Incremental quaternion singular value decomposition and its application for low rank quaternion matrix completion

Computing the optimal low rank approximations of quaternion matrices is the key target in many quaternion matrix related problems including color images inpainting and recognition, which can be reconstructed by some dominant singular values of quaternion matrices. However, the singular value decomposition of large-scale quaternion matrices requires expensive storage and computational costs. In this paper, we propose an incremental quaternion singular value decomposition (IQSVD) method for a class of quaternion matrices, where the number of columns far exceeds the number of rows, to improve computing efficiency. What’s more, based on IQSVD, we consider the low rank quaternion matrix completion problem and design a proximal linearized minimization algorithm with convergence guarantee to solve it. Numerical experiments on synthetic data and real-world videos illustrate the efficiency of IQSVD and the proposed proximal linearized minimization algorithm involved IQSVD.