Robust PCA by Manifold Optimization
Robust PCA by Manifold Optimization
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发表时间：
2017-08
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通讯作者：
Teng Zhang;Yi Yang
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作者：
Teng Zhang;Yi Yang
稳健的 PCA 是一种广泛使用的统计过程，用于恢复具有严重损坏的观测值的底层低秩矩阵。这项工作将鲁棒 PCA 问题视为低秩矩阵流形上的非凸优化问题，并提出了两种基于流形优化的算法（针对两个版本的撤销）。结果表明，通过适当设计的初始化，所提出的算法可以保证线性收敛到底层低秩矩阵。与之前基于低秩矩阵的 Burer-Monterio 分解的工作相比，所提出的算法理论上减少了对底层低秩矩阵条件数的依赖。模拟和真实数据示例证实了我们方法的竞争性能。
Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank matrices, and proposes two algorithms (for two versions of retractions) based on manifold optimization. It is shown that, with a proper designed initialization, the proposed algorithms are guaranteed to converge to the underlying low-rank matrix linearly. Compared with a previous work based on the Burer-Monterio decomposition of low-rank matrices, the proposed algorithms reduce the dependence on the conditional number of the underlying low-rank matrix theoretically. Simulations and real data examples confirm the competitive performance of our method.