Classic ADMM Algorithm for Compressed Sensing Problems

Compressed sensing is a technique that leverages signal sparsity for efficient sampling, while the Alternating Direction Method of Multipliers (ADMM) serves as a powerful optimization tool for solving related inverse problems. ADMM decomposes the objective function into manageable subproblems that can be processed in parallel, making it particularly suitable for constrained sparse recovery tasks. In implementation, ADMM typically alternates between updating primal variables, auxiliary variables, and dual variables using proximal operators.

In compressed sensing applications, ADMM commonly splits the original problem into a data fidelity term and a sparsity regularization term. The data fidelity term ensures reconstructed signals align with observed measurements, while sparsity-promoting regularizers (such as L1-norm) enforce signal sparsity through shrinkage operations. By introducing auxiliary variables and dual variables, ADMM alternately updates each variable block—often solved using least-squares for fidelity terms and soft-thresholding for L1 norms—guaranteeing convergence to a global optimum under convex conditions.

ADMM's robustness and flexibility allow it to handle diverse measurement matrices and noise models. The algorithm's step-size parameter controls convergence rates, enabling balance between computational efficiency and reconstruction accuracy. This framework has been extended to multidimensional signals, nonlinear measurements, and other complex compressed sensing scenarios through modular adaptation of subproblem solvers.