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Spring 2014: Research-Interaction-Team on the Phaseless Reconstruction Problem



Organizers:

Radu Balan

 

Kasso Okoudjou

  • Email: rvbalan at math.umd.edu
  • Office: Math building 2308 ; Phone: 301 405 5492
  • Office: CSCAMM (CSIC building) 4131 ; Phone: 301 405 1217

  • Email: kasso at math.umd.edu

Meetings: 2.00pm-3.00pm on Mondays in MATH0104.

Topics:
This RIT focuses on nonlinear inverse problems. In particular we study results for the following setup: alpha:H->R^m where (alpha(x))_k = |(Ax)_k| is the absolute value computed entrywise of a linear transform. This problem has applications to X-Ray chrystallography, signal processing, communications, and machine learning, to name a few.

References:
Spring 2014: Finite Dimensional Case

  • Theory

  • [1] Jameson Cahill, Peter G. Casazza, Jesse Peterson and Lindsay Woodland, Phase Retrieval by Projections, arXiv:1305.6226v3[math.FA], 18 July 2013.
    [2] Yang Wang, Zhiqiang Xu, Phase Retrieval for Sparse Signals, arXiv:1310.0873v2[cs.IT], 4 Oct. 2013.
    [3] James Milgram,Immersing Projective Spaces, Annals Mathematics, vol. 85, no. 3 (1967), 473--482.
    [4] Radu Balan, Stability of Phase Retrievable Frames, arXiv:1308.5465v1[math.FA], 25 Aug. 2013.
    [5] Radu Balan, Dongmian Zou, Phase Retrieval using Lipschitz Continuous Maps, arXiv:1403.2301 [math.FA], 11 Mar. 2014,

  • Algorithms

  • 1. Convex Optimization
    [6] Emmanuel J. Candes, Thomas Strohmer, Vladislav Voroninski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming, Comm.PureAppl.Math., vol. 66, (2013) 1241-1274.
    [7] Laurent Demanet, Paul Hand, Stable optimizationless recovery from phaseless linear measurements, arXiv:1208.1893v2[math.NA], 21 Aug. 2012.
    [8] Emmanuel Candes, Xiaodong Li, Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns, arXiv:1208.6247v2[cs.IT], 17 Sept. 2012.
    [9] Henrik Ohlsson, Allen Y. Yang, Roy Dong, S. Shankar Sastry, CPRL – An Extension of Compressive Sensing to the Phase Retrieval Problem, NIPS 2012.
    [10] Emmanuel J. Candes, Xiaodong Li, Mahdi Soltanolkotabi, Phase Retrieval from Coded Diffraction Patterns, arXiv:1310.3240v2[cs.IT], 6 Nov. 2013.

    2. Iterative Methods
    [11] Afonso S. Bandeira and Dustin G. Mixon, Near-Optimal Phase Retrieval of Sparse Vectors, arXiv:1308.0143v1[cs.IT], 1 Aug. 2013.

    3. Constructions
    [12] Matthew Fickus, Dustin G. Mixon, Aaron A. Nelson, Yang Wang, Phase retrieval from very few measurements, arXiv:1307.7176v1[math.FA], 26 July 2013.
    [13] Bernhard G. Bodmann and Nathaniel Hammen, Stable Phase Retrieval With Low-Redundancy Frames, arXiv:1302.5487v1[math.FA], 22 Feb 2013.

  • Fourier Transform case: X-Ray

  • [14] Albert Fannjiang, Absolute Uniqueness Of Phase Retrieval With Random Illumination, Inverse Problems, 28 (2012).
    [15] Albert Fannjiang, Wenjing Liao, Compressed Sensing Phase Retrieval, ASILOMAR 2011

     

    Schedule (all meetings in MATH 0104 at 2:00pm, unless otherwise said):

    February 3, 2014: Organizatorial Meeting; Radu Balan: Real Case - the necessary and sufficient condition; 2n-1 result
    February 10, 2014: Radu Balan, The complex case (I): Notations: C^n -> R^2n scalar product transfer; sets of symmetric matrices, Sym(H) and S^p,q(H).
    February 17, 2014: Radu Balan, The complex case (II): Necessary and Sufficient conditions in terms of R(x) matrix;
    February 28, 2014: -- Radu Balan, The complex case (III) - stability bounds as in [4];
    March 3, 2014: Xuemei Chen: PhaseLift Algorithm from [6] (I) canceled: University was closed due to snow
    March 14, 2014: Xuemei Chen: PhaseLift Algorithm from [6] (I) -- NOTE CHANGE OF DAY
    March 17, 2014: No meeting (spring break)
    March 24, 2014: Xuemei Chen: PhaseLift Algorithm from [6] (II)
    March 31, 2014: Max Scharrenbroich: PhaseLift Algorithm with low redundancy [8] (I)
    April 7, 2014: No meeting
    April 14, 2014: Max Scharrenbroich: PhaseLift Algorithm with low redundancy [8] (II)
    April 21, 2014: Ming Zhong: Phase Retrieval for Sparse Signals [2] (I)
    April 28, 2014: Ming Zhong: Phase Retrieval for Sparse Signals [2] (II)
    May 5, 2014: Dongmian Zou: Lipschitz Constant in Phaseless Reconstruction [5]
    May 12, 2014: