Near-Optimal Recovery Performance of PhaseLift for Phase Retrieval from Coded Diffraction Patterns

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 $\mathbf{w} \in \mathbb{R}^m$, existing recovery bounds scale on the order of $\|\mathbf{w}\|_2$, 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 $\|\mathbf{w}\|_2/\sqrt m$. 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 $O(\log n \cdot \|\mathbf{w}\|_2/\sqrt m)$, and further show that there exists a noise vector for which the error lower bound exceeds $O\bigl(\frac{1}{\sqrt{\log n}} \cdot \frac{\|\mathbf{w}\|_2}{\sqrt m}\bigr)$. Here, $n$ is the dimension of the signals we aim to recover. Moreover, for mean-zero sub-Gaussian noise vector $\mathbf{w} \in \mathbb R^m$ with sub-Gaussian norm $\sigma$, we establish a bound of order $O\bigl(\sigma \sqrt{\frac{n \log^4 n}{m}}\bigr)$, 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.