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Phase Retrieval by the Inverse Power Method

  • Qi Luo
  • Hongxia Wang
  • Jianyun Chen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10996)

Abstract

Phase retrieval is to recover signals from phaseless linear measurements. The most efficient methods to tackle this problem are nonconvex gradient approaches, which however generally need an elaborate initialized guess to ensure successful reconstruction. The inverse power method is proposed to provide a more accurate initialization. Numerical experiments illustrate the higher accuracy of the proposed method over other initialization methods. And we further demonstrate the iterative use of the initialization method can obtain an even better estimate.

Keywords

Phase retrieval Spectral method Inverse power method 

References

  1. 1.
    Miao, J., Charalambous, P., Kirz, J., Sayre, D.: Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature 400(6742), 342 (1999)CrossRefGoogle Scholar
  2. 2.
    Shechtman, Y., Eldar, Y.C., Cohen, O., Chapman, H.N., Miao, J., Segev, M.: Phase retrieval with application to optical imaging: a contemporary overview. IEEE Sig. Process. Mag. 32(3), 87–109 (2015)CrossRefGoogle Scholar
  3. 3.
    Stefik, M.: Inferring dna structures from segmentation data. Artif. Intell. 11(1–2), 85–114 (1978)CrossRefGoogle Scholar
  4. 4.
    Fienup, C., Dainty, J.: Phase retrieval and image reconstruction for astronomy. Image Recovery Theory Appl. 231, 275 (1987)Google Scholar
  5. 5.
    Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmonic Anal. 20(3), 345–356 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bodmann, B.G., Hammen, N.: Stable phase retrieval with low-redundancy frames. Adv. Comput. Math. 41(2), 317–331 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1(1), 15–22 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Netrapalli, P., Jain, P., Sanghavi, S.: Phase retrieval using alternating minimization. In: Advances in Neural Information Processing Systems, pp. 2796–2804 (2013)Google Scholar
  9. 9.
    Elser, V.: Phase retrieval by iterated projections. JOSA A 20(1), 40–55 (2003)CrossRefGoogle Scholar
  10. 10.
    Candes, E.J., Li, X., Soltanolkotabi, M.: Phase retrieval via wirtinger flow: theory and algorithms. IEEE Trans. Inf. Theory 61(4), 1985–2007 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zhang, H., Liang, Y.: Reshaped wirtinger flow for solving quadratic system of equations. In: Advances in Neural Information Processing Systems, pp. 2622–2630 (2016)Google Scholar
  12. 12.
    Wang, G., Giannakis, G.B., Eldar, Y.C.: Solving systems of random quadratic equations via truncated amplitude flow. IEEE Trans. Inf. Theory 64, 773–794 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wang, G., Giannakis, G., Saad, Y., Chen, J.: Solving most systems of random quadratic equations. In: Advances in Neural Information Processing Systems, pp. 1865–1875 (2017)Google Scholar
  14. 14.
    Chen, Y., Candes, E.: Solving random quadratic systems of equations is nearly as easy as solving linear systems. In: Advances in Neural Information Processing Systems, pp. 739–747 (2015)Google Scholar
  15. 15.
    Chen, P., Fannjiang, A., Liu, G.R.: Phase retrieval by linear algebra. SIAM J. Matrix Anal. Appl. 38(3), 854–868 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer, New York (2013).  https://doi.org/10.1007/978-1-4419-8853-9CrossRefMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of ScienceNational University of Defense TechnologyChangshaPeople’s Republic of China
  2. 2.Beijing Institute of GraphicsBeijingPeople’s Republic of China

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