Convex set-oriented singular value decomposition with bounded uncertainties

Abstract

Singular value decomposition (SVD) has played a crucial role in mathematics and applied mathematics, particularly in the field of practical engineering technology, which is almost indispensable in these fields. However, incomplete information caused by uncertainties from multiple sources and fluctuations in inherent properties during engineering practice can seriously interfere with obtaining singular values. Based on the convex set uncertainty theory, a novel convex set-oriented singular value decomposition (CSSVD) is proposed to overcome the limitations of using the nominal SVD method. The CSSVD aims to accurately and efficiently calculate the uncertainty bounds of singular values and vectors through the bounds and correlations of initial uncertainty parameters. Traditional probability methods rely on a sufficient sample size, and when the sample size is insufficient, accurate results cannot be obtained. However, the CSSVD method is based on set theory and can still demonstrate excellent performance in scenarios with small samples. This study investigates the detailed derivation steps of the proposed CSSVD method, taking into consideration properties such as normalization, orthogonality, and Monte Carlo simulation (MCS) to verify the accurate bounds of convex sets. The proposed method is finally applied to several numerical examples to demonstrate its superiority. This includes verifying singular values and vectors, handling high-order matrices, matrices with close singular values, and rectangular matrices. An application of image denoising is also used to evaluate the effectiveness of the proposed method.