Phase retrieval using random cubatures and fusion frames of positive semidefinite matrices

• Autorzy: Martin Ehler , Manuel Gräf , Franz J. Király
• Języki publikacji: EN

Abstrakty

EN

As a generalization of the standard phase retrieval problem,we seek to reconstruct symmetric rank- 1 matrices from inner products with subclasses of positive semidefinite matrices. For such subclasses, we introduce random cubatures for spaces of multivariate polynomials based on moment conditions. The inner products with samples from sufficiently strong random cubatures allow the reconstruction of symmetric rank- 1 matrices with a decent probability by solving the feasibility problem of a semidefinite program.

• Wydawca: De Gruyter Open
• Czasopismo: Waves, Wavelets and Fractals
• Rocznik: 2015
• Tom: 1
• Numer: 1

Daty

otrzymano

2015-05-29

zaakceptowano

2015-10-06

online

2015-12-31

Twórcy

autor

Martin Ehler

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autor

Manuel Gräf

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autor

Franz J. Király

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Bibliografia

• [1] C. Bachoc and M. Ehler, Tight p-fusion frames, Appl. Comput. Harmon. Anal. 35 (2013), no. 1, 1–15. [Crossref][WoS]
• [2] C. Bachoc and M. Ehler, Signal reconstruction from the magnitude of subspace components, IEEE Trans. Inform. Theory 61 (2015), no. 7, 1–13. [WoS][Crossref]
• [3] R. Balan, Stability of phase retrievable frames, arXiv:1308.5465v1 (2013).
• [4] R. Balan, P. Casazza, and D. Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal 20 (2006), 345– 356. [Crossref]
• [5] A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal. 37 (2014), no. 1, 106–125. [Crossref][WoS]
• [6] F. Barthe, F. Gamboa, L.-V. Lozada-Chang, and A. Rouault, Generalized Dirichlet distributions on the ball and moments, Alea 7 (2010), 319–340.
• [7] B. Bhatia, Matrix analysis, Springer, New York, 1996.
• [8] B. G. Bodmann and N. Hammen, Stable phase retrieval with low-redundancy frames, arXiv:1302.5487v1 (2013). [WoS]
• [9] A. Bondarenko, D. Radchenko, and M. Viazovska, Optimal asymptotic bounds for spherical designs, arXiv:1009.4407v3 (2011).
• [10] A. V. Bondarenko, D. V. Radchenko, and M. S. Viazovska, On optimal asymptotic bounds for spherical designs, arXiv:1009.4407v1 (2010).
• [11] J. Cahill, P. G. Casazza, J. Peterson, and L. Woodland, Phase retrieval by projections, arXiv:1305.6226v3 (2013).
• [12] E. J. Candès, Y. Eldar, T. Strohmer, and V. Voroninski, Phase retrieval via matrix completion, arXiv:1109.0573v2 (2011).
• [13] E. J. Candès and X. Li., Solving quadratic equations via PhaseLift when there are about as many equations as unknowns, Foundations of Computational Mathematics 14 (2014), 1017–1026. [Crossref][WoS]
• [14] E. J. Candès, T. Strohmer, and V. Voroninski, PhaseLift: Exact and stable signal recovery from magnitude measurements via convex programming, Communications on Pure and Applied Mathematics, DOI:10.1002/cpa.21432 66 (2013), no. 8, 1241– 1274. [WoS][Crossref]
• [15] Y. Chikuse, Statistics on special manifolds, Lecture Notes in Statistics, Springer, New York, 2003.
• [16] P. de la Harpe and C. Pache, Cubature formulas, geometrical designs, reproducing kernels, and Markov operators, Infinite groups: geometric, combinatorial and dynamical aspects (Basel), vol. 248, Birkhäuser, 2005, pp. 219–267.
• [17] L. Demanet and P. Hand, Stable optimizationless recovery from phaseless linear measurements, J. Fourier Anal. Appl. 20 (2014), 199–221. [Crossref][WoS]
• [18] M. Ehler, Random tight frames, J. Fourier Anal. Appl. 18 (2012), no. 1, 1–20. [Crossref]
• [19] M. Ehler and M. Gräf, Cubatures and designs in unions of Grassmann spaces, arXiv (2014).
• [20] M. Ehler and S. Kunis, Phase retrieval using time and Fourier magnitude measurements, 10th International Conference on Sampling Theory and Applications, 2013.
• [21] V. Elser and R. P. Millane, Reconstruction of an object from its symmetry-averaged diffraction pattern, Acta Crystallographica Section A 64 (2008), no. 2, 273–279. [WoS]
• [22] J. R. Fienup, Phase retrieval algorithms: a comparison, Applied Optics 21 (1982), no. 15, 2758–2769. [Crossref]
• [23] F. Filbir and H. N. Mhaskar, A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel, J. Fourier Anal. Appl. 16 (2010), no. 5, 629–657. [Crossref][WoS]
• [24] R.W. Gerchberg andW. O. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik 35 (1972), no. 2, 237–246.
• [25] D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory 57 (2011), 1548–1566. [WoS][Crossref]
• [26] D. Gross, F. Krahmer, and R. Kueng, A partial derandomization of PhaseLift using spherical designs, J. Fourier Anal. Appl. 21 (2015), no. 2, 229–266. [WoS][Crossref]
• [27] A. T. James, Distributions of matrix variates and latent roots derived from normal samples, Annals ofMathematical Statistics 35 (1964), no. 2, 475–501. [Crossref]
• [28] A. T. James and A. G. Constantine, Generalized Jacobi polynomials as spherical functions of the Grassmann manifold, Proc. London Math. Soc. 29 (1974), no. 3, 174–192. [Crossref]
• [29] H. König, Cubature formulas on spheres, Adv. Multivar. Approx. Math. Res. 107 (1999), 201–211.
• [30] R. Kueng, H. Rauhut, and U. Terstiege, Low rank matrix recovery from rank one measurements, arXiv:1410.6913 (2014).
• [31] H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comp. 70 (2002), 1113–1130. [Crossref]
• [32] R. J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, New York, 1982.
• [33] F. Philipp, Phase retrieval from 4n-4 measurements: A proof for injectivity, Proc. Appl. Math. Mech. 14 (2014), no. 833-834. [Crossref]
• [34] E. Riegler and G. Tauböck, Almost lossless analog compression without phase information, in Proc. IEEE Int. Symp. Inf. Th. (Hong Kong, China), 2015, pp. 1–5.
• [35] T. Strohmer and R. W. Heath, Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal. 14 (2003), no. 3, 257–275. [Crossref]
• [36] J.A. Tropp, User-friendly tools for random matrices: An introduction., NIPS.
• [37] J.A. Tropp„ User-friendly tail bounds for sums of random matrices, Journal Foundations of Computational Mathematics 12 (2012), no. 4, 389–434. [WoS]
• [38] I. Waldspurger, A. d’Aspremont, and S. Mallat, Phase recovery, maxcut and complex semidefinite programming, arXiv:1206.0102v2 (2012).
• [39] T. Wong, Generalized Dirichlet distribution in Bayesian analysis, Appl. Math. Comput. 97 (1998), no. 2-3, 165–181. [Crossref]

Identyfikatory

DOI

10.1515/wwfaa-2015-0005