  Singular value thresholding (SVT) plays an important role in the well-known
robust principal component analysis (RPCA) algorithms which have many
applications in computer vision and recommendation systems. In this paper, we
formulate and study a weighted singular value thresholding (WSVT) problem,
which uses a combination of the nuclear norm and a weighted Frobenius norm. We
present an algorithm to numerically solve WSVT and establish the convergence of
the algorithm. As a proof of concept, we apply WSVT with a simple choice of
weight learned from the data to the background estimation problem in computer
vision. The numerical experiments show that our method can outperform RPCA
algorithms. This indicates that instead of tackling the computationally
expensive $\ell_1$ norm employed in RPCA, one may switch to the weighted
Frobenius norm and achieve about the same or even better performance.
