On the Convex Geometry of Weighted Nuclear Norm Minimization

Low-rank matrix approximation, which aims to construct a low-rank matrix from an observation, has received much attention recently. An efficient method to solve this problem is to convert the problem of rank minimization into a nuclear norm minimization problem. However, soft-thresholding of singular values leads to the elimination of important information about the sensed matrix. Weighted nuclear norm minimization (WNNM) has been proposed, where the singular values are assigned different weights, in order to treat singular values differently. In this paper the solution for WNNM is analyzed under a particular weighting condition using the connection between convex geometry and compressed sensing algorithms. It is shown that the WNNM is convex where the weights are in non-descending order and there is a unique global minimizer for the minimization problem. KeywordsWeighted nuclear norm minimization, Convex geometry, Low-rank matrix approximation, Linear inverse problem, Phase Transition.