PERTURBATION-THEORY FOR ORTHOGONAL PROJECTION METHODS WITH APPLICATIONS TO LEAST-SQUARES AND TOTAL LEAST-SQUARES

The stabilized versions of the least squares (LS) and total least squa res (TLS) methods are two examples of orthogonal projection methods co mmonly used to ''solve'' the overdetermined system of linear equations AX approximate to B when A is nearly rank-deficient, In practice, whe n this system represents the noisy version of an exact rank-deficient, zero-residual problem, TLS usually yields a more accurate estimate of the exact solution. However, current perturbation theory does not jus tify the superiority of TLS over LS. In this paper we establish a mode l for orthogonal projection methods by reformulating the parameter est imation problem as an equivalent problem of nullspace determination. W hen the method is based on the singular value decomposition of the mat rix [A B], the model specializes to the well-known TLS method. We deri ve new lower and upper perturbation bounds for orthogonal projection m ethods in terms of the subspace angle, which shows how the perturbatio n of the approximate nullspace affects the accuracy of the solution. I n situations where TLS is typically used, such as in signal processing where the noise-free compatible problem is exactly rank-deficient, ou r upper bounds suggest that the TLS perturbation bound is usually smal ler than the one for LS, which means that TLS is usually more robust t han LS under perturbations of all the data. Also, the bounds permit a comparison between the LS and TLS solutions, as well as for any two co mpeting orthogonal projection methods. We include numerical simulation s to illustrate our conclusions.