

Spring 2014: ResearchInteractionTeam 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: 2.00pm3.00pm 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 XRay 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), 473482.
[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) 12411274.
[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, NearOptimal 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 LowRedundancy Frames, arXiv:1302.5487v1[math.FA], 22 Feb 2013.
Fourier Transform case: XRay
[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; 2n1 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:
