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In this paper, we consider the sparse phase retrieval problem, recovering an s -sparse signal x ♮ ∈ R n from m phaseless samples y i = | 〈 x ♮ , a i 〉 | for i = 1 , ... , m. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an efficient second-order algorithm for sparse phase retrieval. Our proposed algorithm is theoretically guaranteed to give an exact sparse signal recovery in finite (in particular, at most O (log ⁡ m + log ⁡ (‖ x ♮ ‖ 2 / | x min ♮ |)) steps, when { a i } i = 1 m are i.i.d. standard Gaussian random vector with m ∼ O (s log ⁡ (n / s)) and the initialization is in a neighborhood of the underlying sparse signal. Together with a spectral initialization, our algorithm is guaranteed to have an exact recovery from O (s 2 log ⁡ n) samples. Since the computational cost per iteration of our proposed algorithm is the same order as popular first-order algorithms, our algorithm is extremely efficient. Experimental results show that our algorithm can be several times faster than existing sparse phase retrieval algorithms. [ABSTRACT FROM AUTHOR]

We develop a primal dual active set with continuation algorithm for solving the ℓ 0 -regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples 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, a finite step global convergence of the overall algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and its convergence behavior. [ABSTRACT FROM AUTHOR]