  The nuclear norm minimization (NNM) is commonly used to approximate the
matrix rank by shrinking all singular values equally. However, the singular
values have clear physical meanings in many practical problems, and NNM may not
be able to faithfully approximate the matrix rank. To alleviate the
above-mentioned limitation of NNM, recent studies have suggested that the
weighted nuclear norm minimization (WNNM) can achieve a better rank estimation
than NNM, which heuristically set the weight being inverse to the singular
values. However, it still lacks a rigorous explanation why WNNM is more
effective than NMM in various applications. In this paper, we analyze NNM and
WNNM from the perspective of group sparse representation (GSR). Concretely, an
adaptive dictionary learning method is devised to connect the rank minimization
and GSR models. Based on the proposed dictionary, we prove that NNM and WNNM
are equivalent to L1-norm minimization and the weighted L1-norm minimization in
GSR, respectively. Inspired by enhancing sparsity of the weighted L1-norm
minimization in comparison with L1-norm minimization in sparse representation,
we thus explain that WNNM is more effective than NMM. By integrating the image
nonlocal self-similarity (NSS) prior with the WNNM model, we then apply it to
solve the image denoising problem. Experimental results demonstrate that WNNM
is more effective than NNM and outperforms several state-of-the-art methods in
both objective and perceptual quality.
