This paper introduces the weighted nuclear norm minimization (WNNM) problem, a significant extension of the standard nuclear norm minimization (NNM) used in low-rank matrix approximation. Unlike NNM, which equally weights all singular values, WNNM allows for different weights to be assigned to each singular value, providing more flexibility in handling practical problems where singular values have clear physical meanings. The authors analyze the solutions of WNNM under different weighting conditions and propose an efficient algorithm for solving it. They apply the WNNM algorithm to image denoising, exploiting the nonlocal self-similarity (NSS) of images. Experimental results show that the proposed WNNM algorithm outperforms state-of-the-art denoising algorithms such as BM3D in both quantitative measures (PSNR) and visual perception quality, demonstrating its superior performance in preserving local structures and reducing artifacts.This paper introduces the weighted nuclear norm minimization (WNNM) problem, a significant extension of the standard nuclear norm minimization (NNM) used in low-rank matrix approximation. Unlike NNM, which equally weights all singular values, WNNM allows for different weights to be assigned to each singular value, providing more flexibility in handling practical problems where singular values have clear physical meanings. The authors analyze the solutions of WNNM under different weighting conditions and propose an efficient algorithm for solving it. They apply the WNNM algorithm to image denoising, exploiting the nonlocal self-similarity (NSS) of images. Experimental results show that the proposed WNNM algorithm outperforms state-of-the-art denoising algorithms such as BM3D in both quantitative measures (PSNR) and visual perception quality, demonstrating its superior performance in preserving local structures and reducing artifacts.