Subspace Phase Retrieval

In recent years, phase retrieval has received much attention in many fields including statistics, applied mathematics and optical engineering. In this paper, we propose an efficient algorithm, termed Subspace Phase Retrieval (SPR), which can accurately recover a $n$-dimensional $k$-sparse complex-valued signal given its $\mathcal O(k\log^3 n)$ magnitude-only Gaussian samples. This offers a significant improvement over many existing methods that require $\mathcal O(k^2 \log n)$ or more samples. Also, our sampling complexity is nearly optimal as it is very close to the fundamental limit $\mathcal O(k \log \frac{n}{k})$ for sparse phase retrieval.

[1]  Jonathan Scarlett,et al.  Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors , 2021, NeurIPS.

[2]  Patrick Rebeschini,et al.  Nearly Minimax-Optimal Rates for Noisy Sparse Phase Retrieval via Early-Stopped Mirror Descent , 2021, ArXiv.

[3]  Zhiqiang Xu,et al.  Sparse Phase Retrieval Via PhaseLiftOff , 2020, IEEE Transactions on Signal Processing.

[4]  Fan Wu,et al.  Hadamard Wirtinger Flow for Sparse Phase Retrieval , 2020, AISTATS.

[5]  Yu Xia,et al.  The recovery of complex sparse signals from few phaseless measurements , 2019, Applied and Computational Harmonic Analysis.

[6]  Huiping Li,et al.  Phase retrieval from Fourier measurements with masks , 2021, Inverse Problems & Imaging.

[7]  G. A. Young,et al.  High‐dimensional Statistics: A Non‐asymptotic Viewpoint, Martin J.Wainwright, Cambridge University Press, 2019, xvii 552 pages, £57.99, hardback ISBN: 978‐1‐1084‐9802‐9 , 2020, International Statistical Review.

[8]  Hongbin Li,et al.  PhaseEqual: Convex Phase Retrieval via Alternating Direction Method of Multipliers , 2020, IEEE Transactions on Signal Processing.

[9]  Lan V. Truong,et al.  Support Recovery in the Phase Retrieval Model: Information-Theoretic Fundamental Limit , 2019, IEEE Transactions on Information Theory.

[10]  Jian-Feng Cai,et al.  Toward the Optimal Construction of a Loss Function Without Spurious Local Minima for Solving Quadratic Equations , 2018, IEEE Transactions on Information Theory.

[11]  Ziyang Yuan,et al.  Phase Retrieval via Sparse Wirtinger Flow , 2017, J. Comput. Appl. Math..

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

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

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

[15]  Tom Goldstein,et al.  PhasePack: A phase retrieval library , 2017, 2017 51st Asilomar Conference on Signals, Systems, and Computers.

[16]  Michael I. Jordan,et al.  How to Escape Saddle Points Efficiently , 2017, ICML.

[17]  Yingbin Liang,et al.  Reshaped Wirtinger Flow for Solving Quadratic System of Equations , 2016, NIPS.

[18]  John Wright,et al.  A Geometric Analysis of Phase Retrieval , 2016, 2016 IEEE International Symposium on Information Theory (ISIT).

[19]  Yonina C. Eldar,et al.  Phase Retrieval: An Overview of Recent Developments , 2015, ArXiv.

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

[21]  Yonina C. Eldar,et al.  Phase Retrieval with Application to Optical Imaging: A contemporary overview , 2015, IEEE Signal Processing Magazine.

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

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

[24]  Alexandre d'Aspremont,et al.  Phase recovery, MaxCut and complex semidefinite programming , 2012, Math. Program..

[25]  Vladimir N. Temlyakov,et al.  Sparse Approximation and Recovery by Greedy Algorithms , 2013, IEEE Transactions on Information Theory.

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

[27]  Xiaodong Li,et al.  Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns , 2012, Found. Comput. Math..

[28]  Philippe Rigollet,et al.  Complexity Theoretic Lower Bounds for Sparse Principal Component Detection , 2013, COLT.

[29]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[30]  Yonina C. Eldar,et al.  Phase Retrieval: Stability and Recovery Guarantees , 2012, ArXiv.

[31]  Jian Wang,et al.  Generalized Orthogonal Matching Pursuit , 2011, IEEE Transactions on Signal Processing.

[32]  Emmanuel J. Candès,et al.  PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming , 2011, ArXiv.

[33]  Michael B. Wakin,et al.  Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property , 2009, IEEE Transactions on Information Theory.

[34]  Deanna Needell,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, ArXiv.

[35]  Olgica Milenkovic,et al.  Subspace Pursuit for Compressive Sensing Signal Reconstruction , 2008, IEEE Transactions on Information Theory.

[36]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[37]  R. Tibshirani,et al.  Sparse Principal Component Analysis , 2006 .

[38]  Fan Chung Graham,et al.  Concentration Inequalities and Martingale Inequalities: A Survey , 2006, Internet Math..

[39]  Ken Kreutz-Delgado,et al.  The Complex Gradient Operator and the CR-Calculus ECE275A - Lecture Supplement - Fall 2005 , 2009, 0906.4835.

[40]  øöö Blockinø.  Phase retrieval, error reduction algorithm, and Fienup variants: A view from convex optimization , 2002 .

[41]  Gordon L. Stuber,et al.  Principles of mobile communication (2nd ed.) , 2001 .

[42]  Gordon L. Stüber.  Principles of mobile communication , 1996 .

[43]  Robert W. Harrison,et al.  Phase problem in crystallography , 1993 .

[44]  B. Dwork.  Generalized Hypergeometric Functions , 1990 .

[45]  Rick P. Millane,et al.  Phase retrieval in crystallography and optics , 1990 .

[46]  J. Borwein,et al.  Two-Point Step Size Gradient Methods , 1988 .

[47]  J R Fienup,et al.  Reconstruction of an object from the modulus of its Fourier transform. , 1978, Optics letters.

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

[49]  A. Walther.  The Question of Phase Retrieval in Optics , 1963 .