Analyzing the Weighted Nuclear Norm Minimization and Nuclear Norm Minimization based on Group Sparse Representation

The nuclear norm minimization (NNM) tends to over-shrink the rank components and treats the different rank components equally, thus limits its capability and flexibility in practical applications. Recent advances have suggested that the weighted nuclear norm minimization (WNNM) is expected to be more appropriate than NNM. However, it still lacks a plausible mathematical explanation why WNNM is more appropriate than NNM. In this paper, we analyze the WNNM and NNM from a point of the group sparse representation (GSR). Firstly, an adaptive dictionary for each group is designed. Then we show mathematically that WNNM is more appropriate than NNM. We exploit the proposed scheme to two typical low level vision tasks, including image deblurring and image compressive sensing (CS) recovery. Experimental results have demonstrated that the proposed scheme outperforms many state-of-the-art methods.