PhaseMax: convex phase retrieval via basis pursuit. We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a ”non-lifting” relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.

References in zbMATH (referenced in 15 articles , 1 standard article )

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  1. Aghasi, Alireza; Ahmed, Ali; Hand, Paul; Joshi, Babhru: BranchHull: convex bilinear inversion from the entrywise product of signals with known signs (2020)
  2. Li, Huiping; Li, Song; Xia, Yu: PhaseMax: stable guarantees from noisy sub-Gaussian measurements (2020)
  3. Ma, Cong; Wang, Kaizheng; Chi, Yuejie; Chen, Yuxin: Implicit regularization in nonconvex statistical estimation: gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution (2020)
  4. Zhang, Teng: Phase retrieval using alternating minimization in a batch setting (2020)
  5. Ahmed, Ali; Aghasi, Alireza; Hand, Paul: Simultaneous phase retrieval and blind deconvolution via convex programming (2019)
  6. Bahmani, Sohail: Estimation from nonlinear observations via convex programming with application to bilinear regression (2019)
  7. Bahmani, Sohail; Romberg, Justin: Solving equations of random convex functions via anchored regression (2019)
  8. Cai, Jian-Feng; Liu, Haixia; Wang, Yang: Fast rank-one alternating minimization algorithm for phase retrieval (2019)
  9. Hand, Paul; Voroninski, Vladislav: An elementary proof of convex phase retrieval in the natural parameter space via the linear program PhaseMax (2019)
  10. Mondelli, Marco; Montanari, Andrea: Fundamental limits of weak recovery with applications to phase retrieval (2019)
  11. Zhang, Richard Y.; Sojoudi, Somayeh; Lavaei, Javad: Sharp restricted isometry bounds for the inexistence of spurious local minima in nonconvex matrix recovery (2019)
  12. Chang, Huibin; Lou, Yifei; Duan, Yuping; Marchesini, Stefano: Total variation-based phase retrieval for Poisson noise removal (2018)
  13. Goldstein, Tom; Studer, Christoph: PhaseMax: convex phase retrieval via basis pursuit (2018)
  14. Sun, Ju; Qu, Qing; Wright, John: A geometric analysis of phase retrieval (2018)
  15. Xiang, Jianhong; Yue, Huihui; Yin, Xiangjun; Ruan, Guoqing: A reweighted symmetric smoothed function approximating (L_0)-norm regularized sparse reconstruction method (2018)