Truncated amplitude flow with coded diffraction patterns

Abstract The phase retrieval problem, which involves Fourier transforms with random masks, has a wide range of applications in signal and image processing, such as diffraction imaging. In recent years, various first-order methods have been explored, including truncated amplitude flow (TAF). These methods are considered to be simple and effective for solving the phase retrieval problem. However, existing convergence results for TAF mainly focus on random Gaussian measurements, which is very far from practical scenarios. In this paper, we analyze the convergence of a revised TAF (RTAF) for the phase retrieval problem. We focus on coded diffraction patterns composed of Fourier transforms and random masks. Then we justify the linear convergence behavior of such algorithm when near the true solution. By utilizing the local initializer found by a non-truncated spectral method, we establish the exact convergence rates in both the noiseless case and the bounded noise case. Further, we have shown that a general signal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="bold-italic">x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> can be estimated on an optimal sampling order (specifically, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> ). Moreover, our demonstrations indicate that sharp stability can be maintained even when the observations are contaminated by bounded noise. Our experiments have revealed that the RTAF algorithm can reliably estimate the original signal using fewer measurements, aligning with our theoretical analysis. Additionally, numerical experiments have highlighted that the RTAF algorithm outperforms the truncated Wirtinger flow and truncated Riemannian gradient descent algorithms in terms of efficiency and stability, showcasing its superior performance over state-of-the-art algorithms.