PhaseLift for Coded Diffraction Patterns: Optimal Sampling Rate

2026-08-03

AI Summary

This work resolves a long-standing open problem in structured phase retrieval by establishing the optimal sampling rate required for exact reconstruction of complex-valued signals from coded diffraction patterns. Under the standard random mask model, the authors prove for the first time that PhaseLift recovers any n-dimensional complex signal—up to a global phase—with only O(log n) random masks, achieving the information-theoretic lower bound Ω(log n). This yields a total sampling complexity of O(n log n). The analysis hinges on several key technical innovations: an approximate dual certificate constructed via an enhanced golfing scheme, adaptive mask allocation, and a dimension-independent truncation threshold. The proposed method attains this optimal sampling efficiency with a failure probability that decays polynomially in the signal dimension.

Abstract

Recovering a complex-valued signal from coded diffraction patterns, namely the Fourier intensities obtained after modulating the signal with a collection of masks, is a fundamental structured phase retrieval problem arising in diffraction imaging and related applications. Despite its practical importance, the theoretical analysis of this structured framework remains scarce. In the standard random mask model, the optimal sampling rate achievable by computationally tractable recovery methods has remained open. In this paper, we establish the optimal sampling rate for the PhaseLift feasibility program. More precisely, PhaseLift achieves exact recovery of an unknown signal $\pmb{x}_0\in\mathbb{C}^n$, up to a global phase, from $\mathcal{O}(\log n)$ random masks, with polynomially decaying failure probability. Since $Ω(\log n)$ masks are necessary to identify certain signals under the erasure mask ensemble, our result thereby achieves the optimal mask complexity. Equivalently, PhaseLift attains the optimal total sampling rate of $m=\mathcal{O}( n\log n)$ scalar intensity measurements. The proof is based on an approximate dual certificate construction via a refined golfing scheme that combines adaptive mask allocation with a dimension-independent truncation threshold.

Problem

• phase retrieval
• coded diffraction patterns
• sampling rate
• Fourier intensities
• mask complexity

Innovation

• PhaseLift
• coded diffraction patterns
• optimal sampling rate
• phase retrieval
• dual certificate