Abstract
We address the problem of recovering an n-vector from m linear measurements lacking sign or phase information. We show that lifting and semidefinite relaxation suffice by themselves for stable recovery in the setting of m=O(nlogn) random sensing vectors, with high probability. The recovery method is optimizationless in the sense that trace minimization in the PhaseLift procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem. The optimizationless perspective allows for a Douglas-Rachford numerical algorithm that is unavailable for PhaseLift. This method exhibits linear convergence with a favorable convergence rate and without any parameter tuning.
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Balan, R., Bodmann, B., Casazza, P.G., Edidin, D.: Painless reconstruction from magnitudes of frame vectors. J. Fourier Anal. Appl. 15(4), 488–501 (2009)
Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345–356 (2006)
Bandeira, A.S., Singer, A., Spielman, D.A.: A Cheeger inequality for the graph connection Laplacian. To appear in SIAM J. Matrix Anal. Appl.
Beck, A., Teboulle, M.: A fast iterative Shrinkage-Thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Candes, E.J., Eldar, Y.C., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6(1), 199–225 (2011)
Candes, E.J., Li, X.: Solving quadratic equations via phaselift when there are about as many equations as unknowns. To appear in Found. Comput. Math. (2012)
Candes, E.J., Strohmer, T., Voroninski, V.: PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66(8), 1241–1274 (2013)
Chai, A., Moscoso, M., Papanicolaou, G.: Array imaging using intensity-only measurements. Inverse Probl. 27(1), 015005 (2011)
Combettes, P., Pesquet, J.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York (2010)
Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421–439 (1956)
Fienup, J.R.: Phase retrieval algorithms: a comparison. Appl. Opt. 21(15), 2758–2769 (1982)
Gerchberg, R., Saxton, W.: A practical algorithm for the determination of phase from image and diffraction plane pictures. Optik 35, 237–246 (1972)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semi-definite programming. J. ACM 42, 1115–1145 (1995)
Griffin, D., Lim, J.: Signal estimation from modified short-time Fourier transform. IEEE Trans. Acoust. Speech Signal Process. 32(2), 236–243 (1984)
Harrison, R.W.: Phase problem in crystallography. J. Opt. Soc. Am. A 10(5), 1045–1055 (1993)
Miao, J., Ishikawa, T., Shen, Q., Earnest, T.: Extending X-ray crystallography to allow the imaging of non-crystalline materials, cells and single protein complexes. Annu. Rev. Phys. Chem. 59, 387–410 (2008)
Millane, R.P.: Phase retrieval in crystallography and optics. J. Opt. Soc. Am. A 7, 394–411 (1990)
Raguet, H., Fadili, J.M., Peyre, G.: A generalized forward-backward splitting. SIAM J. Imaging Sci. 6(3), 1199–1226 (2013)
Recht, B., Fazel, M., Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)
Singer, A.: Angular synchronization by eigenvectors and semidefinite programming. Appl. Comput. Harmon. Anal. 30(1), 20–36 (2011)
Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Eldar, Y.C., Kutyniok, G. (eds.) Compressed Sensing: Theory and Applications. Camb. Univ Press, Cambridge (2010)
Waldspurger, I., d’Aspremont, A., Mallat, S.: Phase recovery, Maxcut, and complex semi-definite programming. Preprint (2012). arXiv:1206.0102
Walther, A.: The question of phase retrieval in optics. Opt. Acta 10, 41–49 (1963)
Acknowledgements
The authors acknowledge generous funding from the National Science Foundation, the Alfred P. Sloan Foundation, TOTAL S.A., and the Air Force Office of Scientific Research. The authors would also like to thank Xiangxiong Zhang for helpful discussions.
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Communicated by Peter Casazza.
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Demanet, L., Hand, P. Stable Optimizationless Recovery from Phaseless Linear Measurements. J Fourier Anal Appl 20, 199–221 (2014). https://doi.org/10.1007/s00041-013-9305-2
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DOI: https://doi.org/10.1007/s00041-013-9305-2