The robust principal component analysis (RPCA) decomposes a data matrix into a low-rank part and a sparse part. There are mainly two types of algorithms for RPCA. The first type of algorithm applies regularization terms on the singular values of a matrix to obtain a low-rank matrix. However, calculating singular values can be very expensive for large matrices. The second type of algorithm replaces the low-rank matrix as the multiplication of two small matrices. They are faster than the first type because no singular value decomposition (SVD) is required. However, the rank of the low-rank matrix is required, and an accurate rank estimation is needed to obtain a reasonable solution. In this paper, we propose algorithms that combine both types. Our proposed algorithms require an upper bound of the rank and SVD on small matrices. First, they are faster than the first type because the cost of SVD on small matrices is negligible. Second, they are more robust than the second type because an upper bound of the rank instead of the exact rank is required. Furthermore, we apply the Gauss-Newton method to increase the speed of our algorithms. Numerical experiments show the better performance of our proposed algorithms.

中文翻译：

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鲁棒的主成分分析的快速算法

健壮的主成分分析（RPCA）将数据矩阵分解为低阶部分和稀疏部分。RPCA算法主要有两种。第一种算法将正则化项应用在矩阵的奇异值上以获得低秩矩阵。但是，对于大型矩阵，计算奇异值可能会非常昂贵。第二种算法将低秩矩阵替换为两个小矩阵的乘法。它们比第一种更快，因为不需要奇异值分解（SVD）。但是，需要低秩矩阵的秩，并且需要准确的秩估计以获得合理的解决方案。在本文中，我们提出了结合两种类型的算法。我们提出的算法在小矩阵上要求秩和SVD的上限。第一，它们比第一种更快，因为在小型矩阵上SVD的成本可忽略不计。其次，它们比第二种类型更健壮，因为需要等级上限而不是确切等级。此外，我们应用高斯-牛顿法来提高算法的速度。数值实验表明，我们提出的算法具有更好的性能。