Phase retrieval with PhaseLift algorithm
Tóm tắt
This paper provides a contemporary overview of phase retrieval problem with PhaseLift algorithm and summarizes theoretical results which have been derived during the past few years. Based on the lifting technique, the phase retrieval problem can be transformed into the low rank matrix recovery problem and then be solved by convex programming known as PhaseLift. Thus, stable guarantees for such problem have been gradually established for measurements sampled from sufficiently random distribution, for instance, the standard normal distribution. Further, exact recovery results have also been set up for masked Fourier measurements which are closely related to practical applications.
Tài liệu tham khảo
S Bahmani, J Romberg. Efficient compressive phase retrieval with constrained sensing vectors, IEEE Neural Information Processing Systems, 2015: 523–531.
S Bahmani, J Romberg. A flexible convex relaxation for phase retrieval, Electron J Stat, 2017, 11(2): 5254–5281.
R Balan, B G Bodmann, P G Cassazza, D Edidin. Painless reconstruction from magnitudes of frame coefficients, J Fourier Anal Appl, 2015, 15(4): 488–501.
R Balan, B G Bodmann, P G Cassazza, D Edidin. Fast algorithms for signal reconstruction without phase, Proc SPIE, 2007, 6701: 67011L–67011L–9.
R Balan, P Casazza D Edidin. On signal reconstruction without noisy phase, Appl Comp Harm Anal, 2006, 20: 345–356.
B Baykal. Blind channel estimation via combining autocorrelation and blind phase estimation, IEEE Transactions on Circuits and Systems, 2004, 51(6): 1125–1131.
R Berinde, A C Gilbert, P Indyk, H Karloff, M J Strauss. Combining geometry and combinatorics: A unified approach to sparse signal recovery, 2008 Forty-sixth Annual Allerton Conference on Communication, Control, and Computing, 2008: 798–805.
T Cai, X Li, Z Ma. Optimal rates of convergence for noisy sparse phase retrieval via thresholded Wirtinger flow, Ann Statist, 2016, 44(5): 2221–2251.
J F Cai, K Wei. Solving systems of phaseless equations via Riemannian optimization with optimal sampling complexity, arXiv preprint arXiv:1809.02773, 2018.
E J Candès, Y C Eldar, T Strohmer, V Voroninshi. Phase retrieval via matrix completion, SIAM J Imaging Sci, 2013, 6(1): 199–225.
E J Candès, X Li. Solving quadratic equations via phaselift when there are about as many equations as unknowns, Found Comput Math, 2014, 14(5): 1017–1026.
E J Candès, X Li, M Soltanolkotabi. Phase retrieval from coded diffraction patterns, Appl Comput Harmon Anal, 2015, 39(2): 277–299.
E J Candès, X Li, M Soltanolkotabi. Phase retrieval via Wirtinger Flow: theory and algorithms, IEEE Trans Inform Theory, 2015, 61(4): 1985–2007.
E J Candès, Y Plan. Matrix completion with noise, IEEE Proc, 2010, 98(6): 925–936.
E J Candès, T Tao. Decoding by linear programming, IEEE Trans Inform Theory, 2005, 51(12): 4203–4215.
E J Candès, B Recht. Exact matrix completion via convex optimization, Found Comput Math, 2009, 9(6): 717–772.
E J Candès, T Stromher, V Voroninshi. PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming, Comm Pure Appl Math, 2013, 66(8): 1241–1274.
Y Chen, E J Candès. Solving random quadratic systems of equations is nearly as easy as solving linear systems, Comm Pure Appl Math, 2015, 70(5): 739–747.
C C Chen, J Miao, C W Wang, T K Lee. Application of the optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method(GHIO), Phys Rev B, 2007, 76: 064113.
J C Dainty, J R Fienup. Array imaging using intensity-only measurements, phase retrieval and image reconstruction for astronomy, H. Stark (Ed), Image Recovery: Theory and Application, Academic Press, San Diego, 1987: 231–275.
A Fannjiang, W Liao. Phase retrieval with random phase illumination, J Opt Soc Amer A, 2012, 29(9): 1847–1859.
J R Fienup. Reconstruction of an object from the modulus of its Fourier transform, Opt Lett, 1978, 3(1): 27–29.
J R Fienup. Phase retrieval algorithms: a comparison, 1982, Appl Opt, 21(15): 2758–2769.
J Finkelstein. Pure-state informationally complete and “really” complete measurements, Phys Rev A, 2004, 70(5): 052107.
R W Gerchberg, W O Saxton. A Practical algorithm for the determination of phase from image and diffraction plane pictures, Optik, 1972, 35(2): 237–246.
M X Goemans, D P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM, 1995, 42(6): 1115–1145.
T Goldstein, C Studer. PhaseMax: convex phase retrieval via basis pursuit, IEEE Trans Inform Theory, 2018, 64(4):2675–2689.
D Gross. Recovering low-rank matrices from few coefficients in any basis, IEEE Trans Inform Theory, 2011, 57(3): 1548–1566.
D Gross, F Krahmer, R Kueng. A partial de-randomization of PhaseLift using spherical designs, J Fourier Anal Appl, 2015, 21(2): 229–266.
D Gross, F Krahmer, R Kueng. Improved recovery guarantees for phase retrieval from coded diffraction patterns, Appl Comput Harmon Anal, 2017, 42(1): 37–64.
D Gross, Y K Liu, S T Flammia, S Becker, J Eisert. Quantum-state tomography via compressed sensing, Phys Rev Lett, 2010, 105(15): 150401.
M H Hayes. The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform, IEEE Transactions on Acoustics, Speech and Signal Processing, 1982, 30(2): 140–154.
E M Hofstetter. Construction of time-limited functions with specified autocorrelation functions, IEEE Trans Inform Theory, 1964, 10(2): 119–126.
S Hoggar. t-Designs in projective spaces, Eur J Comb, 1982, 3(3): 233–254.
M J Humphry, B Kraus, A C Hurst, A M Maiden, J M Rodenburg. Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging, Nature Communications, 2012, 3(1): 730.
M Iwen, A Viswanathan, Y Wang. Robust sparse phase retrieval made easy, Appl Comput Harmon Anal, 2017, 42(1): 135–142.
K Jaganathan, Y Eldar, B Hassibi. Phase retrieval with masks using convex optimization, IEEE International Symposium on Information Theory, 2015: 1655–1659.
K Jaganathan, Y Eldar, B Hassibi. Phase Retrieval: An Overview of Recent Developments, arXiv preprint arXiv: Information Theory, 2015.
A Klappenecker, M Rotteler. Mutually unbiased bases are complex projective 2-designs, IEEE International Symposium on Information Theory, 2005, 1: 1740–1744.
V Koltchinskii, S Mendelson. Bounding the smallest singular value of a random matrix without concentration, International Mathematics Research Notices, 2015, 23: 12991–13008.
H König. Cubature formulas on spheres, Advances in Multivariate Approximation, Proceedings of the 3rd International Conference on Multivariate Approximation Theory, 1999: 201–211.
F Kramher, Y Liu. Phase retrieval without small-ball probability assumptions, IEEE Trans Inform Theory, 2018, 64(1): 485–500.
R Kueng, H Rauhut, U Terstiege. Low rank matrix recovery from rank one measurements, Appl Comput Harmon Anal, 2014, 42(1): 88–116.
J M Landsberg. Tensors: Geometry and Applications, American Mathematical Society (AMS), 2012.
G Lecué, S Mendelson. Compressed sensing under weak moment assumptions, arXiv preprint arXiv:1401.2188, 2014.
X Li, V Voroninski. Sparse signal recovery from quadratic measurements via convex programming, SIAM J Math Anal, 2013, 45(5): 3019–3033.
H Li, S Li. Phase retrieval from coded diffraction patterns with noise, Submitted, 2019.
H Li, S Li, Y Xia. PhaseMax: Stable guarantees from noisy sub-gaussian measurements, Anal Appl, Online, 2019.
S Marchesini. Phase retrieval and saddle-point optimization, J Opt Soc Amer A, 2007, 24(10): 3289–3296.
S Marchesini. A unified evaluation of iterative projection algorithms for phase retrieval, Rev Sci Instrum, 2007, 78(1): 011301.
K Maryia, K Richard, R Holger, U Terstiege. Stable low rank matrix recovery from null space property, Information and Inference: A Journal of the IMA, 2016, 5(4): 405–441.
S Mendelson. A remark on the diameter of random sections of convex bodies, arXiv preprint arXiv: 1312.3608, 2014.
S Mendelson. Learning without concentration, J Assoc Comput Mach, 2015, 62(3): 21.
S Mendelson. Learning without concentration for general loss functions, Probability Theory and Related Fields, 2018, 171(1): 459–502.
R Millane. Phase retrieval in crystallography and optics, J Opt Soc Amer A, 1990, 7(3): 394–411.
M L Moravec, J K Romberg, R G Baraniuk. Compressive phase retrieval, Proceedings of SPIE, 2007, 6701: 6701201–67012011.
P Netrapalli, P Jain, S Sanghavi. Phase retrieval using alternating minimization, IEEE Trans Signal Process, 2015, 63(18): 4814–4826.
A Neumaier. Combinatorial Configurations in Terms of Distances, Department of Mathematics Memorandum, Eindhoven, 1981.
K A Nugent, A G Peele, H N Chapman, A P Manusco. Unique phase recovery for non-periodic objects, Phys Rev Lett, 2003, 91(20): 203902.
H Ohlsson, A Yang, R Dong, S Sastry. CPRL-an extension of compressive sensing to the phase retrieval problem, IEEE Neural Information Processing Systems, 2012: 1367–1375.
A L Patterson. A Fourier series method for the determination of the components of interatomic distances in crystals, Phys Rev, 1934, 46(5): 372–376.
A L Patterson. Ambiguities in the X-ray analysis of crystal structures, Phys Rev, 1944, 46: 195–201.
R Pedarsani, D Yin, K Lee, K Ramchandran. PhaseCode: fast and efficient compressive phase retrieval based on sparse-graph codes, IEEE Trans Inform Theory, 2017, 63(6): 3663–3691.
L Rabiner, B H Juang. Fundamentals of speech recognition, Prentice Hall, 1993.
B Recht, M Fazel, P Parrilo. Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev, 2010, 52(3): 471–501.
B Sanderson. Immersions and embeddings of projective spaces, Proc Lond Math Soc, 1964, 1: 137–153.
J L C Sanz. Mathematical considerations for the problem of Fourier transform phase retrieval from magnitude, SIAM J Appl Math, 1985, 45(4): 651–664.
Y Shechtman, A Beck, Y C Eldar. GESPAR: efficient phase retrieval of sparse signals, IEEE Trans Signal Process, 2014, 62(4): 928–938.
Y Shechtman, Y C Eldar, A Szameit, M Segev. Sparsity based sub-wavelength imaging with partially incoherent light via quadratic compressed sensing, 2011, Optics Express, 19(16): 14807–14822.
M Stefik. Inferring DNA structures from segmentation data, Artificial Intelligence, 1978, 11(1): 85–114.
Y Tan, R Vershynin. Phase retrieval via randomized Kaczmarz: theoretical guarantees, arXiv preprint arXiv:1706.09993, 2018.
J R Tropp. Convex Recovery of a Structured Signal from Independent Random Linear Measurements, arXiv preprint arXiv: Information Theory, 2014: 67–101.
I Waldspurger, A d’Aspremont, S Mallat. Phase recovery, maxcut and complex semidefinite programming, Math Program, 2015, 149(1): 47–81.
A Walther. The question of phase retrieval in optics, Journal of Modern Optics, 1963, 10(1): 41–49.
G Wang, G B Giannakis, Y C Eldar. Solving systems of random quadratic equations via truncated amplitude flow, IEEE Trans Inform Theory, 2018, 64(2): 773–794.
G Wang, G B Giannakis, Y Saad, J Chen. Phase retrieval via reweighted amplitude flow, IEEE Trans Signal Process, 2018, 66(11): 2818–2833.
G Wang, G B Giannakis, J Chen. Scalable solvers of random quadratic equations via stochastic truncated amplitude flow, IEEE Trans Signal Process, 2017, 65(8): 1961–1974.
Y Wang, Z Q Xu. Phase retrieval for sparse signals, Appl Comput Harmon Anal, 2014, 37(3): 531–544.
G Wang, L Zhang, G B Giannakis, M Akcakaya, J Chen. Sparse phase retrieval via truncated amplitude flow, IEEE Trans Signal Process, 2018, 66(2): 479–491.
K Wei. Solving systems of phaseless equations via Kaczmarz methods: a proof of concept study, Inverse Problem, 2015, 31(12): 125008–125030.
G Zauner. Quantendesigns: Grundzülge einer nichtkommutativen Designtheorie, PhD dissertation, University of Vienna, 1999.
H Zhang, Y Zhou, Y Liang, Y Chi. Reshaped Wirtinger flow and incremental algorithm for solving quadratic system of equations, arXiv preprint arXiv:1605.07719, 2016.
H Zhang, Y Chi, Y Liang. Provable non-convex phase retrieval with outliers: Median truncated Wirtinger flow, arXiv preprint arXiv:1603.03805, 2016.
G Zheng, R Horstmeyer, C Yang. Wide-field, high-resolution Fourier pty-chographic microscopy, Nature Photonics, 2013, 7: 739–745.