  In recent years, the nuclear norm minimization (NNM) problem has been
attracting much attention in computer vision and machine learning. The NNM
problem is capitalized on its convexity and it can be solved efficiently. The
standard nuclear norm regularizes all singular values equally, which is however
not flexible enough to fit real scenarios. Weighted nuclear norm minimization
(WNNM) is a natural extension and generalization of NNM. By assigning properly
different weights to different singular values, WNNM can lead to
state-of-the-art results in applications such as image denoising. Nevertheless,
so far the global optimal solution of WNNM problem is not completely solved yet
due to its non-convexity in general cases. In this article, we study the
theoretical properties of WNNM and prove that WNNM can be equivalently
transformed into a quadratic programming problem with linear constraints. This
implies that WNNM is equivalent to a convex problem and its global optimum can
be readily achieved by off-the-shelf convex optimization solvers. We further
show that when the weights are non-descending, the globally optimal solution of
WNNM can be obtained in closed-form.
