Robust PCA Using Nonconvex Rank Approximation and Sparse Regularizer

We consider the robust principal component analysis (RPCA) problem where the observed data are decomposed to a low-rank component and a sparse component. Conventionally, the matrix rank in RPCA is often approximated using a nuclear norm. Recently, RPCA has been formulated using the nonconvex $$\ell _{\gamma }$$-norm, which provides a closer approximation to the matrix rank than the traditional nuclear norm. However, the low-rank component generally has sparse property, especially in the transform domain. In this paper, a sparsity-based regularization term modeled with $$\ell _1$$-norm is introduced to the formulation. An iterative optimization algorithm is developed to solve the obtained optimization problem. Experiments using synthetic and real data are utilized to validate the performance of the proposed method.