Sparse phase retrieval: Convex algorithms and limitations

We consider the problem of recovering signals from their power spectral densities. This is a classical problem referred to in literature as the phase retrieval problem, and is of paramount importance in many fields of applied sciences. In general, additional prior information about the signal is required to guarantee unique recovery as the mapping from signals to power spectral densities is not one-to-one. In this work, we assume that the underlying signals are sparse. Recently, semidefinite programming (SDP) based approaches were explored by various researchers. Simulations of these algorithms strongly suggest that signals upto O(n1/2- ϵ) sparsity can be recovered by this technique. In this work, we develop a tractable algorithm based on reweighted ℓ1-minimization that recovers a sparse signal from its power spectral density for significantly higher sparsities, which is unprecedented. We also discuss the limitations of the existing SDP algorithms and provide a combinatorial algorithm which requires significantly fewer ”phaseless” measurements to guarantee recovery.

[1]  Babak Hassibi,et al.  Recovery of sparse 1-D signals from the magnitudes of their Fourier transform , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

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

[3]  Tetsunao Matsuta,et al.  国際会議開催報告:2013 IEEE International Symposium on Information Theory , 2013 .

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

[5]  Yonina C. Eldar,et al.  Simultaneously Structured Models With Application to Sparse and Low-Rank Matrices , 2012, IEEE Transactions on Information Theory.

[6]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

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

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

[9]  Peter G. Casazza,et al.  Phase retrieval , 2015, SPIE Optical Engineering + Applications.

[10]  J. Franklin,et al.  Ambiguities in the X‐ray analysis of crystal structures , 1974 .

[11]  Stephen P. Boyd,et al.  Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices , 2003, Proceedings of the 2003 American Control Conference, 2003..

[12]  Richard G. Baraniuk,et al.  Compressive phase retrieval , 2007, SPIE Optical Engineering + Applications.

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

[14]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[15]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[16]  Yonina C. Eldar,et al.  Sparsity Based Sub-wavelength Imaging with Partially Incoherent Light via Quadratic Compressed Sensing References and Links , 2022 .

[17]  Babak Hassibi,et al.  On robust phase retrieval for sparse signals , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[18]  Michael L. Fredman,et al.  Trans-dichotomous algorithms for minimum spanning trees and shortest paths , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[19]  Amitava Ghosh,et al.  QAVA: quota aware video adaptation , 2012, CoNEXT '12.

[20]  B. Bollobás.  The evolution of random graphs , 1984 .

[21]  Wenjing Liao,et al.  Compressed sensing phase retrieval , 2011, 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR).

[22]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2011, SIAM Rev..

[23]  A. L. Patterson.  A Direct Method for the Determination of the Components of Interatomic Distances in Crystals , 1935 .

[24]  Martin Vetterli,et al.  Sparse spectral factorization: Unicity and reconstruction algorithms , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

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

[26]  Justin K. Romberg,et al.  Compressed Sensing Phase Retrieval , 2007 .

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

[28]  Babak Hassibi,et al.  Reconstruction of integers from pairwise distances , 2012, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[29]  A. L. Patterson.  Ambiguities in the X-Ray Analysis of Crystal Structures , 1944 .

[30]  Yonina C. Eldar,et al.  Phase Retrieval via Matrix Completion , 2013, SIAM J. Imaging Sci..

[31]  Xiaodong Li,et al.  Sparse Signal Recovery from Quadratic Measurements via Convex Programming , 2012, SIAM J. Math. Anal..