  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.
