Orthogonal Projection And Total Least-Squares

Overdetermined linear systems often arise in applications such as signal processing and modem communication. When the overdetermined system of linear equations AX approximate to B has no solution, compatibility may be restored by an orthogonal projection method. The idea is to determine an orthogonal projection matrix P by some method M such that [($) over tilde A ($) over tilde B] = P[A B], and ($) over tilde AX = ($) over tilde B is compatible. Denote by X(M) the minimum norm solution to ($) over tilde AX = ($) over tilde B using method M. In this paper conditions for compatibility of the lower rank approximation and subspace properties of ($) over tilde A in relation to the nearest rank-k matrix to A are discussed. We find upper and lower bounds for the difference between the solution X(M) and the SVD-based total least squares (TLS) solution X(SV D) and also provide a perturbation result for the ordinary TLS method. These results suggest a new algorithm for computing a total least squares solution based on a rank revealing QR factorization and subspace refinement. Numerical simulations are included to illustrate the conclusions.