Abstract
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.
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References
Alexeev, B., Bandeira, A.S., Fickus, M., Mixon, D.G.: Phase retrieval with polarization. SIAM J. Imaging Sci. 7, 35–66 (2014)
Backhouse, N.B., Bradley, C.J.: Projective representations of Abelian groups. Proc. Am. Math. Soc. 36, 260–266 (1972)
Balan, R.: Stability of phase retrievable frames. In: Proceedings of SPIE, Wavelets and Sparsity XV, 88580H (2013)
Balan, R., Wang, Y.: Invertibility and robustness of phaseless reconstruction. Appl. Comput. Harmon. Anal. 38, 469–488 (2015)
Balan, R., Zou, D.: On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem. Linear Algebra Appl. 496, 152–181 (2016)
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)
Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20, 345–356 (2006)
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)
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)
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)
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)
Bandeira, A.S., Chen, Y., Mixon, D.G.: Phase retrieval from power spectra of masked signals. Inf. Inference 3, 83–102 (2014)
Bendory, T., Eldar, Y.C.: A least squares approach for stable phase retrieval from short-time Fourier transform magnitude, preprint arXiv:1510.00920 (2015)
Bodmann, B.G., Hammen, N.: Stable phase retrieval with low-redundancy frames. Adv. Comput. Math. 41, 317–33 (2015)
Bodmann, B.G., Casazza, P.G., Edidin, D., Balan, R.: Frames for Linear Reconstruction Without Phase. CISS Meeting, Princeton, NJ (2008)
Bojarovska, I., Flinth, A.: Phase retrieval from Gabor measurements. J. Fourier Anal. Appl. 22, 542–567 (2016)
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)
Candès, E.J., Eldar, Y.C., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6, 199–225 (2013)
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)
Cheng, C.: A character theory for projective representations of finite groups. Linear Algebra Appl. 469, 230–242 (2015)
Cheng, C., Fu, J.: On the rings of projective characters of Abelian groups and Dihedral groups, preprint (2016)
Conca, A., Edidin, D., Hering, M., Vinzant, C.: An algebraic characterization of injectivity in phase retrieval. Appl. Comput. Harmon. Anal. 38, 346–356 (2015)
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)
Eldar, Y.C., Hammen, N., Mixon, D.: Recent advances in phase retrieval. In: IEEE Signal Processing Magazine, pp. 158–162 (September 2016)
Fickus, M., Mixon, D.G., Nelson, A.A., Wang, Y.: Phase retrieval from very few measurements. Linear Algebra Appl. 449, 475–499 (2014)
Jaganathan, K., Eldar, Y.C., Hassibi, B.: Phase retrieval: an overview of recent developments. arXiv:1510.07713
Karpilovsky, G.: The Schur Multiplier. London Mathematical Society Monographs. Clarendon Press, Oxford (1987)
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)
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)
Acknowledgements
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).
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Communicated by Peter G. Casazza.
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Li, L., Juste, T., Brennan, J. et al. Phase Retrievable Projective Representation Frames for Finite Abelian Groups. J Fourier Anal Appl 25, 86–100 (2019). https://doi.org/10.1007/s00041-017-9570-6
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DOI: https://doi.org/10.1007/s00041-017-9570-6