Phase retrieval via smoothed amplitude flow

•A smoothed amplitude flow method for phase retrieval which introduces a novel smooth loss function so as to avoid checking and modifying the gradient components.•The global convergence to the optimal point via the gradient algorithm with an elaborate initialization stage with high probability.•Better numerical performance in terms of the recovery rate and the converging speed compared with other state-of-the-art methods. Phase retrieval (PR) is an inverse problem about recovering a signal from phaseless linear measurements. This problem can be efficiently solved by minimizing a nonconvex amplitude-based loss function. However, this loss function is nonsmooth. To address the nonsmoothness, a series of methods have been proposed by adding truncating, reweighting and smoothing operations to adjust the gradient or the loss function and achieved better performance. But these operations bring about extra rules and parameters that need to be carefully designed. Unlike previous works, we present a smoothed amplitude flow (SAF) method which introduces a novel smooth loss function so as to avoid checking and modifying the gradient components in the gradient descent stage. This novel loss function can be regarded as a smoothed version of the original amplitude-based loss function. We prove that SAF converges geometrically to a global optimal point via the gradient algorithm with an elaborate initialization stage with high probability. Substantial numerical tests empirically illustrate that the proposed loss function is significantly superior to the original amplitude-based loss function. SAF also outperforms other state-of-the-art methods in terms of the recovery rate and the converging speed.