Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm (1510.06895v1)

Abstract: 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.

An Overview of Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm

The paper "Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm" addresses a significant challenge in the field of low-rank matrix recovery, particularly the shortcomings observed when using the nuclear norm as a convex surrogate for the rank function. Traditional approaches utilizing the nuclear norm often yield suboptimal solutions because the nuclear norm is a loose approximation of the original rank function. This paper offers a novel methodology by employing nonconvex surrogates to better approximate the rank function.

Methodology and Algorithm

The authors propose a new approach by using a family of nonconvex surrogates of the $L_0$-norm applied to the singular values of a matrix, leading to a nonconvex nonsmooth minimization problem. To tackle this, they introduce the Iteratively Reweighted Nuclear Norm (IRNN) algorithm. The IRNN algorithm iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, capitalizing on the special properties of nonconvex surrogate functions that allow for a closed-form solution.

These nonconvex surrogates of the $L_0$-norm, including well-known functions like the $L_p$-norm, SCAD, and others, provide a more accurate approximation of the rank function, enhancing recovery tasks in both synthetic and real image data contexts. The IRNN algorithm is noted for its theoretical guarantee of decreasing the objective function value monotonically, with any limit point serving as a stationary point—key for establishing convergence.

Experimental Results

The paper provides comprehensive experiments on both synthetic data and real images, demonstrating the efficacy of the IRNN method compared to state-of-the-art convex algorithms. The numerical results highlight that IRNN consistently achieves better low-rank matrix recovery, suggesting the strong potential of nonconvex models in practical applications.

Implications and Future Directions

The implications of this research are notable both in theoretical and practical domains. Theoretically, this work expands the understanding of low-rank minimization by leveraging nonconvex optimization strategies, suggesting that more accurate rank approximations can be beneficial. Practically, the demonstrated improvements in numerical performance have applications across various domains where low-rank structures are exploited, such as computer vision, signal processing, and machine learning.

The authors propose future work in several directions, including exploring the convergence properties of IRNN further, particularly in multi-block variable settings. Additionally, there is potential for extending these techniques to handle problems with more intricate structure or constraints, such as those arising in nonconvex tensor decomposition.

Overall, the paper makes a significant contribution to the field of low-rank matrix recovery, providing both a novel theoretical framework and a practical algorithm that researchers can leverage to address complex problems more effectively. The concepts and methods introduced have the potential to influence future advancements in related areas of computer science and applied mathematics.