Phase Retrieval via Smooth Amplitude Flow

Abstract

Phase retrieval (PR) is an inverse problem about recovering a signal from phaseless linear measurements. This problem can be effectively solved by minimizing a nonconvex amplitude-based loss function. However, this loss function is non-smooth. To address the non-smoothness, 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 smooth amplitude flow method (SAF) which minimizes a novel loss function, without additionally modifying the gradient or the loss function during gradient descending. Such a new heuristic can be regarded as a smooth version of the original non-smooth amplitude-based loss function. We prove that SAF can converge geometrically to a global optimal point via the gradient algorithm with an elaborate initialization stage with a high probability. Substantial numerical tests empirically illustrate that the proposed heuristic is significantly superior to the original amplitude-based loss function and SAF also outperforms other state-of-the-art methods in terms of the recovery rate and the converging speed. Specially, it is numerically shown that SAF can stably recover the original signal when number of measurements is smaller than the information-theoretic limit for both the real and the complex Gaussian models.

arXiv comment18 pages, 6 figures, two referrences added

References in corpus (8)

• Phase Retrieval via Wirtinger Flow: Theory and Algorithms
• Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
• Symmetries and many-body excited states with neural-network quantum states
• Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution
• Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms
• Solving Large-scale Systems of Random Quadratic Equations via Stochastic Truncated Amplitude Flow
• Phase Retrieval Using Feasible Point Pursuit: Algorithms and Cramér-Rao Bound
• Perturbed Amplitude Flow for Phase Retrieval