Compressive Phase Retrieval of Structured Signal

Compressive phase retrieval is the problem of recovering a structured vector $\boldsymbol{x} \in \mathbb{C}^n$ from its phaseless linear measurements. A compression algorithm aims to represent structured signals with as few bits as possible. As a result of extensive research devoted to compression algorithms, in many signal classes, compression algorithms are capable of employing sophisticated structures in signals and compress them efficiently. This raises the following important question: Can a compression algorithm be used for the compressive phase retrieval problem? To address this question, COmpressive PhasE Retrieval (COPER) optimization is proposed, which is a compression-based phase retrieval method. For a family of compression codes with rate-distortion function denoted by $r(\delta)$, in the noiseless setting, COPER is shown to require slightly more than $\lim\limits_{\delta \rightarrow 0} \frac{r(\delta)}{\log(1/\delta)}$ observations for an almost accurate recovery of $\boldsymbol{x}$.

[1]  Lei Tian,et al.  Compressive Phase Retrieval , 2011 .

[2]  Justin P. Haldar,et al.  Accelerated Wirtinger Flow: A fast algorithm for ptychography , 2018, 1806.05546.

[3]  Yuxin Chen,et al.  Gradient descent with random initialization: fast global convergence for nonconvex phase retrieval , 2018, Mathematical Programming.

[4]  Richard G. Baraniuk,et al.  From Denoising to Compressed Sensing , 2014, IEEE Transactions on Information Theory.

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

[6]  Ali Ahmed,et al.  Robust Compressive Phase Retrieval via Deep Generative Priors , 2018, ArXiv.

[7]  S. Sastry,et al.  Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming , 2011, 1111.6323.

[8]  Richard G. Baraniuk,et al.  BM3D-PRGAMP: Compressive phase retrieval based on BM3D denoising , 2016, 2016 IEEE International Conference on Image Processing (ICIP).

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

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

[11]  Arian Maleki,et al.  From compression to compressed sensing , 2012, 2013 IEEE International Symposium on Information Theory.

[12]  Christos Thrampoulidis,et al.  Phase retrieval via linear programming: Fundamental limits and algorithmic improvements , 2017, 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

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

[14]  Richard G. Baraniuk,et al.  prDeep: Robust Phase Retrieval with Flexible Deep Neural Networks , 2018, ICML 2018.

[15]  Kari Pulli,et al.  FlexISP , 2014, ACM Trans. Graph..

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

[17]  Justin Romberg,et al.  Phase Retrieval Meets Statistical Learning Theory: A Flexible Convex Relaxation , 2016, AISTATS.

[18]  H. Vincent Poor,et al.  Compression-Based Compressed Sensing , 2016, IEEE Transactions on Information Theory.

[19]  Michael Elad,et al.  The Little Engine That Could: Regularization by Denoising (RED) , 2016, SIAM J. Imaging Sci..

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

[21]  Michael Elad,et al.  Restoration by Compression , 2017, IEEE Transactions on Signal Processing.

[22]  Sundeep Rangan,et al.  Compressive Phase Retrieval via Generalized Approximate Message Passing , 2014, IEEE Transactions on Signal Processing.

[23]  Allen Y. Yang,et al.  CPRL -- An Extension of Compressive Sensing to the Phase Retrieval Problem , 2012, NIPS.

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

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

[26]  Babak Hassibi,et al.  Learning without the Phase: Regularized PhaseMax Achieves Optimal Sample Complexity , 2018, NeurIPS.

[27]  A. Maleki,et al.  From compression to compressed sensing , 2012, 2013 IEEE International Symposium on Information Theory.

[28]  Yue M. Lu,et al.  Fundamental limits of phasemax for phase retrieval: A replica analysis , 2017, 2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

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

[30]  Cishen Zhang,et al.  Robust Compressive Phase Retrieval via L1 Minimization With Application to Image Reconstruction , 2013, ArXiv.

[31]  H. Vincent Poor,et al.  Using compression codes in compressed sensing , 2016, 2016 IEEE Information Theory Workshop (ITW).

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