A Comparative Study for the Weighted Nuclear Norm Minimization and Nuclear Norm Minimization

Nuclear norm minimization (NNM) tends to over-shrink the rank components and treats the different rank components equally, thus limits its capability and flexibility. Recent studies have shown that the weighted nuclear norm minimization (WNNM) is expected to be more accurate than NNM. However, it still lacks a plausible mathematical explanation why WNNM is more accurate than NNM. This paper analyzes the WNNM and NNM from the perspective of the group sparse representation (GSR). In particular, an adaptive dictionary for each group is designed to connect the rank minimization and GSR models. Then, we prove that the rank minimization model is equivalent to GSR model. Based on that conclusion, we show mathematically that WNNM is more accurate than NNM. To make the proposed model tractable and robust, the alternative direction multiplier method (ADMM) framework is developed to solve the proposed model. We exploit the proposed scheme to three low level vision tasks, including image deblurring, image inpainting and image compressive sensing (CS) recovery. Experimental results demonstrate that the proposed scheme outperforms many state-of-the-art methods in terms of both quantitative measures and visual perception quality.