An interval perturbation method for singular value decomposition (SVD) with unknown-but-bounded (UBB) parameters

Abstract

The singular value decomposition (SVD) is a crucial method with successful practical applications. However, the quantification of uncertainties in SVD perturbation is challenging. To address this issue, this study introduces an interval perturbation method for SVD with unknown-but-bounded (UBB) parameters. Unlike probabilistic approaches that require statistical data on uncertain parameters, this method only necessitates the bounds of uncertain parameters. By using non-probabilistic theory, the proposed method can provide accurate and fast estimation of singular values and vectors. The paper provides a detailed derivation process for obtaining the interval bounds of singular values and vectors using the interval perturbation method. The subinterval method is applied to improve the estimation precision. The effectiveness of the proposed method is demonstrated through four numerical examples and an application example. The robustness of the proposed method is verified by various levels of uncertainties, different cases of subintervals, different matrix scales, close singular values, rectangular matrices, and small but nonzero singular values. The proposed method can be used in engineering numerical calculation inverse problem fields that require SVD with small uncertainties, including dynamic identification, image processing, and signal processing. Therefore, the superiority of the proposed method in accurately and quickly estimating singular values and vectors with uncertainties can be applied in the aforementioned research fields.