本文研究一类低秩矩阵优化问题,其中惩罚项为目标矩阵奇异值的ℓp(0＜p＜1)正则函数.基于半阈值函数在稀疏/低秩恢复问题中的良好性能,本文提出奇异值半阈值(singular value half thresholding,SVHT)算法来求解ℓp正则矩阵优化问题.SVHT算法的主要迭代利用了子问题的闭式解,但与现有算法不同,其本质上是对目标函数在当前点进行局部1/2近似,而不是局部线性或局部二次近似.通过构造目标函数的Lipschitz和非Lipschitz近似函数,本文证明了 SVHT算法生成序列的任意聚点都是问题的一阶稳定点.在数值实验中,利用模拟数据和实际图像数据的低秩矩阵补全问题对SVHT算法进行测试.大量的数值结果表明,SVHT算法对低秩矩阵优化问题在速度、精度和鲁棒性等方面都表现优异.
Abstract
In this paper,we study the low-rank matrix optimization problem where the penalty term is the ℓp(0＜p＜1)regularization.Inspired by the good performance of the half thresholding function in sparse/low-rank recovery problems,we propose a singular value half thresholding(SVHT)algorithm to solve the ℓp regularized matrix optimization problem.The main iteration in the SVHT algorithm uses the closed-form solution of the subproblem to make a local 1/2 approximation to the objective function at the current point instead of a local linear or local quadratic approximation.By constructing Lipschitz and non-Lipschitz approximate functions of the objective function,we prove that any accumulation point of the sequence generated by the SVHT algorithm is a first-order stationary point of the problem.In numerical experiments,we test the SVHT algorithm through low-rank matrix completion problems using both simulated and real image data.Extensive numerical results show the superior efficiency of the SVHT algorithm for low-rank matrix optimization problems in terms of speed,accuracy,and robustness.
关键词
低秩矩阵优化问题/矩阵ℓp正则/闭式解/奇异值半阈值算法/一阶稳定点
Key words
low-rank matrix optimization problem/matrix ℓp regularization/closed-form solution/singular value half thresholding algorithm/first-order stationary point
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