Efficient Sparse Recovery With Arctangent Regularization: A Novel Iterative Thresholding Algorithm

Several existing works have revealed the effectiveness of arctangent-type penalties in exploiting sparsity for compressed sensing. However, addressing the subproblems associated with the arctangent penalty incurs considerable computational cost. Aiming to reduce complexity, we derive the closed-form proximity operator of an arctangent penalty, which is expressed as hyperbolic functions of sine and cosine in this paper. Accordingly, a computationally-efficient arctangent regularization iterative thresholding (ARIT) algorithm for sparse approximation is proposed. Furthermore, we theoretically prove that under certain conditions, the ARIT algorithm converges to a local minimizer of the arctangent regularization problem with an eventually linear convergence. Extensive experiments are conducted to compare our scheme with conventional iterative thresholding algorithms, demonstrating the former superiority in terms of the probability of successful recovery, rate of support recovery, phase transition, and robustness to noise.