The Regularized Total Least Squares Problem: Theoretical Properties and Three Globally Convergent Algorithms

Total Least Squares (TLS) is a method for treating an overdetermined system of linear equations Ax ≈ b, where both the matrix A and the vector b are contaminated by noise. In practical situations, the linear system is often ill-conditioned. For example, this happens when the system is obtained via discretization of ill-posed problems such as integral equations of the first kind (see e.g., [7] and references therein). In these cases the TLS solution can be physically meaningless and thus regularization is essential for stabilizing the solution. Regularization of the TLS solution was addressed by several approaches such as truncation methods [6, 8] and Tikhonov regularization [1]. In this talk we will consider a third approach in which a quadratic constraint is introduced. It is well known [7, 11] that the quadratically constrained total least squares problem can be formulated as a problem of minimizing a ratio of two quadratic function subject to a quadratic constraint: