Global Convergence of Rtlsqep: A Solver of Regularized Total Least Squares Problems via Quadratic Eigenproblems

Abstract

The total least squares (TLS) method is a successful approach for linear problems if both the matrix and the right hand side are contaminated by some noise. In a recent paper Sima, Van Huffel and Golub suggested an iterative method for solving regularized TLS problems, where in each iteration step a quadratic eigenproblem has to be solved. In this paper we prove its global convergence, and we present an efficient implementation using an iterative projection method with thick updates.

Solving Regularized Total Least Squares Problems Based on Eigenproblems

“…In [107,108] this method is called RTL-SQEP (Regularized Total Least Squares via Quadratic Eigenvalue Problems) since in each step the rightmost eigenvalue and corresponding eigenvector of a quadratic eigenproblem has to be determined. Its global convergence was proved in [63], cf. subsection 3.4.1.…”

“…In chapter 3 in subsection 3.4.2 it is shown that a similar technique has been successfully applied for accelerating the RTLS solver in [108] which is based on a sequence of quadratic eigenvalue problems, cf. [60,63]. Another method for RTLS problems presented in [95] which is based on a sequence of linear eigenproblems has also been accelerated substantially in [62,66].…”

“…The global convergence of the first Algorithm 3.1 was proven in [63] and a convergence proof for the second Algorithm 3.2 can be found in [6]. Let us consider the quadratically constrained TLS problem (3.69), i.e.…”

“…The difference between Algorithms 3.1 and 3.2 is highlighted by proving the existence of a negative Lagrange multiplier, see also [60] and the end of subsection 3.4.1. Efficient implementations for solving the sequence of QEPs are discussed in [60,63,66] with the special emphasis on iterative projection methods and updating techniques that accelerates the RTLSQEP method in [108] substantially. This subsection is organized as follows.…”

“…This first-order conditions were used in literature in two ways to solve problem (3.68): In [60,63,66,107,108] for a fixed parameter λ I problem (3.81) is solved for (x, λ L ), which yields a convergent sequence of updates for λ I . A closely related algorithm is presented in [5,6] that solve a similar sequence where in each iteration step the equality constraint in (3.81) is replaced by the original inequality Lx 2 ≤ ∆ 2 .…”