Phase retrieval using alternating minimization in a batch setting

Abstract

This paper considers the problem of phase retrieval, where the goal is to recover a signal z ∈ C n from the observations y i = | a i ⁎ z | , i = 1 , 2 , ⋯ , m . While many algorithms have been proposed, the alternating minimization algorithm is still one of the most commonly used and the simplest methods. Existing works have proved that when the observation vectors { a i } i = 1 m are sampled from a complex normal distribution C N ( 0 , I ) , the alternating minimization algorithm recovers the underlying signal with a good initialization when m = O ( n ) , or with random initialization when m = O ( n 2 ) , and it is conjectured that random initialization succeeds with m = O ( n ) [26] . This work proposes a modified alternating minimization method in a batch setting and proves that when m = O ( n log 5 ⁡ n ) , the proposed algorithm with random initialization recovers the underlying signal with high probability. The proof is based on the observation that after each iteration of alternating minimization, with high probability, the correlation between the direction of the estimated signal and the direction of the underlying signal increases.