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Convex Recovery of a Structured Signal from Independent Random Linear Measurements

  • Joel A. Tropp
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This chapter develops a theoretical analysis of the convex programming method for recovering a structured signal from independent random linear measurements. This technique delivers bounds for the sampling complexity that are similar to recent results for standard Gaussian measurements, but the argument applies to a much wider class of measurement ensembles. To demonstrate the power of this approach, the chapter presents a short analysis of phase retrieval by trace-norm minimization. The key technical tool is a framework, due to Mendelson and coauthors, for bounding a nonnegative empirical process.

Notes

Acknowledgements

JAT gratefully acknowledges support from ONR award N00014-11-1002, AFOSR award FA9550-09-1-0643, and a Sloan Research Fellowship. Thanks are also due to the Moore Foundation.

References

  1. 1.
    D. Amelunxen, M. Lotz, M.B. McCoy, J.A. Tropp, Living on the edge: phase transitions in convex programs with random data. Inf. Inference 3(3), 224–294 (2014). Available at http://arXiv.org/abs/1303.6672 CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Balan, B.G. Bodmann, P.G. Casazza, D. Edidin, Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15(4), 488–501 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    S. Boucheron, G. Lugosi, P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence (Oxford University Press, Oxford, 2013)CrossRefGoogle Scholar
  4. 4.
    T.T. Cai, T. Liang, A. Rakhlin, Geometrizing local rates of convergence for linear inverse problems (April 2014). Available at http://arXiv.org/abs/1404.4408
  5. 5.
    E.J. Candès, T. Strohmer, V. Voroninski, PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66(8), 1241–1274 (2013)CrossRefzbMATHGoogle Scholar
  6. 6.
    V. Chandrasekaran, B. Recht, P.A. Parrilo, A.S. Willsky, The convex geometry of linear inverse problems. Found. Comput. Math. 12(6), 805–849 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    R.Y. Chen, A. Gittens, J.A. Tropp, The masked sample covariance estimator: an analysis via the matrix Laplace transform method. Inf. Inference 1, 2–20 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    K.R. Davidson, S.J. Szarek, Local operator theory, random matrices and Banach spaces, in Handbook of the Geometry of Banach Spaces, vol. I (North-Holland, Amsterdam, 2001), pp. 317–366Google Scholar
  9. 9.
    M. Fazel. Matrix Rank Minimization with Applications. Ph.D. thesis, Stanford University, 2002.Google Scholar
  10. 10.
    R. Foygel, L. Mackey, Corrupted sensing: novel guarantees for separating structured signals. Trans. Inf. Theory 60(2), 1223–1247 (2014)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Y. Gordon, Some inequalities for Gaussian processes and applications. Isr. J. Math. 50(4), 265–289 (1985)CrossRefzbMATHGoogle Scholar
  12. 12.
    Y. Gordon. On Milman’s inequality and random subspaces which escape through a mesh in \(\mathbf{R}^{n}\), in Geometric Aspects of Functional Analysis (1986/1987). Lecture Notes in Mathematics, vol. 1317 (Springer, Berlin, 1988), pp. 84–106Google Scholar
  13. 13.
    D. Gross, Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57(3), 1548–1566 (2011)CrossRefGoogle Scholar
  14. 14.
    V. Koltchinskii, S. Mendelson, Bounding the smallest singular value of a random matrix without concentration (December 2013). Available at http://arXiv.org/abs/1312.3580
  15. 15.
    R. Latała, K. Oleszkiewicz, On the best constant in the Khinchin-Kahane inequality. Studia Math. 109(1), 101–104 (1994)MathSciNetzbMATHGoogle Scholar
  16. 16.
    G. Lecué, S. Mendelson, Compressed sensing under weak moment assumptions (January 2014). Available at http://arXiv.org/abs/1401.2188
  17. 17.
    M. Ledoux, M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes (Springer, Berlin, 1991)CrossRefzbMATHGoogle Scholar
  18. 18.
    S. Mendelson. Empirical processes with a bounded \(\psi _{1}\) diameter. Geom. Funct. Anal. 20(4), 988–1027 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    S. Mendelson, A remark on the diameter of random sections of convex bodies (December 2013). Available at http://arXiv.org/abs/1312.3608
  20. 20.
    S. Mendelson, Learning without concentration. J. Assoc. Comput. Mach. (2014, to appear). 62(3), (2015). Available at http://arXiv.org/abs/1401.0304
  21. 21.
    S. Mendelson, Learning without concentration for general loss functions (October 2014). Available at http://arXiv.org/abs/1410.3192
  22. 22.
    S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17(4), 1248–1282 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    V. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Linear Spaces. Number 1200 in LNM (Springer, New York, 1986)Google Scholar
  24. 24.
    S. Oymak, B. Hassibi, New null space results and recovery thresholds for matrix rank minimization. Partial results presented at ISIT 2011 (2010). Available at http://arXiv.org/abs/1011.6326
  25. 25.
    S. Oymak, B. Hassibi, Sharp MSE bounds for proximal denoising. Partial results presented at Allerton 2012 (March 2013). Available at http://arxiv.org/abs/1305.2714
  26. 26.
    S. Oymak, C. Thrampoulides, B. Hassibi, Simple bounds for noisy linear inverse problems with exact side information (December 2013). Available at http://arXiv.org/abs/1312.0641
  27. 27.
    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry (Cambridge University Press, Cambridge, 1989)CrossRefzbMATHGoogle Scholar
  28. 28.
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)CrossRefzbMATHGoogle Scholar
  29. 29.
    M. Rudelson, R. Vershynin, On sparse reconstruction from Fourier and Gaussian measurements. Commun. Pure Appl. Math. 61(8), 1025–1045 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    M. Sion, On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    M. Stojnic, Various thresholds for 1-optimization in compressed sensing (2009). Available at http://arXiv.org/abs/0907.3666 Google Scholar
  32. 32.
    M. Talagrand, The Generic Chaining. Upper and Lower Bounds of Stochastic Processes. Springer Monographs in Mathematics (Springer, Berlin, 2005)Google Scholar
  33. 33.
    C. Thrampoulides, S. Oymak, B. Hassibi, Simple error bounds for regularized noisy linear inverse problems. Appeared at ISIT 2014 (January 2014). Available at http://arXiv.org/abs/1401.6578
  34. 34.
    J. Tropp, M. Wakin, M. Duarte, D. Baron, R. Baraniuk, Random filters for compressive sampling and reconstruction, in Proceedings of the 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 2006 (ICASSP 2006), vol. 3, May 2006, pp. III–875Google Scholar
  35. 35.
    A.W. van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes. Springer Series in Statistics (Springer, New York, 1996). With applications to statistics.Google Scholar
  36. 36.
    R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing (Cambridge University Press, Cambridge, 2012), pp. 210–268Google Scholar
  37. 37.
    G.A. Watson, Characterization of the subdifferential of some matrix norms. Linear Algebra Appl. 170, 33–45 (1992)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Joel A. Tropp
    • 1
  1. 1.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA

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