论文标题
稳健主成分分析的快速算法，在等级上具有上限
Fast algorithms for robust principal component analysis with an upper bound on the rank
论文作者
论文摘要
鲁棒主成分分析（RPCA）将数据矩阵分解为低级别部分和稀疏部分。 RPCA主要有两种类型的算法。第一种类型的算法将正则化项应用于矩阵的奇异值以获得低级别矩阵。但是，对于大型矩阵而言，计算单数值可能非常昂贵。第二种类型的算法取代了两个小矩阵的乘法。它们比第一种类型快，因为不需要单数值分解（SVD）。但是，需要低级矩阵的等级，并且需要准确的等级估计来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。我们提出的算法需要小矩阵上等级的上限和SVD。首先，它们比第一种类型快，因为小矩阵上的SVD成本可以忽略不计。其次，它们比第二种类型更强大，因为需要等级的上限而不是确切的等级。此外，我们采用高斯 - 纽顿方法来提高算法的速度。数值实验表明我们提出的算法的性能更好。
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.
