CUR and Generalized CUR Decompositions of Quaternion Matrices and their Applications

Abstract

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.

Keywords:

• Color video background subtraction
• CUR decomposition
• generalized CUR decomposition
• quaternion matrix
• robust quaternion principal component analysis

AMS SUBJECT CLASSIFICATIONS:

• 15A09
• 65F20

Acknowledgments

The authors would like to thank the Editor Prof. M. Nashed and two anonymous reviewers for their careful and very detailed comments on our paper.

Notes

1 https://github.com/fleph/RQCUR/tree/main/color_video

2 https://github.com/fleph/RQCUR/tree/main/output

Additional information

Funding

Renjie Xu is supported by Shanghai Municipal Science and Technology Commission under grant 23WZ2501400. Shenghao Feng is supported by the National Natural Science Foundation of China under grant 12271108. Yimin Wei is supported by the National Natural Science Foundation of China under grant 12271108 and the Ministry of Science and Technology of China under grant G2023132005L. Hong Yan is supported by the Hong Kong Research Grants Council (Project 11204821), the Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA) and City University of Hong Kong (Projects 9610034 and 9610460).