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

2025-09-12

AI Summary

This paper investigates the stability of the PhaseLift algorithm for phase retrieval from coded diffraction patterns (CDPs) under additive noise. Addressing the limitation of existing error bounds—namely, their dependence on the ℓ²-norm of the noise vector (|mathbf{w}|_2) without reflecting average noise intensity—the work provides the first rigorous proof of Soltanolkotabi’s conjecture: the optimal error bound scales with the *average* noise magnitude, i.e., (|mathbf{w}|_2 / sqrt{m}). Leveraging tools from convex optimization, random matrix theory, and high-dimensional statistics, the authors derive an upper bound of (O(log n cdot |mathbf{w}|_2 / sqrt{m})) under adversarial noise and (O(sigma sqrt{n log^4 n / m})) under sub-Gaussian noise. Matching minimax lower bounds are established in both settings, differing only by logarithmic factors. These results close a fundamental theoretical gap in the stability analysis of PhaseLift for CDP-based phase retrieval.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $Oigl(frac{1}{sqrt{log n}} cdot frac{|mathbf{w}|_2}{sqrt m}igr)$. 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 $σ$, we establish a bound of order $Oigl(σsqrt{frac{n log^4 n}{m}}igr)$, 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.

Problem

• Analyzing PhaseLift stability under noisy measurements
• Establishing optimal recovery bounds for adversarial noise
• Proving near-optimal performance for sub-Gaussian noise

Innovation

• Proves near-optimal PhaseLift error bound
• Establishes logarithmic factor recovery guarantee
• Provides minimax lower bounds validation