Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 86–100 | Cite as

Phase Retrievable Projective Representation Frames for Finite Abelian Groups

  • Lan Li
  • Ted Juste
  • Joseph Brennan
  • Chuangxun Cheng
  • Deguang HanEmail author


We consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering for the phase-retrieval problem. For a given multiplier \(\mu \) of a finite abelian group G, we show that the representation dimension of any irreducible \(\mu \)-projective representation of G is exactly the rank of the symmetric multiplier matrix associated with \(\mu \). With the help of this result we are able to prove that every irreducible \(\mu \)-projective representation of a finite abelian group G admits a frame vector with the maximal span property, and obtain a complete characterization for all such frame vectors. Consequently the complement of the set of all the maximal span frame vectors for any projective unitary representation of any finite abelian group is Zariski-closed. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.


Phase retrieval Frames Group representations 

Mathematics Subject Classification

Primary 42C15 46C05 20C25 20K01 



The authors thank the referees very much for carefully reading the paper and for several elaborate and valuable suggestions. Deguang Han is partially supported by the NSF Grants DMS-1403400 and DMS-1712602, and Lan Li is partially supported by Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JM1046).


  1. 1.
    Alexeev, B., Bandeira, A.S., Fickus, M., Mixon, D.G.: Phase retrieval with polarization. SIAM J. Imaging Sci. 7, 35–66 (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Backhouse, N.B., Bradley, C.J.: Projective representations of Abelian groups. Proc. Am. Math. Soc. 36, 260–266 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balan, R.: Stability of phase retrievable frames. In: Proceedings of SPIE, Wavelets and Sparsity XV, 88580H (2013)Google Scholar
  4. 4.
    Balan, R., Wang, Y.: Invertibility and robustness of phaseless reconstruction. Appl. Comput. Harmon. Anal. 38, 469–488 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balan, R., Zou, D.: On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem. Linear Algebra Appl. 496, 152–181 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction from the absolute value of the frame coefficients. In: Proceedings of SPIE, vol. 5914, pp. 1–8 (2005)Google Scholar
  7. 7.
    Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20, 345–356 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Balan, R., Casazza, P.G., Edidin, D.: Equivalence of reconstruction from the absolute value of the frame coefficients to a sparse representation problem. IEEE Signal Process. Lett. 14(5), 341–345 (2007)CrossRefGoogle Scholar
  9. 9.
    Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Fast algorithms for signal reconstruction without phase. In: Proceedings of SPIE-Wavelets XII, San Diego, vol. 6701, pp. 670111920–670111932 (2007)Google Scholar
  10. 10.
    Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Painless reconstruction from magnitudes of frame vectors. J. Fourier Anal. Appl. 15, 488–501 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bandeira, A.S., Cahill, J., Mixon, D.G., Nelson, A.A.: Saving phase: injectivity and stability for phase retrieval. Appl. Comput. Harmon. Anal. 37, 106–125 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bandeira, A.S., Chen, Y., Mixon, D.G.: Phase retrieval from power spectra of masked signals. Inf. Inference 3, 83–102 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bendory, T., Eldar, Y.C.: A least squares approach for stable phase retrieval from short-time Fourier transform magnitude, preprint arXiv:1510.00920 (2015)
  14. 14.
    Bodmann, B.G., Hammen, N.: Stable phase retrieval with low-redundancy frames. Adv. Comput. Math. 41, 317–33 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bodmann, B.G., Casazza, P.G., Edidin, D., Balan, R.: Frames for Linear Reconstruction Without Phase. CISS Meeting, Princeton, NJ (2008)CrossRefzbMATHGoogle Scholar
  16. 16.
    Bojarovska, I., Flinth, A.: Phase retrieval from Gabor measurements. J. Fourier Anal. Appl. 22, 542–567 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Candès, E.J., Li, X.: Solving quadratic equations via PhaseLift when there are about as many equations as unknowns. Found. Comput. Math. 14, 1017–1026 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Candès, E.J., Eldar, Y.C., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6, 199–225 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Candès, E.J., Strohmer, T., Voroninski, V.: PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66, 1241–1274 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cheng, C.: A character theory for projective representations of finite groups. Linear Algebra Appl. 469, 230–242 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cheng, C., Fu, J.: On the rings of projective characters of Abelian groups and Dihedral groups, preprint (2016)Google Scholar
  22. 22.
    Conca, A., Edidin, D., Hering, M., Vinzant, C.: An algebraic characterization of injectivity in phase retrieval. Appl. Comput. Harmon. Anal. 38, 346–356 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Eldar, Y.C., Sidorenko, P., Mixon, D.G., Barel, S., Cohen, O.: Sparse phase retrieval from short-time Fourier measurements. IEEE Signal Process. Lett. 22, 638–642 (2015)CrossRefGoogle Scholar
  24. 24.
    Eldar, Y.C., Hammen, N., Mixon, D.: Recent advances in phase retrieval. In: IEEE Signal Processing Magazine, pp. 158–162 (September 2016)Google Scholar
  25. 25.
    Fickus, M., Mixon, D.G., Nelson, A.A., Wang, Y.: Phase retrieval from very few measurements. Linear Algebra Appl. 449, 475–499 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jaganathan, K., Eldar, Y.C., Hassibi, B.: Phase retrieval: an overview of recent developments. arXiv:1510.07713
  27. 27.
    Karpilovsky, G.: The Schur Multiplier. London Mathematical Society Monographs. Clarendon Press, Oxford (1987)zbMATHGoogle Scholar
  28. 28.
    Li, L., Cheng, C., Han, D., Sun, Q., Shi, G.: Phase retrieval from multiple-window short-time Fourier measurements. IEEE Signal Process. Lett. 24, 372–376 (2017)CrossRefGoogle Scholar
  29. 29.
    Nawab, S.H., Quatieri, T.F., Lim, J.S.: Signal reconstruction from short-time Fourier transform magnitude. IEEE Trans. Acoust. Speech Signal Process. 31, 986–998 (1983)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Lan Li
    • 1
  • Ted Juste
    • 2
  • Joseph Brennan
    • 2
  • Chuangxun Cheng
    • 3
  • Deguang Han
    • 2
    Email author
  1. 1.School of ScienceXi’an Shiyou UniversityXi’anChina
  2. 2.Department of MathematicsUniversity of Central FloridaOrlandoUSA
  3. 3.Department of MathematicsNanjing UniversityNanjingChina

Personalised recommendations