Low rank phase retrieval

We study the problem of recovering a low-rank matrix, X, from phaseless measurements of random linear projections of its columns. We develop a novel solution approach, called AltMinTrunc, that consists of a two-step truncated spectral initialization step, followed by a three-step alternating minimization algorithm. We obtain sample complexity bounds for the AltMinTrunc initialization to provide a good approximation of the true X. When the rank of X is low enough, these are significantly smaller than what existing single vector phase retrieval algorithms need. Via extensive experiments, we demonstrate the same for the entire algorithm.

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

[2]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

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

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

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

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

[7]  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.

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

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

[10]  John D. Lafferty,et al.  A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements , 2015, NIPS.

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

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

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

[14]  Yonina C. Eldar,et al.  Sparsity-based single-shot sub-wavelength coherent diffractive imaging , 2011, 2012 IEEE 27th Convention of Electrical and Electronics Engineers in Israel.

[15]  Yonina C. Eldar,et al.  Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless Measurements , 2019, ICML.

[16]  Yonina C. Eldar,et al.  Low-Rank Phase Retrieval , 2016, IEEE Transactions on Signal Processing.

[17]  Prateek Jain,et al.  Low-rank matrix completion using alternating minimization , 2012, STOC '13.