The PhaseLift algorithm is an effective convex method for solving the phase retrieval problem from Fourier measurements with coded diffraction patterns (CDP). While exact reconstruction guarantees are well-established in the noiseless case, the stability of recovery under noise remains less well understood. In particular, when the measurements are corrupted by an additive noise vector w Rᵐ, existing recovery bounds scale on the order of \|w\|₂, which is conjectured to be suboptimal. More recently, Soltanolkotabi conjectured that the optimal PhaseLift recovery bound should scale with the average noise magnitude, that is, on the order of \|w\|₂/ m. However, establishing this theoretically is considerably more challenging and has remained an open problem. In this paper, we focus on this conjecture and provide a nearly optimal recovery bound for it. We prove that under adversarial noise, the recovery error of PhaseLift is bounded by O (n \|w\|₂/ m), and further show that there exists a noise vector for which the error lower bound exceeds O (1 n \|w\|₂ m). Here, n is the dimension of the signals we aim to recover. Moreover, for mean-zero sub-Gaussian noise vector w Rᵐ with sub-Gaussian norm σ, we establish a bound of order O (σn ⁴ n{m}), and also provide a corresponding minimax lower bound. Our results affirm Soltanolkotabi's conjecture up to logarithmic factors, providing a new insight into the stability of PhaseLift under noisy CDP measurements.
Huang et al. (Fri,) studied this question.