Sparse Signal Recovery From Phaseless Measurements via Hard Thresholding Pursuit
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In this paper, we consider the sparse phase retrival 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)) steps, when {a_i}_i=1^m are i.i.d. standard Gaussian random vector with m∼ O(slog(n/s)) and the initialization is in a neighbourhood of the underlying sparse signal. Together with a spectral initialization, our algorithm is guaranteed to have an exact recovery from O(s^2log 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.
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