A Primal Dual Active Set with Continuation Algorithm for the \ell^0-Regularized Optimization Problem
This addresses optimization challenges in compressed sensing, offering an incremental improvement for researchers in signal processing and sparse recovery.
The paper tackles the ℓ⁰-regularized least-squares problem in compressed sensing by developing a primal dual active set with continuation algorithm, establishing finite-step global convergence under conditions like mutual incoherence or restricted isometry properties and demonstrating efficiency and accuracy through numerical examples.
We develop a primal dual active set with continuation algorithm for solving the \ell^0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, the finite step global convergence of the algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and the convergence analysis.