Sparse Phase Retrieval Via Iteratively Reweighted Amplitude Flow

Sparse phase retrieval (PR) aims at reconstructing a sparse signal vector from a few phaseless linear measurements. It emerges naturally in diverse applications, but it is NP-hard in general. Drawing from advances in nonconvex optimization, this paper presents a new algorithm that is termed compressive reweighted amplitude flow (CRAF) for sparse PR. CRAF operates in two stages: Stage one computes an initial guess by means of a new spectral procedure, and stage two implements a few hard thresholding based iteratively reweighted gradient iterations on the amplitude-based least-squares cost. When there are sufficient measurements, CRAF reconstructs the true signal vector exactly under suitable conditions. Furthermore, its sample complexity coincides with that of the state-of-the-art approaches. Numerical experiments showcase improved performance of the proposed approach relative to existing alternatives.

[1]  Chinmay Hegde,et al.  Phase Retrieval Using Structured Sparsity: A Sample Efficient Algorithmic Framework , 2017, ArXiv.

[2]  Arian Maleki,et al.  Optimization-Based AMP for Phase Retrieval: The Impact of Initialization and $\ell_{2}$ Regularization , 2018, IEEE Transactions on Information Theory.

[3]  Tom Goldstein,et al.  PhaseMax: Convex Phase Retrieval via Basis Pursuit , 2016, IEEE Transactions on Information Theory.

[4]  Gang Wang,et al.  Solving Most Systems of Random Quadratic Equations , 2017, NIPS.

[5]  Yuxin Chen,et al.  Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution , 2017, Found. Comput. Math..

[6]  Tom Goldstein,et al.  Linear Spectral Estimators and an Application to Phase Retrieval , 2018, ICML.

[7]  Gang Wang,et al.  Compressive Phase Retrieval via Reweighted Amplitude Flow , 2017, IEEE Transactions on Signal Processing.

[8]  J R Fienup,et al.  Phase retrieval algorithms: a comparison. , 1982, Applied optics.

[9]  Xiaodong Li,et al.  Phase Retrieval via Wirtinger Flow: Theory and Algorithms , 2014, IEEE Transactions on Information Theory.

[10]  R. Gerchberg.  A practical algorithm for the determination of phase from image and diffraction plane pictures , 1972 .

[11]  Gang Wang,et al.  Sparse Phase Retrieval via Truncated Amplitude Flow , 2016, IEEE Transactions on Signal Processing.

[12]  Xiaodong Li,et al.  Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow , 2015, ArXiv.

[13]  Yuxin Chen,et al.  Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems , 2015, NIPS.

[14]  Tengyao Wang,et al.  A useful variant of the Davis--Kahan theorem for statisticians , 2014, 1405.0680.

[15]  Prateek Jain,et al.  Phase Retrieval Using Alternating Minimization , 2013, IEEE Transactions on Signal Processing.

[16]  Feng Ruan,et al.  Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval , 2017, Information and Inference: A Journal of the IMA.

[17]  Yonina C. Eldar,et al.  Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow , 2016, IEEE Transactions on Information Theory.

[18]  Volkan Cevher,et al.  Model-Based Compressive Sensing , 2008, IEEE Transactions on Information Theory.

[19]  Yonina C. Eldar,et al.  GESPAR: Efficient Phase Retrieval of Sparse Signals , 2013, IEEE Transactions on Signal Processing.

[20]  Andrea Montanari,et al.  Fundamental Limits of Weak Recovery with Applications to Phase Retrieval , 2017, COLT.

[21]  Christos Thrampoulidis,et al.  Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms , 2018, ArXiv.

[22]  Vladislav Voroninski,et al.  Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Space , 2016, ArXiv.