On DC based Methods for Phase Retrieval

In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization sub-problems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard $\ell_1$ norm term is added into the minimization mentioned above. The subproblem is approximated by a proximal gradient method which is solved by the shrinkage-threshold technique directly without iterations. In addition, a modified Attouch-Peypouquet technique is used to accelerate the iterative computation. These lead to more effective algorithms than the Wirtinger flow (WF) algorithm and the Gauss-Newton (GN) algorithm and etc.. A convergence analysis of both DC based algorithms shows that the iterative solutions is convergent linearly to a critical point and will be closer to a global minimizer than the given initial starting point. Our study is a deterministic analysis while the study for the Wirtinger flow (WF) algorithm and its variants, the Gauss-Newton (GN) algorithm, the trust region algorithm is based on the probability analysis. In particular, the DC based algorithms are able to retrieve solutions using a number $m$ of measurements which is about twice of the number $n$ of entries in the solution with high frequency of successes. When $m\approx n$, the $\ell_1$ DC based algorithm is able to retrieve sparse signals.

[1]  Zhiqiang Xu,et al.  Phase Retrieval for Sparse Signals , 2013, ArXiv.

[2]  Ronald,et al.  Learning representations by backpropagating errors , 2004 .

[3]  Hédy Attouch,et al.  On the convergence of the proximal algorithm for nonsmooth functions involving analytic features , 2008, Math. Program..

[4]  Bo Wen,et al.  A proximal difference-of-convex algorithm with extrapolation , 2016, Computational Optimization and Applications.

[5]  Bing Zheng,et al.  A New Modified Barzilai–Borwein Gradient Method for the Quadratic Minimization Problem , 2017, J. Optim. Theory Appl..

[6]  Dan Edidin,et al.  An algebraic characterization of injectivity in phase retrieval , 2013, ArXiv.

[7]  Wotao Yin,et al.  A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion , 2013, SIAM J. Imaging Sci..

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

[9]  J. Miao,et al.  High resolution 3D x-ray diffraction microscopy. , 2002, Physical review letters.

[10]  Xiaodong Li,et al.  Phase Retrieval from Coded Diffraction Patterns , 2013, 1310.3240.

[11]  Zhiqiang Xu,et al.  Phase retrieval from the magnitudes of affine measurements , 2018 .

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

[13]  Hedy Attouch,et al.  The Rate of Convergence of Nesterov's Accelerated Forward-Backward Method is Actually Faster Than 1/k2 , 2015, SIAM J. Optim..

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

[15]  R. Balan,et al.  On signal reconstruction without phase , 2006 .

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

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

[18]  O. Zariski,et al.  ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS. , 1958, Proceedings of the National Academy of Sciences of the United States of America.

[19]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[20]  Yonina C. Eldar,et al.  Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms , 2012, SIAM J. Optim..

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

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

[23]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[24]  Jieping Ye,et al.  A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems , 2013, ICML.

[25]  Johan P. Hansen,et al.  INTERSECTION THEORY , 2011 .

[26]  Hédy Attouch,et al.  Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality , 2008, Math. Oper. Res..

[27]  F. Deutsch.  Best approximation in inner product spaces , 2001 .

[28]  Yuejie Chi,et al.  Reshaped Wirtinger Flow and Incremental Algorithm for Solving Quadratic System of Equations , 2016 .

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

[30]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[31]  Laura Waller,et al.  Accelerated Wirtinger Flow for Multiplexed Fourier Ptychographic Microscopy , 2018, 2018 25th IEEE International Conference on Image Processing (ICIP).

[32]  K. Kurdyka.  On gradients of functions definable in o-minimal structures , 1998 .

[33]  T. Willmore.  Algebraic Geometry , 1973, Nature.

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

[35]  Geoffrey E. Hinton,et al.  Learning representations by back-propagating errors , 1986, Nature.

[36]  Cynthia Vinzant,et al.  A small frame and a certificate of its injectivity , 2015, 2015 International Conference on Sampling Theory and Applications (SampTA).

[37]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[38]  D. Mumford.  The red book of varieties and schemes , 1988 .

[39]  Meng Huang,et al.  Phase retrieval from the norms of affine transformations , 2018, ArXiv.

[40]  Zhiqiang Xu,et al.  Generalized phase retrieval : measurement number, matrix recovery and beyond , 2016, Applied and Computational Harmonic Analysis.

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

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

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

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

[45]  L. Liao,et al.  R-linear convergence of the Barzilai and Borwein gradient method , 2002 .

[46]  Zhang Fe.  Phase retrieval from coded diffraction patterns , 2015 .

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

[48]  Changzhi Wu,et al.  A DC PROGRAMMING APPROACH FOR SENSOR NETWORK LOCALIZATION WITH UNCERTAINTIES IN ANCHOR POSITIONS , 2013 .

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

[50]  Bing Gao,et al.  Phaseless Recovery Using the Gauss–Newton Method , 2016, IEEE Transactions on Signal Processing.

[51]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[52]  Le Thi Hoai An,et al.  The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems , 2005, Ann. Oper. Res..