Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm
This work addresses the problem of improving low-rank matrix recovery for applications like image and signal processing, but it is incremental as it builds on existing convex methods with nonconvex extensions.
The paper tackled the suboptimal solutions from nuclear norm relaxation in low-rank matrix recovery by proposing a nonconvex surrogate approach and the Iteratively Reweighted Nuclear Norm (IRNN) algorithm, which enhanced recovery performance in experiments on synthesized data and real images.
The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to perform a family of nonconvex surrogates of $L_0$-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.