  In the past decades, exactly recovering the intrinsic data structure from
corrupted observations, which is known as robust principal component analysis
(RPCA), has attracted tremendous interests and found many applications in
computer vision. Recently, this problem has been formulated as recovering a
low-rank component and a sparse component from the observed data matrix. It is
proved that under some suitable conditions, this problem can be exactly solved
by principal component pursuit (PCP), i.e., minimizing a combination of nuclear
norm and $l_1$ norm. Most of the existing methods for solving PCP require
singular value decompositions (SVD) of the data matrix, resulting in a high
computational complexity, hence preventing the applications of RPCA to very
large scale computer vision problems. In this paper, we propose a novel
algorithm, called $l_1$ filtering, for \emph{exactly} solving PCP with an
$O(r^2(m+n))$ complexity, where $m\times n$ is the size of data matrix and $r$
is the rank of the matrix to recover, which is supposed to be much smaller than
$m$ and $n$. Moreover, $l_1$ filtering is \emph{highly parallelizable}. It is
the first algorithm that can \emph{exactly} solve a nuclear norm minimization
problem in \emph{linear time} (with respect to the data size). Experiments on
both synthetic data and real applications testify to the great advantage of
$l_1$ filtering in speed over state-of-the-art algorithms.

Numerical Analysis;Computer Vision and Pattern Recognition