Complex-valued sparse recovery via double-threshold sigmoid penalty

• A novel sparse recovery method is proposed by introducing the DTHS penalty.
• A general discussion on the unbiasedness of regularization techniques.
• The analysis and comparison of DTHS and other regularization methods.

The thresholding methods based on the generalized iteratively reweighted least squares (IRLS) iteration are discussed under the complex-valued condition in this paper. A new thresholding function (Double-Threshold Sigmoid (DTHS) function) and two associated algorithms (DTHS-1 and DTHS-2) are proposed herein, and their convergence performances are discussed in detail. It is shown that the generalized IRLS algorithm is unbiased if the thresholding penalty can eliminate the undesired perturbation term added on the correlation matrix of the measurement matrix. Compared with the others, the new algorithms are endowed with stability and insensitivity with respect to the regularization parameter by selecting some sound upper thresholds and dividing the iteration procedures into the degraded stage and DTHS stage respectively. Further analyses show that the DTHS-1 algorithm is suitable to deal with the sparse and continuous problems for both of the i.i.d. random matrix and under-resolution PSF matrix. The noise performance of the DTHS-1 algorithm is always superior to that of the IRLS algorithm, especially in the face of the under-resolution PSF matrix.