Error bounds for noisy compressive phase retrieval

This paper provides a random complex measurement matrix and an algorithm for complex phase retrieval of sparse or approximately sparse signals from the noisy magnitudes of the measurements obtained with this matrix. We compute explicit error bounds for the recovery which depend on the noise-to-signal ratio, the sparsity s, the number of measured quantitites m, and the dimension of the signal N. This requires m to be of the order of s ln(N/s). In comparison with sparse recovery from complex linear measurements, our phase retrieval algorithm requires six times the number of measured quantities.