  Low-rank matrix factorization (LRMF) has received much popularity owing to
its successful applications in both computer vision and data mining. By
assuming noise to come from a Gaussian, Laplace or mixture of Gaussian
distributions, significant efforts have been made on optimizing the (weighted)
$L_1$ or $L_2$-norm loss between an observed matrix and its bilinear
factorization. However, the type of noise distribution is generally unknown in
real applications and inappropriate assumptions will inevitably deteriorate the
behavior of LRMF. On the other hand, real data are often corrupted by skew
rather than symmetric noise. To tackle this problem, this paper presents a
novel LRMF model called AQ-LRMF by modeling noise with a mixture of asymmetric
Laplace distributions. An efficient algorithm based on the
expectation-maximization (EM) algorithm is also offered to estimate the
parameters involved in AQ-LRMF. The AQ-LRMF model possesses the advantage that
it can approximate noise well no matter whether the real noise is symmetric or
skew. The core idea of AQ-LRMF lies in solving a weighted $L_1$ problem with
weights being learned from data. The experiments conducted on synthetic and
real datasets show that AQ-LRMF outperforms several state-of-the-art
techniques. Furthermore, AQ-LRMF also has the superiority over the other
algorithms in terms of capturing local structural information contained in real
images.
