一种改进的求解低秩矩阵补全问题的惯性交替方向法  

AN IMPROVED INERTIAL ALTERNATING DIRECTION METHOD FOR LOW RANK MATRIX COMPLETION PROBLEMS

作　　者：闫喜红[1] 唐晓妮 李超 Yan Xihong;Tang Xiaoni;Li Chao(Taiyuan Normal University,Collge of Mathematics and Statistics,Jinzhong 030619,China)

机构地区：[1]太原师范学院数学与统计学院,晋中030619

出　　处：《计算数学》2024年第2期144-155,共12页Mathematica Numerica Sinica

基　　金：国家自然科学基金(11901424);山西省回国留学人员科研教研资助项目(2022-170);山西省科技创新人才团队项目(Nos.202204051002018)资助.

摘　　要：交替方向法是求解矩阵补全问题的经典方法之一.近年来,随着信息的高速发展,需要处理的矩阵规模越来越大.为进一步提高交替方向法求解大规模矩阵补全问题的效率,本文将交替方向法中的一个子问题结合惯性策略进行加速,即利用该子问题的前一步迭代点和前一步的惯性迭代点进行线性组合得到新一步的惯性迭代点,从而提出了一种改进的求解矩阵补全问题的惯性交替方向法。本文在合理的假设条件下,给出了新算法的收敛性证明.最后,通过随机矩阵补全及图像修复实例的数值实验结果验证了新算法的优越性.The alternating direction method is one of the classical methods for solving matrix completion problems.Recently,with the rapid development of information,the size of matrices to be processed is very huge.In order to further improve the efficiency of the alternating direction method for solving the large-size matrix completion problems,we borrow the inertia strategy to solve a subproblem of the alternate direction method.Specifically,we obtain the next inertial iteration point by linear combination of the previous iteration point and the previous inertial iteration point of the subproblem.Thus we propose an improved inertial alternating direction method for low rank matrix completion problems in this paper.The convergence analysis of the new algorithm is given under reasonable assumptions.Finally,the superiority of the new algorithm is verified by numerical experimental results of random matrix completion problems and image restoration examples.

关 键 词：矩阵补全 交替方向法 惯性策略 

分 类 号：O151.21[理学—数学]