📝 SummarXiv

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 $\mathbf{w} \in \mathbb{R}^m$, 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.