Efficient quaternion CUR method for low-rank approximation to quaternion matrix
Peng-Ling Wu, Kit Ian Kou, Hongmin Cai, Zhaoyuan Yu
TL;DR
This work tackles the high cost of quaternion SVD-based low-rank approximations by introducing the QMCUR method, a CUR-style approach that uses actual columns and rows from a quaternion matrix $X$ to form a $CUR$ representation with core $U = C^{\dagger} X R^{\dagger}$. Two sampling strategies—length-based and uniform—determine the selections of columns and rows, with a practical choice of submatrix sizes $m \times k\log k$ and $k\log k \times n$. A perturbation analysis provides a bound on the approximation error in terms of the noise $\mathbf{E}$ and submatrix pseudoinverses, showing linear dependence on the perturbation. Empirical results on synthetic data and color images demonstrate substantial speedups over QSVD while preserving accuracy, highlighting QMCUR's potential for scalable color image processing and other large-scale quaternion matrix tasks. Overall, QMCUR offers a QSVD-avoiding, efficient, and provably reliable framework for LRQA in the quaternion domain.
Abstract
The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested, which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quaternion matrices. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.