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Smoothed Low Rank and Sparse Matrix Recovery by Iteratively Reweighted Least Squares Minimization

机译:通过迭代加权最小二乘最小化平滑低秩和稀疏矩阵恢复

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This paper presents a general framework for solving the low-rank and/or sparse matrix minimization problems, which may involve multiple nonsmooth terms. The iteratively reweighted least squares (IRLSs) method is a fast solver, which smooths the objective function and minimizes it by alternately updating the variables and their weights. However, the traditional IRLS can only solve a sparse only or low rank only minimization problem with squared loss or an affine constraint. This paper generalizes IRLS to solve joint/mixed low-rank and sparse minimization problems, which are essential formulations for many tasks. As a concrete example, we solve the Schatten- norm and -norm regularized low-rank representation problem by IRLS, and theoretically prove that the derived solution is a stationary point (globally optimal if ). Our convergence proof of IRLS is more general than previous one that depends on the special properties of the Schatten- norm and -norm. Extensive experiments on both synthetic and real data sets demonstrate that our IRLS is much more efficient.
机译:本文提出了解决低秩和/或稀疏矩阵最小化问题的通用框架,该问题可能涉及多个非光滑项。迭代重新加权最小二乘(IRLSs)方法是一种快速求解器,它通过交替更新变量及其权重来平滑目标函数并将其最小化。然而,传统的IRLS仅能解决平方损失或仿射约束的仅稀疏或仅低秩的最小化问题。本文概括了IRLS,以解决联合/混合低秩和稀疏最小化问题,这是许多任务的基本公式。举一个具体的例子,我们通过IRLS解决了Schatten-norm和-norm正规化的低秩表示问题,并从理论上证明了导出的解是一个平稳点(如果,则为全局最优)。我们对IRLS的收敛性证明比以前的更为广泛,后者取决于Schattennorm和-norm的特殊属性。在综合和真实数据集上进行的大量实验表明,我们的IRLS效率更高。

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