Linearized Bregman Method for Sparse Signal Recovery in Compressed Sensing

In compressed sensing for sparse signal recovery, the linearized Bregman method serves as a crucial computational technique. This approach is widely employed in domains like image processing to compress and reconstruct signals efficiently. Its primary advantage lies in decomposing complex optimization problems into a sequence of simpler subproblems, significantly accelerating the solving process. Specifically, the method iteratively solves these subproblems to progressively approximate the optimal solution of the original problem, achieving high computational efficiency. From an implementation perspective, the algorithm typically involves two key steps per iteration: a gradient descent update for linearization and a Bregman projection for sparsity enforcement. Code implementations often utilize shrinkage/thresholding operators (e.g., soft-thresholding for L1 regularization) to handle non-smooth terms, while leveraging conjugate gradient methods or preconditioning for linear system solves. The iterative structure allows parallelization and adapts well to large-scale problems, making it applicable to various signal processing tasks including MRI reconstruction and seismic data processing.