Exactly Recovering Low-Rank Matrix in Linear Time via $l_1$ Filter

Recovering a low rank matrix from corrupted data, which is known as Robust PCA, has attracted considerable interests in recent years. This problem can be exactly solved by a combined nuclear norm and $l_1$ norm minimization. However, due to the computational burden of SVD inherent with the nuclear norm minimization, traditional methods suffer from high computational complexity, especially for large scale datasets. In this paper, inspired by digital filtering idea in image processing, we propose a novel algorithm, named $l_1$ Filter, for solving Robust PCA with linear cost. The $l_1$ Filter is defined by a seed, which is a exactly recovered small submatrix of the underlying low rank matrix. By solving some $l_1$ minimization problems in parallel, the full low rank matrix can be exactly recovered from corrupted observations with linear cost. Both theoretical analysis and experimental results exhibit that our method is an efficient way to exactly recovering low rank matrix in linear time.