Linearized Bregman Method for Sparse Signal Recovery in Compressed Sensing

In the field of compressed sensing, sparse signal recovery represents a fundamental challenge. While traditional optimization methods like L1-norm minimization are effective, they often suffer from high computational complexity. The linearized Bregman method has emerged as an efficient alternative that has gained significant attention in recent years.

The core concept of this approach combines Bregman iteration with linearization techniques, utilizing Bregman distance to enhance the convergence properties of conventional gradient descent algorithms. Its primary advantage lies in handling large-scale sparse recovery problems while maintaining low iteration complexity. From an implementation perspective, this typically involves creating a function that initializes parameters such as step size and regularization coefficient before entering the main iteration loop.

The algorithm procedure consists of two key computational steps: First, linearizing the approximation of the objective function through gradient calculations, then updating variables using Bregman iteration. This design enables the algorithm to perform simple matrix-vector multiplications during each iteration, avoiding the need to solve complex optimization subproblems. A typical code implementation would structure these operations within a while-loop that checks convergence criteria, with each iteration updating the solution vector using proximal operator calculations.

Compared to traditional methods, linearized Bregman demonstrates faster convergence rates and superior noise robustness. In practical applications, it proves particularly suitable for sparse reconstruction tasks in medical imaging, radar signal processing, and similar domains. Parameter selection, especially the tuning of step size and regularization coefficients, significantly impacts algorithm performance and typically requires experimental adjustment through cross-validation techniques in code implementations.