Optimally regularized inverse of singular value decomposition and application to signal extrapolation

Abstract

The singular value decomposition (SVD) is an excellent tool to attain reliable and stable least squares parameters estimation in ill-conditioned cases, by discarding small singular values adequately. However, large singular values and small singular values do not always separate neatly. In the present paper, we introduce multiple regularization parameters to modify the Moore-Penrose pseudo-inverse matrix for the purpose of stabilization of ill-posed least squares problems. Optimal values of the regularization parameters can be determined so as to minimize an estimated mean squares error (EMSE) criterion calculated by using only accessible signal data. Thus, the proposed scheme can successfully give threshold conditions whether smaller singular values should be adopted or discarded. The relationship with the optimal truncation of the singular values is also investigated analytically. Effectiveness of the proposed method is discussed in applications to optimal extrapolation of band-limited signals.