Efficient Robust Principal Component Analysis via Block Krylov Iteration and CUR Decomposition

Robust principal component analysis (RPCA) is widely studied in computer vision. Recently an adaptive rank estimate based RPCA has achieved top performance in low-level vision tasks without the prior rank, but both the rank estimate and RPCA optimization algorithm involve singular value decomposition, which requires extremely huge computational resource for large-scale matrices. To address these issues, an efficient RPCA (eRPCA) algorithm is proposed based on block Krylov iteration and CUR decomposition in this paper. Specifically, the Krylov iteration method is employed to approximate the eigenvalue decomposition in the rank estimation, which requires
$O(ndrq+n(rq)^{2})$
for an
$(n\times d)$
input matrix, in which
$q$
is a parameter with a small value,
$r$
is the target rank. Based on the estimated rank, CUR decomposition is adopted to replace SVD in updating low-rank matrix component, whose complexity reduces from
$O(rnd)$
to
$O(r^{2}n)$
per iteration. Experimental results verify the efficiency and effectiveness of the proposed eRPCA over the state-of-the-art methods in various low-level vision applications.