics
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Total least squares (TLS) is a method for treating an overdetermined system of linear equations ${\bf A} {\bf x} \approx {\bf b}$, where both the matrix ${\bf A}$ and the vector ${\bf b}$ are contaminated by noise. Tikhonov regularization of the TLS (TRTLS) leads to an optimization problem of minimizing the sum of fractional quadratic and quadratic functions. As such, the problem is nonconvex. We show how to reduce the problem to a single variable minimization of a function ${\mathcal{G}}$ over a closed interval. Computing a value and a derivative of ${\mathcal{G}}$ consists of solving a single trust region subproblem. For the special case of regularization with a squared Euclidean norm we show that ${\mathcal{G}}$ is unimodal and provide an alternative algorithm, which requires only one spectral decomposition. A numerical example is given to illustrate the effectiveness of our method.