Sparse Signal Recovery from Phaseless Measurements via Hard Thresholding Pursuit

Abstract 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.