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 28 articles , 1 standard article )

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  1. Cai, Jian-Feng; Li, Jingzhi; Lu, Xiliang; You, Juntao: Sparse signal recovery from phaseless measurements via hard thresholding pursuit (2022)
  2. Bahmani, Sohail; Lee, Kiryung: Low-rank matrix estimation from rank-one projections by unlifted convex optimization (2021)
  3. Bonifaci, Vincenzo: A Laplacian approach to (\ell_1)-norm minimization (2021)
  4. Charisopoulos, Vasileios; Chen, Yudong; Davis, Damek; Díaz, Mateo; Ding, Lijun; Drusvyatskiy, Dmitriy: Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence (2021)
  5. Gao, Bing; Liu, Haixia; Wang, Yang: Phase retrieval for sub-Gaussian measurements (2021)
  6. Li, Huiping; Li, Song: Riemannian optimization for phase retrieval from masked Fourier measurements (2021)
  7. Luo, Qi; Lin, Shijian; Wang, Hongxia: Robust phase retrieval via median-truncated smoothed amplitude flow (2021)
  8. Xiao, Zhuolei; Wang, Ya; Gui, Guan: Smoothed amplitude flow-based phase retrieval algorithm (2021)
  9. Zhuang, Zhitao; Wang, Kaixin: The Cramer-Rao lower bound in a non-AWGN model for the affine phase retrieval (2021)
  10. Aghasi, Alireza; Ahmed, Ali; Hand, Paul; Joshi, Babhru: BranchHull: convex bilinear inversion from the entrywise product of signals with known signs (2020)
  11. Bendory, Tamir; Edidin, Dan: Toward a mathematical theory of the crystallographic phase retrieval problem (2020)
  12. Krahmer, Felix; Stöger, Dominik: Complex phase retrieval from subgaussian measurements (2020)
  13. Li, Hui-ping; Li, Song: Phase retrieval with PhaseLift algorithm (2020)
  14. Li, Huiping; Li, Song; Xia, Yu: PhaseMax: stable guarantees from noisy sub-Gaussian measurements (2020)
  15. 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)
  16. Zhang, Teng: Phase retrieval using alternating minimization in a batch setting (2020)
  17. Ahmed, Ali; Aghasi, Alireza; Hand, Paul: Simultaneous phase retrieval and blind deconvolution via convex programming (2019)
  18. Bahmani, Sohail: Estimation from nonlinear observations via convex programming with application to bilinear regression (2019)
  19. Bahmani, Sohail; Romberg, Justin: Solving equations of random convex functions via anchored regression (2019)
  20. Cai, Jian-Feng; Liu, Haixia; Wang, Yang: Fast rank-one alternating minimization algorithm for phase retrieval (2019)

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