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

Abstract: 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 $\mathbf{w} \in \mathbb{R}<sup>m$, 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 $\mathbf{w} \in \mathbb R<sup>m$ with sub-Gaussian norm $\sigma$, we establish a bound of order $$O\bigl(\sigma \sqrt{\frac{n \log⁴ n}{m}}\bigr)$$, 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.