Amplitude-Based Smoothed Sparse Phase Retrieval Algorithm

Sparse phase retrieval mainly solves nonconvex and nonsmooth problems. Aiming at the nonsmooth problem in sparse phase retrieval, we propose a smoothing algorithm which is called sparse smoothed amplitude flow (SPSAF). The proposed SPSAF algorithm is an amplitude-based nonconvex sparse smoothing phase retrieval algorithm. First, the original phase retrieval loss function is smoothed without modifying the gradient in the gradient refinement stage, thereby reducing the computational complexity of the overall algorithm. Secondly, the support of the original signal is estimated by differential analysis of the gaps, and the initialization can be obtained through a carefully designed method based on this support. Finally, we get sparse estimates by gradient descent based on hard thresholding. Numerical experiments show that the proposed SPSAF algorithm has significant improvements in recovery performance, convergence speed, and sampling complexity. Further, the SPSAF algorithm is stable in noisy environments.