Methods for Solving Low-Rank Matrix and Sparse Matrix Problems

This method provides an effective solution for decomposing matrices into low-rank and sparse components. It has gained widespread adoption in image processing due to its computational efficiency and robustness. When dealing with tasks such as image compression, denoising, and image restoration, this approach leverages unique advantages through optimization algorithms like Robust Principal Component Analysis (RPCA). The implementation typically involves convex optimization techniques where the objective function minimizes the nuclear norm (for low-rank component) and L1-norm (for sparse component). Key functions in practical implementations include singular value thresholding for rank minimization and proximal operators for sparse regularization. This methodology significantly reduces computational complexity while enhancing algorithm stability and robustness, making it highly valuable for real-world applications with promising future prospects.