Fast algorithms for robust principal component analysis with an upper bound on the rank

TL;DR

This paper introduces fast RPCA algorithms that leverage an upper bound on the rank, combining the advantages of existing methods while reducing computational costs and improving robustness.

Contribution

The paper proposes novel RPCA algorithms that require only an upper bound on the rank, integrating SVD on small matrices and the Gauss-Newton method for enhanced speed and robustness.

Findings

Algorithms are faster than traditional SVD-based methods.

More robust than low-rank factorization methods requiring exact rank.

Numerical experiments demonstrate superior performance.

Abstract

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.