Weighted Nuclear Norm Minimization for Image Denoising
Most of the Denoisers in Image Processing make a simple assumption - The data has small number of freedom degrees while noise has high number.
Hence if we try to represent the given noisy data with small number of parameters we probably match the data while exclude most of the noise.

In the case above we try to limit the number of Singular Values of the base which represents the patches of the image.
The assumption means that most of the patches can be represented by a dictionary with small number of Singular Values -> Small number of degrees of freedom to create "Crazy Patches".