CUR and Generalized CUR Decompositions of Quaternion Matrices and their Applications

Low-rank approximations of quaternion matrices have garnered interest across various applications, such as color images and signal processing. In this paper, we propose the CUR and generalized CUR decompositions of quaternion matrices and utilize the CUR decomposition of the quaternion matrix to solve the robust quaternion principal component analysis (RQPCA) problem. Through the discrete empirical interpolation method (DEIM) for the subset selection, we present the error analysis of the approximation of the CUR and the generalized CUR decompositions of quaternion matrices. The error bound depends on the conditioning of sampling submatrices. Next, we employ the alternating projection and adopt the CUR decomposition of the quaternion matrix to tackle the RQPCA. The non-convex algorithm runs fast and significantly reduces computational complexity. The performance advantages of our algorithms have been experimentally verified on artificial datasets. The RQCUR significantly affects color video background subtraction in the experiment.